Charge Induction and Polarization on the Surface of a Topological InsulatorDue to an Emergent Gauge Field

Kiminori Hattori+

Department of Systems Innovation, Graduate School of Engineering Science, Osaka University,Toyonaka, Osaka 560-8531, Japan

(Received October 21, 2013; accepted January 9, 2014; published online February 12, 2014)

Exchange interaction with surface magnetization exerts a local U(1) gauge field on Dirac fermions that exist on thesurface of a topological insulator (TI). We show that the emergent gauge field gives rise to charge induction andpolarization on the TI surface via the quantum anomalous Hall (QAH) effect. These phenomena correspond to animplementation of the Laughlin’s thought experiment [R. B. Laughlin, Phys. Rev. B 23, 5632 (1981)] without inserting amagnetic flux, and enable a local evaluation of the QAH transport on a specific TI surface.

1. Introduction

The nontrivial topology in the band structure of three-dimensional topological insulators (3DTIs) enforces theemergence of 2D surface states in the bulk insultinggap.1–7) In time-reversal symmetric situations, these statesare described at low energies as massless Dirac fermions. Thepredicted Dirac states have been experimentally observed inrepresentative materials such as Bi1�xSbx, Bi2Se3, Bi2Te3, andSb2Te3.8–11) One of the most remarkable effects exhibited bysurface electrons is the quantum anomalous Hall (QAH)effect, i.e., a quantized Hall transport without an externalmagnetic field.3–6) The QAH effect may be understoodprimarily through the effective electromagnetic actionS3D ¼ e2

2h

Rd3r dtE � B for 3DTIs. This action implies a

topological magnetoelectric (TME) coupling, by which amagnetic (electric) field induces an electric (magnetic)polarization.4–6,12–20) For a semi-infinite TI with an openboundary, S3D is reduced to the Chern–Simons term S2D ¼e2

4h

Rd2r dt "���A�@�A� on the 2D surface.4,12) This action

in turn describes a half-quantization of the surface Halleffect. Inversely, the QAH effect can account for theTME effects, reflecting the bulk-boundary correspond-ence.4–6,12,20)

These topological effects become observable when surfacestates are gapped by an external perturbation that breakstime-reversal symmetry, such as magnetic doping or contactwith a ferromagnet to induce exchange coupling to surfaceelectrons.4–6,21–26) In this situation where Dirac fermions arecoupled to magnetism, we show that there exists a new classof topological effects that consists of charge induction andpolarization without electric fields or electrostatic potentials.These effects are driven by a pure gauge mechanism, and inthis sense are qualitatively different from the electromagneticresponse of TIs argued in previous studies.12–20,27–29) Thedriving gauge field can be locally generated on the surface byrotating the exchange field that also acts as a gap generator.The physical mechanism of surface charging is in closeanalogy to Laughlin’s gauge argument for the quantum Halleffect; the latter is put in terms of wave function transporton the surface of a fictitious cylinder exposed to a radialmagnetic field, as an additional magnetic flux threadingthe cylinder is changed.30) For TIs, surface magnetizationgenerically creates such a gauge-induced effect without fluxinsertion.

2. Theoretical Analysis

Consider 2D electrons on the TI surface subjected toa Zeeman or exchange field. The relevant single-particleHamiltonian is given by

H ¼ vFez � ð� � pÞ þm � �; ð1Þwhere vF is the Fermi velocity, ez is the unit vector normal tothe surface, � ¼ ð�x; �y; �zÞ is the vector of Pauli matricesin spin space, p ¼ �ih�r is the momentum operator, andm ¼ ðmx;my; mzÞ represents the exchange field in energyunits. The out-of-plane exchange field mz produces a massgap of size 2jmzj in the Dirac dispersion of surface states. Themassive surface states exhibit the QAH effect when thechemical potential lies in the gap. On the other hand, the in-plane exchange field mk ¼ ðmx;my; 0Þ correlates to the U(1)gauge field Am ¼ ðez �mkÞ=evF.27–29) This is easily shownby rewriting Eq. (1) in the form

H ¼ vFez � ½� � ðpþ eAmÞ� þ mz�z; ð2Þwhich contains a minimal coupling to the gauge field. Asimilar formulation is also derived from the Zeeman couplingin a 3DTI as discussed in Appendix A. The equivalencebetween exchange and minimal couplings yields two differ-ent perspectives to illuminate the physical implications forsurface charging. In the following, we will first formulatethe response of the system’s charge to the exchange field.Subsequently, the underlying physics will be uncovered interms of the QAH transport driven by the gauge field.

