Zigzag edge modes in a Z2 topological insulator: Reentrance and completely flat spectrum

Ken-Ichiro Imura,1,2 Ai Yamakage,1 Shijun Mao,1,3 Akira Hotta,1 and Yoshio Kuramoto1

1Department of Physics, Tohoku University, Sendai 980-8578, Japan2Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan

3Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China�Received 31 May 2010; revised manuscript received 30 July 2010; published 27 August 2010�

The spectrum and wave function of helical edge modes in Z2 topological insulator are derived on a squarelattice using Bernevig-Hughes-Zhang �BHZ� model. The BHZ model is characterized by a “mass” termM�k�=�−Bk2. A topological insulator realizes when the parameters � and B fall on the regime, either 0�� /B�4 or 4�� /B�8. At � /B=4, which separates the cases of positive and negative �quantized� spin Hallconductivities, the edge modes show a corresponding change that depends on the edge geometry. In the �1,0�edge, the spectrum of edge mode remains the same against change in � /B, although the main location of themode moves from the zone center for � /B�4, to the zone boundary for � /B�4 of the one-dimensional �1D�Brillouin zone. In the �1,1�-edge geometry, the group velocity at the zone center changes sign at � /B=4 wherethe spectrum becomes independent of the momentum, i.e., flat, over the whole 1D Brillouin zone. Furthermore,for � /B�1.354, the edge mode starting from the zone center vanishes in an intermediate region of the 1DBrillouin zone, but reenters near the zone boundary, where the energy of the edge mode is marginally below thelowest bulk excitations. On the other hand, the behavior of reentrant mode in real space is indistinguishablefrom an ordinary edge mode.

DOI: 10.1103/PhysRevB.82.085118 PACS number�s�: 72.10.�d, 72.15.Rn, 73.20.Fz

I. INTRODUCTION

Insulating states of nontrivial topological order have at-tracted much attention both theoretically and experimentally.A topological insulator has a remarkable property of beingmetallic on the surface albeit insulating in the bulk. Recentlymuch focus is on a specific type of topological insulators,1

which are said to be “Z2 nontrivial.”2 The latter occurs as aconsequence of interplay between a specific type of spin-orbit interaction and band structure.3 Such systems are in-variant under time reversal and show Kramers degeneracy.From the viewpoint of an experimental realization, the origi-nal idea of Kane and Mele �KM� �Refs. 2 and 3� was oftencriticized for being unrealistic since it relies on a relativelyweak spin-orbit coupling in graphene. In order to overcomesuch difficulty, Bernevig, Hugues and Zhang �BHZ� pro-posed an alternative system,4 which is also Z2 nontrivial butnot based on graphene. BHZ model was intended to describelow-energy electronic properties of a two-dimensional �2D�layer of HgTe/CdTe quantum well. Conductance measure-ment in a ribbon geometry5 showed that the system exhibitsindeed a metallic surface state, which is also called helicaledge modes.

This paper highlights the spectrum and wave function ofsuch helical edge modes in the BHZ model. Respecting ap-propriately the crystal structure of original HgTe/CdTe quan-tum well, one can safely implement it as a tight-bindingmodel on a 2D square lattice.1,4 Note, however, that in con-trast to KM model which can be represented as a purelylattice model, in BHZ an internal spin-1/2 degree of freedom,stemming from the s-type �6 and p-type �8 orbitals, resideson each site of the square lattice in addition to the real elec-tronic spin. We also mention that in the continuum limit withvanishing topological mass term, KM model has two valleys�K and K�� whereas BHZ has a single valley �at ��. Another

idea which we can borrow from graphene study is the sensi-bility of edge spectrum on different types of edge structure,i.e., either zigzag of armchair type.6,7 This applies also to thehelical edge modes of BHZ topological insulator in a ribbongeometry since the edge spectrum is predominantly deter-mined by how the 2D bulk band structure is “projected” ontothe one-dimensional �1D� edge axis. Indeed, the structure ofBHZ helical edge modes has been extensively studied in Ref.1 in the �1,0�-edge geometry and in the tight-binding imple-mentation. However, in the practical experimental setup,5

this is certainly not the only one which is relevant to deter-mine the transport characteristics at the edges. Here, in thispaper our main focus is on the other representative geometry,the �1,1� edge. In the �1,1� geometry, as a consequence of thespecific way how “hidden” Dirac cones �or valleys� in the 2Dspectrum is projected onto the �1,1� axis, edge modes showsome unexpected behaviors.

In order to motivate further the present study, let us firstrecall the importance of edge modes in the quantum Hallstate under magnetic field that exhibits a finite and quantized�charge� Hall conductivity �xy

c . In realistic samples with aboundary, quantization of Hall conductivity is attributed todissipationless transport due to a gapless chiral edge mode.In contrast to charge Hall effect, a finite spin Hall currentdoes not need breaking of the time reversal symmetry. In thequantized spin Hall �QSH� effect, the Chern number in thebulk takes an integral value, and correspondingly there ap-pears integral pairs of gapless edge modes. On the otherhand, the Z2 topological insulator is characterized by an oddnumber of gapless modes per edge that are robust againstweak perturbations preserving the time-reversal symmetry.

The existence of such gapless edge states is generallyguaranteed by a general theorem under the name of bulk/edge correspondence.8,9 The BHZ model has a convenientfeature that the location of the minimum energy gap can be

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controlled by changing the parameters in the model. In par-ticular, the sign of spin Hall conductivity changes discontinu-ously as the mass parameter of the model is varied. Hence,the corresponding change in edge spectrum should provideuseful information on the bulk/edge correspondence. Fur-thermore, understanding of the nature of edge modes underspecific edge geometries should serve as possible applicationof topologically protected phenomena in nanoarchitectonics.We take the representative cases of the �1,0� and �1,1� edges,which we call also the straight and zigzag edges,respectively.

This paper is organized as follows: in Sec. II, we clarifyour motivations to study the lattice version of a Z2 topologi-cal insulator �BHZ model�, implemented as a nearest-neighbor �NN� tight-binding model. Explicit form of theBHZ lattice hamiltonian is introduced in Sec. III. It is dem-onstrated that by considering a lattice model, one naturallytakes into account hidden Dirac cones, the latter lacking inthe analyses based on the continuum Dirac model. In Secs.IV and V, we study the detailed structure of gapless edgemodes under two different edge geometries: straight and zig-zag edges. Section V is the highlight of this paper, demon-strating that the zigzag edge modes of BHZ lattice modelshow unique features. We first point out that at �=4B, a pairof completely flat spectrum appears at E=0; besides the edgewave function can be trivially solved. We then show, in ahalf-empirical way, that this analytic exact solution at �=4B can be generalized to the case of an arbitrary � /B� �0,8� �this idea is schematically represented in Fig. 1�.Using the exact solution thus constructed, we highlight thenature of reentrant edge modes, another unique feature of theedge modes in the �1,1�-edge geometry. The reentrant edgemodes possess two contrasting characters in real and mo-mentum spaces: though well distinguished in real space, theylive in an extremely small energy scale in the spectrum. Sec-tion VI is devoted to conclusions. The gapless edge modes ofBHZ topological insulator are also treated in the frameworkof continuum Dirac model in Appendix.

II. STATEMENT OF THE PROBLEM

It has been well recognized that the quantized spin Hallconductivity is determined by wave functions of Bloch elec-trons over the entire Brillouin zone �BZ�. On the other hand,

only the electronic states near the Fermi level are relevant tothe change in the Hall conductivity when the Fermi level isshifted. Simplified effective models are useful for the lattercase since various theoretical techniques can be employed inthe low-energy range. In this paper, we work mainly with thelattice version of the BHZ model and make some commentsin the low-momentum limit.

A. Continuum vs lattice theories

Why do we have to go back to a lattice model? Firstbecause we need to recover the correct absolute value of spinHall conductance �xy

s . The latter is defined as the differenceof Hall conductance for up and down �pseudo� spins multi-plied by −� / �2e�:

�xys = −

�

2e��xy

↑ − �xy↓ � . �1�

In QSH systems, the spin Hall conductance is quantized tobe an integer in units of e / �2��. This is completely in par-allel with the quantization of charge Hall conductance inunits of e2 /h in quantized Hall system. In both cases, suchintegers are topological invariants and protected againstweak perturbations.34

On the other hand, if one calculates, using Kubo formula,the contribution of a single Dirac fermion, e.g., of the con-tinuous Dirac model at the � point4 to spin Hall conductance,then this gives half of the value expected from the topologi-cal quantization.10–13 In order to be consistent, it is naturallyassumed that there must be even number of Dirac cones.18

However, the low-energy effective theory with which we arestarting contains obviously a single Dirac cone.27 As we willsee explicitly, such trivial discrepancy is naturally resolvedby considering a lattice version of the BHZ model.4

Another aspect motivating us to employ the lattice versionof BHZ model is the fact the idea of an edge states is a realspace concept and we need a priori to go back to real spaceto give an unambiguous definition to it. In this paper, wehighlight the detailed structure of gapless edge modes undera specific edge geometry. Clearly, introduction of the latterneeds a precise description in real space. Recall also herethat edge modes of graphene nanoribbon exhibit contrastingbehaviors in zigzag and armchair edge geometries: e.g., thesystem becomes either metallic or semiconducting in thearmchair geometry, depending on Nr mod 3, with Nr beingthe number of rows whereas a completely flat edge modeappears in the zigzag geometry.6,7 Where does the differencecomes from? In momentum space, the question is how thebulk Dirac cone structure look like when viewed from theedge. Note that in the zigzag geometry the flat edge modeconnects the two Dirac points �DP�: K and K� whereas in thearmchair geometry these two points are projected onto thesame point at the edge. In a topological insulator, this bulk toedge projection is even a more subtle issue since not all theDirac cones are explicit �see Table I�. In a sense, zigzag edgein the square lattice BHZ model �see Fig. 8� plays the fol-lowing double role: it resembles the zigzag edge in grapheneat �=4B whereas it may rather resemble the armchair edgeat �=0 and at �=8B.

trivial phasetrivial phasetopological phase

k

Δ/B

Δ/B=4

0

4

8

trivial phasetrivial phase

extraporateno Δ/B dependence

flat edge modeexactly soluble at each k

FIG. 1. �Color online� Recipe �conceptual� for constructing theexact edge wave function in the zigzag edge geometry—a half-empirical way.

