LMU MT U M

E B

TMP eoretical and Mathematical Physics

Masterrsquos Thesis

Jordan Wigner Transformations andantum Spin Systems on Graphs

Tobias Ried

Thesis Adviser Prof Dr Simone WarzelSubmission Date 31st August 2013

First Reviewer Prof Dr Simone WarzelSecond Reviewer Prof Dr Peter MuumlllerThesis Defence 9th September 2013

A

e Jordan Wigner transformation is an important tool in the study of one-dimensional quantum spin systems Aer reviewing the one-dimensional JordanWigner transformation generalisations to graphs are discussed As a first res-ult it is proved that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph To avoid the difficultiescaused by statistical gauge fields an indirect transformation due to W andS is introduced and applied to a particular quantum spin system on asquare laice ndash the constrained Gamma matrix model In a random exteriormagnetic field a locality estimate on the Heisenberg evolution ndash a zero-velocityLieb Robinson type bound in disorder average ndash is proved for the constrainedGamma matrix model using localisation results for the two-dimensional Ander-son model

A

Firstly I would like to thank Professor Simone Warzel for supervising thisthesis roughout my studies of physics and mathematical physics her enthu-siasm for those subjects has been a constant source of inspiration I am deeplygrateful for her guidance and support

I am also indebted to Professor Peter Muumlller who agreed to second reviewmythesis

For leing me use one of his grouprsquos offices which enormously facilitatedthe writing process I am thankful to Professor Jan von Del

eTMP programmewould probably not bewhat it is if it were not for RobertHelling His commitment is gratefully acknowledged

My thanks and appreciations also go to Alessandro Michelangeli and every-one else who enriched this final phase of my studies not only on an academicbut also on a personal level

Contents

1 Introduction 511 antum mechanics on combinatorial graphs 612 antum spin systems 7

2 Jordan-Wigner transformation 1121 e Jordan Wigner transformation in one dimension 11

211 Jordan Wigner transformation 11212 Extension to the infinite chain 14

22 Generalisations to graphs 17221 Examples and the existence of special JordanWigner trans-

formations 19222 e two-dimensional Jordan Wigner transformation 23223 A Jordan Wigner transformation for tree graphs 26

23 e WS approach 27231 Link operators and their algebra 28

3 e Gamma matrix model (GMM) 3731 Jordan Wigner transformation for the constrained GMM 3832 Dynamical Localisation and Lieb-Robinson bounds 45

321 e Anderson model on ℤ119889 45322 Lieb Robinson bounds and exponential clustering 49

4 Conclusion and Outlook 61

A Some missing calculations in Srsquos theorem 63

Bibliography 67

C

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

First Reviewer Prof Dr Simone WarzelSecond Reviewer Prof Dr Peter MuumlllerThesis Defence 9th September 2013

A

e Jordan Wigner transformation is an important tool in the study of one-dimensional quantum spin systems Aer reviewing the one-dimensional JordanWigner transformation generalisations to graphs are discussed As a first res-ult it is proved that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph To avoid the difficultiescaused by statistical gauge fields an indirect transformation due to W andS is introduced and applied to a particular quantum spin system on asquare laice ndash the constrained Gamma matrix model In a random exteriormagnetic field a locality estimate on the Heisenberg evolution ndash a zero-velocityLieb Robinson type bound in disorder average ndash is proved for the constrainedGamma matrix model using localisation results for the two-dimensional Ander-son model

A

Firstly I would like to thank Professor Simone Warzel for supervising thisthesis roughout my studies of physics and mathematical physics her enthu-siasm for those subjects has been a constant source of inspiration I am deeplygrateful for her guidance and support

I am also indebted to Professor Peter Muumlller who agreed to second reviewmythesis

For leing me use one of his grouprsquos offices which enormously facilitatedthe writing process I am thankful to Professor Jan von Del

eTMP programmewould probably not bewhat it is if it were not for RobertHelling His commitment is gratefully acknowledged

My thanks and appreciations also go to Alessandro Michelangeli and every-one else who enriched this final phase of my studies not only on an academicbut also on a personal level

Contents

1 Introduction 511 antum mechanics on combinatorial graphs 612 antum spin systems 7

2 Jordan-Wigner transformation 1121 e Jordan Wigner transformation in one dimension 11

211 Jordan Wigner transformation 11212 Extension to the infinite chain 14

22 Generalisations to graphs 17221 Examples and the existence of special JordanWigner trans-

formations 19222 e two-dimensional Jordan Wigner transformation 23223 A Jordan Wigner transformation for tree graphs 26

23 e WS approach 27231 Link operators and their algebra 28

3 e Gamma matrix model (GMM) 3731 Jordan Wigner transformation for the constrained GMM 3832 Dynamical Localisation and Lieb-Robinson bounds 45

321 e Anderson model on ℤ119889 45322 Lieb Robinson bounds and exponential clustering 49

4 Conclusion and Outlook 61

A Some missing calculations in Srsquos theorem 63

Bibliography 67

C

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

A

e Jordan Wigner transformation is an important tool in the study of one-dimensional quantum spin systems Aer reviewing the one-dimensional JordanWigner transformation generalisations to graphs are discussed As a first res-ult it is proved that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph To avoid the difficultiescaused by statistical gauge fields an indirect transformation due to W andS is introduced and applied to a particular quantum spin system on asquare laice ndash the constrained Gamma matrix model In a random exteriormagnetic field a locality estimate on the Heisenberg evolution ndash a zero-velocityLieb Robinson type bound in disorder average ndash is proved for the constrainedGamma matrix model using localisation results for the two-dimensional Ander-son model

A

Firstly I would like to thank Professor Simone Warzel for supervising thisthesis roughout my studies of physics and mathematical physics her enthu-siasm for those subjects has been a constant source of inspiration I am deeplygrateful for her guidance and support

I am also indebted to Professor Peter Muumlller who agreed to second reviewmythesis

For leing me use one of his grouprsquos offices which enormously facilitatedthe writing process I am thankful to Professor Jan von Del

eTMP programmewould probably not bewhat it is if it were not for RobertHelling His commitment is gratefully acknowledged

My thanks and appreciations also go to Alessandro Michelangeli and every-one else who enriched this final phase of my studies not only on an academicbut also on a personal level

Contents

1 Introduction 511 antum mechanics on combinatorial graphs 612 antum spin systems 7

2 Jordan-Wigner transformation 1121 e Jordan Wigner transformation in one dimension 11

211 Jordan Wigner transformation 11212 Extension to the infinite chain 14

22 Generalisations to graphs 17221 Examples and the existence of special JordanWigner trans-

formations 19222 e two-dimensional Jordan Wigner transformation 23223 A Jordan Wigner transformation for tree graphs 26

23 e WS approach 27231 Link operators and their algebra 28

3 e Gamma matrix model (GMM) 3731 Jordan Wigner transformation for the constrained GMM 3832 Dynamical Localisation and Lieb-Robinson bounds 45

321 e Anderson model on ℤ119889 45322 Lieb Robinson bounds and exponential clustering 49

4 Conclusion and Outlook 61

A Some missing calculations in Srsquos theorem 63

Bibliography 67

C

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

A

Firstly I would like to thank Professor Simone Warzel for supervising thisthesis roughout my studies of physics and mathematical physics her enthu-siasm for those subjects has been a constant source of inspiration I am deeplygrateful for her guidance and support

I am also indebted to Professor Peter Muumlller who agreed to second reviewmythesis

For leing me use one of his grouprsquos offices which enormously facilitatedthe writing process I am thankful to Professor Jan von Del

eTMP programmewould probably not bewhat it is if it were not for RobertHelling His commitment is gratefully acknowledged

My thanks and appreciations also go to Alessandro Michelangeli and every-one else who enriched this final phase of my studies not only on an academicbut also on a personal level

Contents

1 Introduction 511 antum mechanics on combinatorial graphs 612 antum spin systems 7

2 Jordan-Wigner transformation 1121 e Jordan Wigner transformation in one dimension 11

211 Jordan Wigner transformation 11212 Extension to the infinite chain 14

22 Generalisations to graphs 17221 Examples and the existence of special JordanWigner trans-

formations 19222 e two-dimensional Jordan Wigner transformation 23223 A Jordan Wigner transformation for tree graphs 26

23 e WS approach 27231 Link operators and their algebra 28

3 e Gamma matrix model (GMM) 3731 Jordan Wigner transformation for the constrained GMM 3832 Dynamical Localisation and Lieb-Robinson bounds 45

321 e Anderson model on ℤ119889 45322 Lieb Robinson bounds and exponential clustering 49

4 Conclusion and Outlook 61

A Some missing calculations in Srsquos theorem 63

Bibliography 67

C

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

Contents

1 Introduction 511 antum mechanics on combinatorial graphs 612 antum spin systems 7

2 Jordan-Wigner transformation 1121 e Jordan Wigner transformation in one dimension 11

211 Jordan Wigner transformation 11212 Extension to the infinite chain 14

22 Generalisations to graphs 17221 Examples and the existence of special JordanWigner trans-

formations 19222 e two-dimensional Jordan Wigner transformation 23223 A Jordan Wigner transformation for tree graphs 26

23 e WS approach 27231 Link operators and their algebra 28

3 e Gamma matrix model (GMM) 3731 Jordan Wigner transformation for the constrained GMM 3832 Dynamical Localisation and Lieb-Robinson bounds 45

321 e Anderson model on ℤ119889 45322 Lieb Robinson bounds and exponential clustering 49

4 Conclusion and Outlook 61

A Some missing calculations in Srsquos theorem 63

Bibliography 67

C

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

C

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

Chapter 1

Introduction

Q spin systems are an important playground for studying many-bodyquantum phenomena Due to the fact that the Hilbert space at each laice

site is finite-dimensional they are relatively simple models yet still capture someimportant physical features that would otherwise be too difficult to treat in thegeneral framework is includes the study of quantum dynamics thus focussingon many-body scaering effects

In the study of one-dimensional quantum spin systems a transformationbetween spins (or hard-core bosons) and fermions first introduced by Jand W [WJ28] has proven to be a valuable tool It can be used to transferresults from (non-interacting) fermionic systems to spin systems with nearestneighbour interactions (Heisenberg model 119883119884-model 119883119883119885-model et cetera)

Recent developments in the theory of random Schroumldinger operators in par-ticular the Anderson model make it possible to analyse random quantum spinsystems and look for analogues of dynamical localisation (the notion of spec-tral or eigenfunction localisation does not make sense any longer in many-bodysystems)

is thesis has the following objectives

(i) Review the one-dimensional JordanWigner transformation and discuss gen-eralisations to general (finite connected) simple graphs In particular it willbe proven that the occurrence of statistical gauge fields in the transformedHamiltonian is related to the structure of the graph

(ii) Introduce an indirect Jordan Wigner transformation due to W andS [Wos82 Szc85] and apply it to a particular quantum spin systemndash the constrained Gamma Matrix model

(iii) Present zero-velocity Lieb Robinson bounds (in disorder average) whosevalidity can be used to define localisation of quantum spin systems in ran-

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

C 1 I

dom magnetic fields similar to strong dynamical localisation in the Ander-son model ese can be used to prove exponential decay of (averaged)ground state correlations [HSS12] and therefore provide a simplified ver-sion of the mobility gap concept introduced by H [Has10]

(iv) Prove a zero-velocity Lieb Robinson type bound for the constrained GammaMatrixModel thereby providing another example (next to the isotropic ran-dom 119883119884 chain discussed by H et al [HSS12]) of the above concepts

In the remainder of this introductory chapter some basic notions of quantummechanics on combinatorial graphs and quantum spin systems are reviewed

11 antum meanics on combinatorial graphs

Combinatorial graphs are the underlying laices in tight binding models of elec-trons in solid state physics hard core bosons in optical laices and spin systems(just to name a few applications)

A combinatorial graph120126 = (119881 119864) consists of a set of vertices119881 = 119881(120126) andedges (or links) 119864 = 119864(120126) sub 119909 119910 ∶ 119909 119910 isin 119881 connecting them Two vertices 119909and 119910 are said to be neighbouring 119909 sim 119910 if 119909 119910 isin 119864 e graph 119866 is assumedto be simple ie two vertices are at most connected by one edge and vertices arenot connected to themselves Let Adj(120126) be the adjacency matrix of 120126 ie thesymmetric |119881| times |119881|-matrix with elements

Adj(120126)(119909 119910) =⎧⎪⎨⎪⎩

1 if 119909 sim 1199100 otherwise

A simple path 119979119873 is a graph with vertex set 119881 = 1199071113567 hellip 119907119873 and edges119864 = 1199071113567 1199071113568 1199071113568 1199071113569 hellip 119907119873minus1113568 119907119873 such that 119907119896 ne 119907119895 for all 119896 119895 isin 1 hellip 119873ere is a natural ordering on119979119873 namely 119907119895 ⪯ 119907119896 if and only if 119895 le 119896 ie 119907119895 lieson the path from 1199071113567 to 119907119896

If additionally the first and the last vertex 1199071113567 and 119907119873 are connected by an edgethe resulting graph is called a cycle or simple loop 119966119873+1113568 of length 119873 + 1

Bethe lattice and Cayley tree

e Bethe laice 120121119911 with coordination number 119911 ge 2 is the (infinite) connectedcycle-free graphwith the property that each vertex has exactly 119911 neighbours Fix-ing one particular vertex as root 119900 the collection of its 119911 neighbours is referredto as the first shell Each vertex in the first shell has another (119911minus1) child vertices

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

12

119900

120121119911(119873)

(a)

119900

120139119896(119873)

(b)

Figure 11 (a) Cayley tree 120121119911(119873) and (b) its forward part 120139119896(119873)

combined forming the second shell Proceeding along up to 119873 shells the result-ing tree120121119911(119873) is called Cayley tree (of size119873) e 119899th shell has119977119899 = 119911(119911minus1)119899minus1113568vertices and the Cayley tree 120121119911(119873) has a total number of

119977 = 1 +1198731114012119899=1113568

119977119899 = 1 + 119911(119911 minus 1)119873 minus 1

119911 minus 2 = 119911(119911 minus 1)119873 minus 2119911 minus 2

verticesere is a natural (partial) ordering on 120121119911 defined by

119909 ⪯ 119910 ∶hArr x lies on the unique path from 119900 to 119910

Given two vertices 119909 119910 isin 120121119911 their distance dist(119909 119910) is given by the length of theunique path connecting them In particular the 119899th shell is the set of all vertices119909 with dist(119909 119900) = 119899 and 120121119911(119873) = 119909 isin 120121119911 ∶ dist(119909 119900) le 119873

Sometimes it is customary to consider only the forward part120139119896(119873) = 120121+119911 (119873)

of a Cayley tree ie the regular rooted tree graph with branching number 119896 =119911 minus 1

12 antum spin systemsis section is intended to describe the usual mathematical framework whendiscussing quantum spin systems [BR02 Nac06] Let 120126 be a connected (finite or

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

C 1 I

infinite) graph eg the 119889-dimensional laice ℤ119889 119889 ge 1 or the Bethe laice 120121119911with coordination number 119911 ge 2 (see section 11)

For each 119909 isin 120126 let ℌ119909 = ℌ119909 be a finite dimensional Hilbert spacesup1 Forsimplicity it is assumed that all the Hilbert spaces ℌ119909 have the same dimension119899 ge 2 such that ℌ119909 cong ℂ119899 forall119909 isin 120126 Denote by1200831113567(120126) the set of finite subsets of 120126For Λ isin 1200831113567(120126) define ℌ1113657 =⨂119909isin1113657

ℌ119909Observables of the system at site 119909 are elements of the 119862lowast-algebra 120068119909 =

120069(ℌ119909) of bounded linear operators on ℌ119909 which is isomorphic to the algebra of119899times119899 complex matrices120080119899(ℂ) e algebra of observables for a system in a finiteset Λ is 1200681113657 = 120069(ℌ1113657) = ⨂119909isin1113657

120068119909Since for two disjoint finite sets Λ1113568 cap Λ1113569 = empty one has ℌ11136571113578cup11136571113579 = ℌ11136571113578 otimes ℌ11136571113579

there is a natural embedding of 12006811136571113578 into 12006811136571113578cup11136571113579 given by

12006811136571113578 ni 119860 ↦ 119860 otimes 12079311136571113579 isin 12006811136571113578cup11136571113579

ie 12006811136571113578 cong 12006811136571113578 otimes 12079311136571113579 sube 12006811136571113578cup11136571113579 e 119862lowast-algebra of local observables is defined asthe union of the increasing family (1200681113657)1113657isin1200831113577(120126)

120068111361111136141113602 = 11139961113657isin1200831113577(120126)

1200681113657

and its norm closure is the 119862lowast-algebra of quasi-local observables 120068 = 120068111361111136141113602An interaction (or potential) on the quantum spin system defining a quantum

spin model is a map Φ ∶ 1200831113567(120126) rarr 120068 with the properties that Φ(119883) isin 120068119883and Φ(119883)lowast = Φ(119883) for each 119883 isin 1200831113567(120126) It is called finite range if there is an119877 gt 0 such that for all finite sets 119883 with diam(119883) gt 119877 one has Φ(119883) = 0 eHamiltonian with a finite set Λ and an interaction Φ is given by

1198671113657 = 11986711136621113657 = 1114012

119883sube1113657Φ(119883)

It is a self-adjoint element of 1200681113657

Time evolution

e time evolution (Heisenberg dynamics) of a quantum spin system correspond-ing to an interaction Φ (respectively the Hamiltonian 1198671113657) is given by the one-parameter group (1205721113657119905 )119905isinℝ of automorphisms of 1200681113657

1205721113657119905 (119860) = 11989011136081199051198671113666119860119890minus11136081199051198671113666 119860 isin 1200681113657

It is well-defined by the spectral theorem (1198671113657 being self-adjoint)

sup1e notation 119909 isin 120126 is used throughout this thesis instead of 119909 isin 119881(120126) if there is no roomfor confusion

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

12

Example 11 (e XY model) Let ℌ119909 = ℂ1113569 for each 119909 isin 120126 e anisotropic XYmodel in an external field is defined by the one- and two-body interaction

Φ(119883) =

⎧⎪⎪⎨⎪⎪⎩

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] if 119883 = 119909 119910 anddist(119909 119910) = 11205841199091205901113570119909 if 119883 = 1199090 otherwise

where

1205901113568 = 11141020 11 01114105 1205901113569 = 1114102

0 minusii 0 1114105 1205901113570 = 1114102

1 00 minus11114105

are the Pauli matrices and the real-valued sequences 120583119909 120574119909 and 120584119909 describethe coupling strength anisotropy and external magnetic field strength respect-ively

e Hamiltonian for a finite subset Λ of 120126 is then given by

1198671198831198841113657 = 1114012

119909119910isin11136571113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin1113657