Generally, charge density is expressed as �ðrÞ ¼�e R1

�1 d" fð"Þnðr; "Þ, where nðr; "Þ is the local density ofstates (DOS) and is given by nðr; "Þ ¼ � 1

� ImTrGþðr; r; "Þ;fð"Þ is the Fermi function; and Gþðr; r0; "Þ is the retardedGreen’s function in real space. Analogously, spin density isgiven by ��ðrÞ ¼ h�

2

R1�1 d" fð"Þn�ðr; "Þ with the local spin

DOS n�ðr; "Þ ¼ � 1� ImTr ��Gþðr; r; "Þ. The retarded func-

tion obeys the Dyson equation Gþ ¼ Gþ0 þ Gþ

0 m � �Gþ forthe Hamiltonian of the form H ¼ H0 þm � �, leading to theidentity @

@m�Gþ ¼ Gþ��Gþ. Therefore, the charge-density

response to the exchange field m� is described by @@m�

�ðrÞ ¼e� Im

R1�1 d" fTrðGþ��GþÞrr in matrix notation, such that

ðABÞrr ¼Rd2r0 Aðr; r0ÞBðr0; rÞ. This result is identically

derived from the Kubo formalism as well as the KeldyshGreen’s function formalism.31) In arriving at the above result,we assumed the grand-canonical ensemble, in which the

Journal of the Physical Society of Japan 83, 034704 (2014)

http://dx.doi.org/10.7566/JPSJ.83.034704

034704-1 ©2014 The Physical Society of Japan

system is taken to be in contact with an infinitely largeparticle bath having a fixed chemical potential ®, and hence@@m�

f ¼ 0. The total charge of the system is given byQ ¼ R

d2r �ðrÞ, for which with employing the identity@@" G

þ ¼ �GþGþ, we find a concise relationship

@Q

@m�¼ e

Z 1

�1d" � @f

@"

� �N�ð"Þ; ð3Þ

where N�ð"Þ ¼Rd2r n�ðr; "Þ denotes the spin DOS.

Equation (3) interrelates charge and spin degrees of freedom,and explains that if the system is spin polarized, it is chargedin the presence of the exchange field. This does not contradictcharge conservation. In the grand canonical ensemble, thenumber of electrons contained in the system may changebecause of connection to the reservoir. In what follows, wewill show that charge induction of this kind is realizable in aquantized fashion on the TI surface.

To elucidate charge induction quantitatively, we considera slab-shaped TI as shown in Fig. 1. A surface gap can becreated by magnetic doping or coating the surface with aferromagnetic thin film having perpendicular anisotropy. Theinduced mass changes sign at a magnetic domain wall.For simplicity, we assume the out-of-plane exchange fieldspatially varies as mzðxÞ ¼ m sgnðxÞ on the top surface. Sucha mass boundary, at which a chiral edge state is formed,4–6) iscrucial to charge induction as shown below. The uniform in-plane field mk can be generated by applying an externalmagnetic field Bk parallel to the surface, which rotates themagnetization by an angle sin�1 Bk=BK , where BK denotesthe anisotropy field. Thermal equilibrium with a reservoir canbe established, e.g., by mounting the TI slab on a metallicsubstrate.

