Dirac Lattices

Dirac lattices: Down to High Energy!

Corneliu Sochichiu

SungKyunKwan Univ. (SKKU)

Chisinau, August 13, 2012

C.Sochichiu (SKKU) Dirac Lattices Swansea2012 1 / 36

Outline

1 Motivation & Philosophy

2 The model

3 Low energy limit

4 Emergent Dirac fermion

� Based on: 1112.5937 (v2.0 to come soon), see also 1012.5354


Motivation & Philosophy

QFT’s like Standard Model are relativistic theories, based on Lorentzsymmetry group

Lorentz symmetry is an exact symmetry, no was violation observedapart from. . .

. . . But are they indeed exact symmetries? Why?

The Lorentz symmetry is not compact, and there are critics, claimingthe inconsistency of field theories based on exact Lorentz symmetry[Jizba-Sardigli2011]

An alternative is to consider the high energy QFT models as lowenergy approximations to some non-relativistic model and Lorentzsymmetry as emergent approximate symmetry [. . . Horava2009. . . ]


Motivation & Philosophy

QFT’s like Standard Model are relativistic theories, based on Lorentzsymmetry group

Lorentz symmetry is an exact symmetry, no was violation observedapart from. . . so far

. . . But are they indeed exact symmetries? Why?

The Lorentz symmetry is not compact, and there are critics, claimingthe inconsistency of field theories based on exact Lorentz symmetry[Jizba-Sardigli2011]

An alternative is to consider the high energy QFT models as lowenergy approximations to some non-relativistic model and Lorentzsymmetry as emergent approximate symmetry [. . . Horava2009. . . ]


Emergent Lorentz & Gauge symmetry

Apart from what one can imagine, there are physical examples of emergingLorentz symmetry

Graphene: Since long time it is known that the electron wave functionin the low energy limit is described by relativistic Dirac fermion in2+1 dimensions [Wallace1947]

The low energy theory has an emergent Lorentz and global(nonabelian) gauge invariance. The global gauge invariance can bepromoted to local one by considering the low energy limit of latticedefect fields [CS2011]

Tomonaga–Luttinger liquid. . .

Can the same scenario be applied to high energy particle physics infour dimensions?

“Space diamond” lattice regularization [Creutz2007]


Fermi surface

Consider a Fermi system (Pauli exclusion principle)

In the low energy limit the dynamics is determined by the states nearthe Fermi surface

Fermi surface can take the forms of various geometrical varieties:points, lines, etc

Which of these shapes are stable?

ABS construction [Atiyah-Bott-Shapiro]: Varieties with non-trivial topological(in fact, K-theory) charge [Horava2005,Volovik2011]


Fermi point

“Mathematical Fact’’: Fluctuations around a Fermi point aredescribed by Weyl/Dirac/Majorana particle

Stable and non-stable Fermi points:I stability: no small deformations can lead to disappearance of the Fermi

point (no consistent mass term is possible)I non-stability: Small deformations can lift the Fermi point (one can

generate a consistent mass term)

In the case of a Fermi point, the stability can be provided bynontrivial homotopy class of maps from the sphere surrounding thepoint to the space of energy matrices [Volovik2011]


So, do we live on a Fermi point?

Fermi systems provide a convenient tool for the encoding of thespace-time geometry [Lin-Lunin-Maldacena]

Matrix models and gauge theories lead to Fermi systems or behavelike Fermi systems

The elementary particle spectrum can be seen as quasiparticleexcitations around Fermi surface [Volovik]

Gauge/gravity interactions can be generated dynamically [Sakharov1968]

So, a fermi system is all one needs to build a Universe like ours, but. . .

can we figure out a microscopic theory flowing to the existentparticle models in the IR?








can we figure out a microscopic theory flowing to the existentparticle models in the IR?

OK. . . However, first let’s look at a simpler problem!








can we figure out a the microscopic theory flowing to the existentparticle models in the IR?








can we figure out the microscopic theory flowing to the existentparticle models in the IR?


