Stability of the X-Y-phase of the two-dimensional C4 point group insulator

Bart de Leeuw,1, 2 Carolin Küppersbusch,1, 3 Vladimir Juričić,1 and Lars Fritz11Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,

Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands2Mathematical Institute, Utrecht University, Budapestlaan 6, 3584 CD Utrecht

3Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

Noninteracting insulating electronic states of matter can be classified according to their symme-tries in terms of topological invariants which can be related to effective surface theories. Theseeffective surface theories are in turn topologically protected against the effects of disorder. Topolog-ical crystalline insulators are, on the other hand, trivial in the sense of the above classification butstill possess surface modes. In this work we consider an extension of the Bernevig-Hughes-Zhangmodel that describes a point group insulator. We explicitly show that the surface properties of thisstate can be as robust as in topologically nontrivial insulators, but only if the Sz-component of thespin is conserved. However, in the presence of Rashba spin-orbit coupling this protection vanishesand the surface states localize, even if the crystalline symmetries are intact on average.

I. INTRODUCTION

Soon after the prediction1–3 and subsequent discoveryof the quantum spin Hall insulator4 as a novel state ofelectronic matter with properties protected by topology,it became clear that this state fits into an even granderscheme.5–8 By now many more topological phases havebeen identified both theoretically and experimentally.9–15The classification of noninteracting electronic insulat-ing states with topological order was one of the mile-stones of theoretical condensed matter physics in thelast decade.16–18 Within the so-called ’tenfold periodictable’ states of matter with topological order are classi-fied according to the presence or absence of symmetries ofthe underlying systems,19 such as time-reversal, particle-hole, and chiral.

One of the prominent features of topological electronicsystems is the existence of exotic gapless edge or surfacestates. In particular, these boundary modes can realizeone-dimensional chiral or single two-dimensional Diracfermions, usually ruled out by the fermion doubling the-orem (at least if all the symmetries are preserved). Im-portantly, in the presence of disorder these states can beprotected against localization which leads to unusuallyrobust transport properties. An instructive point of viewon the existence of these robust boundary states is thatthe tenfold periodic table does not require spatial sym-metries such as translations or rotations to be present,which is particularly apparent in the classification schemeusing non-linear σ models.16 This implies that althoughthe calculation of topological invariants can in generalmost easily be accomplished in clean band-insulators, inprinciple there is no need to have a well-defined crys-tal momentum for doing so. On the other hand, elec-tronic states trivial according to the tenfold classifica-tion but still featuring edge or surface modes have beenrecently identified once crystalline symmetries, such astranslations, rotations, reflections and inversions, weretaken into account.20–24 These states are conventionallyreferred to as topological crystalline insulators20 (TCI).

Interestingly, these states, predicted to be realized in Sn-and Pb-based compounds,25 have been reported to be ob-served first in Refs. 26–28, and subsequently have beenexperimentally studied in Refs. 29–31. An urging ques-tion is therefore whether the boundary states in topologi-cal phases where crystal symmetries allow to define topo-logical invariants can enjoy similar protection against theeffects of disorder as they do in tenfold-wise topologicallynontrivial insulators.

To answer this question at least for one concrete ex-ample, we investigate the transport properties of an ex-tension of the well-known Bernevig-Hughes-Zhang (BHZ)model that was recently introduced.21, to which we referas a point group insulator (PGI) in the following. There,it was identified that a rotational symmetry leads to theexistence of a state characterized by a trivial topologicalinvariant in the tenfold sense, but non-trivial in a sensedefined in Sec. II. Within this note we make a compar-ison of this PGI with a standard quantum spin Hall in-sulator (QSHI), for which the BHZ model was originallyformulated.3 For the QSHI we find, as expected and wellknown, very robust transport properties with a quantizedlead-to-lead conductance, both in presence and absenceof disorder (as long as the disorder strength is smallerthan the bulk-gap scale). Importantly, this property isalso robust against breaking of the spin-rotational sym-metry, introduced for instance by Rashba spin-orbit cou-pling, and is a direct consequence of time-reversal sym-metry. For the PGI, on the other hand, we find thatthe conductance is only quantized and robust againstdisorder if the Sz-component of the spin is conserved.However, in the presence of Rashba spin-orbit couplingdisorder localizes, in the Anderson sense, the boundarymodes leading to a vanishing conductance, in agreementwith the specific PGI state being trivial in the sense of theperiodic table. We note here that our results concerningthe stability of the edge modes in a PGI without Rashbaspin-orbit coupling are in agreement with the findings re-cently reported in Refs. 32 and 33, but also show that theresults obtained therein are fine tuned and non generic.