Charge induction reflects the spin DOS, which iscalculated from the retarded Green’s function. The retardedfunction can be constructed from scattering wave functionsby using the McMillan method.32–34) Details of thisderivation are described in Appendix B. The local spinDOS nxðx; "Þ calculated for m > 0 is displayed in Fig. 2(a).The chiral edge state spatially localized around x ¼ 0 appearswithin the mass gap where "2 < m2. It has been confirmed inthe calculation that nx ¼ sgnðmÞn and ny ¼ nz ¼ 0 in thegap, indicating that the edge state carries a perfect �xpolarization.35) This is a corollary of the unidirectionalchiral-state propagation along the y-direction, since electriccurrents are expressed generally as j ¼ evF

h�=2ð�x; �y; �zÞ � ez

for surface Dirac fermions. More quantitatively, the half-metallic property is accounted for by solving the eigenvalueequation H ¼ E . The in-gap solution is written as Er ¼rh�vFq and r ¼ exp½ikyyþ ðimyx� jmxjÞ=h�vF�ð1; rÞt withq ¼ ky þ mx=h�vF and r ¼ sgnðmÞ, showing that the eigens-pinor of �x constitutes the edge-state wave function. In viewof this, the effective edge-Hamiltonian is written as HðrÞ

edge ¼rh�vFqP

ðrÞx , where PðrÞ

x ¼ ð1þ r�xÞ=2 is the projection oper-ator onto the eigenstate of �x. The corresponding DOS isgiven by Nedge ¼ Ly=hvF, where Ly denotes the systemdimension in the y-direction. Figure 2(b) illustrates that thespin DOS varies as Nxð"Þ ¼ sgnðmÞNedgeðm2 � "2Þ. Thispeculiar behavior can be explicitly shown using an analyticexpression of Gþ (see Appendix B for details). Thus, weconclude that the total charge on the TI surface satisfies

@Qsurf

@mx¼ sgnðmÞeNedgeðm2 � �2Þ; ð4Þ

at zero temperature. This is a salient feature showing thatcharge induction occurs only in the QAH regime �2 < m2.The mass inversion assumed here is fundamental to chargeinduction since otherwise surface states are fully gapped andNx vanishes. In addition, it is worth noting that the total spinSx ¼ h�

2

R1�1 d" fð"ÞNxð"Þ is insensitive to mx, implying that

edge occupation stays in equilibrium even in the presence ofthe exchange field.

We are now in a position to incorporate the emergent gaugefield Am into consideration. Equation (4) can be rewritten as

@Qsurf

@�m¼ sgnðmÞ e

2

hðm2 � �2Þ; ð5Þ

in terms of the Aharonov–Bohm (AB) flux �m ¼ AmyLy. TheAB flux adds a phase to the electron wave function, andis reminiscent of the Laughlin’s gauge argument for thequantum Hall effect.30) Imagine that a 2D surface is bent intoa cylinder such that the x-axis is parallel to the axis of thecylinder, and the y-axis is along the circumference. Then, Amcorresponds to a magnetic flux �m ¼ AmyLy piercing thecylinder. Even in a slab geometry as depicted in Fig. 1, theAB flux defined by �m ¼ R

C dr � Am is gauge invariant forthe closed path C along the circumference of the system. Incontrast to the Laughlin cylinder, a notable feature of TIs isthat the gauge field is locally generated by the exchange

m +mx

yz m +m

m||Am

Fig. 1. (Color online) A TI slab covered by a ferromagnetic thin film withperpendicular magnetization. The exchange coupling with the ferromagnetinduces a mass gap in the surface states of the TI. The induced mass �mchanges sign at a domain wall, where a chiral edge state is formed. Thesurface magnetization can be rotated by applying an external magnetic fieldparallel to the surface. The resulting in-plane exchange field mk produces thegauge field Am ¼ ðez �mkÞ=evF.

0 0.5 1-2

-1

0

1

2

x ( )-4 -2 0 2 4

-2

-1

0

1

2

-0.1 0 0.1

nx (1 / m 2 )

(m)

(a) (b)

Nx / Nedge

Fig. 2. (Color online) (a) Local spin DOS nxðx; "Þ and (b) total spin DOSNxð"Þ calculated for mzðxÞ ¼ m sgnðxÞ and mx ¼ 0. In these figures, thecharacteristic length scale of the in-gap mode ‘ ¼ h�vF=m is taken as thelength unit, and the mass m as the energy unit.

