Max Planck Institute for the Physics of Complex Systems

NQS 2014, Kyoto, Nov. 12, 2014

Frank Pollmann

Detecting topological orders

• Different phases of matter are usually understood in terms of spontaneous symmetry breaking

- Magnets: spin rotation and TR

- Crystals: translation and rotation

• Characterized by local order parameter

Quantum phases of matter

Z2

H = �X

j

��z

j

�z

j+1 + g�x

j

�

m=

hSzi

Ordered Disordered g

• symmetric Ising model:

• In the last years several “topological ” phases have been discovered that cannot be described by symmetry breaking

- Quantum Hall effects

- Topological insulators

- Haldane spin chains

- …

• Topological phases have fascinating features: Fractionalization, “anyonic” quasiparticles, protected edge modes, …

[Klitzing ’80, Tsui ’82, Laughlin ’83]

[Haldane ‘83]

[Kane & Mele ’05]

� = ⌫e2

h

Quantum phases of matter

• No complete classification of topological order known yet!

Matter occurs in different phases

“Intrinsic”

“Chiral”

-Spin Liquids -FQHE

-IQHE -Chern Insulator

Anyons

Chiral Edge Modes

“Trivially disordered”

“Symmetry”

“Symmetry protected”

-Topological Insulators -Haldane Phase

Symmetry Fractionalization

“Symmetry broken”“Symmetry Enriched”

Intrinsic topological order

ei�

• Intrinsic topological order: Gapped quantum phases that are robust to any small (local) perturbation

• Characterized by quasiparticle excitations that obey fractional statistics “anyons”

- Topological degeneracy (= number of anyons)

- Topological entanglement entropy :�[Kitaev and Preskill ’06, Levin and Wen ‘06]

[Wen ’90]

S = ↵L� �

A

B

| i =P

↵pp↵ |�A↵ i |�B↵ i

S = �P

↵ p↵ log p↵

Abelian: � = log(

p#anyons)

Intrinsic topological order: Spin Liquid

• Toric code model:

: � = log 21, e,m, f[Kitaev ‘03]

: � = log 21, s+, s�,m

• Double semion model:

!

[Freedman et al ‘04]

Intrinsic topological order: Spin Liquid

• Interpolate between the two fix point models and use exact diagonalization to study the phase diagram

!

!

6

(a) (b)

(c) (d)

Figure 5: Spectral decomposition A of the ground state wavefunctionat the transition points (dashed lines) for different system sizes. Theleft column corresponds to 48 spins and the right column correspondto 75 spins. The top row shows the behavior of A(!,�) at the knownsecond-order transition from the TC phase to the polarized phase at� ⇡ 0.18. The bottom row shows A(!, µ) at the transition from theTC to the DSem phase where a transition occurs at µ = 0.5.

that even away from the fixed points, we should not be ableto distinguish them using TEE. As introduced in Ref. 12, wecalculate the TEE numerically by adding and subtracting theentanglement entropy of different regions such that the lengthdependent contributions cancel out.

Figure 6 shows the TEE as the system is tuned along twodifferent directions in the phase diagram. The blue trianglesshow a transition from the TC phase to a polarized phase23 onincreasing the magnetic field. The red circles show a transitionfrom the TC phase to the DSem phase (with zero magneticfield). We see a dip at the transition point µ = 0.5 due tofinite-size effects, but the TEE remains the same on both sidesof the transition. The triangular inset shows the TEE at allpoints in the phase diagram. The dark areas indicate regionswith high TEE and correspond to topologically ordered phaseswith TEE close to log 2.

Hence, TEE distinguishes the topological regions from thetrivial regions in the phase diagram. However, it cannot dis-tinguish between the different topologically ordered phases.To do that, we turn to extracting the braiding statistics of theexcitations in the topological region.

Figure 6: Variation of the topological entanglement entropy (TEE) asthe system is tuned from the TC fixed point to the DSem fixed point(along � = 0 - red circles, top x-axis) , and from the TC fixed pointto the polarized fixed point (along µ = 0 - blue triangles, bottomx-axis). The inset shows the TEE at all points in the phase diagramwith dark red regions indicating higher TEE. This indicates that thebottom part of the phase diagram is topologically ordered. The sizeof the system considered is 48 spins.