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B. Continuous Dirac model and its boundary conditions

Spin Hall conductance is a topological quantity deter-mined by the global structure of entire 2D Brillouin zone.The helical edge modes, encoding the same information, is,therefore, a priori not derived from a local description in the2D Brillouin zone. One exception to such a general idea isthe study of Ref. 14 �see also Appendix�, in which gaplessedge modes are derived from the continuous model in a stripgeometry by simply applying the condition that all the pseu-dospin components of wave function vanish at theboundary.15 This implies that information about the helicaledge modes is actually encoded in the original �single� Diraccone. This seems to be rather surprising, if one recalls that inthe case of KM model, the distinction between trivial andnontrivial phases is made by a relative sign of the mass gapat K and K� points, which are, of course, macroscopicallyseparated in momentum space. Here, in the BHZ model, thesame distinction is made by relative sign between the mass�k0� term and the k2 term added to the Dirac Hamiltonian.

Motivated by this observation, we investigate the struc-ture of helical edge modes under different boundary condi-tions for the periodic BHZ model, implemented as a squarelattice and NN tight-binding model. In parallel with the armchair and zigzag edges for graphene, we consider �a� usuallyconsidered �1,0�-�straight� boundary,4 as well as �b� �1,1�-�zigzag� boundary for the tight-binding BHZ model.

C. Explicit vs hidden Dirac cones

The idea of focusing on Dirac fermions in the descriptionof quantized Hall effect has appeared in the context of tran-sitions between different plateaus. For describing the transi-tions, half-integer quantization is not a drawback since thedifference of Hall conductance before and after the gap clos-ing is quantized to be an integer in units of e2 /h: 1 /2−�−1 /2�=1 or vice versa. A discrete jump in the Hall conduc-tance across the transition is indeed consistent with countingbased on the emergence of massless Dirac fermions at thetransition.16,21

The absolute value of Hall conductance is, on the otherhand, a winding number, and determined by the vortices.17

Here, each vortex gives, in contrast to a Dirac cone, an inte-gral contribution to the Hall conductance �in units of e2 /h�.An interesting question is whether the total Hall conduc-tance, often expressed as a topological invariant,17 can bealso written as a sum of contributions from emergent Diracelectrons in the spectrum. Our empirical answer is yes,18

indicating that the number of Dirac electrons emergent in thespectrum is always even, reminiscent of the no-go theoremof Nielsen and Ninomiya in 3+1 dimensions.19 It should benoted that here not only explicit Dirac electrons �gapless fora given set of parameters� but also hidden Dirac electrons�massive for that set of parameters� must be taken into ac-count. Such massive Dirac electrons are called “spectators”in Ref. 18, in the sense that they are inactive for the transi-tion. Spectators are indispensable to ensure the correct inte-ger quantization of the Hall conductance.

III. BHZ MODELS

A. BHZ model in the long-wavelength limit

Let us first consider the BHZ model in the long-wavelength limit. The low-energy effective Hamiltonian, de-scribing the vicinity of gap closing at �= �0,0�, is the mini-mal model to capture the physics of a Z2-topologicalinsulator. This effective Hamiltonian is also contrasting tothe prototypical KM model, in that the former describes onlya single Dirac cone. The distinction between the Z2 trivialand nontrivial phases is, therefore, made by adding a k2 termto the usual Dirac Hamiltonian. The explicit form of BHZHamiltonian is implemented by the following 44 matrix:

H�k� = �h�k� 0

0 h��− k� � , �2�

where k= �kx ,ky� is a 2D crystal momentum, here measuredfrom the � point. The lower-right block h��−k� is deducedfrom h�k� by imposing time reversal symmetry.

TABLE I. Nature of four Dirac cones in the BHZ lattice model. The four Dirac cones appear at differentvalues of the tuning parameter �, and at different points of the BZ: �, X, X�, and M. Away from the gapclosing, such Dirac electrons acquires a mass gap. The sign of such mass gap, together with the chirality ,determines their contribution to �xy

�s�= �e2 /h. In the table, only their sign is shown. The symmetry of thevalence orbital is also shown in the parentheses, which is either, s �inverted gap� or p �normal gap�, corre-sponding, respectively, to the parity eigenvalue: �s=+1 or �p=−1. The latter is related to Z2 index as�−1� =�DP�DP.20 We also assume B�0 with no loss of generality.

DP � X1 X2 M �DP�xy�s� �DP�DP

k= �kx ,ky� at the DP �0, 0� �0,� /a� �� /a ,0� �� /a ,� /a�

Mass gap � �−4B �−4B �−8B

Chirality + − − +

��0 −�p� +�p� +�p� −�p� 0 +1

0���4B +�s� +�p� +�p� −�p� 2e2 /h −1

4B���8B +�s� −�s� −�s� −�p� −2e2 /h −1

8B�� +�s� −�s� −�s� +�s� 0 +1

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The bulk energy spectrum: E=E�k� is then given by solv-ing the eigenvalue equation for the upper-left block h�k� ofthe 44 BHZ Hamiltonian, i.e.,

h�k���k� = E��k� . �3�

In order to represent h�k� in a compact form, we introduce ad vector, d= �dx ,dy ,dz�, each component of which is either aneven or an odd function of k: dx,y,z=dx,y,z�k�, whose parity isdetermined by symmetry considerations.4 As far as the low-energy universal properties in the vicinity of � point is con-cerned, we need to keep only the lowest order terms ofdx,y,z�k�, and in this long-wavelength limit, they read explic-itly,

dx�k� = Akx, dy�k� = Aky

dz�k� = � − B�kx2 + ky

2� . �4�

Other parameters which appear in Ref. 4, i.e., C and D areset to be zero, which, however, does not lose any generality.The bulk energy spectrum is thus determined by diagonaliz-ing the following “spin Hamiltonian,” h�k�=d�k� ·�, where�= ��x ,�y ,�z�, are Pauli matrices. Using the standard repre-sentation for �, h�k� reads explicitly as,

h�k� = � dz dx − idy

dx + idy − dz� . �5�

Each row and column of Eq. �5� represent an “orbital spin”associated with the s-type �6 and the p-type �8 orbitals ofthe original three-dimensional band structure of HgTe andCdTe.22 Then, by choosing the “spin quantization axis” in thedirection of d-vector, one can immediately diagonalize theHamiltonian h�k�, i.e.,

h�k�d�k�� = � d�k�d�k�� , �6�

where the eigenvalue E�k�= �d�k� given by,

d�k� = d�k� = �dx2 + dy

2 + dz2. �7�

This implies,

E�k�2 = �2 + �A2 − 2B��k2 + B2k4, �8�

where k2=kx2+ky

2. When ��A2 /2B��0, the dispersion re-lation Eq. �8� represents a wine-bottle structure, �Fig. 2� i.e.,E�k� shows a minimum at a finite value of k. At the criticalvalue �0=A2 /2B, the density of states shows van Hove sin-gularity.

B. BHZ model on square lattice and Dirac-cone interpretation

Lattice version of the BHZ model is implemented as atight-binding Hamiltonian. To construct such a Hamiltonianexplicitly, we replace linear and quadratic dependences inh�k� on kx and ky as in Eq. �5�, by a function which has theright periodicity of the square lattice. This can be imple-mented as,

dx�k� →A

asin�kxa� ,

dy�k� →A

asin�kya� ,

dz�k� → � −2B

a2 �2 − cos�kxa� − cos�kya�� , �9�

where a is the lattice constant. Equation �9� corresponds toregularizing the effective Dirac model on a square latticewith only NN hopping. In this setup, i.e., Eqs. �2� and �5�together with Eq. �9�, the lattice version of BHZ model ac-quires four gap closing points shown in Table I, if one allowsthe original mass parameter � to vary beyond the vicinity of�=0. The new gap closing occurs at different points in theBrillouin zone from the original Dirac cone �� point�,namely, at X1= �� /a ,0�, X2= �0,� /a�, and M = �� /a ,� /a�.The gap closing at M occurs at �=8B whereas the gap clos-ing at X1 and X2 occurs simultaneously when �=4B.