The paper is organized as follows. In Sec. II, we intro-

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duce the model and its phase digram, and in Sec. III, wepresent the results concerning the transport properties inboth QSHI and PGI phases. Our conclusions are drawnin Sec. IV.

II. MODEL, TOPOLOGICAL INVARIANTSAND PHASE DIAGRAM

The model we study was recently introduced in Ref.21 and represents an extension of the BHZ Hamiltonianthat includes next-nearest neighbor hoppings. Its BlochHamiltonian has the generic form

H =∑k

Ψ†k

(H(k) HSO(k)

H†SO(k) H∗(−k)

)Ψk, (1)

where

H(k) = τ · d(k), with d(k) =

sin kx + cos kx sin ky− sin ky + cos ky sin kx

M − 2B [2− cos kx − cos ky]− 4B̃ [1− cos kx cos ky]

, (2)

and τ = (τx, τy, τz) is the vector composed of the Paulimatrices. The wave function is a four component ob-ject, and, motivated by the low-energy band-structureof HgTe/CdTe quantum wells, contains s− and p−likeorbitals with the respective spins Ψk =

(s↑k, p

↑k, s↓k, p↓k

).

The parameter M is related to the offset of the chemicalpotential between the s− and p−like orbitals, while B(B̃) is proportional to the (next-)nearest-neighbor hop-ping between the orbitals of the same type, while bothnearest-neighbor and next-nearest-neighbor hoppings be-tween different orbitals are set to one.

We choose a version of Rashba spin-orbit couplingwhich is not symmetric in the orbitals s and p34 butrespects all other symmetries discussed below. The cor-responding term HSO in the Hamiltonian (1) reads

HSO = R0

(−i sin kx − sin ky 0

0 0

). (3)

The Hamiltonian (1) obeys time reversal symmetry,which is implemented via T = iτ0 ⊗ σyK, where σyacts in spin space and K denotes complex conjuga-tion (τ0 is the identity in orbital space). This sym-metry will be intact hereafter. Additionally, the cleansystem has the discrete rotational C4 symmetry aboutthe axis orthogonal to the crystal plane, represented byR = 1

2 (τ0 (1 + i) + τz (1− i)) ⊗ Rspin, with Rspin =exp(iπσz/4). The invariance of the Hamiltonian (1) un-der this symmetry operation is shown in the Appendix.However, this symmetry will be broken by disorder andthe boundary of the sample.

B̃

B= 1

M

B

� X � Y trivial

80 12

R0

B

12

6

metal

⌫ = 0⌫ = 1 ⌫ = 0

CR = �1 CR = 2 CR = 0

Figure 1: Phase diagram of the clean system. The phasediagram without Rashba spin-orbit coupling is obtainedfrom the calculation of the topological C4 index CR, and thecorresponding Z2 index ν.

A. Topological Characterization of the phases on asquare lattice

Before we study the model (1), we discuss a topologicalcharacterization of the possible topological phases on thesquare lattice. In Ref. 21 it was shown that in the pres-ence of time-reversal symmetry, the distinct topologicalphases on a lattice with a given space group symmetrycan be characterized in terms of the band inversions attime-reversal invariant (TRI) momenta in the Brillouinzone (BZ). The band inversion is given in terms of thesign of the Pfaffian at TRI momenta. In the BHZ modelthe sign of the Pfaffian at a TRI momentum is equalto the sign of the mass term [dz(k) in Eq. (1)] at thatpoint.21,33 When these points are, in addition, related by