C. Braiding Statistics

We extract the braiding statistics of the excitations in thetopologically ordered region. In particular, we obtain the Uand S-matrices which quantify the self statistics (exchangestatistics) and the mutual statistics respectively. An elementof the U -matrix specifies the phase obtained by the many-body wavefunction when we exchange two identical particles.An element of the S-matrix corresponds to the phase obtainedwhen we move one of the particles in a closed path aroundanother particle.

The U and S-matrices of the TC and DSem model canbe calculated exactly.19,27 However, away from these fixedpoints, the Hamiltonian is no longer exactly solvable and wehave to obtain the matrices numerically.

We obtain the U and S-matrices by constructing overlapsof so-called Minimally Entangled States (MES) on non-trivialbipartitions of a torus19,20 (Fig. 7). The MES are ground stateswhich minimize the bipartite entanglement entropy on a givenbipartition of the system. When we consider non-trivial bi-partitions of a torus, it turns out that the MES are eigenstatesof Wilson loop operators defined parallel to the entanglementcuts. Now, take for example the S-matrix. It is the matrixthat transforms the eigenstates of one Wilson loop operator toanother. Thus, it follows that the S-matrix corresponds to acertain unitary transformation in the MES basis.

Topological entanglement distinguishes between the topological and trivial phase

Both topological phases have the same topological entanglement

S. C. Morampudi, C.v. Keyserlingk, and FP  Phys. Rev. B 90, 035117 (2014).N = 48

S

U

• matrix relates MES along independent torus cuts: mutual statistics

• encodes the self statistics

Intrinsic topological order

• Minimally entangled states (MES) for non-trivial bipartitions of the torus

|⌅ii, . . . , |⌅N i |⌅0ii, . . . , |⌅0

N i

[Zhang et al. ’12; Grover et al. ’12]

0

B@⌅01...

⌅0N

1

CA = S ·

0

B@⌅1...

⌅N

1

CA

(a) (b)

A B

Wy

-a

a xy

Intrinsic topological order: Spin Liquid

• Interpolate between the two fix point models and use exact diagonalization to study the phase diagram

!

! The modular matrix distinguishes DS and TC phases

Numerical indicationsfor a 1st order transition

7

Figure 7: Non-trivial bipartitions of the torus. We consider the par-tition on the right and trace over one half of the torus to obtain thereduced density matrix. The yellow lines indicate the long loopsaround the torus characterizing the possible winding sectors.

To obtain the MES, we need to first calculate all the degen-erate ground states of the system. There are four ground statesfor both the TC and the DSem phase on a torus. These canbe characterized by their winding numbers (modulo 2) aroundtwo independent directions of the torus, i.e., by having an evenor odd number of loops winding around the torus. We obtainfour almost degenerate ground states for the Hamiltonian ofEq. (7) by diagonalizing it separately in each winding numbersector. We then obtain the MES (|⌅i) by minimizing the en-tanglement entropy for a linear combination of these groundstates. Since we have four ground states, our parameter spaceconsists of the surface of a 3-sphere plus additional phase fac-tors,

|⌅xi = ⇠1

|00i+ ei�1⇠2

|01i+ ei�2⇠3

|10i+ ei�4⇠4

|11i, (14)

where |⌅xi corresponds to a MES on one of the non-trivialbipartitions of the torus and |↵�i is the ground state in the ↵�winding sector.

