Topological Field Theory of Time-Reversal Invariant Insulators

8/11/2019 Topological Field Theory of Time-Reversal Invariant Insulators

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arXiv:0802.3537

v1

[cond-mat.mes-hall]24Feb2008

Topological Field Theory of Time-Reversal Invariant Insulators

Xiao-Liang Qi, Taylor Hughes and Shou-Cheng ZhangDepartment of Physics, Stanford University, Stanford, CA 94305

We show that the fundamental time reversal invariant (TRI) insulator exists in 4 + 1 dimensions,where the effective field theory is described by the 4 + 1 dimensional Chern-Simons theory and thetopological properties of the electronic structure is classified by the second Chern number. Thesetopological properties are the natural generalizations of the time reversal breaking (TRB) quantumHall insulator in 2+ 1 dimensions. The TRI quantum spin Hall insulator in 2+1 dimensions and thetopological insulator in 3 + 1 dimension can be obtained as descendants from the fundamental TRIinsulator in 4 + 1 dimensions through a dimensional reduction procedure. The effective topologicalfield theory, and theZ2 topological classification for the TRI insulators in 2 +1 and 3 +1 dimensionsare naturally obtained from this procedure. All physically measurable topological response functionsof the TRI insulators are completely described by the effective topological field theory. Our effectivetopological field theory predicts a number of novel and measurable phenomena, the most strikingof which is the topological magneto-electric effect, where an electric field generates a magnetic fieldin the same direction, with an universal constant of proportionality quantized in odd multiples ofthe fine structure constant = e2/c. Finally, we present a general classification of all topologicalinsulators in various dimensions, and describe them in terms of a unified topological Chern-Simonsfield theory in phase space.

Contents

I. Introduction 1

II. TRB topological insulators in 2 + 1dimensions and its dimensional reduction 3A. The first Chern number and topological

response function in (2 + 1)-d 3B. Example: two band models 4C. Dimensional reduction 5D. Z2 classification of particle-hole symmetric

insulators in (1 + 1)-d 7E. Z2 classification of (0 + 1)-d particle-hole

symmetric insulators 10

III. Second Chern number and its physicalconsequences 11A. Second Chern number in (4 + 1)-d non-linear

response 12B. TRI topological insulators based on lattice

Dirac models 13

IV. Dimensional reduction to (3 + 1)-d TRIinsulators 16A. Effective action of (3 + 1)-d insulators 16B. Physical Consequences of the Effective Action

S3D 18C. Z2 topological classification of time-reversal

invariant insulators 19D. Physical properties ofZ2-nontrivial

insulators 22

V. Dimensional reduction to (2 + 1)-d 27A. Effective action of (2 + 1)-d insulators 27B. Z2 topological classification of TRI insulators 30C. Physical properties of the Z2 nontrivial

insulators 30

VI. Unified theory of topological insulators 33A. Phase space Chern-Simons theories 33B. Z2 topological insulator in generic dimensions38

VII. Conclusion and discussions 41

A. Conventions 41

B. Derivation of Eq. (54) 421. Topological invariance of Eq. (53) 422. Adiabatic deformation of arbitraryh(k) to

h0(k) 433. Calculation of correlation function (53) for

h0(k) 43

C. The winding number in the non-linearresponse of Dirac-type models 44

D. Stability of edge theories in genericdimensions 45

References 46

I. INTRODUCTION

Most states or phases of condensed matter can be de-scribed by local order parameters and the associatedbroken symmetries. However, the quantum Hall (QH)state1,2,3,4 gives the first example of topological states ofmatter which have topological quantum numbers differ-ent from ordinary states of matter, and are described inthe low energy limit by topological field theories. Soonafter the discovery of the integer QH effect, the quanti-zation of Hall conductance in units ofe2/h was shownto be a general property of two-dimensional time rever-sal breaking (TRB) band insulators5. The integral of thecurvature of the Berrys phase gauge field defined over the
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magnetic Brillouin zone (BZ) was shown to be a topologi-cal invariant called the first Chern number, which is phys-ically measured as the quanta of the Hall conductance.In the presence of many-body interactions and disorder,the Berry curvature and the first Chern number can bedefined over the space of twisted boundary conditions6.In the long wave length limit, both the integer and thefractional QH effect can be described by the topological

Chern-Simons field theory7

in 2 + 1 dimensions. Thiseffective topological field theory captures all physicallymeasurable topological effects, including the quantiza-tion of the Hall conductance, the fractional charge, andthe statistics of quasi-particles8.

Insulators in 1 + 1 dimensions can also have uniquetopological effects. Solitons in charge density waveinsulators can have fractional charge or spin-chargeseparation9. The electric polarization of these insula-tors can be expressed in terms of the integral of theBerrys phase gauge field in momentum space10,11. Dur-ing an adiabatic pumping cycle, the change of electricpolarization, or the net charge pumped across the 1D in-

sulator, is given by the integral of the Berry curvatureover the hybrid space of momentum and the adiabaticpumping parameter. This integral is quantized to be atopological integer12. Both the charge of the soliton andthe adiabatic pumping current can be obtained from theGoldstone-Wilczek formula13.

In this paper we shall show that the topological effectsin the 1 + 1 dimensional insulator can be obtained fromthe QH effect of the 2 + 1 dimensional TRB insulator bya procedure called dimensional reduction. In this pro-cedure one of the momenta is replaced by an adiabaticparameter, or field, and the Goldstone-Wilczek formula,and thus, all topological effects of the 1 + 1 dimensionalinsulators, can be derived from the 2 + 1 dimensionalQH effect. The procedure of dimensional reduction canbe generalized to the higher dimensional TRI insulatorsand beyond, which is the key result of this paper.

In recent years, the QH effect of the 2 + 1 dimensionalTRB insulators has been generalized to TRI insulatorsin various dimensions. The first example of a topologi-cally non-trivial TRI state in condensed matter contextwas the 4D generalization of the QH effect (4DQH) pro-posed in Ref. 14. The effective theory of this modelis given by the Chern-Simons topological field theory in4 + 1 dimensions15. The quantum spin Hall (QSH) ef-fect has been proposed in 2 + 1 dimensional TRI quan-tum models16,17. The QSH insulator state has a gap

for all bulk excitations, but has topologically protectedgapless edge states, where opposite spin states counter-propagate16,18,19. Recently the QSH state has beentheoretically predicted20 and experimentally observed inHgTe quantum wells21. TRI topological insulators havealso been classified in 3 + 1 dimensions22,23,24. These 3Dstates all carry spin Hall current in the insulating state 25.

The topological properties of the 4+1 dimensional TRIinsulator can be described by the second Chern numberdefined over four dimensional momentum space. On the

other hand, TRI insulators in 2 + 1 and 3 + 1 dimensionsare described by a Z2 topological invariant defined overmomentum space16,22,23,24,26,27,28,29,30. In the presenceof interactions and disorder, the momentum space Z2invariant is not well defined, however, one can define amore general Z2 topological invariant in terms of spin-charge separation associated with a flux31,32. One openquestion in this field concerns the relationship between

the classification of the 4 + 1 dimensional TRI insulatorby the second Chern number and the classification of the3 + 1 and 2 + 1 dimensional TRI insulators by the Z2number.

The effective theory of the 4 + 1 dimensional TRI in-sulator is given by the topological Chern-Simons fieldtheory15,33. While the 2 + 1 dimensional Chern-Simonstheory describes a linear topological response to an ex-ternalU(1) gauge field7,8, the 4 + 1 dimensional Chern-Simons theory describes a nonlinear topological responseto an externalU(1) gauge field. The key outstanding the-oretical problem in this field is the search for the topolog-ical field theory describing the TRI insulators in 2+1 and

3 + 1 dimensions, from which all measurable topologicaleffects can be derived.