1205841199091205901113570119909 (11)

and formally the Hamiltonian of the full system may be wrien as

119867119883119884 = 1114012119909119910isin120126

1113603111360811136181113619(11136231113624)=1113568

120583119909[(1 + 120574119909)12059011135681199091205901113568119910 + (1 minus 120574119909)12059011135691199091205901113569119910] +1114012119909isin120126

1205841199091205901113570119909

Elements in 1200681113657 Λ isin 1200831113567(120126) are 2|1113657| times2|1113657|-matrices corresponding to polyno-mials in the Pauli matrices and the 2times2 identity matrix asi-local observablesare given by uniform limits of such polynomials

For completeness and later convenience the most important properties of thePauli matrices are stated below

(i) e commutator between two Pauli matrices is given by

1114094120590120572 1205901205731114097 = 2i120598120572120573120574120590120574

where 120598120572120573120574 is the Levi Civitagrave tensor

(ii) e anti-commutator between two Pauli matrices is 1114106120590120572 1205901205731114109 = 2120575120572120573120793

(iii) (120590 120572)1113569 = 120793 Tr(120590 120572) = 0

(iv) 120590 120572120590 120573 = 120575120572120573120793 + i120598120572120573120574120590 120574

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

C 1 I

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49

Chapter 2

Jordan-Wigner transformation

T Jordan Wigner transformation introduced by J and W in[WJ28] has proven to be a valuable tool in studying spin chains and their

(quantum) phase transitions in one dimensional systems most prominently inthe application to the quantumXYmodel in one dimension by L et al [LSM61]

21 e JordanWigner transformation in one dimen-sion

Consider the XYmodel onℤ as defined above In particular the following sectionwill be concerned with the Hamiltonians 119867119883119884

[minus119873119873] for 119873 isin ℕ and make use of anequivalence between the Pauli matrices and fermionic operators

is method known as Jordan Wigner transformation dates back to Jand W [WJ28] and was used by L S and M [LSM61] in theexact solution of the anisotropic XY chain e infinite model was studied byA andM [AM85] aer the correspondence between the Pauli and CARalgebras had been given by A [Ara84]

211 Jordan Wigner transformation

For 119873 isin ℕ let

119867119873 ∶= 119867119883119884[minus119873119873] =

119873minus11135681114012119909=minus119873

120583119909[(1 + 120574119909)12059011135681199091205901113568119909+1113568 + (1 minus 120574119909)12059011135691199091205901113569119909+1113568] +1198731114012119909=minus119873

1205841199091205901113570119909 (21)

As a first step towards establishing the Jordan Wigner transformation oneintroduces the set of raising and lowering operators (hard core bosons)

119887lowast119909 = 120590+119909 = 11135681113569 1114100120590

1113568119909 + i12059011135691199091114103 119887119909 = 120590minus119909 = 1113568

1113569 11141001205901113568119909 minus i12059011135691199091114103 (22)

C 2 JW

for each 119909 isin [minus119873119873] cap ℤey are locally anticommuting 119887lowast119909 119887119909 = 120793 commuting on different laice

sites [119887lowast119909 119887119910] = [119887lowast119909 119887lowast119910] = [119887119909 119887119910] = 0 for 119909 ne 119910 and satisfy a hard core condition1198871113569119909 = (119887lowast119909)1113569 = 0 (mixed algebra of raisinglowering operators)

Furthermore they have the properties

[1205901113570119909 119887lowast119909] = 2119887lowast119909 [1205901113570119909 119887119909] = minus21198871199091205901113570119909119887119909 = minus119887119909 1198871199091205901113570119909 = 119887119909 1205901113570119909119887lowast119909 = 119887lowast119909 119887lowast1199091205901113570119909 = minus119887lowast119909

(23)

By definition the raisinglowering operators preserve the local structure andsatisfy the relations

1205901113568119909 = 119887119909 + 119887lowast119909 1205901113569119909 = i(119887119909 minus 119887lowast119909) 1205901113570119909 = 2119887lowast119909119887119909 minus 120793 = 119887lowast119909119887119909 minus 119887119909119887lowast119909 (24)

In particular one has for 119910 ne 119909

12059011135681199091205901113568119910 + 12059011135691199091205901113569119910 = 2(119887lowast119909119887119910 + 119887lowast119910119887119909)12059011135681199091205901113568119910 minus 12059011135691199091205901113569119910 = 2(119887119909119887119910 + 119887lowast119910119887lowast119909)

(25)

ese can be used to express 119867119873 in terms of the 119887 operators

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[119887lowast119909119887119909+1113568 + 119887lowast119909+1113568119887119909 + 120574119909(119887119909119887119909+1113568 + 119887lowast119909+1113568119887lowast119909)] +1198731114012119909=minus119873

120584119909(2119887lowast119909119887119909 minus 120793) (26)

In order to construct fermionic creation and annihilation operators 119888lowast119895 and 119888119895respectively obeying the canonical anti-commutation relations (CARs)

119888119909 119888lowast119910 = 120575119909119910120793 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0 for all 119909 119910 isin [minus119873119873] cap ℤ (27)

one needs to break the local structure of the bosonic operators 119887 defined aboveOne possible choice is to define 119888(lowast)minus119873 ∶= 119887

(lowast)minus119873 and

119888119909 ∶=⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠119887119909 119888lowast119909 = 119887lowast119909

⎛⎜⎜⎝

119909minus11135681114015119910=minus119873

1205901113570119910⎞⎟⎟⎠ 119910 = minus119873 + 1hellip 119873 (28)

e operator 119878119909 = prod119909minus1113568119910=minus119873 120590

1113570119910 is usually referred to as string kink or soliton oper-

atorsup1 (especially in the physics literature) [Fra13]

sup1Let |plusmn⟩ be the two eigenstates of 1205901113568 en 119878119909 creates a kink at laice site 119909

119878119909 (|+⟩ otimes⋯ otimes |+⟩) = |minus⟩ otimes⋯ otimes |minus⟩ otimes |+⟩ otimes⋯ otimes |+⟩

21 T J W

minus119873

119873

119909

119910

Φ(119909)

Φ(119910)

Figure 21 Graphical representation of the laice sites contributing to the phasesΦ(119909) and Φ(119910) in the one dimensional Jordan Wigner transformation ey aredirectly related to the string operator (see remark)

Lemma 21 (Jordan Wigner Transformation) e operators defined in (28) sat-isfy the canonical anticommutator relations (27)

Proof Without loss of generality let 119909 lt 119910 en by relation 23 and the commut-ativity of the 119887 operators on different laice sites one has (see also figure 21 fora graphical representation of which sites contribute to the product in the JordanWigner transformation)

119888119909119888lowast119910 =⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠119887119909 119887lowast119910

⎛⎜⎜⎝

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠=⎛⎜⎜⎝119887lowast119910

119910minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠1205901113570119909119887119909120590111357011990911140591113840111406011138401114061=minus119887119909

⎛⎜⎜⎝

119909minus11135681114015119911=minus119873

1205901113570119911⎞⎟⎟⎠= minus119888lowast119910119888119909

Since 119888lowast119909119888119909 = 119887lowast119909 1114100prod119909minus1113568119911=minus119873 120590

11135701199111114103 1114100prod119909minus1113568

119911=minus119873 12059011135701199111114103 119887119909 = 119887lowast119909119887119909 and the 119887s are locally anti com-

muting the 119888 operators inherit this property By the hard core condition andessentially the same calculation as above one proves that 119888lowast119909 119888lowast119910 = 119888119909 119888119910 = 0for all 119909 119910 isin minus119873hellip 119873

Remark Another equivalent way of constructing fermionic operators 119888 from thehard core bosons 119887 is to define (cf [LSM61] and the discussion in section 22)

119888119909 ∶= 11989011136081113662(119909)119887119909 119888lowast119909 = 119887lowast119909119890minus11136081113662(119909) (29)

with Φ(119909) = 120587sum119909minus1113568119910=minus119873 119887

lowast119910119887119910 and inverse transformation

119887119909 = 119890minus11136081113662(119909)119888119909 119887lowast119909 = 119888lowast11990911989011136081113662(119909) (210)

is is due to the fact that the operators (119887lowast119910119887119910)119910=minus119873hellip119873 are mutually commutingand thus

exp⎛⎜⎜⎝i120587

119909minus11135681114012119910=minus119873

119887lowast119910119887119910⎞⎟⎟⎠=

119909minus11135681114015119910=minus119873

exp 1114100i120587119887lowast1199101198871199101114103 =119909minus11135681114015119910=minus119873

exp 111410011135681113569 i120587(1205901113570119910 + 120793)1114103 =

119909minus11135681114015119910=minus119873

(minus1205901113570119910)

which is unitarily equivalent to the string operator 119878119909 diams

C 2 JW

Using the relations 119888lowast119909119888119909 = 119887lowast119909119887119909 119888lowast119909119888119909+1113568 = minus119887lowast119909119887119909+1113568 and 119888119909119888119909+1113568 = 119887119909119887119909+1113568 one gets

119867119873 =119873minus11135681114012119909=minus119873

2120583119909[minus119888lowast119909119888119909+1113568 minus 119888lowast119909+1113568119888119909 + 120574119909(119888119909119888119909+1113568 + 119888lowast119909+1113568119888lowast119909)] +1198731114012119909=minus119873

120584119909(2119888lowast119909119888119909 minus 120793)

(211)

212 Extension to the infinite ain

e extension of the Jordan Wigner transformation to the (two-sided) infinitechainℤ reveals some subtleties that were studied by A and M in theirdiscussion of ground states of the 119883119884-model on ℤ [Ara84 AM85] It turns outthat the 119862lowast-algebra 120068119862119860119877 generated by 119888119909 119888lowast119909 ∶ 119909 isin ℤ and the 119862lowast-algebra 120068119875generated by the Pauli spin matrices 120590120572119909 ∶ 120572 = 1 2 3 119909 isin ℤ are different 119862lowast-subalgebras of an enlarged 119862lowast-algebra 1114023120068 Only their even parts with respect toa certain automorphism of 1114023120068 coincide e problem is that the infinite productprodminusinfin

119909=1113567 1205901113570119909 is not a quasi-local observablesup2

Define the automorphism Θminus ∶ 120068119875 ↦ 120068119875

Θminus(119860) ∶= lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠119860⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119909⎞⎟⎟⎠

which corresponds to a rotation of all the spins on the le half of the laice 119909 le 0by 120587 around the 1205901113570-axis Indeed a simple application of the (anti-)commutationrelations of the Pauli matrices shows that for 120572 = 1 2

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= 120590120572119909 lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119909=1113567

1205901113570119910⎞⎟⎟⎠

1113569

1114059111384011138401114060111384011138401114061=120793

= 120590120572119909 for 119909 gt 0

Θminus(120590120572119909 ) = lim119873rarrinfin

⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠120590120572119909⎛⎜⎜⎝

minus1198731114015119910=1113567

1205901113570119910⎞⎟⎟⎠= minus120590120572119909 for 119909 le 0

and Θminus(1205901113570119909) = 1205901113570119909 for all 119909 isin ℤ Further one can easily see that Θminus is an involu-tion Θ 1113569

minus = 120793Now let 119879 be such that

1198791113569 = 120793 119879 lowast = 119879 119879119860119879 = Θminus(119860) forall119860 isin 120068119875

sup2Indeed one has 1114063prodminus119873119909=1113567 120590

11135701199091114063 = 1 for all 119899 isin ℕ thus it cannot converge in norm

21 T J W

and define 1114023120068 as the 119862lowast-algebra generated by 120068119875 and 119879 It can be decomposedinto a direct sum 1114023120068 = 1200681198751198791113567 +1200681198751198791113568 = 120068119875 +120068119875119879 and with this decomposition athand the automorphism Θminus can be extended to 1114023120068 by seing

Θminus(1198601113568 + 1198601113569119879) ∶= Θminus(1198601113568) + Θminus(1198601113569)119879 1198601113568 1198601113569 isin 120068119875

Remark Introducing 119879 and the enlarged 119862lowast-algebra 1114023120068 was necessary to adaptthe Jordan Wigner transformation 28 to the case of an infinite chain

If Λ = [minus119873119873] cap ℤ then Θminus(119860) = 1114100prodminus119873119909=1113567 120590

11135701199091114103119860 1114100prodminus119873

119909=1113567 12059011135701199091114103 = 119879119860119879 for all

119860 isin 120068119875 In particular 119879 = prodminus119873119909=1113567 120590

1113570119909 isin 120068119875 so 1114023120068 = 120068119875 and the above reduces to

the discussion in section 211 diams

Having introduced the enlarged algebra 1114023120068 it is possible to define the creationand annihilation operators 119888lowast119909 119888119909 isin 1114023120068 by

119888lowast119909 ∶= 119879119878119909120590+119909 = 119879119878119909 11141021205901113568119909 + i1205901113569119909

2 1114105

119888119909 ∶= 119879119878119909120590minus119909 = 119879119878119909 11141021205901113568119909 minus i1205901113569119909

2 1114105119878119909 =

⎧⎪⎪⎨⎪⎪⎩

prod119909minus1113568119910=1113568 120590

1113570119910 if 119909 ge 2

120793 if 119909 = 1prod119909

119910=1113567 1205901113570119910 if 119909 le 0

Here one can see that 119879 was used to substituteprod1113567119909=minusinfin 120590

1113570119909 In particular 119879 has

all the necessary properties to make the family of operators 119888119909 119888lowast119909 119909 isin ℤ fulfilcanonical anticommutation relations is follows from [119878119909 119878119910] = 0 1198781113569119909 = 120793 andΘminus(119878119909) = 119878119909 for all 119909 119910 isin ℤ Also notice that for 119909 le 119910 one has

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

120590plusmn119910 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1Θminus(120590plusmn119910 ) 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= Θminus(120590plusmn119910 )119878119909

and for 119909 gt 119910

119878119909120590plusmn119910 =

⎧⎪⎪⎨⎪⎪⎩

1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 120590plusmn119910 119909 ge 2

120590plusmn119910 119909 = 11114100prod119909

119911=1113567 12059011135701199111114103 120590plusmn119910 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

minusΘminus(120590plusmn119910 ) 1114100prod119909minus1113568119911=1113568 120590

11135701199111114103 119909 ge 2

120590plusmn119910 119909 = 1120590plusmn119910 1114100prod119909

119911=1113567 12059011135701199111114103 119909 le 0

⎫⎪⎪⎬⎪⎪⎭

= minusΘminus(120590plusmn119910 )119878119909

Hence

119888119909119888lowast119909 = (119879119878119909120590minus119909 )(119879119878119909120590+119909 ) = 119878119909Θminus(120590minus119909 )1198791113569119878119909120590+119909 = 119878119909Θminus(120590minus119909 )119878119909120590+119909 = 119878119909Θminus(120590minus119909 )Θminus(120590+119909 )119878119909= 119878119909 (120793 minus Θminus(120590+119909 )Θminus(120590minus119909 )) 119878119909 = 120793 minus 119878119909Θminus(120590+119909 )119878119909120590minus119909 = 120793 minus 119878119909Θminus(120590+119909 )1198791113569119878119909120590minus119909= 120793 minus 119879119878119909120590+119909119879119878119909120590minus119909 = 120793 minus 119888lowast119909119888119909

C 2 JW

and for 119909 lt 119910 one gets

119888119909119888lowast119910 = 119879119878119909120590minus119909119879119878119910120590+119910 = 119878119909Θminus(120590minus119909 )119878119910120590+119910 = minus119878119909119878119910120590minus119909120590+119910 = minus119878119910119878119909120590+119910120590minus119909= minus119878119910Θminus(120590+119910 )119878119909120590minus119909 = minus119878119910Θminus(120590+119910 )1198791113569119878119909120590minus119909 = minus119879119878119910120590+119910119879119878119909120590minus119909 = minus119888lowast119910119888119909

Similarly it can be proven that 119888119909 119888119910 = 119888lowast119909 119888lowast119910 = 0Let 120068119862119860119877 denote the 119862lowast-subalgebra of 1114023120068 generated by those operators It

follows immediately that

Θminus(119888lowast119909) =⎧⎪⎨⎪⎩

119888lowast119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Θminus(119888119909) =⎧⎪⎨⎪⎩

119888119909 if 119909 ge 1minus119888lowast119909 if 119909 le 0

Next the relation between 120068119875 and 120068119862119860119877 is described To do so one has tointroduce another involutive automorphism Θ of 1114023120068 describing the rotation ofall spins around the 1205901113570-axis by 120587 By the construction of 1114023120068 it suffices to define

Θ(120590120572119909 ) = minus120590120572119909 (120572 = 1 2)Θ(1205901113570119909) = 1205901113570119909Θ(119879) = 119879

for 119909 isin ℤ and extend it to an automorphism on 1114023120068 Accordingly the action onthe fermionic operators is

Θ(119888lowast119909) = minus119888lowast119909 Θ(119888119909) = minus119888119909 (119909 isin ℤ)

Clearly Θ 1113569 = 120793 and by definition Θ leaves 120068119875 and 120068119862119860119877 invariant ie for119860 isin 120068119875119862119860119877 also Θ(119860) isin 120068119875119862119860119877

Θ can now be used to decompose both subalgebras into an even (Θ = 1) andan odd part (Θ = minus1) For an operator 119860 isin 120068119875119862119860119877 one has 119860 = 119860+ + 119860minus where119860plusmn = 1113568

1113569 (119860 plusmn Θ(119860)) and 120068119875119862119860119877 = 120068119875119862119860119877+ + 120068119875119862119860119877minus with 120068119875119862119860119877plusmn = 119860 isin 120068119875119862119860119877 ∶

Θ(119860) = plusmn119860 e relation between the spin and CAR algebras is given by

120068119875+ = 120068119862119860119877+ 120068119875minus = 119879120068119862119860119877minus (212)

Define another involutive automorphism 1114026Θminus ∶ 1114023120068 rarr 1114023120068 by

1114026Θminus(1198601113568 + 1198791198601113569) = 1198601113568 minus 1198791198601113569 11986011135681113569 isin 120068119875

en

Θ11141001114026Θminus(1198601113568 + 1198791198601113569)1114103 = Θ (1198601113568 minus 1198791198601113569) = Θ(1198601113568) minus 119879Θ(1198601113569)= 1114026Θminus (Θ(1198601113568) + 119879Θ(1198601113569)) = 1114026Θminus (Θ(1198601113568 + 1198791198601113569))

22 G

ie Θ1114026Θminus = 1114026ΘminusΘ on 1114023120068Since 1114023120068 = 120068119875 + 119879120068119875 the extended algebra 1114023120068 can be decomposed into