Since the model is defined on the honeycomb lattice, thetransformation of the MES under a 2⇡/3 rotation allows usto calculate the US matrix (Appendix B).19 As shown in theappendix, we can also use the US matrix to calculate the Uand S matrices individually. As an example, we indicate the Uand S-matrices obtained at two points on the transition alongthe line � = 0 with ⌘ = 1 � µ. At µ = 0.25, we obtainmatrices close to the exact ones for the TC model

U0.25

jj

=

0

B@

1.01.01.0�1.0

1

CA+ 10

�1

0

BB@

1.5e�i0.1⇡

0.7e�i0.1⇡

1.0ei0.9⇡

0.8ei0.1⇡

1

CCA

S0.25

=

1

2

0

B@

1.0 1.0 1.0 1.01.0 1.0 �1.0 �1.01.0 �1.0 1.0 �1.01.0 �1.0 �1.0 1.0

1

CA+

Figure 8: Variation of the U -matrix as the system is tuned from theTC phase to the DSem phase. The angle ↵j plotted is related to theU -matrix by Ujj = ei↵j . We see a transition from the fermionicstatistics indicative of the TC phase to the semionic statistics indica-tive of the DSem phase. The region near the transition point is am-biguous due to an additional degeneracy which comes into play. Thesize of the system considered is 48 spins.

10

�1

0

BB@

0.3ei0.9⇡ 0.3ei0.9⇡ 0.3ei0.9⇡ 0.3ei0.9⇡

0.3ei0.9⇡ 0.3ei0.7⇡ 0.4ei0.6⇡ 0.7e�i0.9⇡

0.3ei0.9⇡ 0.2e�i0.7⇡

0.5e�i0.1⇡

0.9e�i0.9⇡

0.3ei0.9⇡ 0.7e�i0.1⇡

0.1ei0.7⇡ 0.4ei0.1⇡

1

CCA

At µ = 0.75, we correspondingly obtain matrices close tothe exact ones for the DSem model.

U0.75

jj

=

0

B@

1.01.0�1i1i

1

CA+ 10

�1

0

B@

1.30.7

0.8ei0.5⇡

0.8e�i0.5⇡

1

CA

S0.75

=

1

2

0

B@

1.0 1.0 1.0 1.01.0 1.0 �1.0 �1.01.0 �1.0 �1.0 1.01.0 �1.0 1.0 �1.0

1

CA+

10

�1

0

BB@

�0.3 �0.3 �0.3 �0.3�0.3 �0.2 0.6ei0.8⇡ 0.7e�i0.8⇡

�0.3 0.7e�i0.3⇡

0.4e�i0.9⇡

0.7ei0.3⇡

�0.3 0.7ei0.3⇡ 0.7e�i0.3⇡

0.5ei0.9⇡

1

CCA

From the plot of the elements of the U matrix (Figure 8),we can identify the characteristic statistics of the quasiparticle

Uj,j = ei↵j

S. C. Morampudi, C.v. Keyserlingk, and FP  Phys. Rev. B 90, 035117 (2014).

Intrinsic topological order: FCI

A. Grushin, J. Motruk, M. P. Zaletel, FP, arXiv:1407.6985

• Density Matrix Renormalization Group (DMRG) : Fractional Chern Insulators - Circumferences up to sites - Fingerprints of topological order - Phase stable as

L = 12

V ! 1

[Neupert et al, Sun et al, Tang et al ’11]

• No complete classification of topological order known yet!

Matter occurs in different phases

“Intrinsic”

“Chiral”

-Spin Liquids -FQHE

-IQHE -Chern Insulator

Anyons

Chiral Edge Modes

“Trivially disordered”

“Symmetry protected”

-Topological Insulators -Haldane Phase

Symmetry Fractionalization

“Symmetry broken”“Symmetry Enriched”

1D symmetry protected topological phases

• Spin-1 Heisenberg chain

- Haldane phase: Gapped in the bulk and no symmetry breaking

- Characterized by S=1/2 excitations at the edges

• Edge spins have been observed in the NMR profile close to the chain ends of Mg-doped Y2BaNiO

[Haldane ‘83]

H =P

j~Sj · ~Sj+1

E E

⇥4

[S.H. Glarum, et al.,]

[Affleck et al. ‘87 ]

AKLT: Each spin-1 splits up into two spin-1/2 = 1p

2(| "#i � | #"i)

… …

1D symmetry protected topological phases

FP, E. Berg, A.M. Turner, and M. Oshikawa, Phys. Rev. B 85, 075125 (2012). FP, A.M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010).