In this paper, we solve this outstanding problem byconstructing topological field theories for the 2 + 1 and3 + 1 dimensional TRI insulators using the procedureof dimensional reduction. We show that the 4 + 1 di-mensional topological insulator is the fundamental statefrom which all lower dimensional TRI insulators can bederived. This procedure is analogous to the dimensionalreduction from the 2 + 1 dimensional TRB topologicalinsulator to the 1 + 1 dimensional insulators. There isa deep reason why the fundamental TRB topologicalinsulator exists in 2 + 1 dimensions, while the funda-

mental TRI topological insulator exists in 4 + 1 dimen-sions. The reason goes back to the Wigner-von Neumannclassification34 of level crossings in TRB unitary quan-tum systems and the TRI symplectic quantum systems.Generically three parameters need to be tuned to ob-tain a level crossing in a TRB unitary system, while fiveparameters need to be tuned to obtain a level crossingin a TRI symplectic system. These level crossing singu-larities give rise to the non-trivial topological curvatureson the 2D and 4D parameter surfaces which enclose thesingularities. Fundamental topological insulators are ob-tained in space dimensions where all these parametersare momentum variables. Once the fundamental TRItopological insulator is identified in 4+1 dimensions, thelower dimensional versions of TRI topological insulatorscan be easily obtained by dimensional reduction. In thisprocedure, one or two momentum variables of the 4+1 di-mensional topological insulator are replaced by adiabaticparameters or fields, and the 4 + 1 dimensional Chern-Simons topological field theory is reduced to topologi-cal field theories involving both the external U(1) gaugefield and the adiabatic fields. For the 3 + 1 TRI insula-tors, the topological field theory is given by that of theaxion Lagrangian, or the 3 + 1 dimensional vacuum


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term, familiar in the context of quantum chromodynam-ics (QCD), where the adiabatic field plays the role of theaxion field or the angle. From these topological fieldtheories, all physically measurable topological effects ofthe 3+1 and the 2+1 dimensional TRI insulators can bederived. We predict a number of novel topological effectsin this paper, the most striking of which is the topo-logical magneto-electric (TME) effect, where an electric

field induces a magnetic field in the same direction, witha universal constant of proportionality quantized in oddmultiples of the fine structure constant = e2/c. Wealso present an experimental proposal to measure thisnovel effect in terms of Faraday rotation. Our dimen-sional reduction procedure also naturally produces theZ2 classification of the 3 + 1 and the 2 + 1 dimensionalTRI topological insulators in terms of the integer sec-ond Chern class of the 4+1 dimensional TRI topologicalinsulators.

The remaining parts of the paper are organized as fol-lows. In Sec. II we review the physical consequences ofthe first Chern number, namely the (2 + 1)-d QH effect

and (1 + 1)-d fractional charge and topological pumpingeffects. We begin with the (2+1)-d time reversal breakinginsulators and study the topological transport properties.We then present a dimensional reduction procedure thatallows us to consider related topological phenomena in(1+ 1)-d and (0+ 1)-d. Subsequently, we define aZ2clas-sification of these lower dimensional descendants whichrelies on the presence of a discrete particle-hole symme-try. This will serve as a review and a warm-up exercisefor the more complicated phenomena we consider in thelater sections. In Secs. III, IV, and V we discuss conse-quences of a non-trivial second Chern number beginningwith a parent (4+1)-d topological insulator in Sec. III. InSecs. IV and V we continue studying the consequences ofthe second Chern number but in the physically realistic(3 + 1)-d and (2+ 1)-d models which are the descendantsof the initial (4+1)-d system. We present effective actionsdescribing all of the physical systems and their responsesto applied electromagnetic fields. This provides the firsteffective field theory for the TRI topological insulators in(3 + 1)-d and (2 + 1)-d. For these two descendants of the(4 + 1)-d theory, we show that theZ2classification of thedecedents are obtained from the 2nd Chern number clas-sification of the parent TRI insulator. Finally, in Sec.VI we unify all of the results into families of topologi-cal effective actions defined in a phase space formalism.From this we construct a family tree of all topological

insulators, some of which are only defined in higher di-mensions, and with topological Z2 classifications whichrepeat every 8 dimensions.

This paper contains many new results on topologicalinsulators, but it can also be read by advanced studentsas a pedagogical and self-contained introduction of topol-ogy applied to condensed matter physics. Physical mod-els are presented in the familiar tight-binding forms, andall topological results can be derived by exact and explicitcalculations, using techniques such as response theory al-

ready familiar in condensed matter physics. During thecourse of reading this paper, we suggest the readers toconsult Appendix A which covers all of our conventions.

II. TRB TOPOLOGICAL INSULATORS IN 2 + 1DIMENSIONS AND ITS DIMENSIONAL

REDUCTION

In this section, we review the physics of the TRBtopological insulators in 2 + 1 dimensions. We shall usethe example of a translationally invariant tight-bindingmodel35 which realizes the QH effect without Landau lev-els. We discuss the procedure of dimensional reduction,from which all topological effects of the 1+1 dimensionalinsulators can be obtained. This section serves as a sim-ple pedagogical example for the more complex case of theTRI insulators presented in Sec. III and IV.

A. The first Chern number and topologicalresponse function in (2 + 1)-d

In general, the tight-binding Hamiltonian of a (2+1)-dband insulator can be expressed as

H=

m,n;,

cmhmncn (1)

with m, n the lattice sites and , = 1, 2,..N the bandindices for aN-band system. With translation symmetryhmn= h

(rm rn), the Hamiltonian can be diagonal-ized in a Bloch wavefunction basis:

H= k

ckh (k) ck (2)

The minimal coupling to an external electro-magneticfield is given byhmn hmneiAmn whereAmnis a gaugepotential defined on a lattice link with sites m, n at theend. To linear order, the Hamiltonian coupled to theelectro-magnetic field is obtained as

Hk

ckh (k) ck+k,q

Ai(q)ck+q/2

h(k)

kickq/2

with the band indices omitted. The DC response of thesystem to external field Ai(q) can be obtained by thestandard Kubo formula:

ij = lim0

i

Qij(+ i) ,

Qij(im) = 1

k,n

tr (Ji(k)G(k, i(n+ m))

Jj(k)G(k, in)) , (3)with the DC current Ji(k) = h(k)/ki, i , j = x, y,

Greens function G(k, in) = [in h(k)]1, and thearea of the system. When the system is a band insulator


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with M fully-occupied bands, the longitudinal conduc-tance vanishes, i.e. xx = 0, as expected, while xy hasthe form shown in Ref. 5:

xy = e2

h

1

2

dkx

dkyfxy(k) (4)

with fxy(k) = ay(k)

kx ax(k)

ky

ai(k) = i occ

k| ki

|k , i= x,y.

Physically, ai(k) is the U(1) component of the Berrysphase gauge field (adiabatic connection) in momentumspace. The quantization of the first Chern number

C1 = 1

2

dkx

dkyfxy(k) Z (5)

is satisfied for any continuous states|k defined on theBZ.

Due to charge conservation, the QH response ji =HijEj also induces another response equation:

ji = HijEj (6)

t

= j= H E= HBt

(B) 0= HB (7)where0 = (B= 0) is the charge density in the groundstate. Equations (6) and (7) can be combined togetherin a covariant way:

j =C12

A (8)

where ,, = 0, 1, 2 are temporal and spatial indices.Here and below we will take the units e = = 1 so thate2/h= 1/2.

The response equations (8) can be described by thetopological Chern-Simons field theory of the externalfieldA:

Seff = C1

4

d2x

dtA

A, (9)

in the sense that Seff/A = j recovers the responseequations (8). Such an effective action is topologicallyinvariant, in agreement with the topological nature ofthe first Chern number. All topological responses of theQH state are contained in the Chern-Simons theory8.

B. Example: two band models

To make the physical picture clearer, the simplest caseof a two band model can be studied as an example35.The Hamiltonian of a two-band model can be generallywritten as

h(k) =

3a=1

da(k)a + (k)I (10)

where Iis the 2 2 identity matrix and a are the threePauli matrices. Here we assume that the a represent aspin or pseudo-spin degree of freedom. If it is a real spinthen the a are thus odd under time reversal. If If theda(k) are odd in k then the Hamiltonian is time-reversalinvariant. However, if any of theda contain a constantterm then the model has explicit time-reversal symmetrybreaking. If the a are a pseudo-spin then one has to

be more careful. Since, in this case,T2

= 1 then onlyy is odd under time-reversal (because it is imaginary)while x, z are even. The identity matrix is even un-der time-reversal and(k) must be even ink to preservetime-reversal. The energy spectrum is easily obtained:E(k) = (k)

a d

2a(k). When

a d

2a(k) > 0 for

all k in the BZ, the two bands never touch each other.If we also require that maxk(E(k)) < mink(E+(k)),so that the gap is not closed indirectly, then a gap al-ways exists between the two bands of the system. Inthe single particle Hamiltonian h(k), the vector d(k)acts as a Zeeman field applied to a pseudospin iof a two level system. The occupied band satisfies(d(k)

)

|, k

=

|d(k)

| |, k

, which thus corresponds

to the spinor with spin polarization in thed(k) direc-tion. Thus the Berrys phase gained by|, k duringan adiabatic evolution along some path C in k-space isequal to the Berrys phase a spin-1/2 particle gains dur-ing the adiabatic rotation of the magnetic field along thepath d(C). This is known to be half of the solid anglesubtended by d(C), as shown in Fig.1. Consequently,the first Chern number C1 is determined by the windingnumber ofd(k) around the origin35,36:

C1 = 1

4

dkx

dkyd d

kx d

ky. (11)

From the response equations we know that a non-zeroC1 implies a quantized Hall response. The Hall effectcan only occur in a system with time-reversal symmetrybreaking so ifC1= 0 then time-reversal symmetry is bro-ken. Historically, the first example of such a two-bandmodel with a non-zero Chern number was a honeycomblattice model with imaginary next-nearest-neighbor hop-ping proposed by Haldane37.