1114023120068 = 120068119862119860119877+ + 120068119862119860119877minus + 119879120068119862119860119877+ + 119879120068119862119860119877minus

120068119875 = 120068119862119860119877+ + 119879120068119862119860119877minus

e local Hamiltonian 119867119873 (21) is quadratic in the spin operators and there-fore lies in the positive subalgebra of120068119875 But the positive subalgebras of the spinand CAR algebras coincide which concludes the discussion of the JordanWignertransformation in the case of an infinite chain

22 Generalisations to graphseLSM ansatz 29 can be used to define a JordanWigner trans-formation on arbitrary (connected finite) graphs In this seing the Hamiltoniantransforms to a more difficult one including statistical gauge fields which in gen-eral cannot be removed by gauge transformations

Let 120126 = (119881 119864) be a finite connected graph and 119888119909119909isin120126 a fermionic field on120126 Define the new operators

119886119909 = 11989011136081113662(119909)119888119909 119886lowast119909 = 119888lowast119909119890minus11136081113662(119909) (213)

with Φ(119909) = 120587sum119911isin120126 120593(119909 119911)119888lowast119911119888119911 = 120587sum119911ne119909 120593(119909 119911)119888

lowast119911119888119911 120593(119909 119911) isin ℝ for all 119909 119911 isin 120126

e laer equality uses that without loss of generality one can assume120593(119909 119909) =0 since 1198901113608120601(119909119909)119888lowast119909119888119909119888119909 = 119888119909

Indeed since 119888lowast119909119888119909 is bounded one has the norm-convergent series represent-ation

1198901113608120572119888lowast119909119888119909119888119909 =⎛⎜⎜⎝120793 +1114012

119896ge1113568

(i120572)119896119896 (119888

lowast119909119888119909)119896

⎞⎟⎟⎠119888119909 = 119888119909 120572 isin ℝ

due to the property 1198881113569119909 = 0 e same applies to 119888lowast1199091198901113608120572119888lowast119909119888119909 = 119888lowast119909

By the canonical anti commutation relations the 119886 operators satisfy

119886119909119886lowast119910 = 11989011136081113662(119909)119888119909119888lowast119910119890minus11136081113662(119910) = 120575119909119910120793 minus 11989011136081113662(119909)119888lowast119910119888119909119890minus11136081113662(119910)

= 120575119909119910120793 minus 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888

lowast119910119888119910111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=1198901113634120587120593(119909119910)119888lowast119910

11989011136081113662(119909)119890minus11136081113662(119910) 1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909111405911138401113840111384011138401113840111384011140601113840111384011138401113840111384011138401114061

=119890minus1113634120587120593(119910119909)119888119909

= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119888lowast119910119890minus11136081113662(119910)11989011136081113662(119909)119888119909= 120575119909119910120793 minus 1198901113608120587(120593(119909119910)minus120593(119910119909))119886lowast119910119886119909

(214)

C 2 JW

for 119909 119910 isin 120126 If

120593(119909 119910) minus 120593(119910 119909) = 120599 mod 2 forall119909 ne 119910 (215)

then the 119886 operators are said to satisfy hard core anyonic statistics ie

119886119909119886lowast119910 = 120575119909119910120793 minus 1198901113608120587120599119886lowast119910119886119909 (216)

e parameter 120599 isin [0 1] interpolates between fermionic (120599 = 0) and hard-corebosonic (120599 = 1) statistics and the hard-core property (119886(lowast)119909 )1113569 = 0 follows from thesame property of the fermionic 119888 operators

In the calculation above the equalities

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910

1198901113608120587120593(119910119909)119888lowast119909119888119909119888119909119890minus1113608120587120593(119910119909)119888lowast119909119888119909 = 119890minus1113608120587120593(119910119909)119888119909

have been used ey follow from the following observation in a basis where

119888lowast119909119888119909 is diagonal 119888lowast119909119888119909 = 11141021 00 01114105

119909

119888119909 = 11141020 01 01114105

119909

and 119888lowast119909 = 11141020 10 01114105

119909

en

1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910119890minus1113608120587120593(119909119910)119888lowast119910119888119910 = 1198901113608120587120593(119909119910)119888lowast119910119888119910119888lowast119910 = 1114102

1198901113608120587120593(119909119910) 00 11114105

11991011141020 10 01114105

119910

= 11141020 1198901113608120587120593(119909119910)0 0 1114105

119910

= 1198901113608120587120593(119909119910)119888lowast119910

An analogous calculation yields the second equalityUnless stated otherwise it shall from now on be assumed that 120599 = 1 so that

the 119886 operators describe hard-core bosons Let 119867 be the Hamiltonian of freefermions on 120126 with nearest neighbour hopping

119867 =1114012119909sim119910(119888lowast119909119888119910 + 119888lowast119910119888119909)

en the transformation 213 yields the unitarily equivalent Hamiltonian

119867 =1114012119909sim119910(119886lowast11990911989011136081113662(119909)119890minus11136081113662(119910)119886119910 + 119886lowast11991011989011136081113662(119910)119890minus11136081113662(119909)119886119909) = 1114012

119909sim119910(119886lowast119909119890minus1113608119860(119909119910)119886119910 + 119886lowast1199101198901113608119860(119909119910)119886119909)

(217)

with a statistical gauge field

119860(119909 119910) = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911 = 120587 1114012119911ne119909119910

1114094120593(119910 119911) minus 120593(119909 119911)1114097 119886lowast119911119886119911 (218)

where it has been taken into account that 119886lowast1199091198901113608120573119886lowast119909119886119909 = 119886lowast119909 and 1198901113608120573119886

lowast119909119886119909119886119909 = 119886119909 for all

120573 isin ℝ due to the hard-core property (119886(lowast)119909 )1113569 = 0

22 G

Statistical gauge transformations

From calculation 214 it follows immediately that transformations of the form119909 = 11989011136081113657(119909)119886119909withΛ(119909) = 120587sum119911ne119909 120582(119909 119911)119886

lowast119911119886119911 leave the particle statistics unchanged

iff

120582(119909 119910) = 120582(119910 119909) mod 2 forall119909 ne 119910 (219)

ey can be used to modify the statistical gauge field 119860(119909 119910) (218) while pre-serving particle statistics Explicitly the Hamiltonian 217 is unitarily equivalentto

119867 =1114012119909sim119910(lowast11990911989011136081113657(119909)119890minus1113608119860(119909119910)119890minus11136081113657(119910)119910 + lowast11991011989011136081113657(119910)1198901113608119860(119909119910)119890minus11136081113657(119909)119909)

= 1114012119909sim119910(lowast119909119890minus1113608(119909119910)119910 + lowast1199101198901113608(119909119910)119909)

(220)

with

(119909 119910) = 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 119886lowast119911119886119911

= 120587 1114012119911ne119909119910

1114094(120593(119910 119911) minus 120593(119909 119911)) + (120582(119910 119911) minus 120582(119909 119911))1114097 lowast119911119911(221)

In principle one would like to use such statistical gauge transformations to get ridof the statistical gauge field in 217 altogether is could be achieved by leing120582(119908 119911) = minus120593(119908 119911) for all 119908 ne 119911 But by condition 219 one would have

minus120593(119908 119911) = minus120593(119911 119908) mod 2 forall119908 ne 119911

in contradiction to 215

120593(119908 119911) minus 120593(119911 119908) = 1 mod 2 forall119908 ne 119911

In the following section it will be shown that the occurrence of statisticalgauge fields in Jordan Wigner transformations has to do with the structure ofthe underlying graph 120126

221 Examples and the existence of special JordanWigner trans-formations

It is an interesting question whether there are Jordan Wigner transformationson a given graph 120126 which map fermions to hard-core bosons without introdu-cing a statistical gauge field 119860 Such fields correspond to higher order non-local

C 2 JW

interactions

119886lowast119909119890minus1113608119860(119909119910)119886119910 = 119886lowast119909⎛⎜⎜⎝1114015119911ne119909119910

119890minus1113608120587(120593(119910119911)minus120593(119909119911))119886lowast119911119886119911⎞⎟⎟⎠119886119910

= 119886lowast119909 1114015119911ne119909119910

1114100120793 + (119890minus1113608120587(120593(119910119911)minus120593(119909119911)) minus 1)119886lowast1199111198861199111114103119886119910

since 1198901113608120572119886lowast119911119886119911 = 1198901113608120572119886lowast119911119886119911 + (120793 minus 119886lowast119911119886119911) 120572 isin ℝEquation 218 shows that 119860(119909 119910) = 0 if and only if

120593(119909 119911) = 120593(119910 119911) mod 2 forall119909 sim 119910 119911 ne 119909 119910

Together with the statistics condition 215 this yields the following system forthe phases 120593(119909 119910) 119909 ne 119910 isin 120126

Adj(120126)(119909 119910)(120593(119909 119911) minus 120593(119910 119911)) = 0 mod 2 forall119911 ne 119909 119910120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 (222)

with the adjacency matrixAdj(120126) of 120126 Leing 119890 denote the number of edges in120126 and 119907 the number of vertices there are in total 119907(119907 minus 1) variables 120593(119909 119910) and119907(119907minus1113568)1113569 equations defining the particle statistics together with 119890(119907 minus 2) equations

for the non-occurrence of a statistical gauge field For the graph to be minimallyconnected one has 119890 ge 119907 minus 1 thus the system will in general be overdeterminedfor 119907 gt 4 since

119907(119907 minus 1)2 minus 119890(119907 minus 2) le 119907(119907 minus 1)

2 minus (119907 minus 1)(119907 minus 2) lt 0 if 119907 gt 4

Definition 22 (Special Jordan Wigner transformations) A transformation ofthe above type satisfying the conditions 222 is henceforth called special JordanWigner transformation in this thesis

Example 23 (Simple Paths) Let 120126 = 119979119873 be a simple path of length119873 with ver-tices 1199071113567 hellip 119907119873 en as in the one dimensional Jordan Wigner transformation

120593(119909 119911) = Θ(119909 119911) ∶=⎧⎪⎨⎪⎩

1 if 119911 ≺ 1199090 if 119911 ⪰ 119909

is a solution to 222 is can be seen by noting that

Θ(119911 119909) = 11141081 if 119909 ≺ 1199110 if 119909 ⪰ 1199111114111 = Θ(119909 119911) + 1 minus 120575119909119911 mod 2 (223)

22 G

1199071113567 119907119873

119979119873

(a)

119966119873

1

2

119873

(b)

1198840

1

2

3(c)

Figure 22 e graphs considered in examples 2ndash4 (a) Simple path119979119873 (b) Cyclegraph 119966119873 (c) 119884-graph

so Θ(119911 119909) minus Θ(119909 119911) = 1 minus 120575119909119911 = 1 mod 2 for 119909 ne 119911 Also if 119909 sim 119910 one has

Θ(119909 119911) minus Θ(119910 119911) = 1114108minus120575119911119910 if 119909 ≺ 119910120575119911119909 if 119910 ≺ 1199091114111 = 0 for 119911 ne 119909 119910

Hence there exists a special Jordan Wigner transformation on simple paths119979119873 which is of course not surprising since this is the exact same case as in onedimension diams

Example 24 (Cycle Graph) Let 120126 be a cycle graph 119966119873 119873 ge 3 then the systemof equations 222 determining the phases 120593(119909 119910) is given by

120593(1 119911) = 120593(2 119911) mod 2 forall119911 ge 3120593(2 119911) = 120593(3 119911) mod 2 forall119911 ge 4 119911 le 1

⋮120593(119873 minus 1 119911) = 120593(119873 119911) mod 2 forall119911 le 119873 minus 2

120593(119873 119911) = 120593(1 119911) mod 2 forall2 le 119911 le 119873 minus 1

together with the statistics conditions

120593(119909 119910) = 120593(119910 119909) + 1 mod 2 forall119909 ne 119910 isin 1 hellip 119873

In particular one has

120593(1 3) = 120593(2 3) = 120593(3 2) + 1 = ⋯ = 120593(119873 2) + 1= 120593(1 2) + 1 = 120593(2 1) = 120593(3 1) = 120593(1 3) + 1

meaning that one can construct the contradiction 0 = 1 out of part of the aboveconditions so there exists no special Jordan Wigner transformation on 119966119873 forany 119873 ge 3 diams

C 2 JW

Example 25 (119884-graph) Let120126 be a 119884-graph (sometimes also called 3-legged stargraph or claw) ie the graph with vertex set 119881 = 0 1 2 3 and edge set 119864 =0 119895 ∶ 119895 = 1 2 3 e system of equations 222 reads (everything mod 2)

120593(0 2) = 120593(1 2)120593(0 3) = 120593(1 3)

120593(0 1) = 120593(2 1)120593(0 3) = 120593(2 3)

120593(0 1) = 120593(3 1)120593(0 2) = 120593(3 2)

120593(119909 119910) = 120593(119910 119909) + 1 forall119909 ne 119910

But then

120593(0 2) = 120593(1 2) = 120593(2 1) + 1 = 120593(0 1) + 1 = 120593(3 1) + 1= 120593(1 3) = 120593(0 3) = 120593(2 3) = 120593(3 2) + 1 = 120593(0 2) + 1

So 0 = 1 contradictionis shows that there cannot be a special Jordan Wigner transformation on

the 119884-graph diams

eorem 26 Let 120126 = (119881 119864) be a finite connected graph en there exists aspecial Jordan Wigner transformation in the above sense if and only if the graph isa simple path 120126 = 119979|119881|minus1113568

Proof If 120126 = 119979|119881|minus1113568 then the special Jordan Wigner transformation is given inexample 23

For the converse assume that 120126 is not a simple path en there are twocases

(i) 120126 is a cycle graph 119966|119881| (ii) 120126 contains a 119884-graphAssume120126 is a cycle graph It was proved in example 24 that the system 222

has no solution so there exists no special Jordan Wigner transformationerefore assume that120126 ne 119979|119881|+1113568 119966|119881| In this case it contains a 119884-subgraph

120126119884 = (1199071113567 1199071113568 1199071113569 1199071113570 1199071113567 119907119895 ∶ 119895 = 1 2 3)Assuming further that the system of equations 222 admits a solution on 120126

the equations in particular have to be true for 120126119884 But as shown in example 25these equations are enough to produce the contradiction

120593(1199071113567 1199071113569) = 120593(1199071113568 1199071113569) = 120593(1199071113569 1199071113568) + 1 = 120593(1199071113567 1199071113568) + 1 = 120593(1199071113570 1199071113568) + 1= 120593(1199071113568 1199071113570) = 120593(1199071113567 1199071113570) = 120593(1199071113569 1199071113570) = 120593(1199071113570 1199071113569) + 1 = 120593(1199071113567 1199071113569) + 1

ie 0 = 1Hence the system 222 cannot have solutions on 120126

22 G

222 e two-dimensional Jordan Wigner transformation

Let Λ = [minus119871 119871]1113569 cap ℤ1113569 be a square in ℤ1113569 centred at 0 and 119888119909 119888lowast119909 ∶ 119909 isin Λ bea family of fermionic annihilationcreation operators on Λ e canonical basisvectors are denoted by 1198901113568 and 1198901113569

As an immediate consequence of theorem 26 there cannot exist a specialJordan Wigner transformation on the two-dimensional laice Λ Neverthelessthere do exist Jordan Wigner transformations leading to statistical gauge fieldsey are solutions to the condition 215 (for 120599 = 1)

120593(119909 119910) minus 120593(119910 119909) = 1 mod 2 forall119909 ne 119910 (224)

and shall be discussed in the following two subsections

e FW solution

Let Arg(119911) = ImLog 119911 isin [minus120587 120587) be the relative angle between 119911 (identifying119911 = 1199111113568 + i1199111113569 with 119911 = (1199111113568 1199111113569) isin Λ) and an arbitrary reference axis en by theproperty

Arg(119911) = Arg(minus119911) + 120587 mod 2120587

one can see that 120593(119909 119911) = 1113568120587Arg(119911 minus 119909) satisfies condition 224 e resulting

Jordan Wigner transformation goes back to F and W [Fra89 Wan92]Under this transformation a Hamiltonian of hard core bosons with nearest-

neighbour hopping119867 =1114012

119909isin11136571114012119895=11135681113569

1114100119887lowast119909119887119909+119890119895 + 119887lowast119909+1198901198951198871199091114103

transforms to the Hamiltonian

119867 =1114012119909isin1113657

1114012119895=11135681113569

1114100119888lowast1199091198901113608119860(119909119909+119890119895)119888119909+119890119895 + 119888lowast119909+119890119895119890minus1113608119860(119909119909+119890119895)1198881199091114103

of fermions with nearest-neighbour hopping coupled to a statistical gauge field119860(119909 119910) defined on the edges (119909 119910) of the laice with

119860(119909 119909 + 119890119895) = 120587 1114012119911ne119909119909+119890119895

1114094120593(119909 + 119890119895 119911) minus 120593(119909 119911)1114097 119888lowast119911119888119911

e statistical gauge field 119860 generates a flux through elementary plaquees119875 of the laiceΛ Let 119875 be such an elementary plaquee with corners 119909 119909+1198901113568 119909+

C 2 JW

Φ(119909)

Φ(119910)

119909

119910

(a)

119909

119909 + 1198901113568

119909 + 1198901113568 + 1198901113569

119909 + 1198901113569

119875

120597119875

(b)

Figure 23 Jordan Wigner transformation in two-dimensional laices (a) elaice sites contributing to the phases in the A solution (b) Elementaryplaquee in Λ

1198901113568 + 1198901113569 and 119909 + 1198901113569 (see figure 23) en the statistical flux through 119875 is given by119861119875 = sumℓisin120597119875119860(ℓ) explicitly

119861119875 = 119860(119909 119909 + 1198901113568) + 119860(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 119860(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569) minus 119860(119909 119909 + 1198901113569)= 120587 1114094120593(119909 + 1198901113569 119909) minus 120593(119909 + 1198901113568 119909)1114097 119888lowast119909119888119909+ 120587 1114094120593(119909 119909 + 1198901113568) minus 120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113568)1114097 119888lowast119909+1198901113578119888119909+1198901113578+ 120587 1114094120593(119909 + 1198901113568 119909 + 1198901113568 + 1198901113569) minus 120593(119909 + 1198901113569 119909 + 1198901113568 + 1198901113569)1114097 119888lowast119909+1198901113578+1198901113579119888119909+119890+1113568+1198901113579+ 120587 1114094120593(119909 + 1198901113568 + 1198901113569 119909 + 1198901113569) minus 120593(119909 119909 + 1198901113569)1114097 119888lowast119909+1198901113579119888119909+1198901113579