3

the C4 rotational symmetry, as is the case with X andY points on the square lattice, the corresponding bandinversions are connected to each other. If there is a bandinversion at X point, there is one at Y point as well, andthis is enforced precisely by the C4 symmetry,21 as it isthe case in the X − Y phase, which is obtained in themodel (1), as detailed below. Notice, however, that thisphase has a trivial Z2 topological invariant, which is alsoconfirmed by our stability analysis. Finally, given thegauge of the Bloch states, the sign of the Pfaffian cannotbe changed without closing the band gap, and thusX−Yphase is topologically distinct from other possible phaseson the square lattice, Γ and M phases, which both havenontrivial Z2 invariant.21

An alternative way to show that the X − Y phase istopologically distinct from the other topological phaseson a square lattice was outlined by Ezawa in Ref. 33,where he derived the form of the mirror Chern number interms of the signs of the Pfaffians at TRI points, Eq. (34)therein. Just by interchanging the mirror operator withthe C4 operator, which is allowed since the only ingredi-ent necessary in this derivation is the commutation of theBloch Hamiltonian with the point-group operation,35 thetopological invariant associated with C4 symmetry is alsogiven in terms of signs of the Pfaffians δki at TRI pointski. Therefore, the topological invariant corresponding toC4 symmetry is

CR =1

2(δΓ + δM − 2δX), (4)

since δX = δY due to C4 symmetry. In the X −Y phase,δX = δY = −1, while δΓ = δM = +1, so CR = 2.On the other hand, in the Γ (M) phase, with δΓ = −1(δM = −1), CR = −1, and therefore both Γ and Mphases are topologically distinct from the X − Y phase.

In the X − Y phase, depending on the boundary, thetwo resulting edge modes lie at different TRI momenta,and this is important for the resulting physics of the edgestates. For instance, when the cut is along one of the twoprincipal crystallographic axes x or y, the two edge statesoriginate from the Dirac cones located at the TRI mo-menta kedge = 0 and kedge = π in the boundary BZ. Onthe other hand, when the cut is along the diagonal, thetwo Dirac cones are both located at a non-TRI momen-tum kedge = π/

√2 . The types of the edges are com-

pletely analogous to the surface states discussed in Ref.37 in the context of three-dimensional TCIs on the rock-salt lattice with the bulk band inversion at the four Lpoints in the BZ. The surface perpendicular to the (111)crystallographic direction in this 3D TCI corresponds toan edge along one of the principal axes in the X − Yphase, since the corresponding edge/surface Dirac conesare separated by a finite momentum, and therefore a fi-nite momentum transfer is needed to mix them. On theother hand, the (001) surface in the 3D TCI is analogousto the edge along one of the diagonals in the X−Y phase,since in both cases the surface/edge Dirac cones lie at thesame momentum, and can therefore hybridize.

These general arguments imply that upon adiabati-cally switching on the Rashba-coupling in the X − Yphase, provided that the bulk gap remains open, thetwo Dirac points cannot be mixed and consequently edgestates are stable, as long as there is no perturbation pro-viding the momentum transfer required to mix the twoDirac cones. As mentioned before, there is one specialsituation in which this is not true, however, and thatis when the two Dirac cones corresponding to the twoKramers pairs of edge states coincide in the edge BZ,corresponding to a diagonal cut, as discussed above. Inthat situation a gap will open due to Rashba spin-orbitcoupling. The most natural candidate to generically mixthe Dirac cones in the presence of Rashba spin-orbit cou-pling is scalar disorder, which we consider in the paper.

B. Phase diagram

The phase diagram for the model (1) is shown forB̃/B = 1 as a function of M/B and R0/B in Fig. 1.The phase diagram for R0 = 0 and all values of B̃/Bwas introduced recently.21 If R0 = 0, the Hamiltonianexplicitly conserves the Sz-component of the spin and wecan deduce a phase diagram by calculating the Chernnumbers in the respective Sz spin sectors. As a func-tion of M/B this leads to three phases. For values ofM/B < 8 there is the Γ-phase which is the standardQSHI from the AII symmetry class with a Z2 topologi-cal index ν = 11. This phase can also be understood asa stack of two time-reversed Chern insulators of Chernnumber38,39 C↑ = −C↓ = −1, which leads to the pair ofhelical boundary modes related to each other by time-reversal symmetry. For values 8 < M/B < 12 we findthe X-Y valley phase, indexed as p4 in Ref. 21, whichwe refer to as PGI. In this parameter regime the time-reversed bands have Chern number C↑ = −C↓ = −2.This results in two pairs of helical edge modes, whereeach of the pairs consists of counterpropagating modesrelated by time-reversal symmetry. This phase is trivialin the sense of the periodic table of electronic topologicalstates, i.e., the Z2-index is trivial (ν = 0), but features anon-trivial topological invariant CR stemming from theC4 symmetry, as shown above. For higher values ofM/Bthe model enters a trivial phase, meaning that both thespin Chern numbers as well as the Z2-index are trivial(C↑,↓ = 0, ν = 0, and CR = 0). All phases possesstime-reversal symmetry and are connected via quantumphase transitions with quantum critical points that aremetallic.