Each time a gap closing occurs, one can reexpand thelattice model with respect to small deviations of k measuredfrom the gap closing. The new effective model in the vicinityof such hidden gap closing falls on the same Dirac form asthe original one at the � point, up to the k2 term. In order toquantify the emergence of such hidden Dirac cones, one stillneeds the following two parameters: �i� the mass gap � �es-pecially, its sign� and �ii� the chirality . The latter is asso-ciated with the homotopy in the mapping: k→d�k�. Notethat in the gap closing at X1 and X2 at �=4B, the role ofk�=kx� iky is interchanged compared with the originalDirac cone at the � point. The former �latter� corresponds to=−1 �=+1�. The missing Dirac partner, in the sense ofRef. 19, is found in this way. Once the explicit form ofeffective Dirac Hamiltonian in the continuum limit is given,one can determine its contribution to �xy

�c,s�. In systems withTRS, i.e., of the form Eq. �2�, the contribution from h�k� to�xy

�c�=�xy↑ +�xy

↓ cancels with that of h��−k�. On the other hand,their contribution to �xy

�s�=−�e /2����xy↑ −�xy

↓ �, remains finite,takes a half-integral value, �1 /2 in units of e /2� �2e2 /h inconductance�. Contribution to �xy

�s� from a Dirac point with amass gap � and chirality is,

FIG. 2. �Color online� Bulk energy spectrum: E=E�kx ,ky� �ver-tical axis� of a BHZ lattice model, here, implemented as a NNtight-binding model on a square lattice: cf. Eq. �9�. The spectrum isshown over the entire Brillouin zone: −� /a�kx, ky �� /a �horizon-tal plane�. Parameters are chosen such that A=B=1 and �=2, i.e.,the system is in the topologically nontrivial phase: 0�� /B�8. Thespectrum shows a typical wine-bottle structure around the � point.

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�xy�s� = sign���

1

2

e

2�, �10�

provided that the Fermi energy is in the gap. This can beverified explicitly by applying the Kubo formula to the con-tinuum model.

As mentioned earlier, such counting based on the con-tinuum Dirac model, is known to describe correctly a dis-crete jump of �xy

�c� in the quantum Hall case. Here, we applythe same logic to QSH case. Imagine that one observes theevolution of �xy

�s�, starting with the trivial insulator phase,where �xy

�s�=0, and varying the mass parameter �. The Fermienergy is always kept in the gap unless there appears a Diraccone. Each time such a gap closing occurs, �xy

�s� shows adiscrete change, which can be attributed to the above Diracfermion argument. In the present model, one can verify ex-plicitly that this is indeed the case.

If one evaluates the spin Hall conductance from TKNNformula,17 �xy

�s� allows for the following representation, interms of Berry curvature integrated over the entire Brillouinzone:

�xy�s� =

e

8�

BZ

d2k

4�

�d

�kx

�d

�kx·

d

d3 , �11�

where d= d and d=d�k� is given, e.g., by Eq. �9�. For suchan explicit choice of d�k�, Eq. �11� is evaluated numerically,and plotted as a function of � /B in Fig. 3. When d�k� isgiven by Eq. �9�, the plotted curve �the solid curve shown inblue in Fig. 3, which looks practically like steps� is compa-rable with the column �DP�xy

�s� of Table I. Note that the ab-solute value of �xy

�s� is susceptible of the concrete implemen-tation of d�k� over the entire Brillouin zone whereas itsparity �whether it is even or odd� in units of e / �2�� remainsthe same. As well known, the latter determines system’s Z2property.2,20

We have seen that �xy�s� takes a finite value �e /2� when

0�� /B�8, i.e., the system is in the topological �invertedgap� phase. This is also consistent with the gapless edgepicture, in which the spin Hall conductance of twice the unitof quantum conductance e2 /h is attributed to two channels ofedge modes, which form a pair of Kramers partners. The

apparent half-integer quantization at the � point, in the senseof Eq. �10�, is compensated by the contribution from missingDirac partner�s�, and as a result, is indeed shifted by one-half, replaced by an expected integral quantization.

IV. STRAIGHT EDGE GEOMETRY

Let us first review the behavior of gapless edge modes inthe straight edge geometry, the latter commensurate with thesquare lattice, and can be chosen either normal to the �1,0� orto the �0,1� direction �as in Fig. 4�. Introducing an edge leadsto breaking of the translational invariance in the directionperpendicular to the edge, inducing a coupling betweenDirac cones.

A. Effective one-dimensional model

In the straight edge geometry shown in Fig. 4, electronsare confined inside a strip between the rows at y=a andy=Nra. The translational invariance along the x axis is stillmaintained, allowing for constructing a 1D Bloch state witha crystal momentum kx

kx,J = �I

eikxII,J , �12�

where kx is measured in units of 1 /a with a being the latticeconstant. I ,J=cI,J

† 0 is a one-body electronic state localizedon site �I ,J�, and cI,J

† is an operator creating such an electron.It is also convenient to introduce ckx,J

† , and express kx ,J askx ,J=ckx,J

† 0. Naturally, the two creation operators are re-lated by Fourier transformation similarly to Eq. �12�, i.e.,ck,J

† =�IeikIcI,J

† .In order to introduce the edges, it is convenient to rewrite

the BHZ tight-binding Hamiltonian in terms of the hoppingbetween neighboring rows. Let us first consider the BHZtight-binding Hamiltonian in real space

�2 0 2 4 6 8 10

�2

�1

0

1

2

��B

Οxy

FIG. 3. �Color online� �xy�s� in units of e /2�, obtained by numeri-

cally evaluating the k integral in Eq. �11�, plotted as a function of� /B.

I

J

(0,0) (1,0)

(1,1)(0,1)

straight edge

FIG. 4. �Color online� Straight edge geometry. The two bound-aries of the strip �=edges� are, here, chosen to be perpendicular tothe �0,1� direction. In this figure, the number of rows in the strip isNr=5.

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H = �I,J

��� − 4B��zcI,J† cI,J + ��xcI+1,J

† cI,J + �ycI,J+1† cI,J + h.c.�� ,

�13�

where 22 hopping matrices, �x and �y, are given explicitlyas,

�x = − iA

2�x + B�z, �y = − i

A

2�y + B�z. �14�

In order to rewrite it in terms of the Bloch state, Eq. �12�, orequivalently, in terms of the corresponding creation and an-nihiration operators, ckx,J

† �ckx,J�, we perform Fourier transfor-mation in the �x-� direction along the edge. Equation �13�thus rewrites,

H = �kx,J

�D�kx�ckx,J† ckx,J + ��J+1,Jckx,J+1

† ckx,J + h.c.�� , �15�

where D�kx�’s are diagonal �on row� components, which readexplicitly,

D�kx� = A�x sin kx + �� − 2B�2 − cos kx���z

� A�x sin kx + ��kx��z. �16�

�J+1,J represents a hopping amplitude in the y direction, i.e.,between neighboring rows. Inside the strip, i.e., for J=1, . . . ,Nr, these amplitudes take the same value as in thebulk, given in Eq. �14�, i.e.,

�J+1,J = �y = − iA

2�y + B�z. �17�

In the tight-binding implementation, a strip geometry canbe introduced by switching off all the hopping amplitudesconnecting sites on the edge of the sample to the exterior ofthe sample. In our straight edge geometry, such outermostrows are located at J=1 and J=Nr. We turn off all the hop-ping amplitudes from J=1 to J=0, and the ones fromJ=Nr to J=Nr+1, i.e.,

�1,0 = �Nr+1,Nr= 0. �18�

This boundary condition, �i� breaks the translational invari-ance in the y direction, and �ii� restricts the Hamiltonianmatrix into NrNr blocks.

B. Spectrum and wave function

Let us construct the eigenvector of the straight edgeHamiltonian, Eq. �15�, with an eigenenergy E. Since Eq. �15�is already diagonal with respect to kx, we diagonalize Eq.�15� for a given kx, to find the energy spectrum E=E�kx�. Thecorresponding eigenvector is thus specified by E and kx, andtakes generally the following form:

E,kx = �j

� j�E,kx�kx, j , �19�

where � j�E ,kx� is a 22 spinor specifying the amplitudeand the pseudospin state of eigenvector on row j. One mightrather regard,

� = ��1

�2

�3

]

� , �20�

as the wave function of the corresponding eigenstate. Theeigenvalue equation,

HE,kx = E�kx�E,kx , �21�

can be rewritten, in terms of the �J�E ,kx�’s, in the form of arecursive equation

D�kx�� j + �y� j+1 + �y†� j−1 = E� j . �22�

All the information on the spectrum and the wave func-tion of both the extended bulk states and the localized edgestates is encoded in Eq. �22� and the boundary conditionwhich we will specify later. Since the recursive relation, Eq.�22�, is linear, its eigenmodes take the form of a geometricseries

� j = � j�0, �23�

where � is a solution of the characteristic equation which wewill derive later. If ��1, Eq. �23� may represent an edgemode. Since the recursive relation, Eq. �22�, is of secondorder, its characteristic equation becomes a quadratic equa-tion, giving two solutions for �, say, �=�1,2. On the otherhand, our recursive equation has also a 22 matrix form, wefirst have to solve a �reduced� eigenvalue equation for �0,assuming that � is given. The reduced eigen value equationfor �0 reads,

�D�kx� + ��y +1

��y

†��0 = E�0. �24�

Using Eq. �14� this can be also rewritten as,23

�D�kx� + iA

2�1

�− ���y + B�1

�+ ���z��0 = E�0, �25�

where D�kx� is given in Eq. �16�. This is a 22 eigenvalueequation, and there are generally two solutions for E and twocorresponding eigenvectors for a given �. Recall here that for0���4, our numerical data �Fig. 5� show that the edgespectrum behaves as E→0 in the limit of kx→0. Hereafter,we will focus only on such edge solutions. Since in the samelimit, D�kx�→��0��z, Eq. �25� reduces to,

���0� −A

2�1

�− ���x + B�1

�+ ����0 = 0. �26�

Note that we have multiplied both sides of Eq. �25� by �z. Itis clear from this expression that �0 can be chosen to be aneigenstate of �x, i.e., �0= x�, where

x+ =1�2�1

1�, x− =

1�2� 1

− 1� . �27�

If one denotes the eigenvalue of �x by s, as �x�0= ��0�s�0, then s=1 corresponds to �0= x+, and s=−1 to �0= x−. Namely, s specifies the eigenspinors given in Eq. �27�.