To be concrete, we shall study a particular two bandmodel introduced in Ref.35, which is given by

h(k) = (sin kx)x+ (sin ky)y

+ (m+ cos kx+ cos ky) z, (12)

This Hamiltonian corresponds to the form (10) with

(k) 0 and d(k) = (sin kx, sin ky, m + cos kx+ cos ky).The Chern number of this system is35

C1 =

1, 0< m < 21, 2< m < 00, otherwise.

(13)

In the continuum limit, this model reduces to the 2 + 1dimensional massive Dirac Hamiltonian

h(k) = kxx+kyy+(m+2)z =

m+ 2 kx ikykx+ iky m 2

.


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C

BZ of k S2 of

d(C)

FIG. 1: Illustration of the Berrys phase curvature in a two-band model. The Berrys phase

HC

A draround a pathC inthe BZ is half of the solid angle subtended by the image pathd(C) on the sphere S2.

In a real space, this model can be expressed in tight-binding form as

H =n

c

n

z

ix

2 cn+x+ c

n

z

iy

2 cn+y+ h.c.

+mn

cnzcn (14)

Physically, such a model describes the quantum anoma-lous Hall effect realized with both strong spin-orbit cou-pling (x and y terms) and ferromagnetic polarization(z term). Initially this model was introduced for its sim-plicity in Ref. 35, however, recently, it was shown that itcan be physically realized in Hg1xMnxTe/Cd1xMnxTequantum wells with a proper amount of Mn spinpolarization38.

C. Dimensional reduction

To see how topological effects of 1+1 dimensional insu-lators can be derived from the first Chern number and theQH effect through the procedure of dimensional reduc-tion, we start by studying the QH system on a cylinder.An essential consequence of the nontrivial topology in theQH system is the existence of chiral edge states. For thesimplest case with the first Chern number C1 = 1, thereis one branch of chiral fermions on each boundary. These

edge states can be solved for explicitly by diagonalizingthe Hamiltonian (14) in a cylindrical geometry. That is,with periodic boundary conditions in they-direction andopen boundary conditions in the x-direction, as shown inFig.2 (a). Note that with this choice ky is still a goodquantum number. By defining the partial Fourier trans-formation

cky(x) = 1

Ly

y

c(x, y)eikyy,

FIG. 2: (a) Illustration of a square lattice with cylindricalgeometry and the chiral edge states on the boundary. Thedefinition of x and y axis are also shown by black arrows.

(b) One-d energy spectrum of the model in Eq. (12) withm = 1.5. The red and black line stands for the left andright moving edge states, respectively, while the blue lines arebulk energy levels. (c) Illustration of the edge states evolutionfor ky = 0 2. The arrow shows the motion of end statesin the space of center-of-mass position versus energy. (d)Polarization of the one-d system versus ky. (See text)

with (x, y) the coordinates of square lattice sites, theHamiltonian can be rewritten as

H = ky,xcky

(x)z ix

2 cky(x + 1) + h.c.

+ky,x

cky(x)[sin kyy+ (m+ cos ky) z] cky(x)

ky

H1D(ky). (15)

In this way, the 2D system can be treated as Ly inde-pendent 1D tight-binding chains, where Ly is the periodof the lattice in the y-direction. The eigenvalues of the1D HamiltonianH1D(ky) can be obtained numerically foreach ky , as shown in Fig. 2(b). An important propertyof the spectrum is the presence of edge states, which liein the bulk energy gap, and are spatially localized at thetwo boundaries: x= 0, Lx.The chiral nature of the edgestates can be seen from their energy spectrum. From Fig.2(b) we can see that the velocity v = E/k is alwayspositive for the left edge state and negative for the rightone. The QH effect can be easily understood in this edgestate picture by Laughlins gauge argument3. Considera constant electric field Ey in the y-direction, which canbe chosen as

Ay = Eyt, Ax= 0.


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The Hamiltonian is written H=

kyH1D(ky+ Ay) and

the current along the x-direction is given by

Jx=ky

Jx(ky) (16)

with Jx(ky) the current of the 1D system. In thisway, the Hall response of the 2D system is determinedby the current response of the parameterized 1D sys-tems H1D(q(t)) to the temporal change of the parame-terq(t) =ky+ Ay(t). The gauge vectorAy correspondsto a flux = AyLy threading the cylinder. During atime period 0 t 2/LyEy, the flux changes from 0to 2. The charge that flows through the system duringthis time is given by

Q =

t0

dtky

Jx(ky)

ky

Px(ky)|t0 (17)

with t = 2/LyEy. In the second equality we usethe relation between the current and charge polarizationPx(ky) of the 1D systems Jx(ky) = dPx(ky)/dt. In theadiabatic limit, the 1D system stays in the ground stateofH1D(q(t)), so that the change of polarization Px(ky)is given by Px(ky) = Px(ky 2/Ly) Px(ky). Thusin the Ly limit Q can be written as

Q= 2

0

dkyPx(ky)

ky. (18)

Therefore, the charge flow due to the Hall current gener-ated by the flux through the cylinder equals the chargeflow through the 1-dimensional system H1D(ky), when

ky is cycled adiabatically from 0 to 2. From the QH re-sponse we know Q= HtEyLy = 2H is quantizedas an integer, which is easy to understand in the 1D pic-ture. During the adiabatic change ofky from 0 to 2,the energy and position of the edge states will change,as shown in Fig.2(c). Since the edge state energy is al-ways increasing(decreasing) withkyfor a state on the left(right) boundary, the charge is always pumped to theleft for the half-filled system, which leads to Q= 1 foreach cycle. This quantization can also be explicitly shownby calculating the polarizationPx(ky), as shown in Fig.2(d), where the jump of Px by one leads to Q =1.In summary, we have shown that the QH effect in the

tight-binding model of Eq. (12) can be mapped to anadiabatic pumping effect12 by diagonalizing the systemin one direction and mapping the momentum k to a pa-rameter.

Such a dimensional reduction procedure is not re-stricted to specific models, and can be generalized to any2D insulators. For any insulator with Hamiltonian (2),we can define the corresponding 1D systems

H1D() =kx

ckxh(kx, )ckx (19)

in which replaces the y-direction momentum ky andeffectively takes the place of q(t). When is time-dependent, the current response can be obtained by asimilar Kubo formula to Eq. (3), except that the sum-mation over all (kx, ky) is replaced by that over only kx.More explicitly, such a linear response is defined as

Jx() = G()d

dt (20)

G() = lim0

i

Q(+ i; )

Q(in; ) = kx,im

tr (Jx(kx; )G1D(kx, i(m+ n); )

h(kx; )

G1D(kx, in; )

1

Lx.

Similar to Eq. (4) of the 2D case, the response coefficientG(k) can be expressed in terms of a Berrys phase gaugefield as

G() =

dkx2

fx(kx, ) (21)

=

dkx2

ax

akx

with the sum rule

G()d= C1 Z. (22)

If we choose a proper gauge so that a is always single-valued, the expression ofG() can be further simplifiedto

G() =

dkx

2ax(kx, )

P()

. (23)

Physically, the loop integral

P() =

dkxax/2 (24)

is nothing but the charge polarization of the 1Dsystem10,11, and the response equation (20) simply be-comes Jx = P/t. Since the polarizationP is definedas the shift of the electron center-of-mass position awayfrom the lattice sites, it is only well-defined modulo 1.Consequently, the change P = P( = 2) P( = 0)through a period of adiabatic evolution is an integer equaltoC1, and corresponds to the charge pumped throughthe system. Such a relation between quantized pumpingand the first Chern number was shown by Thouless12.