Upon turning on the Rashba spin orbit coupling, R0 6=0, all symmetries of the system are preserved, exceptfor spin-rotational. Therefore, we cannot define the spinChern numbers any more (the Z2-index is still well de-fined and calculable, though), but we can still defineCR. At a critical value of the Rashba spin-orbit couplingR0c/B, the bulk band-gap closes, and all the phases be-come metallic as R0 is further increased. To show that

4

-3 -2 -1 1 2 3

-2

-1

1

2

�� phase M

B= 7

B̃

B= 1

R0

B=

1

2

-3 -2 -1 1 2 3

-2

-1

1

2

X � Y valley phase B̃

B= 1

R0

B=

1

2

M

B= 10

(a)

(b)

k

k

E

E

Figure 2: Finite size spectrum on a cylinder geometry in theΓ-phase for parameters indicated in the inset. (a) In theΓ-phase one observes one set of helical modes at eachboundary. (b) In the X-Y valley phase we find two pairs ofhelical modes at each boundary.

the clean system still features edge states without con-served spin, we studied the finite size spectrum on a cylin-der. In the case of the Γ-phase for finite R0 < R0c we findone pair of helical edge states (Fig. 2 (a)) while in theX-Y valley phase we find two pairs (except for diagonalcut), see Fig. 2 (b).

Overall we conclude that finite Rashba coupling doesnot change the system properties in the clean system ineither phase below a respective critical R0c where theinsulator is converted into a metal.

III. TRANSPORT PROPERTIES

One of the most striking properties of topological insu-lator states is not only the existence of boundary modesbut also their stability with respect to disorder. The mostprominent example is the quantum Hall state which hasquantized Hall conductivity σH = e2

h n, with n as an in-teger. The integer n can be viewed in two equivalentways: (i) it is the cumulative Chern number of the bandsbelow the chemical potential or (ii) the number of chiralboundary modes. The conductance then accounts for thenumber of channels at the boundary. Naively, one wouldexpect disorder to localize these modes, but their chiralnature provides an escape route: disorder cannot local-ize them since there is no way of converting a left-moverinto a right-mover and vice versa by virtue of them beingunidirectional.

In the case of a QSHI we do not have chiral chan-nels but instead helical modes. This implies that thereare right- and left-movers on either side of the sam-ple which potentially allows for elastic backscattering

due to scalar disorder. However, the helical modes areKramers’ pairs related by time-reversal symmetry, whichimplies that scalar disorder cannot convert left-moversinto right-movers and vice versa due to the orthogonal-ity of Kramers’ pairs. The situation is schematicallydepicted in Fig. 3 (b). Consequently, edge transportis also ballistic resulting in a quantized conductance.This has also been observed in transport experiments onHgTe/CdTe quantum wells.4

In the case of the PGI we find there are two pairs ofboundary modes at each sample edge, see right-hand-sideof Fig. 2 (b), but usually two pairs of modes are unstablesince any right-mover in general is not mutually orthog-onal to both left-movers and they can thus scatter intoeach other under generic circumstances. So the questionwe address here is in which sense and under which condi-tions the X−Y valley phase, topologically trivial accord-ing to the periodic table, has stable transport propertiesonce disorder is added.

The setup that we probe is shown in Fig. 3 (a) wherewe connect a central sample region to left and right leadsand measure the response coefficient for transport fromthe left lead to the right lead in the linear response regime(we assume µL = µR + δV , δV being small).