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Based on these eigenspinors, one can construct the totalwave function. Of course, one still needs to do determine theallowed values of �. For an eigenstate specified by s, � mustsatisfy,

��0� −A

2�1

�− ��s + B�1

�+ �� = 0. �28�

For s=1,

� =− ��0� � ���0�2 + A2 − 4B2

A + 2B� �1,2�0� . �29�

In Eq. �28�, if � is a solution of this quadratic equation fors=1, then 1 /� satisfies the same equation for s=−1.1 Thusthe general solution becomes a linear combination of thefollowing four basic solutions:

� j = �c+1�1�0� j + c+2�2�0� j�x+

+ �c−1�1�0�−j + c−2�2�0�−j�x− . �30�

Of course, at this point, this is just a solution at only onesingle k point, kx=0. However, a solution in the form of Eq.�30�, with �1,2�0� given in Eq. �29�, can be easily generalizedto satisfy Eq. �22� for an arbitrary, finite kx, after a simplereplacement of parameters. As for the eigenmode of the formof Eq. �23�, one has to solve a reduced eigenvalue equationfor �0, and determine � such that Eq. �25� is satisfied. How-ever, since D�kx� has the structure given in Eq. �16�, theeigenmodes for �0 given as Eq. �27� remain to be valid foran arbitrary kx. This might become clearer, if one decom-poses Eq. �25� into the following set of equations:

A�x sin kx�0 = E�0 �31�

���kx��z + iA

2�1

�− ���y + B�1

�+ ���z��0 = 0, �32�

i.e., Eq. �25� is recovered by adding both sides of Eqs. �31�and �32�. The first equation gives the energy dispersion, E=Es�kx�, if �0 is chosen to be an eigenstate of �x, i.e., �x�0= ��0�s�0, where

Es�kx� = sA sin kx = � A sin kx. �33�

Note that this is an exact edge spectrum valid over the entirerange of kx, as far as the edge solution is possible �see Fig.5�. On the other hand, Eq. �32�, analogous to Eq. �26�, jus-

tifies the previous conjecture: �0= x�. While, in the char-acteristic equation for �, one has to make the simple replace-ment: ��0�→��kx�, i.e., Eqs. �26� and �32� are identical upto this replacement. For s=1, the solution for � reads,

� =− ��kx� � ���kx�2 + A2 − 4B2

A + 2B� �1,2�kx� . �34�

Correspondingly, a general solution for � j can be constructedas,

� j = �c+1�1�kx� j + c+2�2�kx� j�x+

+ �c−1�1�kx�−j + c−2�2�kx�−j�x− , �35�

where the coefficients c�1,2 should be chosen to satisfy theboundary conditions. Equation �35� is smoothly connected toEq. �30� in the limit: kx→0.

C. Illustration of edge spectrum

Three panels of Fig. 5 show the energy spectrum �edge+bulk� for different values of � /B. As for the edge part ofthe spectrum, only a part of Eq. �33� is realized. In order todetermine which part of the spectrum in Eq. �33� is indeedactivated, we discuss below the case of semi-infinite geom-etry in some detail.

Figure 5 also demonstrates one of another specific featureof straight edge mode that the main location of the modemoves from the zone center for ��4B, to the zone boundaryfor ��4B. Thus, the group velocity intersecting with theFermi level reverses its sign, reflecting the sign change in �xy

s

in the bulk. This can be regarded as the concrete expressionof bulk/edge correspondence in the present case.8,9

What kind of a boundary condition should we apply inEq. �35�? Suppose that here our system is semi-infinite, forsimplicity, extended from j=1 to j→�. Such a boundarycondition can be applied, by formally requiring that the wavefunction �Eq. �35�� vanishes at j=0, i.e.,

�0 = �0

0� . �36�

This means that the coefficients c�1,2 in Eq. �35� should bechosen to satisfy,

FIG. 5. �Color online� Energy spectrum �numerical� in the straight edge geometry for different values of � �A=B=1�. The number ofrows Nr is, here, chosen to be Nr=100. The dotted curve is a reference, showing the exact edge spectrum given in Eq. �33�. Starting with theleft panel �=B �spectrum shown in red�, �=4B �center panel, spectrum in green� and �=5B �right, blue�.

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c+1 + c+2 = 0, c−1 + c−2 = 0. �37�

This turns out to be rather an important requirement for de-termining the range of validity of the solution given in Eq.�35� since the wave function � j must be normalizable. In Eq.�35�, only the eigenmodes of the form of Eq. �23� with ��1 should be kept in the solution �to be precise, both �1and �2 must be smaller than 1�. In a strip geometry, anothersolution, consisting of both ��1, describes the edge modelocalized at the opposite end of the system.

In Fig. 6, �1 and �2 are plotted as a function of � /B inthe limit kx→0. When 0�� /B�4, both �1 and �2 aresmaller than 1, namely, both 1 /�1 and 1 /�2 are larger than1. This means that only the first two terms of Eq. �35�, bothcorresponding to x+�s=1�, should be kept in the solution,i.e.,

c+1 = − c+2 � 0, c−1 = c−2 = 0. �38�

Outside this region, either �1 or �2 is larger than 1. When��1 since this implies automatically 1 /��1, the eigen-mode corresponding to the latter is still compatible with theboundary condition at j→�. However, because of theboundary condition at j=0, i.e., Eq. �37�, when �1�1 and�2�1, or vice versa, the only possible choice for the coef-ficients c�1,2 is

c+1 = c+2 = c−1 = c−2 = 0. �39�

Namely, a solution of the type of Eq. �35�, or an edge modecrossing at kx=0 is inexistent. This is consistent with the factthat an edge mode crossing at kx=0 exists only in the region,� /B� �0,4� in the straight edge geometry. �When � /B� �4,8�, the edge modes cross at kx=�, i.e., at the zoneboundary, which is also time-reversal symmetric.�

Coming back to the regime in which edge modes are ex-istent, i.e., � /B� �0,4�, one can clearly see in Fig. 6 thatthere are two different behaviors—a flat region where �1and �2 are degenerate, and the remaining part with twobranches. This is due to the fact that the two solutions for �could be either both real, or a pair of complex numbers con-jugate to each other. In the latter case, the two solutions havethe same absolute value, �1= �2 whereas in the presentcase, one can verify that this degenerate value is independentof �, i.e.,

�1 = �2 =��A − 2B

A + 2B� . �40�

This explains the existence of a flat region in Fig. 6. FromEq. �29� we see that the square root becomes pure imaginaryfor all k provided �−����+, where24

��

B= 2�1 ��1 −

A2

4B2� . �41�

At �=1, the edge solution is expected to merge with thebulk spectrum. This happens, when

cos kx = 1 −�

2B. �42�

Such a behavior becomes clearer by plotting �’s as a func-tion of kx. Figure 7 illustrates this feature at �=2 for A=B=1. One can indeed see that the edge spectra merge with thebulk at kx=km, satisfying Eq. �42�. The latter reduces, at thisvalue of �, to cos km=1−� / �2B�=0, i.e., km= �� /2.

It is also instructive to investigate the nature of edgemodes in real space, i.e., the wave function, and compare itwith the general solution �Eq. �35��. In numerical experi-ments, one has to diagonalize the 2Nr2Nr Hamiltonian ma-trix, equivalent to Eq. �15�. An eigen wave function is, there-fore, obtained as a 2Nr-component vector; here, in thestraight edge geometry, the latter can be chosen to be real.The edge wave function is easily identified if it exists, e.g.,by choosing the lowest-energy eigenmode �0 in the upperband. By investigating the structure of such an edge wavefunction, one can explicitly verify that the eigenmodes arespanned by two eigenspinors given in Eq. �27�. In repeatingsuch numerical experiments for different kx and � /B, onecan naturally distinguish an edge state from a bulk state byfocusing on the spatial distribution of the wave function.Here, what deserves much attention is that one can recognizea one-to-one correspondence between localizability of thewave function �0 and its spinor structure.

V. ZIGZAG EDGE GEOMETRY

Let us turn to the case of a different edge geometry, thezigzag edge geometry, shown schematically in Fig. 8. Asmentioned earlier, the zigzag edge geometry considered hereis, in a sense, analogous to a more popular edge geometry ofgraphene ribbon, named in the same way, but defined on ahexagonal lattice. Here, on a square lattice, a zigzag edge isintroduced, either normal to �1,1� or �1,−1� direction �as inFig. 8�. Electrons in the zigzag edge geometry are, therefore,confined to a strip diagonal in the cartesian coordinates, say,along the �1,1� direction as in Fig. 8.

Intuitively, say, because the zigzag surface is literally,“rough,” one expects that this edge geometry might have astronger tendency to trap electrons in the vicinity of theboundary. We show below, on one hand, that this intuitionfrom the macroscopic world is still valid in the microscopicquantum mechanical world. But just as a result of this stron-ger tendency to keep the electrons in its vicinity, on the otherhand, the zigzag edge shows various curious phenomena,

FIG. 6. �Color online� �1 and �2 plotted as a function � /B inthe limit kx→0.