The Setup of the problem

Tight-binding Hamiltonian

H =∑<xy>

t<xy>a†xay = a† · T · a

x , y are sites of a graph and T is its adjacency matrix

T = ‖t<xy>‖

t<xy> are the transition amplitudes; they can be, in principle,arbitrary depending only on the pair < x , y >, but we will restrictourselves to only those which admit a continuum low energy limit

Which structure of T leads to a Dirac fermion in this limit?


Graph structure

Consider physical restrictions on the adjacency matrix

The graph: a superposition of D-dimensional Bravais lattices with thecommon base {ı}, i = 1, . . . ,D

unit cell consists of p sites labeled by the sublattice index α = 1, . . . , p

each site is parameterized by its Bravais lattice coordinates as well asthe sublattice index:

xαn = xα + ni ı,

The sites inside the cell can be connected in an arbitrary way

Only “neighbor” cells are connected

Therefore the adjacency matrix has a block structureI it could be 2D, 3D, etc. blocks. . .

The block structure is needed in order to define the proper continuum limit


The Hamiltonian

The Hamiltonian can be rewritten as,

H =∑n,i

(a†n+ıΓian + a†nΓ†i an+ı

)+∑n

a†nMan

Γi are the inter-cell adjacency matrix blocks and M is the intra-cell matrix

Now, consider the low energy limit for the theory described by thisHamiltonian

We want to find Γi and M leading to Dirac fermion


Fourier transform & Brillouin zones

Due to the translational invariance of Bravais lattice we can do the Fouriertransform

a(k) =∑n

aneik·n, an =

∫B

dk

(2π)Da(k)e−ik·n

The (normalized) Brillouin zone B :

k = ki ı, −π ≤ ki < π

{ı, i = 1, . . . ,D} : the dual (basis to the) Bravais basis

ı · = δij


Low energy limit

The Hamiltonian in the momentum space description:

H =

∫B

dk

(2π)Da†(k)

(∑i

(Γie

iki + Γ†i e−iki)

+ M

)a(k)

The low energy contribution is given by the modes near the lowestenergy states


Low energy limit


H =

∫B

dk

(2π)Da†(k)

(∑i

(Γie


+ M

)a(k)

The low energy contribution is given by the modes near the lowestenergy states

Recall: This is a fermionic system!


Low energy limit


H =

∫B

dk

(2π)Da†(k)

(∑i

(Γie


+ M

)a(k)

The low energy contribution is given by the modes near the lowestenergy states Fermi surface



Low energy limit


H =

∫B

dk

(2π)Da†(k)

(∑i

(Γie


+ M

)a(k)

The low energy contribution is given by the modes near the lowestenergy states Fermi surface


Assume also: symmetric energy spectrum and half-filling of the Fermisea: EF = 0


(Generalized) Fermi surfaces

Fermi surface is an interface between occupied and non-occupiedstates in a system withe exclusion principle

Generic case: Fermi surface has (spacial) co-dimension oneI In D = 3 it is a 2D Fermi surfaceI In D = 2 it is Tomonaga–Luttinger fermion

It is Fermi point and only Fermi point, which brings to the DiracFermion in the low energy limit.

So, we are interested in systems which lead to point-like Fermisurfaces.

Deformations/corrections can lead to the degeneracy of the Fermisurface down to D − 2, D − 3, etc


Fermi point conditions

Fermi point conditions are

det[h(K ∗)] = 0 (Fermi level)

det[h(K ∗ + k)] = αi (K ∗)ki + O(k2) 6= 0 (point cond.)

The energy matrix h(K ) =

[∑i

(Γie

iKi + Γ†i e−iKi

)+ M

]The Hilbert space splits into energy bands: the groups of statescontinuously connected by the variation of “momentum” k


Fermi point condition

Consider the subspace of states belonging to the gapless bands

The Fermi surface condition becomes h(K ∗) = 0

The Fermi point conditions imply that αi are generators of theD-dimensional Clifford algebra C D

Representation can be reducible

The ‘minimal’ irreducible representation has dimension 2[D2 ]

(Therefore p = N ′2[D2 ])

A way to obtain a reducible representation is by reduction of anirreducible representation of C D′ for some D ′ > D

This algebra can be explicitly constructed. . .