• SPT phases: Hamiltonian and ground state have the same symmetry ( )

• Fractionalization of symmetry operators

!

➡Linear representation in the bulk and projective representations at the edges:

• Cohomology : SPT phases are characterized by cohomology classes (complete classification )

GH = G

H2[G,U(1)]

gh = k : UgUh = ei�ghUk

[Chen et al. ’11, Schuch et al. 11]

[Gu et al. ‘09]

g 2 G

ULg UR

g

1D symmetry protected topological phases

FP and A.M. Turner, Phys. Rev. B 86, 125441 (2012). M. P. Zaletel, R. S. K. Mong,  J. Stat. Mech.  P10007 (2014). (see also Haegemann et al. 2012)

• Density matrix renormalization group (DMRG) is based on efficient representation of GS in1D: Schmidt basis

➡Projective representations can be directly extracted

• stabilizes the Haldane phase

H =P

j�Sj · �Sj+1 +D

Pj(S

zj )

2

Z2 ⇥Z2

Ug

S = 1

O / tr(Ux

Uz

U†x

U †z

)

|0i|0i|0i|0i|0i

[White ‘92]

[Ug]↵↵0 = h�R↵ |

Oj2L

gj |�R↵0i| i ⇡

P�↵=1

pp↵ |�L

↵i ⌦ |�R↵ i

• No complete classification of topological order known yet!

Matter occurs in different phases

“Intrinsic”

“Chiral”

-Spin Liquids -FQHE

-IQHE -Chern Insulator

Anyons

Chiral Edge Modes

“Trivially disordered”

“Symmetry protected”

-Topological Insulators -Haldane Phase

Symmetry Fractionalization

“Symmetry broken”“Symmetry Enriched”

g 2 H • Hamiltonian and ground state have the same symmetry ( )

• Fractionalization of symmetry operations in terms of the anyonic quasiparticles (QP)

➡QP induce projective representations:

!

• Co-homology : Inequivalent projective representations distinguish certain SET’s

Symmetry enriched topological order

GH = G

H2[G,U(1)]

Ug

U †g

gh = k : UgUh = ei�ghUk

[Ran, Wan, … ‘12]

• bosons model state: loop gas of AKLT chains

• and -particles in projective ( ) representation

• obtained by diagonalizing a mixed transfer matrix.

Z2

e fS = 1/2

S = 1

QP 1 e m fUx

Uz

U�1x

U�1z

1 -1 1 -1

U⌃

Symmetry enriched topological order

C.-Y. Huang, X. Chen, and FP, , Phys. Rev. B 81, 064439 (2014). .

Z2 ⇥ Z2 symmetry

U⌃

Z2 ⇥ Z2 symmetry

Z2

e fS = 1/2

(a) (b)

(c) (d)

2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

2 3 4 5 6 7 8-1.0

-0.5

0.0

0.5

1.0

1.5

L

L

1 -MES e -MES m-MES f -MES

1 -MES e -MES m-MES f -MES

(a) (b)

(c) (d)

2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

2 3 4 5 6 7 8-1.0

-0.5

0.0

0.5

1.0

1.5

L

L

1 -MES e -MES m-MES f -MES

1 -MES e -MES m-MES f -MES

Symmetry enriched topological order

QP 1 e m fUx

Uz

U�1x

U�1z

1 -1 1 -1

U⌃

• RVB state on the kagome lattice: spin liquid states (spin 1/2 singlets)

• and -particles in projective ( ) representation

C.-Y. Huang, X. Chen, and FP, , Phys. Rev. B 81, 064439 (2014). .

Characterizing topological orders in quantum matter

“Intrinsic topological order”-Toric Code / Double Semion -Fractional Chern Insulators

Anyons

“Symmetry protected topological phases”

“Symmetry enriched”

Symmetry Fractionalization

Thank You!