In order to study the stability of the edge states withrespect to disorder we resort to the well established non-equilibrium Green-function method, which we implementnumerically in the linear response regime40.

In the clean system we can count the boundary modesassuming ballistic transport based on the finite size spec-tra (see Fig. 2), and find

G =e2

h×{

2 in Γ− phase4 in X−Y valley phase

. (5)

(Note that the result in the X − Y phase does not holdif the edge is cut along the diagonal)

We have modelled disorder by scalar disorder, i.e.,through local variations in the chemical potential. Wehave chosen disorder of the box type with a strengthw, thus the local energy is drawn from [−w,w]. Thispreserves all the symmetries of the system required bythe classification scheme, but it obviously breaks trans-lational and rotational symmetries (locally, but not onaverage). Subsequently, we have checked the localizationtendencies by probing the system (i) with a fixed sam-ple size as a function of increasing disorder strength, and(ii) at fixed disorder strength as a function of increasingsample size.

In order to make better comparisons between the QSHIand PGI , we have chosen parameters such that bulkgaps are approximately the same in both systems. Wefound that a convenient parameter set is given by B̃ = B,M/B = 7 (QSHI) or M/B = 10 (PGI) without Rashbacoupling, while with Rashba coupling R0/B = 1 wechoose M/B = 9 for the PGI (to have comparable gaps).

In the absence of Rashba spin orbit coupling (R0 = 0)with conserved spin, we find that both Γ and the X-Yvalley phase are equally stable against disorder. This

5

µL µR

(a) (b)

(c)

�� phase

X � Y valley phase

R0 = 0

h |Vkk0 | i = 0

R0 6= 0and

h |Vkk0 | i = 0 h |Vkk0 | i = 0

h |Vkk0 | i = 0 h |Vkk0 | i = 0

h |Vkk0 | i 6= 0

h |Vkk0 | i 6= 0

h |Vkk0 | i = 0

h |Vkk0 | i = 0

R0 = 0

R0 6= 0

CB

VB

CB

VB

Figure 3: (a) Experimental setup to probe the transportproperties of a sample sandwiched between two leads kept atdifferent chemical potentials µL and µR. (b)Counterpropagating modes at the boundary of a QSHI. Left-and right-movers are Kramers’ partners and are protectedagainst backscattering both for R0 = 0 and R0 6= 0.Vkk′ = V0 denotes a featureless elastic scatterer which mixesall momenta. The zero overlap between thecounterpropagating modes stems from additional quantumnumbers of the band structure. (c) Two sets ofcounterpropagating modes at the boundary of a PGI. Whilethe modes crossing at the time-reversal invariant momentacannot scatter into each other due to time-reversalsymmetry for both R0 = 0 and R0 6= 0, modes from differenttime-reversal invariant momenta can scatter if R0 6= 0.

is not very surprising since both in the Γ-phase as wellas in the X-Y valley phase we can think of the systemas two time-reversed Chern insulators with Chern num-bers C↑,↓ = ∓1 for the QSHI and C↑,↓ = ∓2 for the PGIand the left- and right-moving channels do not mix, seeFig. 3 (b) and (c). In Fig. 4, we display a plot of theconductance of a sample of size 80 × 80 with the chem-ical potential in the bulk gap as a function of disorderstrength. Our results show that both the QSHI and thePGI phases are stable against disorder, since the conduc-tance is quantized. In Fig. 5 we plot conductance at fixeddisorder strength as a function of the system size (we al-ways study systems of transverse size 80 sites), and findagain that both QSHI and PGI are stable.

In the presence of Rashba spin-orbit coupling (R0 6= 0),we find that the features of the QSHI phase are un-changed, as expected. However, the protection of theconductance is lost in the case of the PGI. This can ex-plicitly seen in Fig. 4, where the conductance as a func-tion of disorder strength at fixed sample size decreases,as soon as Rashba spin orbit coupling is switched on.Furthermore, in Fig. 5 we observe that the conductanceas a function of the system size decreases and eventuallywould vanish if we made the sample long enough. Thissignals the localization of the modes in agreement withthe absence of topological protection. As discussed be-

Figure 4: Conductance of finite size system as a function ofincreasing disorder strength. In the case where theSz-component of the spin is a conserved quantity both theQSHI and PGI are equally stable. When the Rashbaspin-orbit coupling is switched on, the QSHI is still equallystable, while the protection for the PGI is lost.