ΣI -basis

The inter-cell adjacency matrices Γi and intra-cell adjacency Mshould be elements of Clifford algebra

They can be expanded in terms of a ‘holomorphic’ Clifford algebrabasis consisting of matrices ΣI , I = 1, . . . ,D ′/2, for some even

D ′ ≥ D, Γi = ΓiIΣI , Γ†i = ΓiIΣ†I ⇒ M = mI (ΣI + Σ†I )

The ΣI matrices are satisfying the algebra

{ΣI ,ΣJ} = {Σ†I ,Σ†J} = 0, {ΣI ,Σ

†J} = δIJI


Fermi point condition in ΣI -basis

In terms of the ΣI -basis the Fermi point equation

hI (K ) = 0, hI (K ) ≡∑i

ΓiI eiKi + mI .

should admit only point-like solutions −→ K ∗i , i = 1, 2, . . .Such a solution will be described below. So far assume it exists. . .


Some properties of Fermi points

Reality condition: If K ∗ is a solution −K ∗ is a solution too

We can associate a topological charge to every point K ∗i ,

Ni =Γ(D2 + 1

)pDπ

D2

∫SD−1α

tr(h−1dh

)∧(D−1),

Due to compactness of the momentum space, the total charge shouldvanish, ∑

i

Ni = 0


Minimal case

Each pair ±K ∗i will contribute one fermionic species leading todegeneracy and enhancement of the global nonabelian internalsymmetry in the low energy limit. The minimal solution contains asingle pair ±K ∗

Therefore, the minimal nontrivial configuration consists of two Fermipoints ±K∗ or o single point (the degenerate case) K∗ = 0

Let’s concentrate on the (non-degenerate) minimal case


Emergent Dirac fermion: Dirac matrices

In the vicinity of ±K ∗

αi (±K ∗) = iΓiI

(cosK ∗i (ΣI − Σ†I )± sinK ∗i i(ΣI + Σ†I )

),

and ki = KI ∓ K ∗IIntroduce index associated with the sign of ±K ∗

αi = iΓiI

(cosK ∗i (ΣI − Σ†I )⊗ I + sinK ∗i i(ΣI + Σ†I )⊗ σ3

)


Embedding into Cartesian basis

αi are linear combinations of matrices βa, a = 1, . . . ,D ′ forming thestandard basis of the Clifford algebra C D′

β2I−1 = −(ΣI + Σ†I )⊗ σ3, β2I = i(ΣI − Σ†I )⊗ I

i.e. αi = ξai βa

with ξ2I−1i = ΓiI sinKi , ξ2I

i = ΓiI cosKi .

With matrices βa we can associate a Cartesian coordinate system


Emerging geometry

The matrices βa satisfy Clifford algebra relations

{βa, βb} = 2δabI

ξai can be regarded as vielbein coefficients for the embedding of aD-dimensional plane into RD′ , where the Clifford algebra is defined.

Induced metric: gij = ξai ξaj

Introducing ‘Cartesian momentum’: qa′

= ξa′

i ki , a′ = 1, . . . ,D

The low energy Hamiltonian takes the form

H = J

∫dDq

(2π)Dψ†(q)βa′q

a′ψ(q) J =√

det ‖gij‖


The low energy action

1 Introduce γ0 = iD′/2β1β2 · · ·β2Np s.t. (γ0)2 = −1

2 Introduce the Dirac conjugate ψ = ψ†γ0

3 Make a (continuous) inverse Fourier transform to real space and getthe low energy action

Sl .e. = −i∫

dD+1x ψγµ∂µψ, γa′

= γ0βa′

That’s all!We got:

A Dirac fermion

An induced geometry






Sl .e. = −i∫


= γ0βa′


X A Dirac fermion

An induced geometry

Something is left behind, however. . .