Ari M. Turner, Johns HopkinsErez Berg, Weizman Masaki Oshikawa, ISSP

Siddhardh Morampudi, MPIPKSJohannes Motruk, MPIPKS Adolfo Grushin, MPIPKS

Xie Chen, Caltech Ching-Yu Huang, MPIPKS / Stony brook

Many-body dynamics out of equilibrium

We also offer individual fellowships (phd, postdoc, sabbatical). Applications are accepted continuously. For details, please check www.pks.mpg.de/visitors

International Focus Workshop11 - 14 March 2015

Significant advances in controlling coherent many-body systems in both condensed matter and ultra-cold atomic systems have raised important questions regarding their dynamics. This workshop brings together theorists and experimentalists studying dynamical phenome-na and nonequilibrium properties of many-body systems.

Applications are welcome and should be made by using the application form on the conference web page (please see URL below). The number of attendees is limited. The registration fee for the international workshop is 120 Euro and should be paid by all participants. Costs for accommodation and meals will be covered by the Max Planck Institute. Limited funding is available to partially cover travel expenses. Please note that childcare is available upon request.

Main Topics:

� Many-body localization� Driven systems � Quench dynamics � Floquet physics� Adiabatic quantum computing

Scientific coordinators:

Jens H. BardarsonMPIPKS Dresden

Sid ParameswaranUC Irvine

Frank PollmannMPIPKS Dresden

Organisation:

Mandy LocharMPIPKS Dresden

Max

Pla

nck

Inst

itute

fo

r th

e P

hys

ics

of

Co

mp

lex

Sys

tem

s

For further information please contact:Visitors Program – Mandy LocharMPI for the Physics of Complex SystemsNöthnitzer Str. 38, D-01187 DresdenTel: +49-351-871-1933Fax: +49-351-871-2199mbdoeq15@pks.mpg.dewww.pks.mpg.de/~mbdoeq15/

Please apply by January 10, 2015 for full consideration.

Invited participants:(* to be confirmed)

D. Abanin (CA)L. Arrachea (AR)B. Bauer (US)I. Bloch (DE)A. Chandran (CA)L. F. Cugliandolo* (FR)A. Das (IN)A. Eckardt (DE)S. Gopalakrishnan (US)A. Green (UK)A. G. Grushin (DE)F. Huveneers (FR)J. Kjäll (SE)M. Kolodrubetz (US)C. Laumann (US)A. Lazarides (DE)M. Lukin (US)F. Marquardt (DE)

F. Heidrich-Meisner (DE)J. Meyer (FR)A. Mirlin (DE)A. Mitra (US)S. Miyashita (JP)J. E. Moore (US)M. Mueller (IT)S. Mukerjee (IN)R. Nandkishore (US)V. Oganesyan (US)D. Pekker* (US)A. Potter (US)M. Rigol (US)M. S. Rudner (DK)A. Scardicchio (IT)K. Sengupta (IN)M. Serbyn (US)S. Sondhi (US)M. Troyer* (CH)R. Vasseur (US)D. Weld (US)

image courtesy of Johan J. Ingles-Le Nobel http://extreme-macro.co.uk/

supported by the Helmholtz VI "New states of matter and their excitations"

1D symmetry protected topological phases

FP, E. Berg, A.M. Turner, and M. Oshikawa, Phys. Rev. B 85, 075125 (2012). FP, A.M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010).

Which symmetries can stabilize topological phases?

• Example : Rotation about single axis

- Redefining removes the phase

• Example : Phase for pairs

!

- Phases cannot be gauged away: Distinct topological phases

Zn

Rn = 1 ) UnR = ei�1

Z2 ⇥Z2

UR = e�i�/nUR

Rx

Rz

= Rz

Rx

) Ux

Uz

= ei�xzUz

Ux

�xz

= 0,⇡

• Numerical extraction of the topological entanglement entropy using a general partition

Intrinsic topological order

A B

C

�� = SA + SB + SC � SBC � SAC � SAC + SABC

Results does not depend on the particular ground state chosen!

[Kitaev and Preskill ’06, Levin and Wen ‘06]