J. Phys. Soc. Jpn. 83, 034704 (2014) K. Hattori

034704-2 ©2014 The Physical Society of Japan

coupling on the TI surface instead of inserting a thin solenoidthat confines a physical magnetic flux. It is found fromEq. (5) that in the QAH regime, an elementary charge e isintroduced on the TI surface when �m changes by a fluxquantum �0 ¼ h=e. The unit flux �m ¼ �0 corresponds tothe exchange field mx ¼ 1=Nedge, which becomes vanishinglysmall as Ly ! 1, indicating that a weak external field Bk issufficient for charge induction for a macroscopically largesystem.

As suggested above, charge induction is intimately relatedto the QAH effect. This effect induces local charge �H ¼�xyBmz and current jH ¼ �xyEm � ez in response to theelectromagnetic fields Em ¼ �@tAm and Bm ¼ r � Amarising from a spatiotemporally varying Am. The QAHcontributions �H and jH satisfy a generalized continuityequation @t�H þ r � jH ¼ gH. The source term gH ¼ Em �ðez � r�xyÞ is nonvanishing when the QAH conductivity�xy ¼ � sgnðmzÞ e22h varies in space. Because of the quantizednature of �xy, the change in �xy is allowed only at a massboundary. Assuming the mass inversion prescribed bymzðxÞ ¼ m sgnðxÞ, excess charge generated at x ¼ 0 buildsup in the 2D bulk region of the TI surface, and can bedescribed as

Qbulk ¼Zd2r dt gHðr; tÞ ¼ sgnðmÞ e

2

h�m: ð6Þ

Comparing Eq. (5) with Eq. (6), we see the equalityQsurf ¼ Qbulk. The total surface charge Qsurf ¼ Qbulk þQedge consists of the bulk and edge components. For thesystem keeping in contact with a reservoir, charge con-servation is generally described by Qsurf þ Qres ¼ 0, whereQres represents the reservoir charge. Therefore, the equalitymentioned above indicates that electrons are substantiallytransferred between the bulk and the reservoir; Qbulk þQres ¼ 0. The vanishing edge contribution Qedge ¼ 0 isconsistent with the absence of induced spin described earlier,which tends to minimize the total energy of the system bysuppressing the Zeeman energy mk � �. The situation ishowever different for a TI disconnected from the reservoir, inwhich charge conservation should hold between the bulkand the edge; Qbulk þ Qedge ¼ 0. This is nothing but anargument showing the bulk-edge correspondence. The chargetransfer between the bulk and the edge predicts

Qedge ¼ � sgnðmÞ e2

h�m; ð7Þ

for the canonical system. It should be emphasized here thatthe induced charge is fully described by a static and uniformAm for which Em ¼ Bm ¼ 0. In this sense, the chargeinduction and transfer we address are driven by the puregauge.

3. Numerical Calculation

A further microscopic insight into charge induction can begained by numerically solving a lattice model of a 3DTI witha slab configuration. For this purpose, we employ a simplemodel consisting of H ¼ H0 þ Hm on a cubic lattice withtwo orbitals per site denoted as ¡ and ¢. The unperturbedpart is expressed as H0 ¼

Pr;r0

yr trr0 r0 with r ¼

ð�r"; �r#; �r"; �r#Þt. The hopping matrix is defined bytrr0 ¼ �t�z � i��x�� for r ¼ r0 � a�, "0�z for r ¼ r0, and 0

otherwise, where �� is the Pauli matrix acting in orbitalspace, and a� ¼ ae� is the unit lattice vector.6,7,16,17) H0

describes a strong TI having surface states with a singleDirac point in the parameter range where 2t < "0 < 6t.16,17)