= 1205871113569 119888lowast119909119888119909 + 120587

1113569 119888lowast119909+1198901113578119888119909+1198901113578 +

1205871113569 119888lowast119909+1198901113578+1198901113579119888119909+1198901113578+1198901113579 minus

11135701205871113569 119888

lowast119909+1198901113579119888119909+1198901113579

e A solution

In his PhD thesis M A proposed another solution to the two dimensionalJordan Wigner transformationsup3 [Azz93] It uses a more natural generalisation ofthe one-dimensional solution with a phase function taking only the values 0 or

sup3Apparently unaware of this result S published the same solution a few years later[Sha95] His conclusions were wrong in several points though [BA01]

22 G

1120593(119909 119911) = Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 (225)

Here 119909 = (1199091113568 1199091113569) 119911 = (1199111113568 1199111113569) and Θ is the step function

Θ(119909119895 minus 119911119895) =⎧⎪⎨⎪⎩

1 if minus 119871 le 119911119895 lt 1199091198950 if 119909119895 le 119911119895 le 119871

To check that condition 224 is indeed satisfied one only needs to recall the prop-erty 223 of the step function to see that120593(119911 119909) = (Θ(1199091113568 minus 1199111113568) + 1 minus 12057511990911135781199111113578)(1 minus 12057511990911135781199111113578) + (Θ(1199091113569 minus 1199111113569) + 1 minus 12057511990911135791199111113579)12057511990911135781199111113578

= Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135781199111113578)1113569 + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578 + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + (1 minus 12057511990911135781199111113578) + (1 minus 12057511990911135791199111113579)12057511990911135781199111113578= 120593(119909 119911) + 1 minus 1205751199091113578119911111357812057511990911135791199111113579 = 120593(119909 119911) + 1 if 119909 ne 119911

en one has

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119888lowast119911119888119911 = 120587

⎛⎜⎜⎝

1199091113578minus111356811140121199111113578=minus119871

11987111140121199111113579=minus119871

119888lowast(11991111135781199111113579)119888(11991111135781199111113579) +1199091113579minus111356811140121199111113579=minus119871

119888lowast(11990911135781199111113579)119888(11990911135781199111113579)⎞⎟⎟⎠

which results in a statistical gauge field of the form

119860(119909 119909 + 1198901113568) = 120587⎛⎜⎜⎝

1198711114012

1199111113579=1199091113579+1113568119888lowast(11990911135781199111113579)119888(11990911135781199111113579) +

1199091113579minus111356811140121199111113579=minus119871

119888lowast(1199091113578+11135681199111113579)119888(1199091113578+11135681199111113579)⎞⎟⎟⎠

119860(119909 119909 + 1198901113569) = 0in the transformed fermionic Hamiltonian and thus the statistical flux

119861119875 = 120587 1114100119888lowast119909+1198901113579119888119909+1198901113579 minus 119888lowast119909+1198901113578119888119909+11989011135781114103 It is interesting to note that even though the two Jordan Wigner transform-

ations produce a different flux 119861119875 the two transformed Hamiltonians are stillunitarily equivalent and the corresponding statistical gauge transformation (inthe sense of 219) is given by

120582(119909 119910) = minus 1113568120587Arg(119911 minus 119909) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578) + Θ(1199091113569 minus 1199111113569)12057511990911135781199111113578

Remark Arsquos solution 225 has an immediate generalisation to higher di-mensional laices for instance in three dimensions Λ = [minus119871 119871]1113570 cap ℤ1113570 thefunction 120593(119909 119911) would read [BA01 Koc95 HZ93]

120593(119909 119911) = Θ(1199091113570 minus 1199111113570)(1 minus 12057511990911135801199111113580) + Θ(1199091113568 minus 1199111113568)(1 minus 12057511990911135781199111113578)12057511990911135801199111113580+ Θ(1199091113569 minus 1199111113569)(1 minus 12057511990911135791199111113579)1205751199091113578119911111357812057511990911135801199111113580

diams

C 2 JW

119900

120579=1

119909 = (3 5)

Figure 24 Polar coordinates on the Cayley tree

223 A Jordan Wigner transformation for tree graphs

Having discussed the JordanWigner transformation for two-dimensional laicesit is straightforward to construct such a transformation for Cayley trees 119861119911(119873)by representing each vertex in terms of suitable polar coordinates and using amodification of the A solution Invoking theorem 26 also in this casethere will be a statistical gauge field in the transformed Hamiltonian

To this aim one assigns each vertex 119909 of the tree the coordinates 119909 = (119903119909 120579119909)where 119903119909 = dist(119909 119900) isin 0 hellip 119873 denotes the shell in which 119909 is situated and120579119909 isin 1 hellip 119977119903119909 is the ldquoanglerdquo measured from some reference path (see figure24) en given a family of hard-core bosonic operators 119887 on120121119911(119873) it is possibleto define Jordan-Wigner fermions via 119888119909 = 11989011136081113662(119909)119887119909 with

Φ(119909) = 1205871114012119911ne119909120593(119909 119911)119887lowast119911119887119911 (226)

= 120587⎛⎜⎜⎝119887lowast119900119887119900 +

119903119909minus11135681114012119903119911=1113568

119977119903119909

1114012120579119911=1113568

119887lowast(119903119911120579119911)119887(119903119911120579119911) +119977119903119909

1114012120579119911=120579119909+1113578

119887lowast(119903119909120579119911)119887(119903119909120579119911)⎞⎟⎟⎠ (227)

ie 120593(119909 119911) = Θ(119903119909 minus 119903119911)(1 minus 120575119903119909119903119911) + Θ(120579119909 minus 120579119911)120575119903119909119903119911 e verification of condition224 is done by a calculation analogous to the one in the two-dimensional case

eHamiltonian of nearest neighbour hopping hard-core bosons is then unit-arily equivalent to

119867 =1114012119909sim1199101114100119888lowast1199091198901113608119860(119909119910)119888119910 + 119888lowast119910119890minus1113608119860(119909119910)1198881199091114103

23 T WS

with statistical gauge field (assuming 119909 sim 119910 119909 ≺ 119910)

119860(119909 119910) = 120587120579119909minus11135681114012120579119911=1113568

119886lowast(119903119909120579119911)119886(119903119909120579119911) + 120587119977119903119910

1114012120579119911=120579119910+1113568

119886lowast(119903119910120579119911)119886(119903119910120579119911)

23 e WS approa

An alternative approach to a generalisation of the one-dimensional JordanWignertransformation⁴ is based upon the observation that the Pauli matrices 1205901113568 and 1205901113569are generators of a 2times2matrix representation of the Clifford algebra ℭ120105(2) since120590120572 120590120573 = 2120575120572120573120793

Clifford algebras in a nutshell

Let 119889 isin ℕ e Clifford algebra ℭ120105(2119889) is the algebra generated by 2119889 elementssatisfying the Clifford algebra relation

Γ 119886 Γ 119887 = Γ 119886Γ 119887 + Γ 119887Γ 119886 = 2120575119886119887120793 119886 119887 isin 1 hellip 2119889 (228)

us the Clifford algebra consists of

120793Γ 119886 119886 = 1hellip 2119889Γ 119886Γ 119887 1 le 119886 lt 119887 le 2119889Γ 119886Γ 119887Γ 119888 1 le 119886 lt 119887 lt 119888 le 2119889⋮Γ 1113568⋯Γ 1113569119889

e special element Γ 1113569119889+1113568 = (minusi)119889Γ 1113568⋯Γ 1113569119889 with the property (Γ 1113569119889+1113568)1113569 = 120793anticommutes with all the other generators of the Clifford algebra Γ 119886 Γ 1113569119889+1113568 =0 for all 119886 = 1hellip 2119889

One can show that the antisymmetric products of two Γrsquos

Γ 119886119887 = 111356811135691113608 [Γ

119886 Γ 119887] = minusiΓ 119886Γ 119887 1 le 119886 lt 119887 le 2119889 (229)

generate a representation of SO(2119889) and for 1 le 119886 lt 119887 le 2119889+1 a representationof SO(2119889 + 1) [Geo99]

⁴N [Nam50] used a similar argument in the one dimensional case which is whyWand S [Wos82 Szc85] refer to this method as the ldquoNambu trickrdquo

C 2 JW

eClifford algebraℭ120105(2119889) can be represented by the algebra1200801113569119889(ℂ) of 2119889times2119889matrices and generators satisfying the above relation for their anticommutatorFor 119889 = 1 these are usually represented by the two Pauli matrices 1205901113568 and 1205901113569 (inthis case the element Γ 1113570 is represented by 1205901113570 since 1205901113570 = minusi12059011135681205901113569) whereas in thecase 119889 = 2 a representation of the ℭ120105(4) generators is given by the Dirac Gammamatrices Γ 1113568 hellip Γ 1113571

For later use some elementary properties of the elements Γ 119886119887 are stated here

[Γ 119886 Γ 119887119888] = 0 for 119886 ne 119887 119888 (230)Γ 119886 Γ 119886119887 = 0 for 119886 ne 119887 (231)[Γ 119886119887 Γ 119888119889] = 0 for (119886 119887) ne (119888 119889) (232)Γ 119886119887 Γ 119886119888 = 0 for 119887 ne 119888 and 119886 ne 119887 119888 (233)

231 Link operators and their algebra

Let 119867 be the Hamiltonian of free fermions described by fermionic creation andannihilation operators 119888119909 119888lowast119909 on a finite symmetric digraph⁵ 120131 = (Λ 119864)

119867 = 2120583 1114012119909119910isin119864

1114100119888lowast119909119888119910 + 119888lowast1199101198881199091114103 +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793) (234)

e parameter 120583 isin ℝ models the hopping strength to the nearest neighbourlaice site and 120584119909 sub ℝ describe an on-site external potential in which thefermions move It will later be assumed that the external potential is given byiid random variables (see chapter 32)

e fermionic operators can be expressed in terms of two self-adjoint Major-ana operators for each site

120585119909 = 119888lowast119909 + 119888119909 and 120578119909 = minusi(119888119909 minus 119888lowast119909) (235)

with the properties

120577lowast119909 = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578 (236)

that is the operators generate a representation of ℭ120105(2|Λ|) In particular 1205771113569119909 = 120793for all 119909 isin Λ and 120577 = 120585 120578

en substituting 119888lowast119909 = 11135681113569 (120585119909 minus i120578119909) and 119888119909 =

11135681113569 (120585119909 + i120578119909) in the Hamiltonian

119867 yields

119867 = 1205831114012119909119910isin119864

1114100i120585119909120578119910 minus i1205781199091205851199101114103 +1114012119909isin1113657

120584119909i120585119909120578119909

⁵at is a directed graph 120131 = (Λ 119864) with the property that if (119909 119910) isin 119864 then also (119910 119909) isin 119864

23 T WS

120585

120578

Figure 25 Example of the double graph for a two-dimensional square laice

To allow a more compact description of the products of two Majorana fer-mions on neighbouring vertices and an easier framework for the proof of histheorem S used the notion of a double laice (see figure 25) in his pa-per [Szc85]

Definition 27 e (directed) double laicegraph 1114026120131 is the directed graph withvertex set 1114026Λ = Λ times 120585 120578 and edge set where (119909120577 119910120589) = ((119909 120577) (119910 120589)) isin ifone of the following holds (i) 119909119910 isin 119864 or (ii) 119909 = 119910 and 120577 ne 120589

Operators defined on the edges of the (double) graph are usually called linkoperators Given a family of link operators 119878(ℓ) ∶ ℓ isin and a path 120574 = ℓ1113568∘⋯∘ℓ119899of length 119899 isin ℕ one defines the associated path operator as

119878(120574) = (minusi)119899minus1113568119878(ℓ1113568)⋯119878(ℓ119899)

(for a degenerate path consisting of only one vertex 119907 one sets 119878(119907) = i120793)

Definition 28 (Link algebra) Let 119878(ℓ) ∶ ℓ isin be a family of operators definedon the edges of the double laice (link operators) ey satisfy the link algebra ifthey have the following properties

(i) 119878(ℓ)lowast = 119878(ℓ) (119878(ℓ))1113569 = 120793 for all ℓ isin

(ii) 119878(ℓ) 119878(ℓprime) = 0 if the edges ℓ ℓprime isin have one common vertex and[119878(ℓ) 119878(ℓprime)] = 0 otherwise

(iii) Tr 1114100prod119909isin1113657 119878(119909120585 119909120578)1114103 = 0

(iv) If 120574 is closed then 119878(120574) = i120793

Example 29 e operators on 1114026120131 defined by 119878(119909120577 119910120589) = i120577119909120589119910 where 120577 120589 isin 120585 120578and 120585119909 120578119909 being a family of Majorana fermions on Λ as above satisfy the linkalgebra

C 2 JW

e Hamiltonian 119867 expressed in terms of these link operators is given by

119867 = 1205831114012119909119910isin119864

1114100119878(119909120585 119910120578) minus 119878(119909120578 119910120585)1114103 +1114012119909isin1113657

120584119909119878(119909120585 119909120578)

e first two defining properties (i)-(ii) follow immediately from the Major-ana algebra 236 To see that (iv) holds let 120574 = ℓ1113568 ∘ ⋯ ∘ ℓ119873 be a closed path oflength 119873 in 1114026120131 ℓ119894 = ((119909119894 120577119894) (119909119894+1113568 120577119894+1113568)) 119894 = 1hellip 119873 minus 1 ℓ119873 = ((119909119873 120577119873) (1199091113568 1205771113568))en

119878(120574) = (minusi)119873minus1113568(i1205771113568119909111357812057711135691199091113579)(i1205771113569119909111357912057711135701199091113581)⋯ (i120577119873minus1113568119909119873minus1113578120577119873119909119873 )(i12057711987311990911987312057711135681199091113578)= (minusi)119873minus1113568i11987312057711135681199091113578(12057711135691199091113579)1113569⋯(120577119873119909119873 )111356912057711135681199091113578 = i120793

Finally one has

Tr1114015119909isin1113657

119878(119909120585 119909120578) = Tr1114015119909isin1113657(i120585119909120578119909) = Tr1114015

119909isin1113657(2119888lowast119909119888119909 minus 120793) =1114015

119909isin1113657Trℌ119909(2119888

lowast119909119888119909 minus 120793) = 0

since the operators 2119888lowast119909119888119909 minus 120793119909isin1113657 are mutually commuting with eigenvalues plusmn1of the same multiplicity diams

S now proved in [Szc85] that the properties of the link operators(definition 28) determine the link algebra uniquely up to unitary isomorphismsis result can be used to prove an implicit version of the Jordan Wigner trans-formation by finding a family of operators satisfying the link algebra e mainresult is a representation of the link algebra in terms of higher-dimensional gammamatrices with appropriate constraints that reduce the dimensionality to the re-quired two per laice site describing fermionic degrees of freedom e precisestatement of the theorem and its proof are presented in the following e nextchapter will then deal with an application of this method to a square laice

eorem 210 (Szczerba [Szc85]) Let 120131 = (Λ 119864) be a directed graph and 1114026120131 =(Λ ) its associated double graph en given a set 119878prime(ℓ) ∶ ℓ isin of link operatorson a finite-dimensional Hilbert space ℌ satisfying the link algebra there exists afamily of Majorana operators 120585119909 120578119909 ∶ 119909 isin Λ on ℌ obeying the relations 236 suchthat

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120577 120589 isin 120585 120578 (237)

If 119878(ℓ) ∶ ℓ isin denotes the link operators from example 29 there is a unitarytransformation 119880 with

119880119878(ℓ)119880 lowast = 119878prime(ℓ) for all ℓ isin (238)

23 T WS

e proof of theorem 210 proceeds in two steps by reduction of the problemto a rooted spanning tree of 120131 where the natural ordering on rooted trees canbe used to construct the required operators 120585 and 120578 and later extended to thewhole graphLemma 211 Let 120139 = (Λ 120139) be a directed tree on Λ with an arbitrary vertexfixed as root = 119900120585 and let 119878prime(ℓ) ∶ ℓ isin 120139 be a family of link operators ona finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced bythe condition

Tr1114100119878prime(120574119900120578)1114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 0 (239)

where 120574119909120577 denotes the unique path from the root of 1114026120139 to the vertex 119909120577 (property(iv) holds automatically since a tree does not contain any cycles) en thereexists a family of operators 120585119909 120578119909 ∶ 119909 isin Λ obeying the Clifford relations 236such that

119878prime(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin 120139 120577 120589 isin 120585 120578 (240)

Proof [Szc85] Consider the family of path operators 119878prime(120574) ∶ ne isin 120139 eyhave the properties

119878prime(120574)lowast = 119878prime(120574)119878prime(120574) 119878prime(120574) = 2120575 for ne (241)

A proof of this and some other useful properties of the path operators are presen-ted in appendix A Defining 119878 = i |1113657|minus1113568119878prime(120574119900120578)prod119909ne119900 119878

prime(120574119909120585)119878prime(120574119909120578) by means of241 one has

119878lowast = (minusi)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103119878prime(120574119900120578) = (minusi)|1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120578)119878prime(120574119909120585)1114103

= (minusi)|1113657|minus1113568119878prime(120574119900120578)(minus1)|1113657|minus111356811141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103 = 119878

and

1198781113569 = i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103i |1113657|minus1113568119878prime(120574119900120578)11141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)1114103

= (minus1)|1113657|minus1113568119878prime(120574119900120578)111356911141001114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)11141031113569

= (minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)119878prime(120574119909120578)119878prime(120574119909120585)119878prime(120574119909120578)

= (minus1)|1113657|minus1113568(minus1)|1113657|minus11135681114015119909ne119900

119878prime(120574119909120585)1113569119878prime(120574119909120578)1113569 = 120793

C 2 JW

Further by assumption 239 Tr119878 = 0 erefore there exists a unitary auto-morphism119882 of ℌ that diagonalises 119878

119882119878119882 lowast = 1114102minus120793 00 1207931114105

e corresponding eigenspaces are denoted by ℌplusmn Now for each ne the pathoperators 119878prime(120574) commute with S [119878 119878prime(120574)] = 0 so that in the ONB where 119878 isdiagonal the path operators are block diagonal

119882119878prime(120574)119882 lowast = 1114102119879minus 00 119879+ 1114105

ne

By the properties 241 it follows immediately that for all ne one has

(119879 )lowast = 119879119879 119879 = 2120575

isin + minus (242)

It is a basic result in the theory of operator algebras (cf eorem 525 in [BR02])that the operators 119879 are uniquely determined up to unitary transformationsHence there exists a unitary isomorphism 119881 ∶ ℌminus rarr ℌ+ such that