Similar to the QH case, the current response can leadto a charge density response, which can be determinedby the charge conservation condition. When the param-eter has a smooth spatial dependence = (x, t), theresponse equation (20) still holds. From the continuityequation we obtain

t = Jx

x =

2P()

xt

= P()x

(25)


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in which is defined with respect to the backgroundcharge. Similar to Eq. (8), the density and current re-sponse can be written together as

j= P((x, t))x

(26)

where , = 0, 1 are time and space. It should be

noted that only differentiation with respect to x, t ap-pears in Eq. (26). This means, as expected, the currentand density response of the system do not depend onthe parametrization. In general, when the Hamiltonianhas smooth space and time dependence, the single par-ticle Hamiltonian h(k) becomes h(k,x,t) h(k, (x, t)),which has the eigenstates|; k,x,t with the band in-dex. Then relabelling t ,x, k as qA, A = 0, 1, 2 we candefine the phase spaceBerrys phase gauge field

AA = i

; qA| qA

|; qA

FAB = AAB BAA (27)

and the phase space current

jPA = 1

4ABCFBC. (28)

The physical current is obtained by integration over thewavevector manifold:

j=

dkjP =

dk

22F2 (29)

where , = 0, 1. This recovers Eq. (26). Note that wecould have also looked at the component jk = dk j

Pk

but this current does not have a physical interpretation.Before moving to the next topic, we would like to applythis formalism to the case of the Dirac model, which re-produces the well-known result of fractional charge in theSu-Schrieffer-Heeger (SSH) model9, or equivalently theJackiw-Rebbi model39. To see this, consider the follow-ing slightly different version of the tight-binding model(12):

h(k, ) = si n kx+ (cos k 1) z+m (sin y+ cos z) (30)

withm >0. In the limitm 1, the Hamiltonian has thecontinuum limit h(k, )

kx + m (sin y+ cos z),

which is the continuum Dirac model in (1 + 1)-d, witha real mass m cos and an imaginary mass m sin . Asdiscussed in Sec. IIB, the polarization

dkxax/2 is

determined by the solid angle subtended by the curved(k) = (sin k, m sin , m cos + cos k 1), as shown inFig. 3. In the limit m1 one can show that the solidangle () = 2 so that P() /2, in which case Eq.(26) reproduces the Goldstone-Wilczek formula13 :

j = . (31)

FIG. 3: Illustration of the d(k, ) vector for the 1D Diracmodel (30). The horizontal blue circle shows the orbit ofd(k)vector in the 3D space for k [0, 2) with fixed. The redcircle shows the track of the blue circle under the variationof . The cone shows the solid angle () surrounded by thed(k) curve, which is 4 times the polarization P().

Specifically, a charge Q = (d/dx)(dx/2) =((+) ())/2 is carried by a domain wall ofthe field. In particular, for an anti-phase domainwall, (+) () = , we obtain fractional chargeq= 1/2. Our phase space formula (28) is a new result,and it provides a generalization of the Goldstone-Wilczekformula to the most general one-dimensional insulator.

D. Z2 classification of particle-hole symmetricinsulators in (1 + 1)-d

In the last subsection, we have shown how the firstChern number of a Berrys phase gauge field appearsin an adiabatic pumping effect and the domain wallcharge of one-dimensional insulators. In these cases,an adiabatic spatial or temporal variation of the single-particle Hamiltonian, through its parametric dependenceon (x, t), is required to define the Chern number. Inother words, the first Chern number is defined for a pa-rameterized familyof Hamiltoniansh(k,x,t), rather thanfor a single 1D Hamiltonian h(k). In this subsection, wewill show a different application of the first Chern num-ber, in which a Z2 topological classification is obtained

for particle-hole symmetric insulators in 1D. Such a re-lation between Chern number and Z2 topology can beeasily generalized to the more interesting case of secondChern number, where a similarZ2 characterization is ob-tained for TRI insulators, as will be shown in Sec. IV CandV B.

For a one-dimensional tight-binding HamiltonianH=mn c

mh

mn(k)cn , the particle-hole transformation is

defined bycm Ccm, where the charge conjugationmatrixCsatisfiesCC= I and CC= I. Under periodic


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8

boundary conditions the symmetry requirement is

H =k

ckh(k)ck =k

ckCh(k)Cck

Ch(k)C= hT(k). (32)From Eq. (32) it is straightforward to see the symme-

try of the energy spectrum: ifEis an eigenvalue ofh(0),

so isE. Consequently, if the dimension ofh(k) is odd,there must be at least one zero mode with E= 0. Sincethe chemical potential is constrained to vanish by thetraceless condition ofh, such a particle-hole symmetricsystem cannot be gapped unless the dimension ofh(k) iseven. Since we are only interested in the classification ofinsulators, we will focus on the case with 2Nbands perlattice site.

Now consider two particle-hole symmetric insulatorswith Hamiltoniansh1(k) andh2(k), respectively. In gen-eral, a continuous interpolation h(k, ), [0, ] be-tween them can be defined so that

h(k, 0) = h1(k), h(k, ) =h2(k) (33)

Moreover, it is always possible to find a properparametrization so thath(k, ) is gapped for all [0, ].In other words, the topological space of all 1D insulatingHamiltoniansh(k, ) is connected, which is a consequenceof the Wigner-Von Neumann theorem34.

Suppose h(k, ) is such a gapped interpolation be-tween h1(k) and h2(k). In general,h(k, ) for (0, )doesnt necessarily satisfy the particle-hole symmetry.For [, 2], define

h(k, ) = C1h(k, 2 )CT . (34)We choose this parameterization so that if we replaced by a momentum wavevector then the correspond-ing higher dimensional Hamiltonian would be particle-hole symmetric. Due to the particle-hole symmetry ofh(k, = 0) and h(k, = ), h(k, ) is continuous for [0, 2], and h(k, 2) = h(k, 0). Consequently, theadiabatic evolution of from 0 to 2 defines a cycle ofadiabatic pumping in h(k, ), and a first Chern numbercan be defined in the (k, ) space. As discussed in Sec.IIC,the Chern number C[h(k, )] can be expressed as awinding number of the polarization

C[h(k, )] =

d

P()

P() =

dk2

E(k)


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9

h2(k)h(k,)

h(k,)

h1(k)

h1(k)

h2(k)

g1(k)

g2(k)

(a) (b)

=

FIG. 4: Illustration of the interpolation between two particle-hole symmetric Hamiltonians h1(k) andh2(k).

h is given by

C[h] C[h] = 2

0

d

P()

P

()

. (38)

Define the new interpolations g1

(k, ) and g2

(k, ) as

g1(k, ) =

h(k, ), [0, ]

h(k, 2 ), [, 2]

g2(k, ) =

h(k, 2 ), [0, ]

h(k, ), [, 2] (39)

g1(k, ) andg2(k, ) are obtained by recombination of thetwo paths h(k, ) and h(k, ), as shown in Fig. 4. Fromthe construction ofg1 and g2, it is straightforward to seethat

C[g1] =

0

dP()

P

()

C[g2] =

2

d

P()

P

()

. (40)

Thus C[h] C[h] = C[g1] + C[g2]. On the otherhand, from Eq. (37) we know C[g1] = C[g2], so thatC[h] C[h] = 2C[g1]. SinceC[g1] Z, we obtain thatC[h]C[h] is even for any two interpolations h(k, ) andh(k, ) betweenh1(k) andh2(k). Intuitively, such a con-clusion simply comes from the fact that the Chern num-ber C[h] andC[h] can be different only if there are sin-gularities between these two paths, while the positions ofthe singularities in the parameter space are always sym-

metric under particle-hole symmetry, as shown in Fig.4.Based on the discussions above, we can define the rel-

ative Chern parity as

N1[h1(k), h2(k)] = (1)C[h(k,)], (41)

which is independent of the choice of interpolationh(k, ), but only determined by the Hamiltoniansh1(k), h2(k). Moreover, for any three particle-hole sym-metric Hamiltonians h1(k), h2(k), h3(k), it is easy to

prove that the Chern parity satisfies the following asso-ciative law:

N1[h1(k), h2(k)]N1[h2(k), h3(k)] = N1[h1(k), h3(k)].

(42)

Consequently, N1[h1(k), h2(k)] = 1 defines an equiva-lence relation between any two particle-hole symmetric

Hamiltonians, which thus classifies all the particle-holesymmetric insulators into two classes. To define thesetwo classes more explicitly, one can define a vacuumHamiltonian as h0(k) h0, where h0 is an arbitrarymatrix which does not depend on k and which satisfiesthe particle-hole symmetry constraint Ch0C =hT0.Thus h0 describes a totally local system, in which thereis no hopping between different sites. Taking such atrivial system as a reference Hamiltonian, we can defineN1[h0(k), h(k)]N1[h(k)] as a Z2 topological quantumnumber of the Hamiltonian h(k). All the Hamiltoniansh(k) with N1[h0(k), h(k)] = 1 are classified as Z2 trivial,while those with N1[h0(k), h(k)] = 1 are considered asZ2 nontrivial. (Again, this classification doesnt depend

on the choice of vacuum h0, since any two vacua areequivalent.)

Despite its abstract form, such a topological character-ization has a direct physical consequence. For a Z2 non-trivial Hamiltonian h1(k), an interpolation h(k, ) canbe defined so that h(k, 0) = h0, h(k, ) = h1(k), and theChern number C[h(k, )] is an odd integer. If we studythe one-dimensional system h(k, ) with open boundaryconditions, the tight binding Hamiltonian can be rewrit-ten in real space as

hmn() = 1

L keik(xmxn)h(k, ),1 m, n L.