Sl .e. = −i∫


= γ0βa′


X A Dirac fermion

X An induced geometry

Something is left behind, however. . .


Moduli space for ΓI

Recall, ΓI should satisfy the condition that equations

hI (K ) = 0, hI (K ) ≡∑i

ΓiI eiKi + mI

have the only solutions for isolated ±K ∗.What are the ΓI ’s?


A Mechanical analogy

. . . we can find a mechanical analogy for the Fermi point condition interms of arm-and-hinge mechanism

Consider [D ′/2] (D + 1)-gons with sides ΓiI . (I = 1, . . . , [D ′/2] countspolygons, and i = 1, . . . .D counts sides within one polygon.) The orienta-tion of the sides with number i in the complex plane is the same for everypolygon and is given by the factor eiKi .Then,

The Fermi level condition corresponds to the closure of the polygon

The point-like nature corresponds to its rigidity

Think about [D ′/2] superposed hinge mechanisms in the two-dimensionalplane, each with D arms of lengths ΓiI and orientation Ki , as well asone horizontal arm of length mI . The arms with the same number i indifferent mechanisms counted by I are kept parallel. The Fermi pointcondition implies that the whole hinge mechanism is (i) closed, and (ii)rigid


Arms and hinges

1

K1

K2

K3

K6

|Γ1I/mI |

|Γ2I/mI |K4

|mI |

K1

K2

K3

K6

|Γ1I |

|Γ2I |K4

K5

(a) (b)

K5

Figure : The (D + 1)-gon representing the equation∑

i ΓiI eiKi + mI for mI 6= 0.

The lengths of the sides are given by |ΓiI |, the angles to the horizontal are Ki . Inthe case of mI = 0 the (D + 1)-gon degenerates to a D-gon. (a) A heptagon forthe problem in D = 6. (b) A set of two heptagons with parallel sites, solving theFermi point problem can be regarded as finding a rigid hinge mechanism. In thiscase more heptagons are needed to make the mechanism rigid.


D = 2 case

1

Γ11eiK1

α

β

γ

Γ12eiK2

Figure : D = 2 situation. There is a unique triangle you can construct with giventhree site lengths. The angles of the triangle are related to the momenta in thefollowing ways: α = π − K1, β = π − K2 + K1 and γ = π + K2.

A single hinge system is needed (a triangle is uniquely determined by thelengths of its sites) ∑

i

ΓieiKi + m = 0


D = 2 case: the solution

Triangle sine rule

Γ1

| sinK2|=

Γ2

| sinK1|=

m

| sin(K2 − K1)|

Leads to solution for Γi

Γ1 =m sinK ∗2

sin(K ∗1 − K ∗2 ), Γ2 =

m sinK ∗1sin(K ∗2 − K ∗1 )

with Fermi points at ± (K ∗1 ,K∗2 )

More generally one can have congruent hinge mechanism based on theabove one:

ΓiI = ηIΓi , mI = ηI

The resulting system is equivalent (upon coordinate transformation) tographene


D = 3 case

3∑i=1

ΓiI eiKi + mI = 0, I = 1, 2

means closure of a solid quadrilateral

1

Γ′1e

iK1Γ′3e

iK3

Γ11eiK1

Γ12eiK1

Γ21eiK2

Γ21eiK2

Γ31eiK3

Γ32eiK3

Figure : The quadrilaterals corresponding to each polygon equation can beobtained from a single master triangle with sides 1, Γ′

1 and Γ′3, by cutting the

upper angle by the side Γ2I eiK2 . The dotted segments correspond to η1,3.


D = 3 case: the solution

The elementary geometry problem has the solution

Γ1I = mIΓ′1 − η1I =

mI sinK ∗3 + ξI sin(K ∗2 − K ∗3 )

sin(K ∗1 − K ∗3 )

Γ2I = −η2I = −ξI

Γ3I = mIΓ′3 − η3I =

−mI sinK ∗1 + ξI sin(K ∗1 − K ∗2 )

sin(K ∗1 − K ∗3 )


Arbitrary D: the Holomorphic Ansatz

For general polygon equation∑i

ΓiI eiKi + mI = 0, I = 1, . . . ,D ′/2,

consider the AnsatzΓiI = Γ′I δi ,2I−1 + Γ′′I δi ,2I .