The Dirac dispersion is gapped by the mass term writtenas Hm ¼ P

r2surf yrmn�n r, where �n ¼ en � � and en is the

surface normal.17) We consider the nonuniform exchangefield mnðrÞ ¼ m sgnðxÞ to create two mass domains separatedat x ¼ 0 where mass is sign-reversed. The gauge fieldAm ¼ ey�m=Ly is assumed to be nonzero only on the topsurface. The associated minimal coupling is implementedby the Peierls substitution: trr0 ! trr0 exp½ð2�i=�0ÞAm �ðr0 � rÞ�. Parameters used in the calculation are "0 ¼ 4t and� ¼ m ¼ t. The system size was chosen to be Lx ¼ 24a andLy ¼ Lz ¼ 12a. For simplicity, we set a ¼ 1 in the following.

The charge density �ðrÞ induced by the gauge field Amcan be calculated from exact numerical diagonalization of thelattice Hamiltonian. The results of calculation are summariz-ed in Fig. 3. The Fermi level is kept fixed at � ¼ 0 forsimulating the grand-canonical situation. In this situation, theTI surface is charged when the gauge field is switched on, asshown in Figs. 3(a) and 3(b). The induced charge is localizedaround the left and right ends of the top surface symmetri-cally. This charge imbalance is interpreted as being due todivergent Hall currents that flow along �x during adiabati-cally ramping up the AB flux from 0 to�m. As expected for aconstant ®, charge fluctuations are relatively small aroundx ¼ 0 where a chiral edge state is formed. The total chargeQtot ¼

Pr �ðrÞ contained in the TI slab varies discontin-

uously with �m, reflecting a discrete DOS formed in thefinite-sized system. However, the increment of a singlecharge e with adding �0 to �m is in reasonable agreementwith Eq. (5). It is also confirmed in the figure that Qtot isvery close to the surface charge Qsurf ¼

Pr2surf �ðrÞ,

verifying that the 3D bulk region of the TI slab is irrelevantto charge induction.

Next, we address the canonical system at half filling. Inthis case, ® varies with �m. As illustrated in Figs. 3(c) and3(d), the bulk and edge regions on the TI surface areoppositely charged while the total surface charge Qsurf

remains negligibly small. This observation manifests thecharge transfer between the bulk and the edge. The edgecharge Qedge, which is evaluated by integrating �ðrÞ over aportion of the surface in the range jxj < Lx=4, shows a linearvariation with �m as predicted by Eq. (7). The chiral edgestate is uniformly charged not only on the top surfacesubjected to the driving gauge field but also on the otherfield-free faces, implying that a dissipationless edge currentencircles the TI slab. This suggests that magnetic manipu-lation of electric currents is feasible on a TI surface.

As demonstrated above, an inhomogeneous charge dis-tribution on the TI surface stems from the QAH currentjH ¼ �xyEm � ez. Since j ¼ @tP, one readily finds that theelectric polarization P ¼ �xyez � Am is established in thepresence of the gauge field Am. In a homogeneous system, Pis uniform for a uniform Am. The uniform polarization can beviewed as charge pumping.4) The charge pumped from theleft end to the right end is described by

Qpump ¼ PxLy ¼ sgnðmÞ e2

2h�m; ð8Þ

J. Phys. Soc. Jpn. 83, 034704 (2014) K. Hattori

034704-3 ©2014 The Physical Society of Japan

indicating that a half charge e=2 is pumped with changing�m by �0. On the other hand, in a heterogeneous systemcomprised of mass domains, P points to opposite directionsin two adjacent domains having opposite signs of mass. Theantiparallel polarization provides an alternative picture ofcharge transfer in an isolated TI.