As discussed in Sec. IIC, there are mid-gap end statesin the energy spectrum of hmn() as a consequence ofthe non-zero Chern number. When the Chern num-ber C[h(k, ) ] = 2n 1, n Z, there are valuesLs [0, 2), s = 1, 2, ..2n 1 for which the Hamilto-nianhmn(s) has zero energy localized states on the leftend of the 1D system, and the same number ofRs valueswhere zero energy states are localized on the right end,as shown in Fig. 5. Due to the particle-hole symme-try betweenhmn() andhmn(2 ), zero levels alwaysappear in pairs at and 2 . Consequently, whenthe Chern number is odd, there must be a zero level at = 0 or = . Since = 0 corresponds to a trivialinsulator with flat bands and no end states, the localizedzero mode has to appear at = . In other words, onezero energy localized state (or an odd number of suchstates) is confined at each open boundary of a Z2 non-trivial particle-hole symmetric insulator.

The existence of a zero level leads to an importantphysical consequencea half charge on the boundary ofthe nontrivial insulator. In a periodic system when thechemical potential vanishes, the average electron densityon each site is nm =

c

mcm

= N when there


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10

12

E

E

=0

20Z2 odd insulator

=

vacuum

=0

vacuum

=0

(a)

(c)

(b)

x

(1)

(2)

FIG. 5: (a) Schematic energy spectrum of a parameterizedHamiltonian hmn() with open boundary conditions. Thered (blue) lines indicate the left (right) end states. The val-ues with zero-energy leftedge states are marked by the solidcircles. (b) Illustration to show that the open boundary of aZ2 nontrivial insulator is equivalent to a domain wall between = (nontrivial) and = 0 (trivial vacuum). (c) Illustra-tion of the charge density distribution corresponding to twodifferent chemical potentials1(red) and2 (blue). The areabelow the curve (1) and (2) is +e/2 and e/2, respec-tively, which shows the half charge confined on the boundary.

are Nbands filled. In an open boundary system, definem() =

c

mcm

Nto be the density deviation

with respect to Non each site. Then particle-hole sym-metry leads to m() =m(). On the other hand,when is in the bulk gap, the only difference between

and is the filling of the zero levels localized on eachend|0Land|0R, so that

lim0+

(m() m()) =

|m|0L|2

for the sites m that are far away enough from the rightboundary. Thus we have

m m(0+) = 1/2 where

the summation is done around the left boundary so thatwe do not pick up a contribution from the other end. Insummary, a chargee/2 (e/2) is localized on the bound-ary if the zero level is vacant (occupied), as shown in Fig.5.

The existence of such a half charge can also be under-stood by viewing the open boundary of a topologicallynontrivial insulator as a domain wall between the non-trivial insulator and the vacuum. By defining the inter-polation hmn(), such a domain wall is described by aspatial dependence of with (x +) = , (x) = 0. According to the response formula (25), thecharge carried by the domain wall is given by

Qd= e

+

dxP((x))

x =e

0

dP(). (43)

By using Eq.. (37) we obtain

Qd=e

2

20

dP() = e

2C[h(k, )]. (44)

It should be noted that an integer charge can always beadded by changing the filling of local states, which meansQd is only fixed modulo e. Consequently, ae/2 chargeis carried by the domain wall if and only if the Chernnumber is odd, i.e., when the insulator is nontrivial.

E. Z2 classification of(0 + 1)-d particle-holesymmetric insulators

In the last subsection, we have shown how a Z2 clas-sification of (1 + 1)-d particle-hole symmetric insulatorsis defined by dimensional reduction from (2 + 1)-d sys-tems. Such a dimensional reduction can be repeatedonce more to study (0 + 1)-d systems, that is, a single-site problem. In this subsection we will show that a Z2classification of particle-hole symmetric Hamiltonians in(0 + 1)-d is also obtained by dimensional reduction. Al-though such a classification by itself is not as interestingas the higher dimensional counterparts, it does providea simplest example of the dimensional reduction chain(2 + 1)-d (1 + 1)-d (0 + 1)-d, which can be latergeneralized to its higher-dimensional counterpart (4+1)-d (3+1)-d (2 + 1)-d. In other words, theZ2 classifi-cation of the (0 + 1)-d particle-hole symmetric insulatorscan help us to understand the classification of (2 + 1)-dTRI insulators as it is dimensionally reduced from the(4 + 1)-d TRI insulator.

For a free, single-site fermion system with Hamiltonianmatrixh, the particle-hole symmetry is defined as

ChC= hT. (45)

Given any two particle-hole symmetric Hamiltonians h1and h2, we follow the same procedure as the last sub-section and define a continuous interpolation h(), [0, 2] satisfying

h(0) =h1, h() = h2, Ch()C= h(2 )T, (46)

where h() is gapped for all . The Hamiltonianh() isthe dimensional reduction of a (1+1)-d Hamiltonian h(k),with the wavevectork replaced by the parameter . Theconstraint (46) is identical to the particle-hole symmetrycondition (32), so thath() corresponds to a particle-holesymmetric (1 + 1)-d insulator. As shown in last subsec-tion,h() is classified by the value of the Chern parityN1[h()]. If N1[h()] =1, no continuous interpola-tion preserving particle-hole symmetry can be defined be-tweenh() and the vacuum Hamiltonianh() =h0, [0, 2]. To obtain the classification of (0 + 1)-d Hamil-tonians, consider two different interpolations h() andh() between h1 and h2. According to the associativelaw (42), we know N1[h()]N1[h()] = N1[h(), h()],


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11

where N1[h(), h()] is the relative Chern parity be-tween two interpolations. In the following we will proveN1[h(), h()] = 1 for any two interpolations h and h

satisfying condition (46). As a result, N1[h()] is inde-pendent of the choice of interpolation between h1 andh2, so that N0[h1, h2] N1[h()] can be defined as afunction ofh1 and h2. The Z2 quantity N0 defined for(0 + 1)-d Hamiltonians plays exactly the same role as

N1[h(k), h

(k)] in the (1 + 1)-d case, from which a Z2classification can be defined.To proveN1[h(), h()] = 1 for any two interpolations,

first define a continuous deformation g(, ) betweenh()and h(), which satisfies the conditions below:

g(, = 0) = h(), g(, = ) = h()

g(0, ) = h1, g(, ) = h2

Cg(, )C = g(2 , 2 )T. (47)From the discussions in last subsection it is easy to con-firm that such a continuous interpolation is always pos-sible, in which g(, ) is gapped for all and . Inthe two-dimensional parameter space , one can define

the Berry phase gauge field and the first Chern numberC1[g(, )]. By the definition of the Chern parity, wehave N1[h(), h()] = (1)C1[g(,)]. However, the pa-rameterized Hamiltoniang(, ) can be viewed in two dif-ferent ways: it not only defines an interpolation betweenh() andh(), but also defines an interpolation betweeng(0, ) =h1 and g (, ) =h2. Sinceg (0, ) and g (, )are vacuum Hamiltonians without any dependence,they have trivial relative Chern parity, which meansN1[h(), h

()] = N1[g(0, ), g(, )] = N1[h1, h2] = 1.In conclusion, from the discussion above we have

proved that any two interpolationsh() andh() belongto the same Z2 class, so that the Chern parity N1[h()]

only depends on the end points h1and h2. Consequently,the quantity N0[h1, h2] N1[h()] defines a relation be-tween each pair of particle-hole symmetric Hamiltoniansh1 and h2. After picking a reference Hamiltonian h0,one can defineall the Hamiltonians with N0[h0, h] = 1as trivial and N0[h0, h] = 1 as nontrivial. Themain difference between this classification and the onefor (1 + 1)-d systems is that there is no natural choice ofthe reference Hamiltonianh0. In other words, the namestrivial and non-trivial only have relative meaning inthe (0 + 1)-d case. However, the classification is stillmeaningful in the sense that any two Hamiltonians withN0[h1, h2] = 1 cannot be adiabatically connected with-out breaking particle-hole symmetry. In other words, themanifold of single-site particle-hole symmetric Hamilto-nians is disconnected, with at least two connected pieces.

As a simple example, we study 2 2 Hamiltonians. Ageneral 2 2 single-site Hamiltonian can be decomposedas

h= d00 +

3a=1

daa (48)

where 0 = I and 1,2,3 are the Pauli matrices. When

h1

h2

h()

FIG. 6: Illustration of the 22 single-site Hamiltonians. Each

point on the sphere represents an unit vector d = d/|d|, andthe north and south poles correspond to the particle-hole sym-metric Hamiltonians h1,2 =

3, respectively. The blue pathshows an interpolation between h1 andh2 satisfying the con-straint (46), which always encloses a solid angle = 2.