119881119879minus119881 lowast = 119879+ for all ne 119900120585 119900120578 (243)

Here it is important that from 242 it does not follow the existence of such aunitary transformation for = 119900120578 One rather gets from

1114102minus120793 00 1207931114105 = 119882119878119882 lowast = i |1113657|minus1113568 1114102

119879minus119900120578prod119909ne119900 119879minus119909120585119879minus119909120578 0

0 119879+119900120578 prod119909ne119900 119879+119909120585119879+119909120578

1114105

= i |1113657|minus1113568 1114102119879minus119900120578prod119909ne119900 119879

minus119909120585119879minus119909120578 0

0 119879+119900120578119881 prod119909ne119900 119879+119909120585119879+119909120578119881 lowast1114105

the relations

i |1113657|minus11135681114100119879minus1199001205781114015119909ne119900

119879minus119909120585119879minus1199091205781114103 = minus120793

i |1113657|minus11135681114100119879+1199001205781198811114015119909ne119900

119879+119909120585119879+119909120578119881 lowast1114103 = 120793

and thus by (119879119900120578)1113569 = 120793

119879+119900120578 = minus119881119879minus119900120578119881 lowast (244)

23 T WS

is is the key observation in the proof of this lemma since this unitary iso-morphism can now be used to construct the desired operators Also the abovestep reveals one drawback of the WS method it is only used thatsuch a 119881 exists but not how it is constructed (which will be quite difficult if thenumber of laice sites |Λ| is very high)

Now define the operator

120578119900 = 11141020 119881 lowast

119881 0 1114105 (245)

with the properties

(120578119900)lowast = 120578prime119900(120578119900)1113569 = 120793

120578119900119882119878prime(120574119900120578)119882 lowast = 0[120578119900119882119878prime(120574)119882 lowast] = 0 for all ne 119900120585 119900120578

(246)

e first two equalities follow immediately from the definition of 120578119900 for the thirdequality one uses

120578119900119882119878prime(120574119900120578)119882 lowast = 11141020 119881 lowast

119881 0 1114105 1114102119879minus119900120578 00 119879+119900120578

1114105 = 11141020 119881 lowast119879+119900120578

119881119879minus119900120578 0 1114105

= 11141020 minus119879minus119900120578119881 lowast

minus119879+119900120578119881 0 1114105 = 1114102minus119879minus119900120578 00 minus119879+119900120578

1114105 11141020 119881 lowast

119881 0 1114105 = minus119882119878prime(120574119900120578)119882 lowast120578119900

by means of 244 e same calculation for 119878prime(120574) ( ne 119900120585 119900120578) and 243 yields thefourth equality

Finally define for 119909 ne 119900 the operators

120585119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900 = i120578119900119882119878prime(120574119900120578)119882 lowast

120585119909 = minusi120585119900119882119878prime(120574119909120585)119882 lowast

120578119909 = minusi120585119900119882119878prime(120574119909120578)119882 lowast(247)

Since 1205851113569119900 = minusi119882119878prime(120574119900120578)119882 lowast120578119900i120578119900119882119878prime(120574119900120578)119882 lowast = 120793 the laer two equalities canbe used to represent a path operator as 119882119878prime(120574119909120577)119882 lowast = i120585119909120577119909 for 120577 = 120585 120578 andcollecting all the previous properties one gets

(120577119909)lowast = 120577119909120577119909 120589119910 = 2120575119909119910120575120577120589 with 120577 120589 isin 120585 120578

It remains to show that each link operators can be represented as productof two of these operators ereto let ℓ = (119909120577 119910120589) isin and assume first that

C 2 JW

119909120577 = 119900120585 en 119878prime(ℓ) = 119878prime(120574119910120589) = 119882 lowasti120585119900120589119910119882 For all other edges with 119909120577 ne 119900120585one has 120574119909120577 ∘ (119909120577 119910120589) = 120574119910120589 and therefore by definition of the path operators119878prime(120574119910120589) = minusi119878prime(120574119909120577)119878prime(119909120577 119910120589) Since 119878prime(120574119909120577)1113569 = 120793 it follows that

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowasti120585119900120577119909119882119882 lowasti120585119900120589119910119882= 119882 lowast(minusi120585119900120577119909120585119900120589119910)119882 = 119882 lowasti1205851113569119900120577119909120589119910119882 = 119882 lowasti120577119909120589119910119882

is concludes the proof

With lemma 211 at hand it is straightforward to prove the theorem for gen-eral graphs

Proof of Srsquos theorem 210 [Szc85] Fix an arbitrary point 119909120585 equiv 119900 and let 1114026120139be a directed spanning tree of 1114026120131with root 119900 By lemma 211 applied to the family119878prime(ℓ) ∶ ℓ isin 120139 of link operators on the spanning tree there exists a family ofMajorana operators defined on all the vertices of 1114026120131 such that119882119878prime(ℓ)119882 lowast = i120577119909120589119910for any ℓ = (119909120577 119910120589) isin 120139 Indeed the only property that needs to be checkedin order to be able to apply this lemma is the trace equality 239 is followsdirectly from property (iii) (definition 28) of operators satisfying the link algebrasince 119878prime(119909120585 119909120578) = i119878prime(120574119909120585)119878prime(120574119909120578) as argued at the end of the proof of lemma 211is reduces point (iii) of the link algebra to 239

Using these Majorana fermions one can define new link operators on thewhole double laice by

(ℓ) = i120577119909120589119910 ℓ = (119909120577 119910120589) isin

ese operators are unitarily equivalent to the link operators 119878prime(ℓ) for ℓ isin 120139Furthermore for any edge in 120131 one has

119878prime(119909120577 119910120589) = i119878prime(120574119909120577)119878prime(120574119910120589)(119909120577 119910120589) = i(120574119909120577)(120574119910120589)

(248)

where 120574119909120577 and 120574119910120589 denote the unique paths from the root 119900 to 119909120577 (119910120589 respectively)along the directed spanning tree 120139 erefore the corresponding path operatorsare products of link operators to edges in 120139 only which implies

119882119878prime(120574119909120577)119882 lowast = (120574119909120577)

By means of 248 it then follows that

119878prime(ℓ) = i119878prime(120574119909120577)119878prime(120574119910120589) = i119882 lowast(120574119909120577)119882119882 lowast(120574119910120589)119882 = 119882 lowasti(120574119909120577)(120574119910120589)119882= 119882 lowast(ℓ)119882

for all edges ℓ isin is concludes the proof that the link algebra (definition 28)defines the link operators uniquely up to unitary transformations (119880 = 119882 lowast)

23 T WS

e reason why this result can be thought of as a generalisation of the usualJordan Wigner transformation is explained in context of the ldquoGamma MatrixModelrdquo in the following chapter

C 2 JW

Chapter 3

e Gamma matrix model (GMM)

I order to demonstrate the applicability of the WS method thequantum spin system on a square laiceΛ = [1 119871]1113569capℤ1113569 with periodic bound-

ary conditions and Hamiltonian (Gamma matrix model)

1198671113654 = 1205831114012119909isin1113657

1114100Γ 1113568119909Γ 11135691113572119909+1198901113578 + Γ 1113570119909Γ 11135711113572119909+1198901113579 minus Γ 11135681113572119909 Γ 1113569119909+1198901113578 minus Γ 11135701113572119909 Γ 1113571119909+11989011135791114103 +1114012119909isin1113657

120584119909Γ 1113572119909 (31)

with nearest-neighbour interaction strength 120583 isin ℝ and in an external mag-netic field described by 120584119909119909isin1113657 sub ℝ is discussed e Hamiltonian acts onℌ =⨂

119909isin1113657ℂ1113571

1198671113654 can be interpreted physically as the Hamiltonian of spin-32 particlesfixed at the sites of the laice Λ with short-range quadrupole-octupole interac-tions A similar Hamiltonian (Gammamatrix model) has been discussed recentlyby Y et al [YZK09] and W et al [WCF12] in the context of a spin- 11135701113569generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices withconnectivity greater than three It is argued in [YZK09] that the ground state oftheir Hamiltonian is an algebraic spin liquid ie a spin liquid whose fermionicexcitations (spinons) are gapless

If one defines the matrices ([MNZ04])

1198781113568 =

⎛⎜⎜⎜⎜⎝

radic11135701113569

radic11135701113569 1

1 radic11135701113569

radic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113569 =

⎛⎜⎜⎜⎜⎝

minusiradic11135701113569iradic11135701113569 minusi

i minusiradic11135701113569iradic11135701113569

⎞⎟⎟⎟⎟⎠

1198781113570 =

⎛⎜⎜⎜⎝

11135701113569 1113568

1113569minus 11135681113569

minus 11135701113569

⎞⎟⎟⎟⎠

the Gamma matrices can be represented by symmetric bilinear combinations of

C 3 T G (GMM)

the SU(2) spin- 11135701113569 matrices namely

Γ 1113568 = 11987611135691113570 = 1113568radic11135701198781113569 1198781113570 = 1205901113570 otimes 1205901113569

Γ 1113569 = 11987611135681113570 = 1113568radic11135701198781113568 1198781113570 = 1205901113570 otimes 1205901113568

Γ 1113570 = 11987611135681113569 = 1113568radic11135701198781113568 1198781113569 = 1205901113569 otimes 120793

Γ 1113571 = 119876(1113569) = 1113568radic11135701114100(1198781113568)1113569 minus (1198781113569)11135691114103 = 1205901113568 otimes 120793

Γ 1113572 = 119876(1113567) = (1198781113570)1113569 minus 11135721113571120793 = 120590

1113570 otimes 1205901113570

Hence the Γ 119886 matrices may be interpreted as spin- 11135701113569 quadrupole (or nematic) op-erators 119876 (the Γ 1198861113572 operators corresponding to spin octupole operators) [MNZ04Wu06 TZY06 WCF12] or alternatively as two-orbit spin- 11135681113569 operators [Wen03WCF12]

31 JordanWigner transformation for the constrainedGMM

In the discussion of the Gamma matrix model it is useful to introduce a familyof (elementary) plaquee (flux) operators For a vertex 119909 isin Λ the elementaryplaquee 119875 equiv 119875119909 is defined as the square with corners 119909 119909 + 1198901113568 119909 + 1198901113568 + 1198901113569 119909 + 1198901113569Its boundary 120597119875 will always be supposed to be positively oriented Given sucha plaquee one can define

119882119875 = (Γ 1113568119909Γ 1113569119909+1198901113578)(Γ 1113570119909+1198901113578Γ 1113571119909+1198901113578+1198901113579)(Γ 1113569119909+1198901113578+1198901113579Γ 1113568119909+1198901113579)(Γ 1113571119909+1198901113579Γ 1113570119909)= minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579

(32)

Additionally for a fixed value of 1199091113569 (respectively 1199091113568) one can define two global(flux) operators

119882119883 equiv 119882119883(1199091113569) =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

119882119884 equiv 119882119884(1199091113568) =11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568)

(33)

e collection of the plaquee and the global (flux) operators will be denotedby 119882120572 ∶ 120572 isin 119883 119884 119875 ∶ 119875 elementary plaquee ey have the followingimportant propertiesLemma 31 e family 119882120572120572isin119883119884119875∶119875 elementary plaquee satisfies the algebra

119882 lowast120572 = 119882120572 1198821113569

120572 = 120793[119882120572119882120573] = 0 [119882120572 1198671113654] = 0Tr(1198821205721113578⋯119882120572119896) = 0

31 J W GMM

for all 120572 120573 and 120572119894 ne 120572119895 whenever 119894 ne 119895

In physics language the operators119882120572 correspond to ℤ1113569 fluxes through theelementary plaquees 119875 and two globalℤ1113569 fluxes [Wen03 YZK09] To make thismore apparent one may write119882120572 = minusexp(iΦ120572)withΦ120572 taking the values 0 and120587

Proof e self-adjointness of the plaquee and global flux operators follows im-mediately from the self-adjointness of Γ 119886 and Γ 119886119887 for 119886 119887 = 1hellip 5 the identity1198821113569

120572 = 120793 from (Γ 119886)1113569 = (Γ 119886119887)1113569 = 120793 for 119886 119887 = 1hellip 5Given two elementary plaquees 119875 and 119876 that have no vertex in common

the property [119882119875119882119876] = 0 is trivial If they have one common vertex the com-mutator is zero as a consequence of equation 232 eg if 119875 and 119876 have theirlower le respectively upper right corner 119909 in common then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)= (minusΓ 11135681113570119909minus1198901113578minus1198901113579Γ 11135701113569119909minus1198901113579Γ 11135691113571119909 Γ 11135711113568119909minus1198901113578)(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

In case they have two vertices in common then the commutation property fol-lows from equation 233 For example if the boom two vertices of 119875 119909 and119909 + 1198901113568 and the top two vertices of 119876 coincide then

119882119875119882119876 = (minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579)(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )= (minus1)1113569(minusΓ 11135681113570119909minus1198901113579Γ 11135701113569119909+1198901113578minus1198901113579Γ 11135691113571119909+1198901113578Γ 11135711113568119909 )(minusΓ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135691113571119909+1198901113578+1198901113579Γ 11135711113568119909+1198901113579) = 119882119876119882119875

As for the global flux operators one has

119882119883119882119884 =11987111140151199111113578=1113568

Γ 1113568(11991111135781199091113579)Γ1113569(1199111113578+11135681199091113579)

11987111140151199111113579=1113568

Γ 1113570(11990911135781199111113579)Γ1113571(11990911135781199111113579+1113568) = 119882119884119882119883

since the only non-trivial commutators are those between Gamma matrices atthe intersection point 119909 = (1199091113568 1199091113569) of the two global loops

120574119883 = 120574119883(1199091113569) = (1 1199091113569) rarr (2 1199091113569) rarr ⋯ rarr (119871 1199091113569) rarr (1 1199091113569)120574119884 = 120574119884(1199091113568) = (1199091113568 1) rarr (1199091113568 2) rarr ⋯rarr (1199091113568 119871) rarr (1199091113568 1)

where the commutation relation follows from the short calculation

Γ 1113569119909Γ 1113568119909Γ 1113571119909Γ 1113570119909 = Γ 1113571119909Γ 1113570119909Γ 1113569119909Γ 1113568119909

Given119882119883 and a plaquee operator119882119875 there are three possible situations

(i) 119875 and the loop 120574119883 have no common vertex in which case [119882119883 119882119875] = 0

C 3 T G (GMM)

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

1113568

1113569

1113570

1113571

Figure 31 Sketch of the assignment of the Gamma matrices to the edges of thesquare laice Λ

(ii) e boom edge of 119875 (119909 119909+1198901113568) lies in 120574119883 en [119882119883 119882119875] = 0 follows from[Γ 1113569119909Γ 1113568119909Γ 1113569119909+1198901113578Γ 1113568119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = minus[Γ 11135681113569119909 Γ 11135681113569119909+1198901113578 Γ 11135681113570119909 Γ 11135701113569119909+1198901113578] = 0 these terms beingthe only operators in 119882119883 and 119882119875 that do not commute trivially Indeedone has by (233)

Γ 11135681113569119909 Γ 11135681113569119909+1198901113578Γ 11135681113570119909 Γ 11135701113569119909+1198901113578 = Γ 11135681113569119909 Γ 11135681113570119909 Γ 11135681113569119909+1198901113578Γ 11135701113569

119909+1198901113578 = 1114100minusΓ11135681113570119909 Γ 11135681113569119909 1114103 1114100minusΓ 11135701113569119909+1198901113578Γ 11135681113569

119909+11989011135781114103= Γ 11135681113570119909 Γ 11135701113569119909+1198901113578Γ 11135681113569119909 Γ 11135681113569119909+1198901113578

(iii) In case the top edge of 119875 lies in 120574119883 then [119882119883 119882119875] = 0 can be shown by asimilar calculation to the one in (ii)

Analogously it follows that [119882119884 119882119875] = 0 for all elementary plaquees 119875In the same manner using (230)-(233) one can prove that the flux operators

commute with the Hamiltonian 1198671113654 by showing that the commutators of eachsummand in 1198671113654 with 119882120572 are zero distinguishing the cases where they havenone one or two vertices in common

e proof of the trace property is based on the identities TrΓ 119886 = 0 andTrΓ 119886119887 = 0 (Γ 119886119887 being antisymmetric) as well as the tensor product structureof the flux operators Since the trace property is not essential in the remainderof this thesis its rather lengthy proof is omied here

In the laice Λ there are |Λ| = 1198711113569 elementary plaquees 119875 which give riseto |Λ| minus 1 independent plaquee operators119882119875 since there is the constraint

1114015119875 elem plaq

119882119875 = 120793

which follows from the assumption of periodic boundary conditions and the factthat (Γ 119886)1113569 = 120793 As for the global flux operators defining 119882119883 equiv 119882119883(1199091113569) and119882119884 equiv 119882119884(1199091113568) for fixed 1199091113569 and 1199091113568 determines119882119883119884(11991011135691113568) for any other 1199101113569 and 1199101113568is follows from119882119883(1199101113569) = 119882119883prod119875isin119982 (11990911135791199101113579)

119882119875 where119982 (1199091113569 1199101113569) denotes the setof all elementary plaquees in the strip 119911 isin Λ ∶ 1199111113569 isin [min(1199091113569 1199101113569)max(1199091113569 1199101113569)]

31 J W GMM

Similarly 119882119884(1199101113568) = 119882119884prod119875isin119982 (11990911135781199101113578)119882119875 erefore there are altogether |Λ| + 1

independent elements in 119882120572Since they all commutewith theHamiltonian eigenstates of1198671113654 can be chosen

to be eigenstates of the 119882120572 as well such that the Hilbert space ℌ splits into sec-tors ℌ119908120572 of fixed flux configurations ie 119882120572120595119908120572 = 119908120572120595119908120572 for 120595119908120572 isin ℌ119908120572and for each 120572

ℌ = 1114008119908120572isinplusmn1113568|1113666|+1113578

ℌ119908120572 (34)

e orthogonal projections onto the respective subsectors are given by

Ξ119908120572 = 1114102120793 + 119908119909119882119883

2 1114105 1114102120793 + 119908119910119882119884

2 1114105 1114015119875sub120131

119875 elem plaq

1114102120793 + 119908119875119882119875

2 1114105

e projection onto the sector with 119908120572 = 1 forall120572 will be of particular import-ance and is henceforth just denoted by Ξ = Ξ1113568 In particular one has

ℌ = Ξℌ oplus (120793 minus Ξ)ℌ

In view of the previous chapter the connection to link operators is describednext Let 120131 = (Λ 119864) be the symmetric digraph corresponding to the squarelaice and let 1114026120131 be its directed double laice For an edge ℓ isin one defines thefamily