The electric polarization on the TI surface with a uniformmass can be numerically evaluated in the present scheme bysetting mnðrÞ ¼ m. In this case, charge induction is inactiveso that there is no essential difference between the grand-canonical and canonical situations. The results of calculationare plotted in Figs. 3(e) and 3(f ). Charge accumulation ofopposite polarities on the opposite sides of the top surfaceconstitutes an electric dipole px ¼ QpumpLx in the presence ofthe gauge field. As predicted in Eq. (8), the pumped chargeQpump grows linearly with the AB flux. In addition to chargeinduction and transfer, surface polarization is a testablesignature of the QAH effect in the absence of domainboundaries and chiral edge states. Moreover, the uniformpolarization signifies that all the electrons on the top surfaceare uniformly delivered in the direction perpendicular tothe gauge field. This is in exact analogy to the Laughlincylinder.30) The present observation demonstrates that thisthought experiment is naturally implemented on a flat TIsurface without flux insertion.

4. Summary

In summary, we have shown that an emergent gauge fieldcoupled to surface magnetization gives rise to chargeinduction and polarization on a TI surface via the QAHeffect. These gauge-induced phenomena are distinct fromconventional TME effects driven by electromagnetic fields.The predicted surface charging should be observable in

experiment using electrostatic or Kelvin probe force mi-croscopy,36) enabling a local evaluation of the QAH effectoperative on a specific surface.

Acknowledgement

This work was supported by a Grant-in-Aid for ScientificResearch (No. 24540322) from the Japan Society for thePromotion of Science.

Appendix A: Zeeman Interaction in a 3DTI

The Dirac model representing an isotropic 3DTI is givenby

H0 ¼ Dk2 þ �zðM0 � Bk2Þ þ A�x� � k; ðA:1Þin momentum space. The bulk Hamiltonian is mapped ontothe lattice version employed for the numerical calculation inthis study with the parameters t ¼ �B=a2, � ¼ A=2a, and"0 ¼ M0 þ 6t. In terms of Eq. (A01), it is not evident thatthe Zeeman interaction Hm ¼ m � � is renormalized into aminimal coupling by the Peierls substitution k ! k þ e

h�Am.

Nevertheless, the renormalization of this kind is found inthe low-energy Hamiltonian describing surface states withinthe bulk gap. Diagonalizing the bulk Hamiltonian underan open boundary condition and projecting the totalHamiltonian H ¼ H0 þ Hm onto the subspace of surfacestates,7,37) we obtain the effective surface Hamiltonianexpressed as

Hsurf ¼ h�vFez � ð� � kÞ þ ðmx�x þmy�yÞ þ mz�z; ðA:2Þwhere h�vF ¼ A

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

pand ¼ D=B. Equation (A02) is

essentially identical to Eq. (1) and naturally explains theequivalence between Zeeman and minimal couplings forsurface electrons, except for the trivial factor ©, which can be

-10-5

05

10

-5

0

5

x (a)

z (a

)-5

0

5

y (a)

-5

0

5

y (a)-0.02

00.02

(e / a2 )

m / 0

Q(e

)

-2

-1

0

1

2

-2 -1 0 1 2

totalsurf

y (a)

-5

0

5

x (a)

z (a

)

-10-5

05

10

-5

0

5

y (a)-5

0

5(e / a

2 )

-0.020

0.02

-2

-1

0

1

2

-2 -1 0 1 2

surfedge

m / 0

Q(e

)

y (a)

-5

0

5

x (a)

z (a

)

-10-5

05

10

-5

0

5

y (a)-5

0

5(e / a

2 )

-0.020

0.02

(a)

(b)

(c)

(d)

(e)

(f)

Qpu

mp

(e)

m / 0

-1

0

1

-2 -1 0 1 2

Fig. 3. (Color online) Local surface charge �ðrÞ at �m ¼ �0 (upper three figures) and integrated charge Q and pumped charge Qpump versus �m (lowerthree figures) calculated for a TI slab with massive surface states. (a) and (b) show the results obtained for the grand-canonical system with � ¼ 0, while (c) and(d) display those derived for the canonical system at half filling. In these two cases, the exchange field normal to the surface is assumed to be mnðrÞ ¼ m sgnðxÞ.Figures (e) and (f ) show the results calculated for the uniform field mnðrÞ ¼ m. In this case, the grand-canonical and canonical situations are equivalent to eachother.

