Method to preserve the chiral-symmetry protection of the zeroth Landau level on a two-dimensional lattice A. Don s Vela1G. Lemut1J. Tworzyd lo2and C. W. J. Beenakker1

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Method to preserve the chiral-symmetry protection of the zeroth Landau level

on a two-dimensional lattice

A. Don´ıs Vela,1G. Lemut,1J. Tworzyd lo,2and C. W. J. Beenakker1

1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

2Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02–093 Warszawa, Poland

(Dated: September 2022)

The spectrum of massless Dirac fermions on the surface of a topological insulator in a perpendicu-

lar magnetic ﬁeld Bcontains a B-independent “zeroth Landau level”, protected by chiral symmetry.

If the Dirac equation is discretized on a lattice by the method of “Wilson fermions”, the chiral sym-

metry is broken and the zeroth Landau level is broadened when Bhas spatial ﬂuctuations. We

show how this lattice artefact can be avoided starting from an alternative nonlocal discretization

scheme introduced by Stacey. A key step is to spatially separate the states of opposite chirality in

the zeroth Landau level, by adjoining +Band −Bregions.

I. INTRODUCTION

A. Objective

We address, in a diﬀerent context, a problem origi-

nating from lattice gauge theory: How to place fermions

on a lattice in a way that respects both gauge invariance

and chiral symmetry [1–8]. Our context is topological in-

sulators [9], three-dimensional (3D) materials having an

insulating bulk and a conducting surface, with massless

Dirac fermions as the low-energy excitations. The Lan-

dau level spectrum of massless Dirac fermions is anoma-

lous, the zeroth Landau level is a ﬂat band pinned to zero

energy irrespective of the magnetic ﬁeld strength [10, 11].

Our objective is to model the surface states on a two-

dimensional (2D) lattice, without breaking the chiral

symmetry that protects the zeroth Landau level from

broadening by disorder. Let us introduce the problem

in some detail.

B. Zeroth Landau level

In a magnetic ﬁeld B, perpendicular to the surface of

the topological insulator, Landau levels form at energies

En=±~ω√n,n∈N, with ω∝√B. The zeroth Landau

level E0= 0 is magnetic-ﬁeld independent [12–15]. If the

perpendicular ﬁeld strength has spatial ﬂuctuations, for

example, because of ripples on the surface, all Landau

levels are broadened except the zeroth Landau level [16].

The E= 0 ﬂat band is protected by a chiral symmetry,

a unitary and Hermitian operator Cthat anti-commutes

with the Hamiltonian [17]. Indeed, the massless 2D Dirac

Hamiltonian

HD=v~kxσx+v~kyσy(1)

anticommutes with the Pauli matrix σz, and this sym-

metry is preserved if one introduces a space-dependent

vector potential by ~k7→ ~k−eA(r).

Topological considerations [18–20] then enforce the ex-

istence of an N-fold degenerate eigenstate at E= 0, with

FIG. 1. Slab of a topological insulator in a perpendicular

magnetic ﬁeld B. Landau levels form on the top and bottom

surface at energy |E| ∝ √n,n= 0,1,2, . . ., symmetrically

arranged around E= 0. The density of states (DOS) of the

zeroth Landau level is not broadened by a spatially ﬂuctuating

B, provided that the slab thickness dis suﬃciently large that

the two surfaces are decoupled.

Nthe number of ﬂux quanta through the surface. The

ﬂat band has a deﬁnite chirality, meaning that it is an

eigenstate of C=σzwith eigenvalue ±1 determined by

the sign of the magnetic ﬁeld.

If we consider a topological insulator in the form of

a slab (see Fig. 1), the top and bottom surfaces each

support a zeroth Landau level, of opposite chirality. The

two ﬂat bands will mix and split if the slab is so thin that

the wave functions of opposite surfaces overlap, but in

thick slabs this breakdown of the topological protection

is exponentially small in the ratio of slab thickness and

penetration depth.

C. 2D lattice formulation

A numerical simulation of the 3D system is costly, it

would be more eﬃcient to retain only the surface degrees

of freedom. If we discretize the 2D surface on a square

lattice (lattice constant a), the Hamiltonian must be pe-

riodic in the momentum components with period 2π/a.

The sin ak dispersion has the proper periodicity, but it

suﬀers from fermion doubling [8]: a spurious massless

degree of freedom appears at k=π/a.

We contrast two lattice formulations that avoid

arXiv:2210.01463v2 [cond-mat.mes-hall] 8 Jan 2023

fermion doubling: an approach due to Wilson [2] with

a sine+cosine dispersion, and an approach due to Stacey

[5] with a tangent dispersion.

In Wilson’s approach [2] the discretized Dirac Hamil-

tonian is

HWilson = (~v/a)X

α=x,y

σαsin akα

+ ∆σzX

α=x,y

(1 −cos akα).(2)

The cosine term ∝∆σzavoids fermion doubling, the only

low-energy excitations are near k= 0, but it breaks chiral

symmetry: HWilson no longer anticommutes with C=σz.