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e.g., completely flat edge modes, and the reentrance of edgemodes in k space, etc.

Clearly, the translational invariance along the �1,1� axis ismaintained, on which x axis is introduced, together with theconserved momentum k=kx in this direction. Accordingly, yaxis is chosen to be in the �1,−1� direction. It may be alsouseful to redefine the indices I and J such that the latticepoints in the original square lattice are located at �x , y�= �Ia /�2,Ja /�2�, where I is an even �odd� integer for J: even�odd�. In the zigzag edge geometry, the spectrum E=E�k�and the wave function � are determined by the followingrecursive equation:

�� − 4B��z� j + �� j+1 + �†� j−1 = E� j �43�

for � j, analogous to Eq. �22� in the straight edge geometry. Inorder to derive Eq. �43�, we first rewrote the tight-bindingHamiltonian �15� in the new labeling, and then considered aBloch state along the x axis, analogous to Eq. �12� but with acrystal momentum k conjugate to x, i.e., k=kx. As was thecase in Eq. �22�, � describes, here, in Eq. �43� the hoppingbetween adjacent rows, and reads explicitly as,

� = iA

2e−ik�x − i

A

2eik�y + 2B�z cos k . �44�

Note that here the crystal momentum k is measured in unitsof 1 / ��2a� so that the zone boundary is always given by k=�.25 To find an edge solution, we first express the solutionof Eq. �43�, in the form of a geometric series, written for-mally in the same as Eq. �23�. Recall that � is �generally� a�complex� number of, for an edge mode, amplitude smallerthan unity �with the understanding that the edge mode islocalized in the vicinity of j=0�. �0 is a two componenteigenvector of the following reduced eigenvalue equation:

��� − 4B��z + �� +1

��†��0 = E�0. �45�

From the analogy to the straight edge case, one may express� as

� = iA

2ck��x − �y� +

A

2sk��x + �y� + 2Bck�z �46�

and rewrite Eq. �45� into the following explicit form:

FIG. 7. �Color online� Upper panel: �1 and �2 plotted as afunction of kx at �=2 �A=B=1�. Reference lines are at kx /�= �0.5, and at �=1 and �=1 /�3—as for the latter, cf. Eq. �40�.�=1 corresponds indeed to the point at which the edge modesmerge with the bulk spectrum �central panel, Nr=50�. Lower panel:an enlarged image of the central panel, plotted also for a system oflarger size: Nr=100.

0 (1,1)-edge

3322

11

zigzagedge

FIG. 8. �Color online� Zigzag or �1,1�-edge geometry �here,chosen to be normal to the �1,−1� axis�, defined in terms of theoriginal square lattice, on which the new x and y axes are super-posed in blue �online�. Numbers on these axes are redefined indicesI and J, i.e., �x , y�= �Ia /�2,Ja /�2�. The number of rows Nr is herechosen to be, Nr=6. When Nr is even �odd�, the two edges areinversion asymmetric �symmetric� with respect to the center ofstrip.

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��� − 4B��z + �� +1

���A

2sk��x + �y� + 2Bck�z�

+ �� −1

��i

A

2ck��x − �y���0 = E�0, �47�

where we ck and sk are short-hand notations for, respectively,cos k and sin k. Comparing this form with the straight edgecase, one can see that here one cannot use the same recipefor solving the problem, i.e., solving the problem at, say, k=0 and then extrapolate its solution to general k. It seems notimpossible to proceed in that direction and solve the problemanalytically but here, we choose to take another route, whichis much simpler, to find still an exact solution, but in a half-empirical way.

A. Completely flat edge mode at �=4B

Some concrete examples of such energy spectrum areshown Fig. 9. A pair of gapless edges modes always appeariff 0�� /B�8. In contrast to the straight edge case, how-ever, they appear always in the vicinity of k=0 and intersectsat k=0.

One of the last panels of Fig. 9 shows a unique feature ofedge modes in the zigzag edge geometry. At �=4B, the edgemodes become completely flat, apart from a small finite-sizegap around the zone boundary. We have already seen suchflat edge modes in graphene in the case of zigzag edge ge-ometry �but on a hexagonal lattice�.7,26 In graphene, such flat

edge modes connect 1D projection of K and K� points via the1D BZ boundary. This is a similar behavior to the presentcase, if one regards the former as the limit of vanishing in-trinsic coupling �or topological mass �� in the KM model.One of the differences between the two cases is that here theflat edge modes cover the entire 1D Brillouin zone.

In order to elucidate the nature of flat edge mode at �=4B, first notice that at this value of � the diagonal terms of�the diagonal blocks of the Hamiltonian matrix in� Eq. �43�vanish. This implies, as in graphene nanoribbon in the zigzagedge geometry, the existence of an eigenstate of the form

� = ��1

0

�3

0

�5

]

� , �48�

i.e., an eigenvector satisfying �2j =0 �j=1,2 , . . .�. Here, asemi-infinite geometry is implicit �for approximating a rib-bon of sufficiently large width or Nr; our system extendedfrom j=1 to j=Nr�, with a boundary condition of �0=0.Under this setup, and with the condition of vanishing diago-nal matrix elements, Eq. �43� implies,

FIG. 9. �Color online� Energy spectrum in the zigzag edge geometry for different values of ��A=B=1�. Upper-left panel: �=0.2B�spectrum shown in red�, upper-central: �=0.8B �spectrum in green�, upper-right: �=2B �spectrum in blue�, lower-left: �=3.2B �spectrumin cyan�, and lower central: �=4B �spectrum in magenta�. At �=4B, the edge modes become completely flat and covers the entire Brillouinzone �cf. case of graphene in the zigzag edge geometry �Refs. 7 and 26��. Notice that here the horizontal axe is suppressed to make the edgemodes legible. Note that at �=4B the bulk spectrum is also gapless; the completely flat edge modes indeed touch the bulk continuum at thezone boundary �projection of 2D Dirac cones�. Compare this panel with the corresponding panel of straight edge case: Fig. 5. The numberof rows Nr is here chosen to be Nr=100. These five plots are superposed in the lower-right panel to show that the edge spectra at differentvalues of � are, in contrast to the straight edge case, not on the same curve. Even in the long-wavelength limit: k→0, their slopes aredifferent.

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��2j+1 + �†�2j−1 = 0�j = 1,2, . . .� . �49�

Simultaneously, it should also have a vanishing eigenenergyE=0 for consistency.

Clearly, Eq. �49� has a solution of the form of a geometricseries, here, for �2j−1 �j=1,2 ,3 , . . .�

�2j+1 = � j�1. �50�

Note that here � plays, roughly, the role of �2 but their pre-cise relation will become clearer when the entire problem issolved. In order to proceed, we recall that � can be writtenexplicitly as,

� = � 2BckA

2�1 − i��sk − ck�

A

2�1 + i��ck + sk� − 2Bck

� . �51�

Then, by assuming a solution of the form of Eq. �50�, Eq.�49� can be reduced to the following eigenvalue problem for�1:

− �−1�†�1 = ��1, �52�

with the eigenvalues,

�� =A2 + 8B2ck

2 � 2Ack�A2sk

2 + 8B2ck2

�2ck2 − 1�A2 − 8B2ck

2 , �53�

and the corresponding eigenvector, u�, i.e.,

− �−1�†u� = ��u�, �54�

given explicitly as,

u� = ����1 − i�1

� . �55�

The coefficient �� is a function of k, which takes preciselythe following form:

���k� = −1

4B�A tan k � �A2 + 8B2 tan2 k� . �56�

Notice that �−=−1 / �2�+�, and the two eigenspinors u� areorthogonal: u−

†u+=0. A general solution in the form of Eq.�48� can be thus constructed by applying −�−1�†, recur-sively, to

�1 = c+u+ + c−u− �57�

and the result is,

FIG. 10. �Color online� �− plotted as a function of k at A=1 butfor different values of B, i.e., B=1,0.6,0.4,0.35,0.2 and B=0, cor-responding, respectively, the colors: red, green, blue, cyan, ma-genta, and orange �upper panel�. Edge spectrum at �=4B and B=0.2 �A=1, Nr=100, lower panel�. Two reference lines are at k= �0.438977. . ., the value of k at which �− vanishes at B=0.2. Theaxes are suppressed in the lower panel so as to highlight the com-pletely flat edge modes.

FIG. 11. �Color online� The ratio, � j =c2j+2 /c2j �j=1,2 ,3 , . . .�plotted �red points with filling to the horizontal axis� at k=0.05 for� /B=0.25 �upper panel� and � /B=0.30 �lower panel�. A=B=1,Nr=100. In the upper panel, the blue line corresponds to the “the-oretical” value, � j =0.759908. . . whereas in lower panel, the plotsare fitted by a curve, � j =r sin�j+1�� /sin j�, with the choice ofparameters, r=0.691189. . . and �=0.133449. . .

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�2j+1 = c+�+j ��+�1 − i�

1� + c−�−

j ��−�1 − i�1

� . �58�

In this construction, the two eigenvectors u� always have theform of Eq. �55�. This feature remains when � is away from4B at which the edge modes are no longer completely flat, orrather even in the regime in which the edge spectrum is notflat at all.