J. Phys. Soc. Jpn. 83, 034704 (2014) K. Hattori

034704-4 ©2014 The Physical Society of Japan

absorbed into the definition of the gauge field Am. Althoughan isotropic model is assumed here for simplicity, Eq. (A02)is also applicable in the long-wavelength limit to realmaterials with uniaxial anisotropy.

Appendix B: Retarded Green’s Function

Here, we formulate the retarded Green’s function for the TIsurface with the mass given by mzðxÞ ¼ m sgnðxÞ. Because oftranslational invariance along y, the 2D Hamiltonian of ourconcern is reduced to HðxÞ ¼ e�ikyyHðx; yÞeikyy for a plane-wave solution. In two separate spatial regions x < 0 andx > 0 (labeled by � ¼ 1; 2, respectively), the eigenfunctionsare given by

�ð�Þ� ðxÞ ¼

h�vFðq� ikÞ"�m�

1

0@

1A exp i �kþ my

h�vF

� �x

� �; ðB:1Þ

where m ¼ �m1 ¼ m2, q ¼ ky þ mx=h�vF, and h�vFk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"2 � ðh�vFqÞ2 �m2

p. Assembling these local eigenmodes,

the right- and left-going scattering wave functions arerepresented as

RðxÞ ¼ ½�ðþÞ1 ðxÞ þ rR�ð�Þ

1 ðxÞ�ð�xÞ þ tR�ðþÞ2 ðxÞðxÞ; ðB:2Þ

LðxÞ ¼ tL�ð�Þ1 ðxÞð�xÞ þ ½�ð�Þ

2 ðxÞ þ rL�ðþÞ2 ðxÞ�ðxÞ; ðB:3Þ

respectively, in the electron-like region where " > 0. Thereflection and transmission coefficients, rL,R and tL,R, can beeasily determined from the continuity of wave functions atx ¼ 0. The retarded function obeys the equation of motion:½"� HðxÞ�Gþðx; x0Þ ¼ ðx� x0Þ. The solution is found to be

Gþ��0 ðx; x0Þ ¼ C½ R�ðxÞ � L�0 ðx0Þðx� x0Þ

þ L�ðxÞ � R�0 ðx0Þðx0 � xÞ�; ðB:4Þwhere � ðxÞ ¼ ðxÞjmy!�my and � ¼ ";# denotes the spinindex. The normalization constant C ¼ �ðqmþ ik"Þ=2ðh�vFkÞ2 is derived from the boundary condition for Gþðx; x0Þat x ¼ x0. The retarded function in the hole-like region where" < 0 is obtained by the simple replacement �ð�Þ

� ðxÞ !�ð�Þ� ðxÞ, or equivalently k! �k. The present formulation

equally deals with propagating and evanescent modes underthe condition sgnð"Þ Im k > 0 to establish the asymptoticbehavior of scattering waves Rðþ1Þ ¼ Lð�1Þ ¼ 0. Atthis level of analysis, it is clear that the equal-positioncorrelation function Gþðx; xÞ is independent of my so that mydoes not affect local physical quantities. It is also found fromEq. (B04) that Gþ

"#ðx; xÞ ¼ Gþ#"ðx; xÞ, and thereby ny ¼

Ny ¼ 0. The spin DOS Nx can be represented as Nxð"Þ ¼Ly2�

R1�1 dky Axðky; "Þ in terms of the spectral function

Axðky; "Þ ¼ � 1

�Im

Z 1

�1dxTr �xG

þðx; x; ky; "Þ; ðB:5Þ

in momentum space, which is calculated from Eq. (B04) to be

Axðky; "Þ ¼ 1

�sgnð"ÞqLx � mj"j

"2 � ðh�vFqÞ2� �

� ½"2 � ðh�vFqÞ2 � m2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"2 � ðh�vFqÞ2 � m2

pþ sgnðmÞ½"� sgnðmÞh�vFq�� sgnð"Þh�vFq½"2 � ðh�vFqÞ2 �m2�: ðB:6Þ

Note that only the second term describing a chiral edge modeis relevant in the gap. Integrating over ky, one finds

Nxð"Þ ¼ sgnðmÞNedgeðm2 � "2Þ: ðB:7ÞAs shown by this analytical result, Nx is unaffected by mx.