C = 1, particle-hole symmetry requires ChC=hT,from which we obtain d0 = d1 = d2 = 0. Thus h = d33,in which d3 = 0 so as to make h gapped. Conse-quently, we can see that the two Z2 classes are simplyd3 > 0 and d3 < 0. When an adiabatic interpolationh() = d0()

0 +a da()

a is defined from d3 > 0 to

d3 < 0, the spin vector d() has to rotate from the northpole to the south pole, and then return along the im-age path determined by the particle-hole symmetry (46),as shown in Fig. 6. The topological quantum numberN0[h1, h2] is simply determined by the Berrys phase en-closed by the pathda(), which is when h1 andh2 areon different poles, and 0 otherwise. From this example

we can understand the Z2 classification intuitively. InSec. V Bwe show that the Z2 classification of (2 + 1)-dTRI insulatorsthe class that corresponds to the QSHeffectis obtained as a direct analog of the (0 + 1)-d casediscussed above.

III. SECOND CHERN NUMBER AND ITSPHYSICAL CONSEQUENCES

In this section, we shall generalize the classificationof the (2 + 1)-d TRB topological insulator in terms ofthe first Chern number and the (2 + 1)-d Chern-Simonstheory to the classification of the (4+1)-d TRI topologicalinsulator in terms of the second Chern number and the(4 + 1)-d Chern-Simons theory. We then generalize thedimensional reduction chain (2+1)-d (1+1)-d (0+1)-d to the case of (4 + 1)-d (3+ 1)-d (2 + 1)-d for TRIinsulators. Many novel topological effects are predictedfor the TRI topological insulators in (3 +1)-d and (2 +1)-d.


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12

W

q

W,qw,k

m

n

r

FIG. 7: The Feynman diagram that contributes to the topo-

logical term (52). The loop is a fermion propagator, and thewavy lines are external legs corresponding to the gauge field.

A. Second Chern number in (4 + 1)-d non-linearresponse

In this subsection, we will show how the second Chernnumber appears as a non-linear response coefficient of(4 + 1)-d band insulators in an externalU(1) gauge field,which is in exact analogy with the first Chern number asthe Hall conductance of a (2 + 1)-d system. To describesuch a non-linear response, it is convenient to use thepath integral formalism. The Hamiltonian of a (4 + 1)-d

insulator coupled to a U(1) gauge field is written as

H[A] =m,n

cmh

mne

iAmncn+ h.c.

+m

A0mcmcm. (49)

The effective action of gauge field A is obtained by thefollowing path integral:

eiSeff[A] =

D[c]D[c]ei

Rdt[

Pmc

m(it)cmH[A]]

= det

(it A0m) mn hmneiAmn

(50)

which determines the response of the fermionic systemthrough the equation

j(x) =Seff[A]

A(x). (51)

In the case of the (2 + 1)-d insulators, the ef-fective action Seff contains a Chern-Simons term(C1/4)A

A as shown in Eq. (9) of Sec. IIA,in which the first Chern number C1 appears as the co-efficient. For the (4 + 1)-d system, a similar topologicalterm is in general present in the effective action, whichis the second Chern-Simons term:

Seff= C2242

d4xdtAAA (52)

where,,,, = 0, 1, 2, 3, 4. As shown in Refs. 33,40,41, the coefficient C2 can be obtained by the one-loopFeynman diagram in Fig. 7, which can be expressed inthe following symmetric form:

C2 = 2

15

d4kd

(2)5Tr

G

G1

q

G

G1

q

G

G1

q

G

G1

q

G

G1

q

(53)

in which q = (, k1, k2, k3, k4) is the frequency-

momentum vector, andG(q) = [+ i h(ki)]1 is thesingle-particle Greens function.

Now we are going to show the relation between C2defined in Eq. (53) and the non-abelian Berrys phasegauge field in momentum space. To make the statement

clear, we first write down the conclusion: For any (4 + 1)-d band insulator with single parti-

cle Hamiltonianh(k), the non-linear response coef-ficient C2 defined in Eq. (53) is equal to the sec-ond Chern number of the non-abelian Berrys phasegauge field in the BZ, i.e.:

C2 = 1

322

d4kijktr [fijfk] (54)

withfij = iaj jai + i [ai, aj ] ,

ai (k) = i , k|

ki|, k

wherei, j,k, = 1, 2, 3, 4.

The indexinai refers to the occupied bands, there-

fore, for a general multi-band model,ai is a non-abelian

gauge field, and fij is the associated non-abelian fieldstrength. Here we sketch the basic idea of Eq. (54),and leave the explicit derivation to Appendix B. Thekey point to simplify Eq. (53) is noticing its topologi-cal invariance i.e. under any continuous deformation ofthe Hamiltonian h(k), as long as no level crossing oc-curs at the Fermi level, C2 remains invariant. Denotethe eigenvalues of the single particle Hamiltonian h(k)as (k), = 1, 2,...,Nwith (k) +1(k). When Mbands are filled, one can always define a continuous de-


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13

E

Generic Insulator Flat Band Model

E

G

FIG. 8: Illustration showing that a band insulator with arbi-trary band structure i(k) can be continuously deformed toa flat band model with the same eigenstates. Since no levelcrossing occurs at the Fermi level, the two Hamiltonians aretopologically equivalent.

formation of the energy spectrum so that(k) G for Mand(k) Efor > M(withE> G), whileall the corresponding eigenstates |, k remain invariant.In other words, each Hamiltonian h(k) can be contin-

uously deformed to some flat band model, as shownin Fig. 8. Since both Eq. (53) and the second Chernnumber in Eq. (54) are topologically invariant, we onlyneed to demonstrate Eq. (54) for the flat band models,of which the Hamiltonians have the form

h0(k) = G

1M

|, k , k| + E>M

|, k , k|

GPG(k) + EPE(k). (55)Here PG(k) (PE(k)) is the projection operator to theoccupied (un-occupied) bands. Non-abelian gauge con-nections can be defined in terms of these projection op-erators in a way similar to Ref. 42. Correspondingly,the single particle Greens function can also be expressedby the projection operators PG, PE, and Eq. (54) canbe proved by straight-forward algebraic calculations, asshown in AppendixB.

In summary, we have shown that for any (4+1)-d bandinsulator, there is a (4 + 1)-d Chern-Simons term ( 52)in the effective action of the external U(1) gauge field,of which the coefficient is the second Chern number ofthe non-abelian Berry phase gauge field. Such a relationbetween Chern number and Chern-Simons term in theeffective action is an exact analogy of the TKNN formulain (2+1)-d QH effect. By applying the equation of motion(51), we obtain

j = C282

AA (56)

which is the non-linear response to the external field A.For example, consider a field configuration :

Ax = 0, Ay = Bzx, Az = Ezt, Aw = At = 0 (57)wherex, y, z, w are the spatial coordinates and t is time.The only non-vanishing components of the field curvature

are Fxy = Bz and Fzt =Ez, which according to Eq.(56) generates the current

jw = C242

BzEz.

If we integrate the equation above over the x, y dimen-sions (with periodic boundary conditions and assumingEz is does not depend on (x, y)), we obtain

dxdyjw = C242

dxdyBz

Ez C2Nxy

2 Ez (58)

whereNxy =

dxdyBz/2 is the number of flux quantathrough the xy plane, which is always quantized to bean integer. This is exactly the 4DQH effect proposed inRef. 14. Thus, from this example we can understanda physical consequence of the second Chern number: Ina (4 + 1)-d insulator with second Chern number C2, aquantized Hall conductance C2Nxy/2 in the zw planeis induced by magnetic field with flux 2Nxy in the per-pendicular (xy) plane.

Similar to the (2+1)-d case, the physical consequences

of the second Chern number can also be understood bet-ter by studying the surface states of an open-boundarysystem, which for the (4 + 1)-d case is described by a(3 + 1)-d theory. In the next subsection we will studyan explicit example of a (4 + 1)-d topological insulator,which helps us to improve our understanding of the phys-ical picture of the (4 + 1)-d topology; especially, afterdimensional reduction to the lower-dimensional physicalsystems.

B. TRI topological insulators based on latticeDirac models

In section IIB, we have shown that the model intro-duced in Ref. 35realizes the fundamental TRB topolog-ical insulator in (2 + 1)-d, and it reduces to the Diracmodel in the continuum limit. Generalizing this con-struction, we propose the lattice Dirac model to be therealization of the fundamental TRI topological insulatorin (4 + 1)-d. Such a model has also been studied in thefield theory literature40,43. The continuum Dirac modelin (4 + 1)-d dimensions is expressed as

H=

d4x

(x)i (ii) (x) + m0

(59)

with i = 1, 2, 3, 4 the spatial dimensions, and , =

0, 1,.., 4 the five Dirac matrices satisfying the Cliffordalgebra

{, } = 2I (60)with Ithe identity matrix44.