As a result of substitution, the polygon equations split into D ′/2independent triangular equations,

Γ′I eiK2I−1 + Γ′′I e

iK2I + mI = 0.

Mimics the canonical form of the rotational matrix, which in an appropriatebasis is a composition of elementary rotations of two-dimensional planes


The solution

We know how to solve the triangular equation. . .

Γ′I =mI sinK ∗2I

sin(K ∗2I−1 − K ∗2I ), Γ′′I =

mI sinK ∗2I−1

sin(K ∗2I − K ∗2I−1)

I = 1, . . .D ′/2Embedding functions:

ξ2I−12I−1 =

mI sinK ∗2I sinK ∗2I−1

sin(K ∗2I−1 − K ∗2I ), ξ2I−1

2I =mI sin2 K ∗2I

sin(K ∗2I−1 − K ∗2I ),

ξ2I2I−1 =

mI sinK ∗2I−1 cosK ∗2I−1

sin(K ∗2I − K ∗2I−1), ξ2I

2I =mI sinK ∗2I−1 cosK ∗2I

sin(K ∗2I − K ∗2I−1),


Induced metricInduced metric is given by 2× 2 blocks(

g2I−1,2I−1 g2I−1,2I

g2I ,2I−1 g2I ,2I

)where

g2I−1,2I−1 =m2

I sin2 (K1) (cos (2K2I−1)− cos (2K2I ) + 2)

1− cos[2(K ∗2I−1 − K ∗2I )],

g2I−1,2I =m2

I sin(K2I−1)(2 sin3 (K ∗2I ) + cos (K ∗2I ) sin

(2K ∗2I−1

))1− cos[2(K ∗2I−1 − K ∗2I )]

,

g2I ,2I−1 =m2

I sin(K ∗2I−1

) (2 sin3 (K ∗2I ) + cos (K ∗2I ) sin (2K2I−1)

)1− cos[2(K ∗2I−1 − K ∗2I )]

,

g2I ,2I =

m2I (5 + cos(4K ∗2I )− 4 cos(2K ∗2I ) cos2(K ∗2I−1)− 2 cos(2K ∗2I−1)

4{1− cos[2(K ∗2I−1 − K ∗2I )]}.


Conclusion & Outlook

We considered conditions under which a discrete model on a graphproduces a Dirac fermion in the low energy limit

These conditions translate to algebraic equations on the adjacencymatrix

We found the general solutions for the case of D = 2, 3 lattices and a‘holomorphic’ solution in the general case. However, not clearwhether this solution is a general one.

As a ‘bonus’ we got induced geometry in the low energy theory

Next step would be to consider Fermi systems which generate‘desired’ symmetries, e.g. that of Standard model.

Dynamical adjacency matrix. The gauge and gravity degrees offreedom should be expected to emerge in the low energy limitthrough the Sakharov mechanism


Backup

Expand the point condition h(K ∗ + k)|Zero subspace = αi (K∗)ki 6= 0

αi (K∗) = i

(Γie

iK∗i − Γ†i e−iK∗i

)∣∣∣Zero subspace


ExampleHaving the solution in 3D, we can easily write a (random) example oflattice model, generating a 3 + 1 dimensional Dirac fermion in the lowenergy limit

m+ξ√3

−ξ

−m+ξ√3

m

−m

The Legend:

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

n

n + l1

n + l2

n + l3

Figure : Symbolic representation of the lattice for the model described byHamiltonian. The model is local in the sense that interaction is limited to thenearest unit cells. The hopping amplitudes are given by different types of lines, asexplained in the legend. The last two line types in the legend correspond tointernal lines of the cell.


Dirac Lattices

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