Figure 5: Conductance of finite size systems at fixeddisorder for increasing system sizes. For R0 = 0 we find thatboth QSHI as well as PGI are equally stable, while for finiteR0 the conductance of the PGI decreases as a function ofthe system size.

fore, this can be traced back to the loss of orthogonalityof left- and right-movers at the sample edges. We haveexplicitly checked the loss of orthogonality under scatter-ing from a structurless impurity (Vkk′ = V0) of the left-and right-movers in the finite size spectrum (as also dis-cussed in Fig. 3 (c)). We can therefore conclude that theabsence of protection against backscattering leads to thelocalization of the boundary modes.

6

IV. CONCLUSIONS

In this paper we have studied the stability of the edgestates of a topological system outside the tenfold classifi-cation of the topological insulator states with respect todisorder. We considered a special instance of a PGI wherediscrete rotational C4 symmetry guarantees the existenceof edge states in a clean system. If the Sz-component ofthe spin is a conserved quantity in this system we findthat the boundary modes are protected against localiza-tion due to disorder. This protection against disorderis lost if Rashba spin-orbit coupling is present. Conse-quently, the system indeed behaves like a trivial insulatorin the sense of the topological classification of electronicstates. For the future it is an interesting prospect tostudy the localization properties of other systems withboundary modes which are outside the tenfold classifica-tion of topological states, such as the recently observedthree-dimensional TCI phase featuring band-inversionsat symmetry-related L−points in the Brillouin zone.26–28

V. ACKNOWLEDGEMENT

We thank Michael Wimmer and Jason Frank for help-ful discussions. We acknowledge funding from the DFGFR 2627/3-1 (C.K. and L.F.). This work is part of theD-ITP consortium, a program of the Netherlands Or-ganisation for Scientific Research (NWO) that is fundedby the Dutch Ministry of Education, Culture and Sci-ence (OCW). V. J. acknowledges financial support fromNWO.

Appendix A: Invariance of the Hamiltonian underC4 rotation

In this Appendix we explicitly show that the Hamil-tonian in Eq. (1) in the main text is invariant under C4

rotations. C4 rotation is represented by

R = Rorb ⊗Rspin, (A1)

with

Rorb = 12 [(1 + i)τ0 + (1− i)τz], (A2)

Rspin = exp(iπ4σz) = 1√2(1 + iσz). (A3)

This symmetry operation acts on the components of themomentum as

kx,y → ±ky,x. (A4)

First, the Hamiltonian (1) in the main text withoutthe Rashba term (R0 = 0) can be conveniently rewrittenas

H0 = dx(k)τx ⊗ σz + (dy(k)τy + dz(k)τz)⊗ σ0, (A5)and is clearly invariant under C4 spin rotation. The in-variance under C4 rotation in the orbital space is a con-sequence of the fact that

R†orbτx,yRorb = ∓τy,x, (A6)

and the transformation of the momentum under C4 givenby Eq. (A4). Namely, the form of the C4 transforma-tion yields dx,y(k) → ∓dy,x(k), while dz(k) is invariant,which together with Eq. (A6), implies the invariance ofH0 under the C4 rotation.

To show that C4 is a symmetry of the Rashba Hamil-tonian, we conveniently rewrite it as

HR =1

2R0(τ0 + τz)⊗ (σy sin kx − σx sin ky). (A7)

The orbital part of this Hamiltonian is invariant under C4

rotation, since the matrices τ0 and τz commute with eachother. Under C4 rotation, Pauli matrices σx,y transformas

R†spinσx,yRspin = ±σy,x. (A8)

Transformation of the momentum under C4, Eq. (A4),implies that sin kx,y → ± sin ky,x under the same opera-tion. This, together with Eq. A8, implies the invarianceof the spin part of the Hamiltonian (A7), which showsthe invariance of the Hamiltonian (A7) under C4 trans-formation.