The lattice (tight-binding) version of this model iswritten as

H =n,i

n

c0 ii

2

n+i+ h.c.


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14

+mn

n0n (61)

or in momentum space,

H =k

k

i

sin kii +

m + c

i

cos ki

0

k.

(62)

Such a Hamiltonian can be written in the compact form

H =k

k

da(k)ak (63)

with

da(k) =

m+ c

i

cos ki

, sin kx, sin ky, sin kz, sin kw

a five-dimensional vector. Similar to the (2 + 1)-d two-band models we studied in Sec. IIB, a single parti-

cle Hamiltonian with the form h(k) = da(k)a

has twoeigenvaluesE(k) =

a d2a(k), but with the key dif-

ference that here both eigenvalues are doubly degenerate.When

a d

2a(k) d2(k) is non-vanishing in the whole

BZ, the system is gapped at half-filling, with the twobands with E=E(k) filled. Since there are two occu-pied bands, anSU(2)U(1) adiabatic connection can bedefined42,45,46. Starting from the Hamiltonian (62), onecan determine the single particle Greens function, andsubstituting it into the expression for the second Chernnumber in Eq. (53). We obtain

C2 = 3

82 d4kabcdedaxdbydczddwde (64)which is the winding number of the mapping da(k)da(k)/ |d(k)| from the BZ T4 to the sphere S4 and

a,b,c,d,e = 0, 1, 2, 3, 4. More details of this calculationare presented in AppendixC.

Since the winding number (64) is equal to the sec-ond Chern number of the Berrys phase gauge field, it istopologically invariant. It is easy to calculateC2 in thelattice Dirac model (61). Considering the lattice Diracmodel with a fixed positive parametercand tunable mass

termm,C2(m) as a function ofm can change only if theHamiltonian is gapless, i.e., if

a d

2a(k, m) = 0 for some

k. Its easy to determine that C2(m) = 0 in the limit

m +, since the unit vector da(k) (1, 0, 0, 0, 0)in that limit. Thus we only need to study the change ofC2(m) at each quantum critical points, namely at criticalvalues ofm where the system becomes gapless.

The solutions of equation

a d2a(k, m) = 0 lead to five

critical values ofm and correspondingk points as listedbelow:

m =

4c, k= (0, 0, 0, 0)2c, k P[(, 0, 0, 0)]

0, k P[(, 0, , 0)]2c, k P[(,,, 0)]4c, k= (, , , )

(65)

in which P[k] stands for the set of all the wavevectorsobtained from index permutations of wavevector k. Forexample, P[(, 0, 0, 0)] consists of (, 0, 0, 0), (0, , 0, 0),(0, 0, , 0) and (0, 0, 0, ). As an example, we can studythe change ofC2(m) around the critical valuem = 4c.In the limit m+ 4c 2c, the system has its minimalgap at k = 0, around which the da(k) vector has theapproximate form da(k)

(m,kx, ky, kz, kw) +o(

|k|),

withm m+ 4c. Taking a cut-off 2 in momen-tum space, one can divide the expression (64)ofC2 intolow-energy and high-energy parts:

C2 = 3

82

|k|

d4k+

|k|>

d4k

abcdedaxdbydczddwde C(1)2 (m, ) + C(2)2 (m, ).

Since there is no level-crossing in the region|k

|>, the

jump ofC2 at m = 0 can only come from C(1)2 . In thelimit|m| < 2, the continuum approximation ofda(k) can be applied to obtain

C(1)2 (m, )

3

82

|k|

d4k m

(m2 + k2)5/2

which can be integrated and leads to

C2m=0+

m=0 = C(1)2

m=0+

m=0 = 1. (66)

From the analysis above we see that the change of thesecond Chern number is determined only by the effec-tive continuum model around the level crossing wavevec-tor(s). In this case the continuum model is just the Diracmodel. Similar analysis can be carried out at the othercritical ms, which leads to the following values of the


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15

second Chern number:

C2(m) =

0, m < 4c or m >4c1, 4c < m < 2c

3, 2c < m


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16

5 0 55

0

5

pz(arb. unit)

E(pz

)(arb.unit)

F

FIG. 10: Illustration of the surface Landau level spectrumgiven by Eq. (70). Each level in the figure is Nxy fold degen-erate. The solid circles are the occupied states of the zerothLandau level, and the red open circle shows the states thatare filled when the gauge vector potential Az is shifted adia-batically from 0 to 2/Lz.

edge of the 4D lattice Dirac model are chiral fermions,

the currentIw carries away chiral charge, leading to thenon-conservation of chirality on the 3D edge.

IV. DIMENSIONAL REDUCTION TO(3 + 1)-DTRI INSULATORS

As shown in Sec. IIC, one can start from a (2 + 1)-d TRB topological insulator described by a Hamilto-nian h(kx, ky), and perform the procedure of dimen-sional reduction by replacing ky by a parameter . Thesame dimensional reduction procedure can be carried outfor the (4 + 1)-d TRI insulator with a non-vanishingsecond Chern number. From this procedure, one ob-tains the topological effective theory of insulators in(3 + 1)-d and (2 + 1)-d. Specifically, for TRI insula-tors a general Z2 topological classification is defined.Compared to the earlier proposals of the Z2 topologicalinvariant16,22,23,24,27,28,29,30, our approach provides a di-rect relationship between the topological quantum num-ber and the physically measurable topological response ofthe corresponding system. We discuss a number of the-oretical predictions, including the TME effect, and pro-pose experimental settings where these topological effectscan be measured in realistic materials.

A. Effective action of(3 + 1)-d insulators

To perform the dimensional reduction explicitly, in thefollowing we show the derivation for the (4 + 1)-d Diracmodel (61). However, each step of the derivation is ap-plicable to any other insulator model, so the conclusionis completely generic.

The Hamiltonian of Dirac model (61) coupled to anexternalU(1) gauge field is given by

H[A] =n,i

n

c0 ii

2

eiAn,n+in+i+h.c.

+mn

n0n. (72)

Now consider a special Landau-gauge configurationsatisfying An,n+i = An+ w,n+w+i, n, which is transla-tionally invariant in the w direction. Thus, under peri-odic boundary conditions the w-direction momentumkwis a good quantum number, and the Hamiltonian can be

rewritten as

H[A] =kw,x,s

x,kw

c0 is

2

eiAx,x+sx+s,kw+ h.c.

+kw,x,s

x,kw

sin(kw+ Ax4) 4

+ (m + c cos(kw+ Ax4)) 0

x,kw

where x stands for the three-dimensional coordinates,Ax4 Ax,x+ w, and s = 1, 2, 3 stands for the x, y, z di-rections. In this expression, the states with different kwdecouple from each other, and the (4 + 1)-d Hamiltonian

H[A] reduces to a series of (3 + 1)-d Hamiltonians. Pickone of these (3 + 1)-d Hamiltonians with fixed kw andrenamekw+ Ax4 = x, we obtain the (3 + 1)-d model

H3D[A, ] =x,s

x

c0 is

2

eiAx,x+sx+s+ h.c.

+x,s

x

sin x4 + (m + c cos x)

0

x

(73)

which describes a band insulator coupled to an electro-magnetic field Ax,x+s and an adiabatic parameter field

x.Due to its construction, the response of the model (73)toAx,x+s and x fields is closely related to the responseof the (4 + 1)-d Dirac model (61) to theU(1) gauge field.To study the response properties of the (3 + 1)-d system,the effective action S3D[A, ] can be defined as

expiS3D[A,] =

D[]D[]ei

Rdt[

Px x(iAx0)xH[A,]].