1198781113654(119909120585 (119909 + 1198901113568)120585) = minus1198781113654((119909 + 1198901113568)120585 119909120585) = Γ 1113568119909Γ 1113569119909+11989011135781198781113654(119909120585 (119909 + 1198901113569)120585) = minus1198781113654((119909 + 1198901113569)120585 119909120585) = Γ 1113570119909Γ 1113571119909+11989011135791198781113654(119909120578 (119909 + 1198901113568)120578) = minus1198781113654((119909 + 1198901113568)120578 119909120578) = Γ 11135681113572119909 Γ 11135691113572

119909+1198901113578

1198781113654(119909120578 (119909 + 1198901113569)120578) = minus1198781113654((119909 + 1198901113569)120578 119909120578) = Γ 11135701113572119909 Γ 11135711113572119909+11989011135791198781113654(119909120585 119909120578) = minus1198781113654(119909120578 119909120585) = Γ 1113572119909

(35)

of link operatorsIn terms of these operators the Hamiltonian can be wrien as

1198671113654 = 1205831114012119909isin1113657

11141011198781113654(119909120585 (119909 + 1198901113568)120578) + 1198781113654(119909120585 (119909 + 1198901113569)120578)

minus 1198781113654(119909120578 (119909 + 1198901113568)120585) minus 1198781113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091198781113654(119909120585 119909120578)(36)

e link operators 35 clearly satisfy properties (i) and (ii) of the link algebra(see definition 28) Also 1198781113654(ℓminus1113568) = minus1198781113654(ℓ) for all ℓ isin by definition

C 3 T G (GMM)

For an elementary plaquee 119875 in Λ times 120585 one has

1198781113654(120597119875) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568 + 1198901113569)120585)sdotsdot 1198781113654((119909 + 1198901113568 + 1198901113569)120585 (119909 + 1198901113569)120585)1198781113654((119909 + 1198901113569)120585 119909120585)

= (minusi)1113570119882119875 = i119882119875

Furthermore for the loop

120574119909 = (119909120585 (119909 + 1198901113568)120585) rarr ((119909 + 1198901113568)120585 (119909 + 1198901113568)120578) rarr ((119909 + 1198901113568)120578 119909120578) rarr (119909120578 119909120585)

with arbitrary corner 119909 isin Λ

1198781113654(120574119909) = (minusi)11135701198781113654(119909120585 (119909 + 1198901113568)120585)1198781113654((119909 + 1198901113568)120585 (119909 + 1198901113568)120578)1198781113654((119909 + 1198901113568)120578 119909120578)1198781113654(119909120578 119909120585)= (minusi)1113570Γ 1113568119909Γ 1113569

119909+1198901113578Γ 1113572119909+1198901113578(minusΓ 11135681113572119909 Γ 11135691113572

119909+1198901113578)(minusΓ 1113572119909) = (minusi)1113570Γ 1113568119909Γ 11135681113572119909 Γ 1113572119909 iΓ11135691113572119909+1198901113578Γ 11135691113572119909+1198901113578= (minusi)1113569(minusΓ 1113568119909Γ 1113572119909Γ 11135681113572119909 )120793119909+1198901113578 = i(Γ 11135681113572119909 )1113569120793119909+1198901113578 = i120793

Substituting (1 harr 3) and (2 harr 4) proves the identity 1198781113654(119909) = i120793 for the loop

119909 = (119909120585 (119909 + 1198901113569)120585) rarr ((119909 + 1198901113569)120585 (119909 + 1198901113569)120578) rarr ((119909 + 1198901113569)120578 119909120578) rarr (119909120578 119909120585)

For the global loop 120574119883 = (1 1199091113569)120585 rarr (2 1199091113569)120585 rarr⋯rarr (119871 1199091113569)120585 rarr (1 1199091113569)120585 (with 1199091113569fixed) one has

1198781113654(120574119883) = (minusi)119871minus11135681198781113654((1 1199091113569)120585 (2 1199091113569)120585)⋯1198781113654((119871 minus 1 1199091113569)120585 (119871 1199091113569)120585)1198781113654((119871 1199091113569)120585 (1 1199091113569)120585)= (minusi)119871minus1113568Γ 1113568(11135681199091113579)Γ

1113569(11135691199091113579)⋯Γ 1113568(119871minus11135681199091113579)Γ

1113569(1198711199091113579)Γ

1113568(1198711199091113579)Γ

1113569(11135681199091113579) = (minusi)

119871minus1113568119882119883

Analogously 1198781113654(120574119884) = (minusi)119871minus1113568119882119884 for a global loop in 1199091113569-direction at fixed 1199091113568erefore the link operators 1198781113654(ℓ) satisfy property (iv) of the link algebra if

119882119875 = 120793 for all elementary plaquees 119875 119882119883 = 119882119884 = 120793 and 119871 isin 4ℕ In thiscase 1198781113654(120574) = i120793 for all closed loops in the double laice 1114026120131 since any such loopoperator can be decomposed into a product of loop operators corresponding toelementary plaquees or either of the two global loop operators119882119883119884

e identities 119882120572 = 120793 forall120572 isin 119883 119884 119875 hold exactly in the sector ℌ1113568 of theHilbert space

Lemma 32 eprojection operatorΞ commuteswith all link operators [Ξ 1198781113654(ℓ)] =0 for all ℓ isin

Proof By the very same argumentation as in lemma 31 it follows that [119882120572 1198781113654(ℓ)] =0 and thus [120793+119882120572

1113569 1198781113654(ℓ)] = 0 for all 120572 and ℓ isin is implies [Ξ 1198781113654(ℓ)] = 0 forall ℓ isin

31 J W GMM

By lemma 32 the link operators leave the space ℌ1113568 = Ξℌ = ranΞ invariantIt follows that the family of constrained link operators

11987811136541113658(ℓ) = Ξ1198781113654(ℓ)Ξ11140701113617111360011136131113658 ℓ isin (37)

inherits properties (i) (ii) and (iv) from the corresponding ones of the operators1198781113654(ℓ)

Any other closed path in 1114026120131 can be decomposed into a product of the aboveelementary loops erefore property (iv) holds for the family of constrainedoperators 11987811136541113658(ℓ) ∶ ℓ isin e constrained Hamiltonian is given by

11986711136541113658 = 1205831114012

119909isin1113657111410111987811136541113658(119909120585 (119909 + 1198901113568)120578) + 11987811136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11987811136541113658(119909120578 (119909 + 1198901113568)120585) minus 11987811136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

12058411990911987811136541113658(119909120585 119909120578)(38)

It still remains to check the trace property (iii) By the identity

1114015119909isin1113657

Γ 1113572119909 =1114015119909isin1113657(minusi)1113569Γ 1113568119909Γ 1113569119909Γ 1113570119909Γ 1113571119909 = (minus1)|1113657|11141011114015

119909isin1113657Γ 1113568119909Γ 1113569119909111410411141011114015

119909isin1113657Γ 1113570119909Γ 11135711199091114104

= (minus1)|1113657|(minus1)1113569(119871minus1113568)111410111987111140151199091113578=1113568

11987111140151199091113579=1113568

Γ 1113568(11990911135781199091113579)Γ

1113569(1199091113578+11135681199091113579)11141041114101

11987111140151199091113579=1113568

11987111140151199091113578=1113568

Γ 1113570(11990911135781199091113579)Γ1113571(11990911135781199091113579+1113568)1114104

= 111410111987111140151199091113578=1113568

119882119884(1199091113568)1114104111410111987111140151199091113579=1113568

119882119883(1199091113569)1114104

it follows that on the subspace Ξℌ one has Ξ1114100prod119909isin1113657 Γ11135721199091114103Ξ1114071

1113617111360011136131113658= 120793 and thus

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11987811136541113658(119909120585 119909120578)⎞⎟⎟⎠= Tr1113658ℌ

⎛⎜⎜⎝Ξ11141001114015

119909isin1113657Γ 11135721199091114103Ξ1114071

1113658ℌ

⎞⎟⎟⎠= Tr1113658ℌ 11141001207931113658ℌ1114103 = rkΞ = 2|1113657|minus1113568

Hence the family of constrained link operators does not fulfil the link algebraand Srsquos theorem is not directly applicable erefore an auxiliary Hilbertspace ℌ1113567 cong ℂ1113569 and an operator Γ1113567 on ℌ1113567 with the properties

(Γ1113567)lowast = Γ1113567 (Γ1113567)1113569 = 120793 Trℌ1113577Γ1113567 = 0

needs to be introduced (eg Γ1113567 = 12059011135701113567) and the definition of the link operators hasto be slightly modified

C 3 T G (GMM)

Let ℌ = ℌ1113567 otimes ℌ and fix a vertex 119907120585 in the double laice 1114026120131 For concretenessthe choice 119907120585 = (1 1)120585 is made here en the modified link operators are definedas

1113654(ℓ) =⎧⎪⎨⎪⎩

120793 otimes 1198781113654(ℓ) if 119907120585 notin ℓΓ1113567 otimes 1198781113654(ℓ) if 119907120585 isin ℓ

(39)

By the identities i119882119875 = 1198781113654(120597119875) and i119882119883119884 = 1198781113654(120574119883119884) the plaquee andglobal flux operators can be naturally extended to ℌ Since in a closed loop con-taining 119900120585 there are exactly two adjoining edges and Γ 11135691113567 = 120793 this extension istrivial 1114026119882120572 = 120793 otimes119882120572

Further the restrictions 11136541113658(ℓ) to Ξℌ = (120793otimesΞ)ℌ = ℌ1113567otimesΞℌ are link operatorssatisfying (i) (ii) and (iv) of the link algebra Property (iii) now does hold due tothe introduction of the Γ1113567 factor Indeed

Tr1113658ℌ⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= Trℌ1113577Γ1113567 sdot Tr1113658ℌ

⎛⎜⎜⎝1114015119909isin1113657

11136541113658(119909120585 119909120578)⎞⎟⎟⎠= 0

Srsquos theorem 210 hence applies to the Hamiltonian

11140261198671113654 = 1205831114012119909isin1113657

11141011113654(119909120585 (119909 + 1198901113568)120578) + 1113654(119909120585 (119909 + 1198901113569)120578)

minus 1113654(119909120578 (119909 + 1198901113568)120585) minus 1113654(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin1113657

1205841199091113654(119909120585 119909120578)

= 120793 otimes 1198671113654 + 120583(Γ1113567 minus 120793) otimes 1114100Γ 1113568119907Γ 11135691113572119907+1198901113578 + Γ 1113568119907minus1198901113578Γ 11135691113572119907 + Γ 1113570119907Γ 11135711113572119907+1198901113579 + Γ 1113570119907minus1198901113579Γ 11135711113572119907

minus Γ 11135681113572119907 Γ 1113569119907+1198901113578 minus Γ 11135681113572119907minus1198901113578Γ 1113572119907 + Γ 11135701113572119907 Γ 1113571119907+1198901113579 + Γ 11135701113572119907minus1198901113579Γ 11135711199071114103 + 120584119907(Γ1113567 minus 120793) otimes Γ 1113572119907

constrained to the flux sector ℌ1113567 otimes ℌ1113568Since the operator Γ1113567 otimes 120793 commutes with 120793 otimes 1198671113654 as well as with the other

terms in 11140261198671113654 eigenstates 120595 of 11140261198671113654 can be classified by the property (Γ1113567 otimes 120793)120595 =plusmn120595 In particular in the sector of the Hilbert space where Γ1113567 otimes 120793 = 120793 otimes 120793 onehas 11140261198671113654 = 120793 otimes 1198671113654

is may help in the understanding of the physics of the extended Gammamatrix model is however not essential in the further discussion which is basedon Srsquos theorem and therefore relies crucially on the validity of the linkalgebra

Corollary 33 (JordanWigner transformation for the Gammamatrix model) e(extended) Gamma matrix model 11140261198671113654 with constraints

32 D L LR

(i) 119882119875 = 120793 for all elementary plaquees 119875 and

(ii) 119882119883 = 119882119884 = 120793

is unitarily equivalent to free fermions on the square laice 120131 = (Λ 119864) (with peri-odic boundary conditions)

119867 = 2120583 1114012119909119910isin119864

(119888lowast119909119888119910 + 119888lowast119910119888119909) +1114012119909isin1113657

120584119909(2119888lowast119909119888119909 minus 120793)

In particular there exists a unitary transformation 119880 such that

111402611986711136541113658 = 119880119867119880 lowast (310)

where 111402611986711136541113658 = Ξ11140261198671113654Ξ1114070

1113617111360011136131113658

Remark As illustrated by S in his paper [Szc85] the above constructionwith link operators expressed as products of Dirac Gamma matrices also workson a laice with different vertex degrees at least if they are all even In this casethe Dirac Gamma matrices have to be replaced by generalised Gamma matrices(generators of ℭ120105(deg(119909))) diams

32 Dynamical Localisation and Lieb RobinsonBoundsfor the GMM

e correspondence between the constrained Gamma matrix model and free fer-mions on the laice Λ can be used to get insights in the behaviour of the systemwhen the on-site magnetic field is not deterministic but given by iid randomvariables e theory of free fermions in a random on-site potential on a planarlaice is well-understood in terms of localisation properties which imply certainlocality bounds for the constrained Gamma matrix model

321 e Anderson model on ℤ119889

Let 119907119909119909isinℤ119889 (119889 ge 1) be a family of real-valued independent and identically dis-tributed (iid) random variables on a probability space (Ω120073ℙ) with commondistribution 1198751113567

ℙ119907119909 isin 119860 = 119875(119860) for all Borel sets 119860 sub ℝ and 119909 isin ℤ119889

It is further assumed that 119875 is absolutely continuous with respect to Lebesguemeasure on ℝ with bounded density and compact support 119875( d120584) = 120588(120584) d120584with 120588 isin 119871infin1113567 (ℝ)

C 3 T G (GMM)

e Anderson model on ℤ119889 is the random Hamiltonian119867120596 = 119870 + 120582119881120596 on ℓ1113569(ℤ119889) (311)

where 119870 = minusΔminus2119889 is up to a constant shi the graph Laplacian onℤ119889 explicitlyfor 119891 isin ℓ1113569(ℤ119889)

(119870119891)(119909) = minus 1114012119910isinℤ119889

dist(119909119910)=1113568

119891(119910)

the random potential 119881120596 is the multiplication operator by the iid random vari-ables

(119881120596119891)(119909) = 119907119909119891(119909)and 120582 ge 0 models the disorder strength

It is a classical result in the theory of random Schroumldinger operators that thespectrumof119867120596 (aswell as its spectral components) is almost surely deterministic120590(119867120596) = Σ 120590(119867120596) = Σ ( isin 119901119901 119886119888 119904119888) is follows from the ergodicity of119867120596 with respect to laice translations In the iid case considered here one has120590(119867120596) = [minus2119889 2119889] + supp119875 [Pas80 KS80 KM82]Definition 34 (Localisation Types) One says that the Anderson model exhibits(i) spectral localisation in the energy regime 119868 sub ℝ if there is ℙ-almost surely

only pure point spectrum in 119868 ie119868 cap Σ 119886119888 = 119868 cap Σ 119904119888 = empty as

(ii) exponential eigenfunction localisation in 119868 away from some random centresof localisation 120585120596 if for eigenfunctions 120595120596 of 119867120596

|120595120596(119909)| le 119862120596119890minus120578|119909minus120585120596 | ℙ minus almost surelywith inverse localisation length 120578 = 120578(119868) gt 0

(iii) (strong) dynamical localisation in the energy interval 119868 if

120124⎡⎢⎣sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569⎤⎥⎦le 119862119890minus120578|119909minus119910| (312)

with 119862 = 119862(119868) lt infin and inverse localisation length 120578 = 120578(119868) gt 0 Here120575119909119909isinℤ119889 denotes the canonical orthonormal basis of ℓ1113569(ℤ119889) 120575119909(119910) = 120575119909119910 forall 119909 119910 isin ℤ119889 and 119875119868(119867120596) is the spectral projection of 119867120596 onto 119868

In this definition dynamical localisation is the strongest notion of localisa-tion and if it holds in some energy interval 119868 then the spectrum in 119868 is aspure point Dynamical localisation also implies that for energies in 119868 there isno quantum transport in the sense that all moments of the position operatorsup1 |119883|

sup1(|119883|120595)(119909) ∶= |119909|120595(119909) for 119909 isin ℤ119889

32 D L LR

are finite for all times

sup119905isinℝ

|119883|119901 119890minus1113608119905119867120596119875119868(119867120596)1205951113569 lt infin ℙ minus almost surely

for all 119901 gt 0 and 120595 isin ℓ1113569(ℤ119889) compactly supported [Sto11]One way of proving strong dynamical localisation in the Anderson model

on ℤ119889 is via the fractional moments method (or AM method[AM93]) which proceeds via exponential bounds on the fractional moments ofthe Greenrsquos function 119866120596 of the random Hamiltonian 119867120596

119866120596(119909 119910 119911) = ⟨1205751199091

119867120596 minus 119911120575119910⟩ 119911 isin ℂ ⧵ ℝ

Restrictions of the Hamiltonian or the Greenrsquos function to subsets Λ sub ℤ119889 aredenoted by 1198671113657

120596 and 1198661113657120596(119909 119910 119911)By a rank-two perturbation argument using Kreĭnrsquos formula one can prove

for all 119904 isin (0 1) the a priori boundsup2

120124119909119910 1114070119866120596(119909 119910 119911)1114070119904 le 119862119904(120588)

120582119904 (313)

for some constant 119862119904(120588) lt infin and all 119909 119910 isin ℤ119889 119911 isin ℂ ⧵ ℝ 120582 gt 0 [Sto11 AW13]Making use of the resolvent identity

1198661113657120596(119909 119910 119911) = 119866119909120596 (119909 119909 119911)120575119909119910 + 1198661113657120596(119909 119909 119911) 1114012119906∶119889(119906119909)=1113568

1198661113657⧵119909120596 (119906 119910 119911)

and noting that the Greenrsquos function 1198661113657⧵119909120596 does not depend on 119881(119909) one canshow that for 119909 ne 119910

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 1114012

119906∶119889(119906119909)=1113568120124 111409411140701198661113657120596(119909 119909 119911)1114070

119904 11140701198661113657⧵119909120596 (119906 119910 119911)11140701199041114097

le 119862119904120582119904

1114012119906∶119889(119906119909)=1113568

120124 11140701198661113657⧵119909120596 (119906 119910 119911)1114070119904

efirst inequality is a straightforward application of Jensenrsquos inequality whereasin the second step first a disorder average over 119881(119909) was taken using the a pri-ori bound 120124119909 11140701198661113657120596(119909 119909 119911)1114070