Another remark, concerning the behavior of Eq. �58� isthat under the choice of signs in Eq. �53�, �+ is alwayslarger than 1 whereas �−�1 except at the zone boundary.This can be easily verified either numerically, i.e., by plotting�� as a function of k, or by showing �− =1 /�+ using di-rectly the expression for �� in Eq. �53�. In numerical simu-lation for systems of a finite number of rows, both of thesetwo solutions play a role giving rise to a pair of edgesolutions.28

In Fig. 10, the upper panel shows �− plotted as a functionof k at A=1 but for different values of B, naturally assuming�=4B. When B is smaller than a critical value Bc=0.35. . . �−changes its sign �has a zero� at intermediate k.29 This contin-ues to be the case even in the limit B vanishes. The lowerpanel shows the edge spectrum at �=4B, B=0.2, and A=1.Note that the zero of �− corresponds to the value of k atwhich the bulk spectrum focuses onto a single point.

B. Wave functions in special cases of parameters

As we will describe in detail in the next subsection, our“recipe” for constructing the exact edge wave function andsimultaneously its spectrum, lies in “extrapolating” the exactsolution available at �=4B to a general value of � /B� �0,8� �recall also Fig. 1�. To complete this program, weneed to refer to some results of the numerical experimentsperformed for a system of finite number of rows. We havealready seen the spectrum of such systems in Fig. 9; here wefocus on the behavior of wave function, i.e., the behavior of� j as a function of j.

In the zigzag edge geometry, it is remarkable that �notonly� the edge wave function �but also the bulk wave func-tion!� has the following particular form:

� = �c1�1 − i�

c2

c3�1 − i�c4

c5�1 − i�c6

]

� , �59�

when that eigenstate represents an edge mode, Eq. �A2� fur-ther simplifies

�� = �c2���1 − i�

1�

c4���1 − i�1

�c6���1 − i�

1�

]

� , �60�

i.e., for a given set of parameters A, B �and �� as well as fora fixed k, � j =c2j−1 /c2j is a constant �=��. The ratio, on theother hand,

� j =c2j+2

c2j, �61�

is a measure of, to what extent the edge mode is localized inthe vicinity of a boundary, say, at j=1.

We have extensively studied such characteristic behaviorsof the edge wave function in numerical experiments. In Fig.11, results of such analyses are shown. For the choice ofparameters such that k /�=0.05, A=B=1 and two differentvalues of � /B: �=0.25 and �=0.30 for comparison. At thisvalue of k /�=0.05, we first verified that the wave function� j takes indeed the form of Eq. �60�, with �� approximatelygiven by,

�+ = 0.687705 . . . , �− = − 0.727056. . . �62�

As in the straight edge case, for k corresponding to an edgemode, i.e., for such a state that are localized in the vicinity ofeither of the two boundaries, this value of �� is commonpractically to all j in the strip, as far as a finite amplitudeexists. For bulk states which are well extended into the inte-rior of the sample, the wave function takes no longer theform of Eq. �60� but keeps still a characteristic form as Eq.�59�.

In the two panels of Fig. 11, � j is plotted as a function ofj for two different values of � /B. At �=0.25, � j saturates atrows away enough from the boundary at j=1 �but not tooclose to the other edge, either�, to a value close to � j=0.759908. . . �blue line�, a value which is later “derived”�see Fig. 12�. Let us assume,30 as in the straight edge case,that the wave function in the zigzag edge geometry takes thefollowing form:

� j = �c+1�1j + c+2�2

j �u+ + �c−1�1−j + c−2�2

−j�u−,

where the eigenspinors u� are always given by Eq. �55�, thelatter found analytically in the limit � /B→4. We have ex-tensively verified the validity of this hypothesis in numericalexperiments. The coefficients c�1,2 are susceptible of sys-tem’s geometry. Here, in a strip geometry, they satisfy,

c+1 = − c+2 � 0, c−1 = c−2 = 0. �63�

for one edge mode and

c+1 = c+2 = 0, c−1 = − c−2 � 0 �64�

for the other. Under this hypothesis, such a behavior as seenin the upper panel of Fig. 11 ��=0.25� is interpreted as aconsequence of two “real solutions” for �, which are also

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both smaller than unity �cf. Fig. 12�.On the other hand, at �=0.30 �in the lower panel of Fig.

11� � j shows an oscillatory behavior. This implies, with thesame hypothesis as above, i.e., the wave function, � j, givenas in Eq. �62�, with the choice of coefficients as Eq. �64�, apair of complex solutions for �. Indeed, the plots at �=0.30 are nicely fitted by a curve, of the form,

� j = �sin�j + 1��

sin j�= r�cos � + sin � cot�J��� , �65�

with the choice of parameters, �=0.691189. . . and �=0.133499. . ., which will be also �a posteriori� justified �seeFig. 13�.

Comparing these two contrasting cases, notice that thecoefficients ��, estimated to be such as Eq. �62�, are com-mon to the two cases, i.e., independent of � /B. One canindeed verify �by changing the parameter � /B in numericalexperiments� that the eigenspinors u� remains always thesame, as far as the state describes an edge mode �see Fig.14�; only � j changes as a function of � /B.

In Fig. 14, � j =c2j−1 /c2j �j=1,2 ,3 , . . .� is plotted for thelowest-energy eigenmode �0 �in the upper band� at differentvalues of � /B. One can see that in the range of k at which�0 is expected to represent an edge mode the plotted pointsfall roughly on the theoretical curve for �−�k�—cf. Eq.�56�—apart from a small disagreement close to the zoneboundary �k=��. This is indeed a key discovery allowing usto proceed to the next step, of extrapolating the earlier exactsolution at �=4B to an arbitrary value of � /B.

C. Derivation of exact edge wave functions

Let us reformulate the recipe for constructing the exactedge wave function and simultaneously its spectrum in thezigzag edge geometry, which has already been briefly out-lined in the introduction �recall also Fig. 1�. �1� We have seenin the previous subsection that the edge wave function � inthe zigzag edge geometry always takes, as far as it describesa localized edge mode, the form of Eq. �60� with a parameter�� depending only on k �and A, B�. All our numerical dataagree with the hypothesis that ��, consequently the reducedtwo-component eigenvector u�, is independent of �. �2� Onthe other hand, we know that the problem can be solvedexactly at �=4B. We have seen, in particular, that the wavefunction � can be constructed from the same set of spinorsu� with a choice of parameters �� given analytically as afunction of k in Eq. �56�.

Taking also into account the fact that the edge modes,gapless at k=0 and characterizing the topological insulator,evolves continuously to the completely flat edge mode at �=4B, one can deduce, from these two observations, that thesolution of the eigenvalue equation for �0, i.e., Eq. �45� foran arbitrary � should be given, indeed, by u�, defined as inEq. �55�, with the parameter ���k� obtained analytically inthe limit: �=4B �recall Fig. 14�.

Thus, for a general value of �, only � and E are unknown�recall that the edge spectrum is no longer flat for a general��. But, clearly, they are solutions of

FIG. 12. �Color online� Theoretical value �derived later� of � �itsmagnitude, �� plotted as a function k for � /B=0.25 �A=B=1�. Atk /�=0.05 there are two possible solutions for �: �−1=0.759908. . .and �−2=0.628683. . ., the larger value of which determines large-jbehavior of � j =c2j+2 /c2j. Merger with bulk occurs when �−1=1,i.e., at k /�=0.263808. . . Close to the zone boundary �k /��0.900237. . .�, reentrance of edge solution occurs �see Fig. 18 fordetails�.

FIG. 13. �Color online� Same as Fig. 12 for � /B=0.3 �A=B=1�. On the reference line at k=0.05, ��0.685038�0.0919993i��=0.691189. . . ,Arg���= �0.133499. . .�.

FIG. 14. �Color online� � j =c2j−1 /c2j �j=1,2 ,3 , . . .� calculatedin a strip geometry �here, Nr=100� is plotted against the theoreticalcurve for �−�k� �shown in black� given in Eq. �56� up to j=Nr /4 forthe lowest-energy eigenmode at different values of � /B=0.1,0.25,0.5,1 ,1.35 �A=B=1 fixed�, corresponding, respectively,to the colors: red, green, blue, cyan, and magenta. Reference linesindicate the regime of k in which the edge modes disappear at thegiven value of � /B.

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��� − 4B��z + �� +1

��†����1 − i�

1� = E���1 − i�

1� ,

�66�

where the 22 matrix � is given explicitly as Eq. �51�.31 Tofind exactly the value of � and E, one has only to solve thisset of equations, and at the end of the calculation, substitutethe analytic expression for a, i.e., �� given in Eq. �56� ob-tained in the limit of �=4B. Clearly, Eq. �66� are a set ofcoupled equations, linear in E and quadratic in �. We expect,therefore, two sets of solutions for �� ,E�, which are given asa function of a. To each of these two sets of solutions, onesubstitutes either �=�+ or �=�−. There exist, therefore, foursets of solutions, in general.

Unfortunately, the analytic formula for these four sets ofsolutions are too lengthy to be shown here. Instead, we plot-ted these four solutions in Fig. 15, for different values of� /B.