1 C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005).2 C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).3 B. A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314,1757 (2006).

4 M. König, et al., Science 318, 766 (2007).5 J.E. Moore, L. Balents, Phys. Rev. B 75, 121306 (2007).6 L. Fu, C. L. Kane, Phys. Rev. B 74, 195312 (2006).7 L. Fu, C. L. Kane, Phys. Rev. Lett. 98, 106803 (2007).

8 L. Fu, C. L. Kane, Phys. Rev. B 76, 045302 (2007).9 D. Hsieh, et al., Nature 452, 970 (2008).

10 D. Hsieh, et al., Science 323, 919 (2009).11 Y. Xia, et al., Nature Phys. 5, 398 (2009).12 H. Zhang, et al., Nature Phys. 5, 438 (2009).13 Y. L. Chen, et al., Science 325, 178 (2009).14 M. Z. Hasan, C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).15 X. L. Qi, S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

7

16 A. P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig,Phys. Rev. B 78, 195125 (2008).

17 A. Kitaev, AIP Conf. Proc. 22, 1132 (2009).18 X.-L. Qi, T. Hughes, S.-C. Zhang, Phys. Rev. B 78, 195424

(2008).19 S. Ryu, A. P. Schnyder, A. Furusaki, A. Ludwig, New J.

Phys., 12, 065010 (2010).20 L. Fu, Phys. Rev. Lett. 106, 106802 (2011).21 R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, Nature

Phys. 9, 98 (2013).22 C.-K. Chiu, H. Yao, and S. Ryu, Phys. Rev. B 88, 075142

(2013).23 T. Morimoto and A. Furusaki, Phys. Rev. B 88, 125129

(2013).24 K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014).25 T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, L. Fu,

Nat. Comm. 3, 982 (2012)26 Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T.

Takahashi, K. Segawa, and Y. Ando, Nature Phys. 8, 800(2012).

27 S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I.Belopolski, J. D. Denlinger, Y. J. Wang, H. Lin, L. A.Wray, G. Landolt, B. Slomski, J. H. Dil, A. Marcinkova, E.Morosan, Q. Gibson, R. Sankar, F. C. Chou, R. J. Cava,A. Bansil, and M. Z. Hasan, Nature Commun. 3, 1192(2012).

28 P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczer-bakow, M. Szot, E. Lusakowska, T. Balasubramanian, B.M. Wojek, M. H. Berntsen, O. Tjernberg, and T. Story,

Nature Mater. 11, 1023 (2012).29 Y. Tanaka, T. Sato, K. Nakayama, S. Souma, T. Taka-

hashi, Z. Ren, M. Novak, K. Segawa, and Y. Ando, Phys.Rev. B 87, 155105 (2013).

30 Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C.Dhital, M. Neupane, S. Xu, Y. J. Wang, R. Sankar, F.Chou, A. Bansil, M. Z. Hasan, S. D. Wilson, L. Fu and V.Madhavan, Science 341, 6153 (2014).

31 I. Zeljkovic, Y. Okada, C. Y. Huang, R. Sankar, D. Walkup,W. Zhou, M. Serbyn, F. Chou, W. F. Tsai, H. Lin, A.Bansil, L. Fu, M. Z. Hasan, and V. Madhavan, NaturePhys. 10, 572 (2014).

32 H. Jiang, H. Liu, J. Feng, Q. Sun, and X. C. Xie, Phys.Rev. Lett. 112, 176601 (2014).

33 M. Ezawa, New J. Phys. 16, 065015 (2014).34 D. G. Rothe, R. W. Reinthaler, C.-X. Liu, L. W.

Molenkamp, S.-C. Zhang, and E. M. Hankiewicz, New J.Phys. 12, 065012 (2010).

35 An analogous derivation can be carried out based on theresults in Ref. 36.

36 C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B86, 115112 (2012).

37 J. Liu, W. Duan, and L. Fu, Phys. Rev. B 88, 241303(R)(2013).

38 D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. denNijs, Phys. Rev. Lett. 49, 405 (1982).

39 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).40 S. Datta, Electronic Transport in Mesoscopic Systems,

Cambridge University Press (1997).