119904 le 119862119904120582minus119904 Iterating this expansion one can prove thefollowing theorem [AW13] due to A and M [AM93]

sup2120124119909119910[sdot] = 120124 1114094 sdot 1114070 119881(119906)119906notin1199091199101114097 denotes the conditional expectation with 119881(119906)119906notin119909119910 fixed

C 3 T G (GMM)

eorem 35 (Complete localisation for high disorder) For any disorder strength120582 satisfying

120582 gt (2119889119862119904)1113568119904 (314)

for some 119904 isin (0 1) there is a constant 119862 lt infin such that for allΛ sub ℤ119889 and 119909 119910 isin Λ

sup119911isinℂ⧵ℝ

120124 11140701198661113657120596(119909 119910 119911)1114070119904 le 119862119890minus120578119889(119909119910) (315)

uniformly in Λ with a localisation length 120578 lt log( |120582|1199041113569119889119862119904)

e connection to the definition of dynamical localisation is established bythe eigenfunction correlator 119876120596(119909 119910 sdot) of 119867120596 which is defined as the total vari-ation of the spectral measure associated with two sites 119909 119910 isin ℤ119889 at is for anyBorel set 119868 sub ℝ

119876120596(119909 119910 119868) = sup119865isin119966 (ℝ)119865infinle1113568

1114070⟨120575119909 119875119868(119867120596)119865(119867120596)120575119910⟩1114070

It has the important property that

sup119905isinℝ

1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩11140701113569 le sup

119905isinℝ1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 119876120596(119909 119910 119868)

where the first inequality simply follows from 1114070⟨120575119910 119890minus1113608119905119867120596119875119868(119867120596)120575119909⟩1114070 le 1 and thesecond one by the particular choice 119865(119867120596) = 119890minus1113608119905119867120596

Under the present assumptions on the single-site distribution for each 119904 isin(0 1) there is a constant 119888119904(120588) lt infin such that for any bounded open set 119868 sub ℝthe eigenfunction correlator averaged over the disorder satisfies [AW13]

120124119876120596(119909 119910 119868) le 119888119904(120588) sup|120578|gt1113567

1114009119868120124 1114070119866120596(119909 119910 119864 + i120578)1114070

119904 d119864 (316)

Puing everything together it follows that for large disorder 120582 gt (2119889119862119904)1113568119904the Andersonmodel exhibits strong dynamical localisation throughout the entirespectrum

Of course the above is just a sketch of the most basic localisation result inthe discrete Anderson model Much more could be said about different disorderregimes and far more general methods than the one used here are nowadays athand A more detailed account on the topic can be found in the notes by K[Kir07] and S [Sto11] as well as in the book by A and W[AW13] which is still in preparation at the time of writing of this thesis

32 D L LR

322 Lieb Robinson bounds and exponential clustering

Consider a quantum spin system on a finite vertex set Λ as described in section12 en for two local observables (cf the definition in section 12) 119860 isin 12006811136531113578 119861 isin 12006811136531113579 (Ω1113568 and Ω1113569 finite disjoint Ω1113568 cupΩ1113569 sub Λ) one has

[119860 119861] ∶= [119860 otimes 1207931113657⧵11136531113578 119861 otimes 1207931113657⧵11136531113579] = 0

Lieb-Robinson bounds are bounds on the commutator aer time evolution of oneof the operators of the form

1114063 11140941205721113657119905 (119860) 1198611114097 1114063 le 1198621113653111357811136531113579119860119861119890minus120578(1113603111360811136181113619(1113653111357811136531113579)minus119907119871119877|119905|) (317)

e velocity 119907119871119877 depends on the specific interaction Φ and is finite for a fairlygeneral class of interactions which are essentially finite-range [LR72] Physic-ally 119907119871119877 is the group velocity with which information or excitations in Ω1113568 canpropagate under the Heisenberg evolution to Ω1113569

Lieb Robinson bounds have been successfully used to prove exponential clus-tering in presence of a spectral gap ie a spectral gap implies exponential decayof ground state correlations [Has04 NS06] Since the converse implication is ingeneral false and the assumption of a spectral gap quite strong one would liketo find weaker requirements on quantum spin systems that allow one to inferexponentially decaying correlation functions

An interesting step in this direction is a result due to H S and S[HSS12] In their paper they prove that zero-velocity Lieb Robinson bounds implyexponentially decaying ground state correlations up to a logarithmic correctionin the gap size e validity of a zero-velocity Lieb Robinson bound has beenproposed by the above authors as a simplified version of Hrsquo definition ofa mobility gap ie a gap to propagating excitations [Has10]

Let 1198671113657 be the Hamiltonian

1198671113657 = 21205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 + 119888lowast119909+1198901198961198881199091114103 +1114012

119909isin1113657120584119909(2119888lowast119909119888119909 minus 120793) 120583 120584119909 isin ℝ

on ℌ = ⨂119909isin1113657

ℂ1113569 Λ sub ℤ119889 finite and connected Using the anticommutationrelations it can be brought into a symmetric form

1198671113657 = 1205831114012119909isin1113657

1198891114012119896=11135681114100119888lowast119909119888119909+119890119896 minus 119888119909119888lowast119909+119890119896 + 119888lowast119909+119890119896119888119909 minus 119888119909+119890119896119888lowast1199091114103 +1114012

119909isin1113657120584119909(119888lowast119909119888119909 minus 119888119909119888lowast119909)

C 3 T G (GMM)

and introducing the vector 119862 = (1198881113568 hellip 119888|1113657| 119888lowast1113568 hellip 119888lowast|1113657|)119905 this can be wrien com-pactly as

1198671113657 = 119862lowastM (1113657)119862 with M (1113657) = 1114102A(1113657) 00 minusA(1113657)1114105 isin ℂ

1113569|1113657|times1113569|1113657|

and A(1113657) isin ℂ|1113657|times|1113657| given by

A(1113657) = 120583Adj(Λ) + diag(1205841113568 hellip 120584|1113657|)

Here Adj(Λ) stands for the adjacency matrix of Λ sub ℤ119889 considered as (un-directed) combinatorial graph Since Adj(Λ) is symmetric A(1113657) is self-adjointthus also (M (1113657))lowast = M (1113657) In particular by the spectral theorem for self-adjointmatrices there is a unitary matrix U that diagonalises A(1113657) UA(1113657)U lowast = D =diag(1205821113568 hellip 120582|119881|) en the unitary matrixW = U oplusU diagonalisesM (1113657)

WM (1113657)W lowast = 1114102D 00 minusD1114105

is fact can be used to introduce creation and annihilation operators for newfermions 119861 = (1198871113568 hellip 119887|1113657| 119887lowast1113568 hellip 119887|1113657|) namely

119861 = W119862

It follows from a straightforward computation that the canonical anticom-mutation relations are in this vector-valued formalism equivalent to

119862119862lowast + 120129(119862119862lowast)119905120129 = 120793 120129 = 11141020 120793120793 01114105

Since 120129 commutes withW it follows that

119861119861lowast + 120129(119861119861lowast)119905120129 = W (119862119862lowast + 120129(119862119862lowast)119905120129)W lowast = 120793

hence the 119887-operators indeed are fermionic operators In terms of these operatorsthe Hamiltonian 1198671113657 can be wrien as

1198671113657 = 119862lowastM119862 = 119861lowast119882M119882 lowast119861 = 119861lowast 1114102D 00 minusD1114105119861

= 1114012119909isin1113657

120582119909(119887lowast119909119887119909 minus 119887119909119887lowast119909) = 1114012119909isin1113657

2120582119909119887lowast119909119887119909 minus 119864(1113657)120793

where 119864(1113657) = sum119909isin1113657 120582119909

32 D L LR

In this form it is particularly easy to calculate the time evolution of the fer-mionic operators For any 119909 isin Λ one has

[1198671113657 119887119909] = minus2120582119909119887119909

and thus by definition of the Heisenberg dynamics

dd119905120572

1113657119905 (119887119909) = i1205721113657119905 ([1198671113657 119887119909]) = minus2i120582119909119887119909

Together with 12057211136571113567 (119887119909) = 119887119909 it follows that 1205721113657119905 (119887119909) = 119890minus11135691113608120582119909119905119887119909 and consequently

1205721113657119905 (119861) = 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861

is result can be used to obtain the time evolution of 119862 for

1205721113657119905 (119862) = 1205721113657119905 (W lowast119861) = W lowast1205721113657119905 (119861) = W lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105 119861 = W

lowast 1114102119890minus111356911136081113685119905 00 1198901113569111360811136851199051114105W119862

= 119890minus111356911136081113694(1113666)119905119862 = 1114102119890minus111356911136081113682(1113666)119905 00 119890111356911136081113682(1113666)1199051114105119862

Introducing the short-hand notation A(1113657)119909119910 (119905) = 1114100119890minus11136081113682(1113666)1199051114103119909119910 one can immediately

write down the Heisenberg evolution of the 119888-operators

1205721113657119905 (119888119909) = 1114012119910isin1113657

A(1113657)119909119910 (minus2119905)119888119910 +1114012119910isin1113657

A(1113657)119909119910 (2119905)119888lowast119910 (318)

It shall from now on be assumed that the family 120584119909119909isin1113657 describing the on-sitepotential in the fermionic Hamiltonian is a family of iid random variables asin section 321 en 1198671113657 is essentially the second quantisation of the Andersonmodel on ℓ1113569(Λ) with disorder strength 120582 = 1113568

1113569120583 In particular it is reasonable todefine

Definition 36 e matricesM (1113657) are dynamically localised if there exist 119862 120578 gt0 such that for all Λ sub ℤ119889 and 119909 119910 isin Λ

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119910(119905)1114071⎤⎥⎦le 119862119890minus120578119889(119909119910) (319)

It has been proved byH et al [HSS12] that dynamical localisation of thematricesM (1113657) implies zero-velocity Lieb-Robinson bounds aer disorder averagein one dimension

C 3 T G (GMM)

Example 37 (Dynamical localisation implies zero-velocity Lieb-Robinson boundsaer disorder average in one dimension) Letℌ =⨂119873

119909=1113568ℂ1113569 and consider the119883119884-

model on ℌ Looking back at section 211 a short calculation shows that in thiscase

A(119873) =

⎛⎜⎜⎜⎜⎝

1205841113568 minus120583minus120583 ⋱ ⋱

⋱ ⋱ ⋱⋱ ⋱ minus120583

minus120583 120584119873

⎞⎟⎟⎟⎟⎠

e following holds

eorem 38 (Hamza Sims Stolz [HSS12]) M (119873) is dynamically localised andthere exist 119862prime 120578 gt 0 such that for all 1 le 119909 lt 119910 and any 119873 ge 119910 one has

120124⎡⎢⎣sup119905isinℝ

1114062[120572119873119905 (119883) 119884]1114062⎤⎥⎦le 119862prime119883119884119890minus120578119889(119909119910) (320)

for all 119883 isin 120068119909 and 119884 isin 120068[119910119873]sup3Furthermore leing 1205951113567 be the almost sure (normalised) ground state of 119867119873

there is a constant 119862 lt infin and a inverse correlation length 120578prime gt 0 such that

120124 1114094 1114070⟨1205951113567 1198831198841205951113567⟩ minus ⟨1205951113567 1198831205951113567⟩⟨1205951113567 1198841205951113567⟩1114070 1114097 le 119862119883 119884119873119890minus120578prime119889(119909119910) (321)

diams

If one instead considers the constrained Gamma matrix model on ℌ1113567 otimes Ξℌin a random exterior magnetic field 120584119909119909isin1113657 which is assumed to satisfy the as-sumptions of subsection 321 it would be interesting to see if similar results alsohold

Using a JordanWigner transformation in the sense of Szczerba (theorem 210)it has been proven in corollary 33 that the GMM-Hamiltonian is unitarily equi-valent to the second quantised version of the Anderson model on ℓ1113569(Λ) In aregime of high disorder 120582 ≫ 1 (cf condition 314) or equivalently low kineticenergy 120583 sim 1120582 ≪ 1 one has complete dynamical localisation which corres-ponds to dynamical localisation of the matrixA(1113657) = 120583Adj(Λ)+diag(1205841113568 hellip 120584|1113657|)

Definition 39 ForΩ sub Λ let1200861113653 be the 119862lowast algebra generated by the set 11136541113658(ℓ) ∶ℓ isin 1114026Ω of link operators corresponding to edges ℓ in the (doubled) set 1114026Ω⁴

sup3In the context of this theorem 1200681113653 refers to the local algebra generated by the Pauli spinmatrices in Ω

⁴Ω being identified in a natural way as subgraph of 120131

32 D L LR

Ι

Ω

Figure 32 On the definition of the relation Ι lt Ω

Lemma 310 For fixed 119909 isin Λ let 119888119909 be a fermionic annihilation operator and119884 isin 1200861113653 such that 119909 notin Ω (assuming for simplicity that Ω is a box subset of Λwith the property 119909119894 lt 119910119894 for all 119910 isin Ω 119894 = 1 2)⁵ en dynamical localisation ofthe matrix 119860(1113657) implies

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦le 4119862 1114102

1 + 119890minus1205781 minus 119890minus120578 1114105

1113569

111405911138401113840111384011140601113840111384011138401114061=∶119862prime

119884119890minus1205781113603111360811136181113619(1199091113653) (322)

with 119862 being the constant from definition 36

Proof Fix the vertex (1 1) as root 119900 of the (directed) spanning tree of Λ depictedin figure 33 (directed away from the root) and complete it to a spanning tree of1114026120131 by adding the (directed) edges (119909120585 119909120578) for 119909 isin Λ e assumptions onΩ implythat 119900 notin Ω

By equation 318 one has

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113657

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] +1114012119911isin1113657

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880] (323)

Since

119888119911 = 11135681113569 (120585119911 + i120578119911) =

11135681113569 (minusi120585119900)119880

lowast(11136541113658(120574119911120585) + i11136541113658(120574119911120578))119880 for 119911 ne 119900 (324)

119888119900 = 11135681113569 (120585119900 + i120578119900) =

11135681113569 (minusi119880

lowast11136541113658(120574119900120578)119880120578119900 + i120578119900) (325)

⁵To shorten notation this shall be wrien as 119909 lt Ω More generally for sets ΙΩ sub ℤ119889 therelation Ι lt Ω is defined by 1199091113568 lt 1199101113568 and 1199091113569 lt 1199101113569 for all 119909 isin Ι and 119910 isin Ω See figure 32

C 3 T G (GMM)

by the construction 247 in Srsquos theorem (119880 = 119882 lowast here as remarked atthe end of the proof of theorem 210) the commutators in 323 can be examinedmore closely

Firstly the properties of 120578119900 (246) imply that for any 119907 119908 ne 119900 the commutator[120578119900 119880 lowast11136541113658(119907120577 119908120589)119880] is zero is follows from the observation that

1114094120578119900 119880 lowast11136541113658(119907120577 119908120589)1198801114097 = 119880 lowast 1114094119880120578119900119880 lowast 11136541113658(119907120577 119908120589)1114097119880= 119880 lowast 1114094119880120578119900119880 lowast i11136541113658(120574119907120577)

11136541113658(120574119908120589)1114097119880

= i119880 lowast 1114094119880120578119900119880 lowast 11136541113658(120574119907120577)1114097 11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577) 1114094119880120578119900119880 lowast 11136541113658(120574119908120589)1114097119880

= i 1114094120578119900 119880 lowast11136541113658(120574119907120577)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

119880 lowast11136541113658(120574119908120589)119880 + i119880 lowast11136541113658(120574119907120577)119880 1114094120578119900 119880 lowast11136541113658(120574119908120589)119880111409711140591113840111384011138401113840111406011138401113840111384011138401114061=1113567 by (246)

= 0

e identity in the second line was proved at the end of the proof of lemma 211and third line is a simple application of Leibnizrsquos rule for commutators erefore119884 being in the local algebra generated by the link operators corresponding toedges in 1114026Ω it follows that [120578119900 119880 lowast119884119880] = 0 (by repeated application of Leibnizrsquosrule if necessary)

Further one has

[120585119900 119880 lowast119884119880] = 1114094minusi119880 lowast11136541113658(120574119900120578)119880120578119900 119880 lowast1198841198801114097

= minusi 1114094119880 lowast11136541113658(120574119900120578)119880119880 lowast1198841198801114097 120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast1198841198801114097

= minusi119880 lowast 111409411136541113658(120574119900120578) 11988411140971114059111384011138401114060111384011138401114061=1113567 since 119900notin1113653

119880120578119900 minus i119880 lowast11136541113658(120574119900120578)119880 1114094120578119900 119880 lowast11988411988011140971114059111384011138401114060111384011138401114061=1113567

= 0

concluding the prove that [119888119900 119880 lowast119884119880] = 0For 119911 ne 119900 the representation 324 yields

[119888119911 119880 lowast119884119880] = minus 11136081113569120585119900 1114094119880

lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880119880 lowast1198841198801114097

minus 11136081113569 [120585119900 119880 lowast119884119880]1114059111384011138401114060111384011138401114061

=1113567

119880 lowast 111410011136541113658(120574119911120585) + i11136541113658(120574119911120578)1114103119880

= minus 11136081113569120585119900119880

lowast 111409411136541113658(120574119911120585) + i11136541113658(120574119911120578) 1198841114097119880

Denote by Ω the set of all vertices in the strip to the right of Ω and of thesame height asΩ (not includingΩ see figure 33) en by definition of the path

32 D L LR

Ω

Ω

119900 = (1 1)

Figure 33 e spanning tree with root of 120131 is shown together with the defini-tion of the set Ω

operators the assumed spanning tree and 119884 isin 1200861113653 it follows that

[11136541113658(120574119911120585) + i11136541113658(120574119911120578) 119884] = 0 for all 119911 notin Ω cupΩ

and therefore

[119888119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩ

Since 119888lowast119910 = 11135681113569 (120585119911 minus i120578119911) for all 119911 isin Λ analogous considerations show that

[119888lowast119911 119880 lowast119884119880] = 0 for all 119911 notin Ω cupΩTaking this into account equation 323 becomes

[1205721113657119905 (119888119909) 119880 lowast119884119880] = 1114012119911isin1113653cup1113653

A(1113657)119909119911(2119905)[119888119911 119880 lowast119884119880] + 1114012119911isin1113653cup1113653

A(1113657)119909119911(minus2119905)[119888lowast119911 119880 lowast119884119880]

By dynamical localisation and the bound [119860 119861] le 2119860 119861 on the commut-