D. Reentrant edge modes

Reentrance of the edge mode is another characteristic fea-ture of the edge mode of zigzag geometry and occurs close tothe zone boundary, k /�=1, when � /B is not too large:� /B�1.354. . .. Very remarkably, the spectrum looks com-pletely “innocent” when this occurs, i.e., the edge mode, say,the lowest energy �=E0� mode in the upper band looks almostcompletely degenerate with the bottom of the �bulk� spec-trum �=E1�, in this regime of k �see Fig. 16�. Existence of anedge mode of such specific character is, on the other hand,nothing exceptional in the zigzag edge geometry. At a value

of �, e.g., � /B=0.25 or � /B=0.30 as in Fig. 11, such reen-trant edge modes are indeed existent. If one focuses on thewave function of, say, the lowest energy mode in the upperband, after touching the lower band at k=0, it continues to bespatially localized when k is small enough, but as the spec-trum merges with the bulk continuum, the wave function alsostarts to penetrate into the bulk. However, close to the zone

FIG. 15. �Color online� Four solutions for � �each curve corresponds to one solution�. Only the magnitude of such solution, which isgenerally a complex number, i.e., � is plotted as a function k at different values of � /B=0.25,, 0.3, 1.35, and 1.45 �A=B=1�.

FIG. 16. �Color online� Merger of the edge mode with bulkcontinuum and “absence” of reentrance in the spectrum. As for thelatter, it turns out later that binding energy of the edge mode is toosmall to be seen at this scale �see Fig. 19�. This is an enlarged imageof the spectrum in the zigzag edge geometry as shown in Fig. 9.Here, the parameters are chosen such that � /B=1.2, A=B=1, andNr=100. Two reference lines at k /�=0.643658. . . �kc1 /� and atk /�=0.826568. . . �kc2 /� introduce three different momentum re-gions: �i� k /�� �0,kc1 /��, �ii� k /�� �kc1 /� ,kc2 /��, and �iii� k /�� �kc2 /� ,1�, corresponding, respectively, to �i� the ordinary edge,�ii� the bulk, and �iii� the reentrant regimes.

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boundary, it starts to be localized again. This is what we callthe reentrance of edge modes.

Figures 17 and 18 highlight the behavior of such reentrantedge modes, naturally in a different regime of k from, say,Fig. 11. Figure 17 shows the behavior of � j at k /�=0.901and k /�=0.903 for �=0.25. At this value of k, the wavefunction � j takes always the form of Eq. �60� but with adifferent set of parameters for �� from the case of Fig. 11,upper panel since it still depends on k. The two plots for � j inFig. 17 show two typical behaviors of the edge wave func-tion, i.e., one corresponding to real and the other to complexsolutions for �. As we have extensively studied in the case ofordinary edge modes �appearing at k /��1�, the two con-trasting behaviors of � j as a function j �in Fig. 11� are natu-rally understood by referring to the theoretical curve of � asa function of k, e.g., such as the one shown in Fig. 12.

What is rather remarkable here, in the case of Fig. 17, isthat this crossover between real and complex solutions oc-curs within a tiny change in k, i.e., from k /�=0.901 in theupper panel to k /�=0.903 in the lower panel. This drasticchange is, however, quite reasonable from the viewpoint ofFig. 18. The upper panel of Fig. 17 shows a monotonic de-cay, which converges asymptotically to a single exponentialdecay. This is consistent with the behavior of theoreticalcurve for � as a function of k in Fig. 18. The latter impliestwo real solutions for � at k /�=0.901: �=0.783429. . . and�=0.956432. . . The latter coincides with the value of � j in

Fig. 17 at which it saturates. At k /�=0.903, on the otherhand, the plots for � j are nicely fitted by the curve, � j=r sin�j+1�� /sin j�, with the choice of parameters, r=0.867804. . . and �=0.140687. . . �see the lower panel of Fig.17�. This is a clear fingerprint that the reentrant edge mode atthis value of k corresponds to a pair of complex solutions for�.

Does the reentrant edge mode really have zero bindingenergy? In order to address this question, we �re�plotted theenergy spectrum �E1−E0, to be precise� but in an enlargedscale roughly by one thousand times in Fig. 19. First, for an“ordinary” edge state, occurring at 0� k /��0.289936�kc1 /� the value E1−E0 is much above the threshold at thisscale. E1−E0 takes a value of order �1 for such ordinaryedge state. As for the reentrant edge mode, Fig. 19 revealsthat it has indeed an extremely small but still a finite bindingenergy. Notice different behaviors of E1−E0 as a function ofNr in the bulk and reentrant regions of k. The former �thelatter� corresponds to k /�� �kc1 /� ,kc2 /��0.8983. . .��k /�� �kc2 /� ,1��. In the bulk region E1−E0 is expected tovanish in the thermodynamic limit. Figure 19 shows indeedthat the binding energy of reentrant edge mode, E1−E0�0.001, is thousand times smaller than that of the ordinaryedge state. This implies the appearance of an extremely smallenergy scale which was not existing in the original Hamil-tonian �cf. Kondo effect�. The reentrance of edge mode isindeed a unique feature, in its contrasting properties in realand momentum space and in the appearance of an extremelysmall energy scale.

VI. CONCLUSIONS

We have highlighted in this paper various unique proper-ties of helical edge modes in Z2 topological insulator. Wehave extensively investigated a lattice version of the BHZmodel, under different edge geometries. One of the specificcharacters of BHZ model is that the spin Hall conductance in

FIG. 17. �Color online� Reentrant edge modes I: � j plotted as afunction of j=1,2 ,3 , . . . at A=B=1 and �=0.25 �as in Fig. 11,upper panel� but for k /�=0.901 �upper panel� and k /�=0.903�lower panel�. In the upper panel, the blue line corresponds to thetheoretical value, �=0.956432 whereas, in the lower panel the plots�in red� are fitted by the curve, � j =r sin�j+1�� /sin j�, with thechoice of parameters, r=0.867804 and �=0.140687.

FIG. 18. �Color online� �Theoretical value of� � plotted as afunction k at �=0.25 �again, as in Fig. 11, upper panel�. Enlargedpicture for k close to the zone boundary: k /��0.9 The reentranceof edge solution occurs at k /�=0.900237. . . whereas two real solu-tions for � is possible when k /��0.901675. . . Two reference linesare at k /�=0.901 and at k /�=0.903, on which � j was plotted inFig. 17. The value of � on these lines are, �=0.783429. . . and �=0.956432. . . on k /�=0.901 �case of real solutions� whereas ��0.859242�0.121601i on k /�=0.903.

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the bulk changes its sign in the middle of topological phase�at �=4B�, i.e., �xy

�s�= �e / �2��, respectively, for 0�� /B�4 and for 4�� /B�8, though both represent a nontrivialvalue. From the viewpoint of bulk-edge correspondence, thisinformation should be also encoded in the edge theory. Wehave seen that the change in �xy

�s� manifests in a very differentway in the �1,0�-�straight� and �1,1�-�zigzag� edge geom-etries. In the �1,0�-edge case, the edge spectrum changes itsglobal structure in the two parameter regimes, i.e., the mainlocation of the mode moves from the zone center for � /B�4, to the zone boundary for � /B�4. As a result, the groupvelocity at the intersection with Fermi level reverses its sign,leading to change in the sign in Landauer conductance at theedge. In the �1,1�-edge case, on the other hand, the edgespectrum is symmetric with respect to �=4B, i.e., neitherchange in the position of gap closing, nor the reversal ofgroup velocity at �=4B. The change in �xy

�s� is here encodedin the swapping of left- and right-going edge modes of thesame spin.

Much of our focuses has been on the analysis of the zig-zag or �1,1�-edge geometry, the latter showing, as a conse-quence of specific way in which the bulk topological struc-ture is projected onto the 1D edge, a number of uniquefeatures, such as the completely flat edge spectrum at �=4B, and the reentrance of edge modes. We have alsoshown, here in a half-empirical way, that the exact edgewave function for zigzag edge geometry can be constructed,by extrapolating the solution at �=4B. The reentrant edgemode, though sharing much of its characteristics with theusual edge mode in real space, introduces a new extremelysmall energy scale which was absent in the original BHZmodel.

ACKNOWLEDGMENTS

K.I., A.Y., and A.H. have been much benefited from use-ful discussions with Jun Goryo on the bulk/edge correspon-dence. K.I. also acknowledges Christoph Brüne, Hartmut

Buhmann, and Laurence Molenkamp for their detailed expla-nation of the experimental situations in STCM �Kyoto�,NGSS-14 �Sendai�, and QHSYST10 �Dresden� conferences.K.I. and A.Y. are supported by KAKENHI �K.I.: Grant-in-Aid for Young Scientists under Grants No. B-19740189 andA.Y.: No. 08J56061 of MEXT, Japan�.

APPENDIX: EDGE SOLUTION IN THELONG-WAVELENGTH LIMIT

Let us first recall that the eigenvector d�k��, which hasappeared in Eq. �6�, is a standard SU�2� spinor, here chosento be single-valued.14,32,33 An eigenvector, corresponding to apositive energy eigenvalue E�0, is d�k�+ with k satisfyingE=E�k�, and represented as,

d�k�+ = �e−i� cos��/2�sin��/2� � =

1

�2d�d − dz��dx − idy

d − dz� .

�A1�

�, � are polar coordinates in d space, satisfying the relationssuch as,

cos � =dz

d, cos � =

dx

�dx2 + dy

2. �A2�

In order to find an edge solution, we focus on a solutionof the form,14

��kx,y� = ���kx�e�y , �A3�

say, in a semi-infinite plane: y�0. The spatial dependence inthe y direction can be taken into account by applying Pierlssubstitution: ky→−i� /�y to ky’s in h�k�.