The alternative approach due to Stacey [5] has a tan-

gent dispersion,

HStacey = (2~v/a)X

α=x,y

σαtan(akα/2).(3)

Fermion doubling is avoided without breaking chiral

symmetry, at the expense of a nonlocal Hamiltonian:

While sines and cosines of momentum only couple nearest

neighboring sites, the tangent of momentum represents a

long-range coupling.

The merit of Stacey’s approach is that the nonlocal

Schr¨odinger equation HStaceyΨ = EΨ can be cast in the

form of a generalized eigenvalue problem,

HΨ = EPΨ,(4)

with local operators H,Pgiven by [21]

H=~v

2aσx(1 + cos aky) sin akx

+~v

2aσy(1 + cos akx) sin aky,(5a)

P=1

4(1 + cos akx)(1 + cos aky).(5b)

Because Hand Pare sparse Hermitian operators, and P

is positive deﬁnite,1the generalized eigenvalue problem

(4) can be solved eﬃciently.

D. Outline

We wish to show that the topological protection of

the zeroth Landau in a 3D topological insulator can be

obtained in a purely 2D formulation. To preserve chiral

symmetry we work with the tangent dispersion, in the

local representation (5).

The ﬁrst step is to introduce the vector potential in a

gauge invariant way — without breaking the locality of

1The operator Pis in general only positive semideﬁnite. It

becomes positive deﬁnite if we choose an odd number of lat-

tice points with periodic boundary conditions in the x– and y–

directions.

the generalized eigenvalue problem. We do this in the

next Section II. In Sec. III we calculate the Landau level

spectrum. The zeroth Landau level contains states of

both chiralities, we show that these can be spatially sep-

arated by adjoining +Band −Bregions. The robustness

of the ﬂat band is assessed in Sec. IV. We conclude in Sec.

II. GAUGE INVARIANT LATTICE FERMIONS

WITH A TANGENT DISPERSION

In Ref. 21 it was shown how the magnetic ﬁeld can

be incorporated in the generalized eigenvalue problem

(5) in a way that is gauge invariant to ﬁrst order in the

ﬂux through a unit cell. Here we will go beyond that

calculation, and preserve gauge invariance to all orders.

For ease of notation we set ~and the lattice constant

aboth equal to unity in most equations that follow. The

electron charge is taken as +e, so that the vector poten-

tial enters in the Hamiltonian as k7→ k−eA.

We recall the deﬁnition of the translation operator,

Tα≡eiˆ

kα=X

n|nihn+eα|.(6)

The sum over nis a sum over lattice sites on the 2D

square lattice, and eα∈ {ex,ey}is a unit vector in the

α-direction. The Peierls substitution ensures gauge in-

variance by the replacement

Tα7→ Tα=X

eiφα(n)|nihn+eα|,

φα(n) = eZn

n+eα

Aα(r)dxα.

(7)

Note that the A-dependent translation operators no

longer commute,

TyTx=e2πiϕ/ϕ0TxTy,(8)

where ϕis the ﬂux through a unit cell in units of the ﬂux

quantum ϕ0=h/e.

One could now apply the Peierls substitution directly

to the Hamiltonian HStacey from Eq. (3), but then one

runs into the obstacle noted in Ref. 21: The transfor-

mation to a local generalized eigenvalue problem only

succeeds to ﬁrst order in A, higher order terms become

nonlocal. Here we therefore follow a diﬀerent route.

We rewrite the operators Hand Pfrom Eq. (5) in

terms of the translation operators (6) and apply the

Peierls substitution (7) at that level. Noting that 1 +

cos kα=1

2(1 + Tα)(1 + T†

α), sin kα=1

2i(Tα−T†

α), we

deﬁne

H=~v

8iaσx(1 + Ty)(Tx− T †

x)(1 + T†

+~v

8iaσy(1 + Tx)(Ty− T †

y)(1 + T†

x),(9a)

P= ΦΦ†,(9b)

Φ = 1

8(1 + Tx)(1 + Ty) + 1

8(1 + Ty)(1 + Tx).(9c)

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Methodtopreservethechiral-symmetryprotectionofthezerothLandaulevelonatwo-dimensionallatticeA.DonsVela,1G.Lemut,1J.Tworzydlo,2andC.W.J.Beenakker11Instituut-Lorentz,UniversiteitLeiden,P.O.Box9506,2300RALeiden,TheNetherlands2FacultyofPhysics,UniversityofWarsaw,ul.Pasteura5,02{093Warszawa,Poland(Dated...

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Method to preserve the chiral-symmetry protection of the zeroth Landau level on a two-dimensional lattice A. Don s Vela1G. Lemut1J. Tworzyd lo2and C. W. J. Beenakker1

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