+hattori@ee.es.osaka-u.ac.jp1) L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).2) J. E. Moore and L. Balents, Phys. Rev. B 75, 121306(R) (2007).3) L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).4) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424

(2008).5) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).6) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).7) H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nat.

Phys. 5, 438 (2009).8) D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z.

Hasan, Nature (London) 452, 970 (2008).9) Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D.

Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 398(2009).

10) Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi,H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z.Hussain, and Z.-X. Shen, Science 325, 178 (2009).

11) D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder,L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J.Cava, and M. Z. Hasan, Phys. Rev. Lett. 103, 146401 (2009).

12) X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323, 1184 (2009).13) A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102,

146805 (2009).14) A. Karch, Phys. Rev. Lett. 103, 171601 (2009).15) R. Li, J. Wang, X.-L. Qi, and S.-C. Zhang, Nat. Phys. 6, 284 (2010).16) G. Rosenberg and M. Franz, Phys. Rev. B 82, 035105 (2010).17) G. Rosenberg, H.-M. Guo, and M. Franz, Phys. Rev. B 82, 041104(R)

(2010).18) W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, 057401 (2010).19) J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, Phys. Rev. Lett.

105, 166803 (2010).20) K. Nomura and N. Nagaosa, Phys. Rev. Lett. 106, 166802 (2011).21) Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D. Qu, J. G. Checkelsky,

L. A. Wray, D. Hsieh, Y. Xia, S.-Y. Xu, D. Qian, M. Z. Hasan, N. P.Ong, A. Yazdani, and R. J. Cava, Phys. Rev. B 81, 195203 (2010).

22) R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang,Science 329, 61 (2010).

23) Y. L. Chen, J.-H. Chu, J. G. Analytis, Z. K. Liu, K. Igarashi, H.-H.Kuo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu, M. Hashimoto, T.Sasagawa, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen,Science 329, 659 (2010).

24) L. A. Wray, S.-Y. Xu, Y. Xia, D. Hsieh, A. V. Fedorov, Y. S. Hor, R. J.Cava, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Phys. 7, 32 (2011).

25) S.-Y. Xu, M. Neupane, C. Liu, D. Zhang, A. Richardella, L. A. Wray,N. Alidoust, M. Leandersson, T. Balasubramanian, J. Sánchez-Barriga,O. Rader, G. Landolt, B. Slomski, J. H. Dil, J. Osterwalder, T.-R.Chang, H.-T. Jeng, H. Lin, A. Bansil, N. Samarth, and M. Z. Hasan,Nat. Phys. 8, 616 (2012).

26) W. Luo and X.-L. Qi, Phys. Rev. B 87, 085431 (2013).27) I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802 (2010).28) T. Yokoyama, J. Zang, and N. Nagaosa, Phys. Rev. B 81, 241410(R)

(2010).29) K. Nomura and N. Nagaosa, Phys. Rev. B 82, 161401(R) (2010).30) R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).31) H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of

Semiconductors (Springer, New York, 2007).32) W. L. McMillan, Phys. Rev. 175, 559 (1968).33) J.-H. Gao, J. Yuan, W.-Q. Chen, Y. Zhou, and F.-C. Zhang, Phys. Rev.

Lett. 106, 057205 (2011).34) G. Tkachov and E. M. Hankiewicz, Phys. Rev. B 88, 075401 (2013).35) D.-H. Lee, Phys. Rev. Lett. 103, 196804 (2009).36) F. Mohn, L. Gross, N. Moll, and G. Meyer, Nat. Nanotechnol. 7, 227

(2012).37) H.-Z. Lu, A. Zhao, and S.-Q. Shen, Phys. Rev. Lett. 111, 146802

(2013).

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