The eigenenergy of such a solution is obtained by asimple replacement: ky→−i� in Eq. �8�, i.e.,

E2 = �2 + �A2 − 2B���kx2 − �2� + B2�kx

2 − �2�2. �A4�

This can be regarded as a quadratic equation with respect to�2. Its two solutions are,

�2 = kx2 +

A2 − 2B�

2B2 �1

2B2�A2�A2 − 4B�� + 4B2E2.

�A5�

For a given set of kx and E, there are two possible values for�2, or equivalently, four possible values for �. Of course, ina semi-infinite plane, say, y�0 the edge solution of the form,Eq. �A3� should decay as y→−�, so only two of such solu-tions are relevant.

We expect that the edge spectrum behaves as E→0 in thelimit of kx→0. Let us parametrize the two solutions in thislimit as

�2 = ����0��2 � P � Q , �A6�

where

P =A2 − 2B�

2B2 , Q =1

2B2�A2�A2 − 4B�� . �A7�

When 0���A2 / �4B���1, P�0, and Q is real. Since a�b as far as b is real, Eq. �A5� represents two positive

FIG. 19. �Color online� Binding energy of the reentrant edgemode: E1−E0 is plotted as a function of k at �=0.3B, A=B=1.Different curves correspond to different size �width� of the system:Nr=100 �blue�, 200 �green�, and 300 �red�. The two reference linesare placed at k /�=0.289936�kc1 /� and k /�=0.8983�kc2 /� �k� �kc1 ,kc2� corresponds to the bulk regime�. The plots reveal anextremely small but still a finite binding energy of the reentrantedge modes.

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solutions for �2, i.e., the wave function represented by Eq.�A3� shows simple exponential damping. On contrary, weexpect a pure imaginary solution for � for an extended statein the bulk. On the other hand, when �1����0=A2 / �2B�,P is always positive but Q becomes purely imaginary. Thustwo solutions for � become complex numbers conjugate toeach other. In this case, the wave function represented by Eq.�A3� shows damped oscillation.

The corresponding eigenvector is obtained by the samereplacement ky→−i�, here in Eq. �A2�, i.e.,

���kx� = �u�

v�� = �A�kx − ��

E − dz��� � � d�kx,− i��+ , �A8�

where dz���=�+B�kx2−�2�. For a given value of kx and E

�0, we thus have identified two solutions characterized bydifferent values of �. In order to construct a general solutionin the presence of a boundary, we need to take a linear com-bination of these two solutions, i.e.,

��y� = c+��+e�+y + c−��−

e�−y � c+�u+

v+�e�+y + c−�u−

v−�e�−y ,

�A9�

where u�, v� are short-hand notations for u��and v��

.We now fix the boundary condition at y=0, which we

choose to be,

��y = 0� = �u+ u−

v+ v−��c+

c−� = �0

0� , �A10�

which implies the following secular equation:

det�A�kx − �+� A�kx − �−�E − dz��+� E − dz��−� � = 0. �A11�

This leads to,

E�kx� = � − B�+�− + Bkx��+ + �−� − Bkx2. �A12�

We have thus identified the two basic equations, Eqs. �A5�and �A12�, for determining the energy spectrum E=E�kx�.

Let us check whether this solution contains the edgemodes. We expect that the edge spectrum behaves, as kx→0, E→0. Recall that in this limit, Eq. �A5� reduces to Eqs.�A6� and �A7�. Equation �37� is also simplified in this limit,as

E = E�0� = � − B�+�0��−

�0�. �A13�

Focusing on the case, ���0��0 and B ,��0, and using theparameterization in Eqs. �A6� and �A7�, one can readilyverify,

�+�0��−

�0� = �P2 − Q2 =�

B. �A14�

Thus Eq. �A13� is safely satisfied.How about the first-order corrections? That is, contribu-

tions of order O�kx� to the energy spectrum, E=E�kx�. Firstnote that there is no O�kx� correction to ��. One can, there-fore, safely replace, at this order, ��’s in Eq. �37� with theirvalues at kx→0, E→0, i.e.,

E = E�0� = � − B�+�0��−

�0� + Bkx��+�0� + �−

�0�� . �A15�

We have already seen that the first two terms cancel whereas

��+�0� + �−

�0��2 = 2a + 2�P2 − Q2 =A2

B2 . �A16�

Thus, the edge spectrum in the continuum limit is deter-mined to be,

E�kx� = � Akx + O�kx2� . �A17�

Remarkably, the slope of the edge spectrum depends only ona single parameter, A. An interesting question is to what ex-tent this conclusion is general? If one calculates the edgespectrum, using a tight-binding model, generally the resultsdepend on the way edges of the sample are introduced withrespect to the lattice. In the case of zigzag edge, in particular,apparently the edge spectrum does not converge to Eq. �A17�even in the long-wavelength limit: kx→0.

1 M. König, H. Buhmann, L. W. Molenkamp, T. Hughes, C.-X.Liu, X.-L. Qi, and S.-C. Zhang, J. Phys. Soc. Jpn. 77, 031007�2008�.

2 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 �2005�.3 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 �2005�.4 B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314,

1757 �2006�.5 M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W.

Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766�2007�.

6 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,and A. K. Geim, Rev. Mod. Phys. 81, 109 �2009�.

7 K. Wakabayashi, Ph.D. thesis, University of Tsukuba, 2000.8 X.-G. Wen, Int. J. Mod. Phys. B 6, 1711 �1992�.9 Y. Hatsugai, Phys. Rev. Lett. 71, 3697 �1993�; Phys. Rev. B 48,

11851 �1993�.

10 S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. 48, 975�1982�.

11 A. J. Niemi and G. W. Semenoff, Phys. Rev. Lett. 51, 2077�1983�.

12 A. N. Redlich, Phys. Rev. Lett. 52, 18 �1984�.13 K. Ishikawa and T. Matsuyama, Nucl. Phys. B 280, 523 �1987�.14 B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu, Phys.

Rev. Lett. 101, 246807 �2008�.15 In the case of graphene �and also KM� zigzag edges, we adopt a

different boundary condition: only the A�B�-sublattice compo-nent of the wave function vanishes at one �the other� boundary.For details, see, Refs. 6 and 7.

16 M. Oshikawa, Phys. Rev. B 50, 17357 �1994�.17 D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs,

Phys. Rev. Lett. 49, 405 �1982�; M. Kohmoto, Ann. Phys. 160,343 �1985�.

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18 Y. Hatsugai, M. Kohmoto, and Y. S. Wu, Phys. Rev. B 54, 4898�1996�.

19 H. B. Nielsen and M. Ninomiya, Phys. Lett. 105B, 219 �1981�;Nucl. Phys. B 185, 20 �1981�; 193, 173 �1981�.

20 L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 �2007�.21 S. Murakami, Prog. Theor. Phys. Suppl. 176, 279 �2008�.22 E. G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C.

R. Becker, G. Landwehr, H. Buhmann, and L. W. Molenkamp,Phys. Rev. B 72, 035321 �2005�.

23 The bulk solutions of Eq. �22� corresponds to the choice, �

=eiky, or �=1 which is consistent with the Bloch theorem. Inthe strip geometry with the periodic boundary condition, ky takesdiscrete values. With the open boundary condition relevant tothe actual strip, ky is no longer a good quantum number. How-ever, if the width of the strip is large, one may roughly interpretthe 1D energy spectra in the strip geometry as composed of themany slices of bulk energy spectrum at different values of ky. Inaddition, a pair of edge modes appear as a characteristic featureof the nontrivial topological property.

24 Real solutions for � appear in the regime: 0����−, and also atthe other end. In the approximation that becomes valid in thesmall wave number, this threshold value is given by �1

=A2 / �4B�. For A=B=1, �−=2−�3=0.2679. . . whereas, �1 is,of course, 1 /4=0.25. If one expands Eq. �41� in powers ofA2 / �4B2�, then at leading order �− coincides with �1.

25 Namely, if one compares it to the straight edge case, e.g., inconsidering the long-wave-length limit, one has to make thecorrespondence between k�2a and kxa.

26 M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusak-abe, J.Phys. Soc. Jpn. 65, 1920 �1996�.

27 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 �1988�.28 When Nr is odd, one solution, corresponding, say, to �−, is lo-

calized in the vicinity of j=1, and indeed has the form of Eq.�48� with �2j+1 given as Eq. �58� and c+=0. The other solution,corresponding to �+, has the same structure of Eq. �48� but with�2j+1 increasing with practically an equal geometric ratio of �+,and naturally localized in the vicinity of the other edge: j=Nr.On the other hand, when Nr is even, the eigenmode of the sys-tem becomes a linear combination of the above two types ofsolutions. This even/odd feature occurs only at � precisely equalto 4B since at this value of � where the edge spectrum becomescompletely flat, the two �generally� counter-propagating edgemodes acquire the same �zero� group velocity, and get mixed.

29 Clearly, at this value of k the edge wave function is extremelylocalized, i.e., onto a single row: j=1 or j=Nr.

30 We leave formal derivation of Eq. �62� to a future publication.Instead, we take it here as a plausible hypothesis fully justifiedby numerical experiments.

31 Inspecting the explicit form of Eqs. �51� and �66�, notice that1� i factors out. So all the coefficients become real.

32 H.-Z. Lu, W.-Y. Shan, W. Yao, Q. Niu, and S.-Q. Shen, Phys.Rev. B 81, 115407 �2010�; W.-Y. Shan, H.-Z. Lu, and S.-Q.Shen, New J. Phys. 12, 043048 �2010�.

33 E. Sonin, arXiv:1006.5218 �unpublished�.34 Integer multiple of 2e2 /h in the language of charge conductance.

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