C 3 T G (GMM)

ator of two operators one gets

120124⎡⎢⎣sup119905isinℝ

1114063 11140941205721113657119905 (119888119909) 119880 lowast1198841198801114097 1114063⎤⎥⎦

le 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦1114063 [119888119911 119880 lowast119884119880] 1114063 + 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911 (minus2119905)1114071⎤⎥⎦1114063 [119888lowast119911 119880 lowast119884119880] 1114063

le 2119880 lowast119884119880 1114012119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(2119905)1114071⎤⎥⎦+ 2119880 lowast119884119880 1114012

119911isin1113653cup1113653

120124⎡⎢⎣sup119905isinℝ

1114071A(1113657)119909119911(minus2119905)1114071⎤⎥⎦

le 4119884 1114012119911isin1113653cup1113653

119862119890minus120578119889(119909119911) le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

with 119862prime = 411986211141001113568+119890minus1205781113568minus119890minus120578 11141031113569

Clearly the same estimate holds for 1205721113657119905 (119888lowast119909) By the homomorphism propertyof 1205721113657119905 and Leibnizrsquos rule

[1205721113657119905 (119888119909119888119910) 119880 lowast119884119880] = [1205721113657119905 (119888119909)1205721113657119905 (119888119910) 119880 lowast119884119880]= 1205721113657119905 (119888119909)[1205721113657119905 (119888119910) 119880 lowast119884119880] + [1205721113657119905 (119888119909) 119880 lowast119884119880]1205721113657119905 (119888119910)

it follows that (assuming 119909 119910 notin Ω)

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119888119909119888119910) 119884]1114062⎤⎥⎦le 119862prime119884119890minus1205781113603111360811136181113619(1199091113653) + 119862prime119884119890minus1205781113603111360811136181113619(1199101113653)

le 119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653)

e inequality remains valid if 119888119909119888119910 is exchanged by 119888lowast119909119888119910 119888119909119888lowast119910 119888lowast119909119888lowast119910 etc ussince the link operators 119878(119909120577 119910120589) can be expressed in terms of no more than fourof the above terms eg

119878(119909120585 119910120585) = i120585119909120585119910 = i(119888lowast119909 + 119888119909)(119888lowast119910 + 119888119910) = i(119888lowast119909119888lowast119910 + 119888119909119888lowast119910 + 119888lowast119909119888119910 + 119888119909119888119910)119878(119909120585 119909120578) = i120585119909120578119909 = i(119888lowast119909 + 119888119909)i(119888lowast119909 minus 119888119909) = 119888lowast119909119888119909 minus 119888119909119888lowast119909

one has

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 4119862prime119884119890minus1205781113603111360811136181113619(1199091199101113653) le 4119862prime(1 + 119890120578)1114059111384011138401114060111384011138401114061

=∶119862Prime

119884119890minus1205781113603111360811136181113619(1199091113653)

32 D L LR

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653)

respectivelyLet 1205731113657119905 be the Heisenberg evolution associated with the Hamiltonian 11140261198671113654

1113658(Λ)

1205731113657119905 (119860) = 1198901113608119905111402611986711136631113667119860119890minus111360811990511140261198671113663

1113667

Lemma 311 Let 119909 119910 isin Λ and 119884 isin 1200861113653 such that 119909 119910 notin Ω 119909 119910 lt Ω Assumefurther that the matrix119860(1113657) is dynamically localised in the sense of definition 36en

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653) (326)

and

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120585 119909120578)) 119884]1114062⎤⎥⎦le 2119862prime119884119890minus1205781113603111360811136181113619(1199091113653) (327)

Proof By Szczerbarsquos theorem 11136541113658(ℓ) = 119880119878(ℓ)119880 lowast for all (directed) edges ℓ isin 1114026120131Since

1114063 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097 1114063 = 1114063119880 lowast 11140941205731113657119905 (11136541113658(ℓ)) 1198841114097119880 1114063

= 1114063119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880 119880 lowast119890minus111360811990511140261198671113663

1113667119880119880 lowast119884119880 minus 119880 lowast119884119880119880 lowast1198901113608119905111402611986711136631113667119880119880 lowast11136541113658(ℓ)119880119880 lowast119890minus111360811990511140261198671113663

11136671198801114063= 1114062 11989011136081199051198671113666119878(ℓ)119890minus11136081199051198671113666119880 lowast119884119880 minus 119880 lowast119884119880 11989011136081199051198671113666119878(ℓ)119890minus111360811990511986711136661114062= 1114063 11140941205721113657119905 (119878(ℓ)) 119880 lowast1198841198801114097 1114063

it follows by lemma 310 that

120124⎡⎢⎣sup119905isinℝ

1114062 [1205731113657119905 (11136541113658(119909120577 119910120589)) 119884]1114062⎤⎥⎦= 120124

⎡⎢⎣sup119905isinℝ

1114062 [1205721113657119905 (119878(119909120577 119910120589)) 119880 lowast119884119880]1114062⎤⎥⎦

le 119862Prime119884119890minus1205781113603111360811136181113619(1199091113653)(328)

and analogously for 11136541113658(119909120585 119909120578)

Having lemmata 310 and 311 at hand the following zero-velocity Lieb Robin-son bound in disorder average can be proved

C 3 T G (GMM)

Proposition 312 Assume that the constrained Gamma Matrix Model 111402611986711136541113658 is dy-

namically localised in the sense of definition 36 en there exist constants 119888 120578 gt 0such that for any two sets ΙΩ sub Λ with Ι lt Ω one has

120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 119888 min(1 |119905|) 119883 119884 119890minus1205781113603111360811136181113619(11136901113653) (329)

for all 119883 isin 1200861113690 and 119884 isin 1200861113653

Proof Part 1 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Let 119883 isin 1200861113690 119884 isin 1200861113653 and define the function

119891(119905) = [1205731113657119905 (119883) 119884]

with derivative

119891prime(119905) = i 11140941205731113657119905 ([111402611986711136541113658 119883]) 1198841114097 = i 11140941205731113657119905 ([11140261198671113654

1113658(Ι) 119883]) 1198841114097 = i 1114094[1205731113657119905 (111402611986711136541113658(Ι)) 1205731113657119905 (119883)] 1198841114097

= minusi 1114094[1205731113657119905 (119883) 119884] 1205731113657119905 (111402611986711136541113658(Ι))1114097 minus i 1114094[119884 1205731113657119905 (11140261198671113654

1113658(Ι))] 1205731113657119905 (119883)1114097

Here 111402611986711136541113658(Ι) only contains the terms in 11140261198671113654

1113658 that do not commute with 119883 e lastline is an application of the Jacobi identity for commutators us 119891 satisfies thedifferential equation

119891prime(119905) = minusi 1114094119891(119905) 1205731113657119905 (111402611986711136541113658(Ι))1114097 + i 1114094[1205731113657119905 (11140261198671113654

1113658(Ι)) 119884] 1205731113657119905 (119883)1114097

Lemma 313 Let 119860(119905) 119905 isin ℝ be a family of linear norm preserving operators⁶in some Banach space 120069 For any function 119861 ∶ ℝ rarr 120069 the solution of

120597119905119884(119905) = 119860(119905)119884(119905) + 119861(119905)

with boundary condition 119884(0) = 0 satisfies the bound

119884(119905) le 1114009119905

1113567119861(119904)d119904

Proof Aproof of this can be found inAppendixA of N et al [NOS06](Lemma A1 therein)

⁶ie the mapping 120574119905 ∶ 120069 rarr 120069 which maps 1199091113567 isin 120069 to the solution 119883(119905) of the differentialequation 120597119883(119905) = 119860(119905)119883(119905) with initial datum 119883(0) = 1199091113567 is an isometry 120574119905(1199091113567) = 1199091113567 for all119905 isin ℝ

32 D L LR

By the above lemma one obtains

119891(119905) = 1114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 1114009|119905|

11135671114063 1114094[119884 1205731113657119904 (11140261198671113654

1113658(Ι))] 1205731113657119904 (119883)1114097 1114063 d119904

le 2119883 1114009|119905|

11135671114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

Since 111402611986711136541113658(Ι) is given by

111402611986711136541113658(Ι) = 1205831114012

119909isin1113690minus111410111136541113658(119909120585 (119909 + 1198901113568)120578) + 11136541113658(119909120585 (119909 + 1198901113569)120578)

minus 11136541113658(119909120578 (119909 + 1198901113568)120585) minus 11136541113658(119909120578 (119909 + 1198901113569)120585)1114104 +1114012119909isin111369012058411990911136541113658(119909120585 119909120578)

where Ιminus = [Ι cup (Ι minus 1198901113568 minus 1198901113569))] cap Λ one has

1201241114063 11140941205731113657119905 (119883) 1198841114097 1114063 le 2119883 1114009|119905|

11135671201241114063 11140941205731113657119905 (11140261198671113654

1113658(Ι)) 1198841114097 1114063 d119904

le 8120583119883 1114012119909isin1113690minus

119862Prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

+ 2max119909isin1113657

|120584119909| 119883 1114012119909isin11136902119862prime 119884119890minus1205781113603111360811136181113619(1199091113653)|119905|

le 1198881113568 |119905| 119883119884119890minus1205781113603111360811136181113619(11136901113653)

Explicitly the constant is 1198881113568 = 8120583119862Prime + 4119862primemax119909isin1113657 |120584119909|Part 2 120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) for all 119905 isin ℝ

By lemma 311 the inequality holds for 119883 being a link operator 119883 = 11136541113658(119909120577 119910120589)with 119909 119910 isin Ι If 119883 is the sum of such link operators the claim follows by triangleinequality If 119883 is the product of link operators corresponding to links withvertices in Ι a straightforward application of Leibnizrsquos rule yields the inequality

It follows that for general 119883 isin 1200861113690 there exists a constant 1198881113569 lt infin such that120124 1114094 1114062 [1205731113657119905 (119883) 119884]1114062 1114097 le 1198881113569 119883119884119890minus1205781113603111360811136181113619(11136901113653) holds for all 119905 isin ℝPart 3 Choosing 119888 = max(1198881113568 1198881113569) concludes the proof of inequality 329

is is a first step in proving zero-velocity Lieb Robinson bounds for the con-strained extended Gamma matrix model e proof still relies on the additionalassumption of the relative positions of the sets Ι and Ω It is also an open ques-tion whether or not the local algebras generated by the link operators cover allphysically interesting observables Nevertheless it has been demonstrated howthe methods in [HSS12] may be applied in the context of an approach via thegeneralised Jordan Wigner transformation due to W and S

C 3 T G (GMM)

Chapter 4

Conclusion and Outlook

I this thesis the Jordan Wigner transformation has been discussed as an im-portant tool in studying properties of one-dimensional quantum spin systems

Various possible extensions of this transformation have been introduced It turnedout that the appearance of statistical gauge fields aer the transformation is re-lated to the structure of the underlying graph In particular one (surprising)result was that only simple path graphs allow for a special Jordan Wigner trans-formation in the sense of definition 22

An alternative approach going back to N in the 1950rsquos and reformu-lated in terms of link operators is based upon the observation that the Pauli spinmatrices are generators of a Clifford algebra Using higher dimensional Cliffordalgebras in expressing the link operators is the main idea behind the WS approach

One of the most basic applications of Srsquos theorem 210 ndash the Gammamatrix model ndash was introduced and its connection to the link operators estab-lished Aer slight modifications the constrained Gamma matrix model allowsfor a description by Jordan Wigner fermions is relation allowed one to studythe constrained GMM in a random external field and prove localisation boundson the Heisenberg evolution ndash a zero-velocity Lieb Robinson type bound in dis-order average (329) Even though the bound presented here relies on additionalassumptions it is nevertheless interesting in that it provides another example ofa quantum spin systemwhere some of the techniques developed by H Sand S can be applied

As hinted at the end of the general discussion of Srsquos theorem andthe Gamma matrix model these methods can also be applied to appropriate sub-spaces of higher dimensional quantum spin systems or quantum spin systemson general graphs (keeping in mind the restriction on the vertex degrees) usinghigher dimensional generalisations of the Gamma matrices

C 4 C O

Appendix A

Some missing calculations inSrsquos theorem

S properties of the link and path operators that were le out in the present-ation of the proof of theorem 210 and lemma 211 are presented here for

completeness since they mainly involve calculations and cannot be found in theoriginal paper [Szc85]

Lemma A1 (Path operators) Let 119878(ℓ) ∶ ℓ isin be a family of link operatorssatisfying properties (i) and (ii) of definition 28 and 119878(ℓminus1113568) = minus119878(ℓ) en thepath operators 119878(120574) (see definition 28 (iv)) have the following properties

(i) 119878(1205741113568 ∘ 1205741113569) = minusi119878(1205741113568)119878(1205741113569) if the vertex at the end of 1205741113568 is the first vertex of1205741113569 and 1205741113568 ∘ 1205741113569 denotes their concatenation

(ii) 119878(1205741113568) 119878(1205741113569) = 0 if 1205741113568 and 1205741113569 are simple paths with exactly one commonedge-point (beginning or end)

(iii) [119878(120574) 119878(ℓ)] = 0 for all closed paths 120574 and ℓ isin

(iv) 119878(120574)lowast = 119878(120574) and (119878(120574)1113569) = 120793 if 120574 is not a closed path

(v) 119878(120574)lowast = minus119878(120574) and (119878(120574)1113569) = minus120793 if 120574 is a closed path

(vi) 119878(120574minus1113568) = minus119878(120574)minus1113568 for all paths 120574

Proof roughout the proof let 120574 = 1205741113568 = ℓ1113568 ∘ ⋯ ∘ ℓ119898 1205741113569 = ℓ119898+1113568 ∘ ⋯ ∘ ℓ119898+119899 for119898 119899 isin ℕ be paths in 1114026120131 with the property that ℓ119898 and ℓ119898+1113568 are neighbouringedges

A A S Srsquo

(i) en 1205741113568 ∘ 1205741113569 = ℓ1113568 ∘ ⋯ℓ119898+119899 and

119878(1205741113568 ∘ 1205741113569) = (minusi)119898+119899minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= (minusi) 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103 1114100(minusi)119899minus1113568119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)1114103= minusi119878(1205741113568)119878(1205741113569)

(ii) Assume additionally that1205741113568 and1205741113569 are simple paths en 119878(ℓ119898) 119878(ℓ119898+1113568) =0 implying

119878(1205741113568)119878(1205741113569) = (minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ1113568)⋯119878(ℓ119898+1113568)119878(ℓ119898)⋯119878(ℓ119898+119899)= minus(minusi)119898+119899minus1113569119878(ℓ119898+1113568)⋯119878(ℓ119898+119899)119878(ℓ1113568)⋯119878(ℓ119898) = minus119878(1205741113568)119878(1205741113569)

the laer equality being a consequence of the fact that all other permuta-tions involved in this calculation involve only link operators to non-neighbouringedges which commute by assumption on the link operators

(iii) Suppose that 120574 is a closed path ie ℓ1113568 and ℓ119898 are neighbouring edges andlet ℓ isin If ℓ and 120574 have no common vertex [119878(120574) 119878(ℓ)] = 0 is trivial Incase they share exactly one common vertex there are two adjoining edgesto ℓ in 120574 say ℓ119896 and ℓ119896+1113568 for some 119896 isin 1 hellip 119898 (identifying 119898 + 1 equiv 1)Hence

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

If 120574 and ℓ have two common vertices ie ℓ equiv ℓ119896 isin 120574 then

119878(120574)119878(ℓ) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119896minus1113568)119878(ℓ119896)119878(ℓ119896+1113568)⋯119878(ℓ119898)119878(ℓ)= (minus1)1113569119878(ℓ)(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) = 119878(120574)119878(ℓ)

(iv) Assume 120574 is not a closed path en

119878(120574)lowast = 1114100(minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)1114103lowast= i119898minus1113568119878(ℓ119898)⋯119878(ℓ1113568)

= i119898minus1113568(minus1)119878(ℓ1113568)119878(ℓ119898)⋯119878(ℓ1113569) = i119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)119878(ℓ1113569)⋯119878(ℓ119898)= 119878(120574)

and

119878(120574)1113569 = (minus1)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ1113568)⋯119878(ℓ119898)= minus(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)⋯119878(ℓ119898)119878(ℓ1113569)⋯119878(ℓ119898)= (minus1)119898minus1113568(minus1)119898minus1113568119878(ℓ1113568)1113569119878(ℓ1113569)1113569⋯119878(ℓ119898)1113569 = 120793

(v) If 120574 is a closed path then the steps in (iv) stay essentially the same apartfrom the fact that 119878(ℓ1113568) 119878(ℓ119898) = 0 instead of [119878(ℓ1113568) 119878(ℓ119898)] = 0 leading toan additional minus sign in both calculations

(vi) Now let 120574 be arbitrary

119878(120574)119878(120574minus1113568) = (minusi)119898minus1113568119878(ℓ1113568)⋯119878(ℓ119898) sdot (minusi)119898minus1113568119878(ℓminus1113568119898 )⋯119878(ℓminus11135681113568 )= (minus1)119898minus1113568(minus1)119898119878(ℓ1113568)⋯119878(ℓ119898)119878(ℓ119898)⋯119878(ℓ1113568) = minus120793

A A S Srsquo

Bibliography

[AM85] H A and T M Ground states of the XY-model Commu-nications in Mathematical Physics 101 (1985) no 2 213ndash245 11 14

[AM93] M A and S M Localization at large dis-order and at extreme energies An elementary derivations Communications inMathematical Physics 157 (1993) no 2 245ndash278 47

[Ara84] H A On the XY-Model on Two-Sided Infinite Chain Publicationsof the RIMS Kyoto University 20 (1984) 277ndash296 11 14

[AW13] M A and S W antum Dynamics with DisorderA Mathematical Introduction In preparation 2013 47 48

[Azz93] M A Interchain-coupling effect on the one-dimensional spin-12antiferromagnetic Heisenberg model Phys Rev B 48 (1993) 6136ndash6140 24

[BA01] B B andM A Generalization of the Jordan-Wigner transformation inthree dimensions and its application to the Heisenberg bilayer antiferromagnetPhys Rev B 64 (2001) 054410 24 25

[BR02] O B and DW R Operator Algebras andantum Stat-istical Mechanics 2 second ed Texts and Monographs in Physics SpringerBerlin Heidelberg 2002 7 32

[Fra89] E F Jordan-Wigner transformation for quantum-spin systemsin two dimensions and fractional statistics Phys Rev Le 63 (1989) 322ndash32523

[Fra13] Field eories of Condensed Maer Physics second ed CambridgeUniversity Press 2013 12

[Geo99] H G Lie Algebras in Particle Physics Frontiers in Physics Perseus BooksAdvanced Book Program 1999 27

[Has04] M B H Locality in antum and Markov Dynamics on Laices andNetworks Phys Rev Le 93 (2004) 140402 49