A Taylor expansion of S3D can be carried out aroundthe field configuration As(x, t) 0, (x, t) 0, whichcontains a non-linear response term directly derived fromthe (4 + 1)-d Chern-Simons action (52):

S3D =G3(0)

4

d3xdtAA. (74)

Compared to the Eq. (52), the field(x, t) = (x, t)0plays the role of A4, and the coefficient G3(0) is de-termined by the same Feynman diagram (7), but evalu-ated for the three-dimensional Hamiltonian (73). Con-sequently, G3(0) can be calculated and is equal to Eq.(53), but without the integration over kw:


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G3(0) = 6

d3kd

(2)4 Tr

G

G1

q

G

G1

q

G

G1

q

G

G1

q

G

G1

0

(75)

whereq = (, kx, ky, kz) .Due to the same calculation

as Sec. IIIAand AppendixB,G3(0) is determined fromthe Berry phase curvature as

G3(0) = 1

82

d3kijktr [fifjk ] , (76)

in which the Berry phase gauge field is definedin the four-dimensional space (kx, ky, kz , 0), i.e.,

ai = i

k, 0; (/ki) k, 0; and a =

i

k, 0; (/0) k, 0; . Compared to the second

Chern number (54), we know that G3(0) satisfies thesum rule

G3(0)d0 = C2 Z, (77)which is in exact analogy with the sum rule of the pump-ing coefficientG1() in Eq. (22) of the (1 + 1)-d system.Recall thatG1() can be expressed as P1()/ , whereP1() is simply the charge polarization. In comparison,a generalized polarization P3(0) can also be defined in(3 + 1)-d so that G3(0) = P3(0)/0. ( Recently, asimilar quantity has also been considered in Ref. 49fromthe point of view of semiclassical particle dynamics.) Theconventional electric polarization P couples linearly tothe external electric field E, and the magnetic polariza-tionM couples linearly to the magnetic field B, however,as we shall show,P3is a pseudo-scalar which couples non-

linearly to the external electromagnetic field combinationEB. For this reason, we coin the term magneto-electricpolarization for P3. To obtainP3(0), one needs to in-troduce the non-Abelian Chern-Simons term:

KA = 1162

ABCDTr

fBC1

3[aB, aC]

aD

, (78)

which is a vector in the four-dimensional parameter spaceq= (kx, ky, kz, 0) and A,B,C,D = x,y,z,.KA satis-fies

AKA = 1322

ABCDtr [fABfCD ]

G3(0) = d3kAKA.When the second Chern number is nonzero, there is anobstruction to the definition of aA, which implies thatKA cannot be a single-valued continuous function inthe whole parameter space. However, in an appropri-ate gauge choice,Ki, i = x, y, z can be single-valued, sothatG3(0) =

d3kK P3(0)/0, with

P3(0) =

d3kK

= 1

162 d3kijkTr fij 1

3[ai, aj ]

ak .(79)

Thus,P3(0) is given by the integral of the non-AbelianChern-Simons 3-form over momentum space. This isanalogous to the charge polarization defined as the in-tegral of the adiabatic connection 1-form over a path inmomentum space.

As is well-known, the three-dimensional integration ofthe Chern-Simons term is only gauge-invariant moduloan integer. Under a gauge transformationai u1aiuiu1iu (u U(M) when M bands are occupied), thechange ofP3 is

P3 = i242

d3kijkTr

u1iu

u1ju

u1ku

,

which is an integer. Thus P3(0), just like P1(), is onlydefined modulo 1, and its change during a variation of0from 0 to 2 is well-defined, and given by C2.

The effective action (74) can be further simplified byintroducingG3 = P3/ . Integration by parts ofS3Dleads to

S3D = 1

4

d3xdtA(P3/)A.

(P3/) can be written as P3, where P3(x, t) =

P3((x, t)) has space-time dependence determined by the field. Such an expression is only meaningful when thespace-time dependence offield is smooth and adiabatic,so that locally can still be considered as a parameter.In summary, the effective action is finally written as

S3D = 1

4

d3xdtP3(x, t)AA. (80)

This effective topological action for the (3 + 1)-d insu-lator is one of the central results of this paper. As weshall see later, many physical consequences can be di-rectly derived from it. It should be emphasized that thiseffective action is well-defined for an arbitrary (3 + 1)-d

insulator Hamiltonianh(k, x, t) in which the dependenceonx, t is adiabatic. We obtained this effective theory bythe dimensional reduction from a (4 + 1)-d system; andwe presented it this way since we believe that this deriva-tion is both elegant and unifying. However, for readerswho are not interested in the relationship to higher di-mensional physics, a self-contained derivation can also becarried out directly in (3 + 1)-d, as we explained earlier,by integrating out the fermions in the presence of theA(x, t) and the (x, t) external fields.


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18

This effective action is known in the field theory lit-erature as axion electrodynamics50,51,52, where the adia-batic fieldP3 plays the role of the axion field53,54. Whenthe P3 field becomes a constant parameter independentof space and time, this effective action is referred to asthe topological term for the vacuum55,56. The axionfield has not yet been experimentally identified, and itremains as a deep mystery in particle physics. Our work

shows that the same physics can occur in a condensedmatter system, where the adiabatic axion field P3(x, t)has a direct physical interpretation and can be accessedand controlled experimentally.

From the discussion above it is clear that 3D TRItopological insulators realize a non-trivial solitonic back-ground field. In Ref. 28 the authors suggest severalcandidate materials which could be 3D topological insu-lators. These 3D materials are topologically non-trivial

because of band inversion mechanism similar to that ofthe HgTe quantum wells20. Ref. 57closely studied thestrained, bulk HgTe. We will keep this system in mindsince it has a simple physical interpretation, and its es-sential physics can be described by the Dirac model pre-sented earlier. We can consider the trivial vacuum out-side the material to have a constant axion field with thevalue = 0 and the interior of a 3D topological in-sulator to have a = background field. The value = does not violate time-reversal (or CP in high-energy language). HgTe is a zero-gap semiconductor andhas no topologically protected features. However, whenstrained, the system develops a bulk insulating gap be-

tween the p-wave light-hole conduction band and thep-wave heavy-hole valence band around the -point.To study the topological features we must also includethe s-wave band which in a conventional material likeGaAs would be a conduction band. Because of the strongspin-orbit coupling in HgTe the band structure is actuallyinverted the s-wave band becomes a valence band. Fora moment we will ignore the heavy-hole band and onlyconsider the light-hole and s-wave band57. The effectiveHamiltonian of these two bands is a massive Dirac Hamil-tonian, but with a negative mass. The negative mass in-dicates a phase shift of in the vacuum angle from itsoriginal unshifted value in the trivial vacuum. The ax-ion domain wall structure at the surface of the topologi-cal insulator traps fermion zero modes which are simplythe topologically protected surface states. If we includethe effects of the heavy-hole band the dispersion of thebulk bands and surface states are quantitatively modi-fied. However, as long as the crystal is strained enoughto maintain the bulk gap the topological phenomena willbe unaffected and the boundary of the 3D topological in-sulator can still be described as an axion domain wall.Thus, this material in condensed matter physics providea direct realization of axion electrodynamics.

B. Physical Consequences of the Effective ActionS3D

In this subsection we present the general physicalconsequences of the effective topological action (80) for(3 + 1)-d insulators coupled to a P3 polarization, and insubsectionIV Cwe focus on its consequences for TRI in-sulators. Since the effective action is quadratic in A,

it describes a linear response to the external electromag-netic fields which depends on the spatial and temporalgradients ofP3. Taking a variation ofS3D[A, ] we obtainthe response equation:

j = 1

2P3A. (81)

The physical consequences Eq. (81) can be understoodby studying the following two cases.

(1) Hall effect induced by spatial gradient of P3.Consider a system in which P3 = P3(z) only depends

on z. For example, this can be realized by the latticeDirac model (73) with = (z). (This type of domain

wall has also been considered in Ref. 58). In this caseEq. (81)becomes

j =zP3

2 A, , , = t ,x, y

which describes a QH effect in the xy plane with theHall conductivity xy = zP3/2, as shown in Fig. 11(a). For a uniform electric field Ex in the x-direction,the Hall current density is jy = (zP3/2)Ex. Thus theintegration over z in a finite range gives the 2D currentdensity in the xy plane:

J2Dy = z2

z1

dzjy = 1

2

z2

z1

dP3Ex.In other words, the net Hall conductance of the regionz1 z z2 is

2Dxy =

z2z1

dP3/2, (82)

which only depends on the change ofP3 in this region,and is not sensitive to any details of the function P3(z).Analogously in the (1 + 1) d case, if we perform thespatial integration of Eq. (25), we obtain the total chargeinduced by the charge polarizationP:

Q= z2

z1dP/2. (83)

By comparing these two equations, we see that the rela-tion betweenP3 and Hall conductance in (3 + 1)-d insu-lators is the same as the relation between charge polar-izationPand the total charge in the (1 + 1)-d case. Asa specific case, a domain wall between two homogeneousmaterials with different P3 will carry Hall conductanceH = P3/2, while the fractional charge carried by adomain wall in (1 + 1)-d is given by Q = P/2.


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19

FIG. 11: (a) Illustration of the Hall effect induced by a spatialgradient of P3. The colors represent different values of P3,which decrease from 0 at the bottom to 1/2 on top. The bluearrow shows the direction of a uniform electric field Ex andthe white arrows show the Hall current density induced, givenby the formula jy = (zP3/2)Ex. (b) Energy spectrum ofthe (3 + 1)-d lattice Dirac model (73) in a magnetic field Bztowards thezdirection. The boundary conditions are periodicin the x and y-directions and open on the z-direction. Thered and black curves show the surface states on the top andbottom surfaces, respectively, each of which is a BzLxLy/2

fold degenerate Landau Level. The parameters of model (73)are chosen to be m = 3, c= 1.

(2) Topological Magneto-electric effect(TME)induced by temporal gradient of P3.

When P3 = P3(t) is spatially uniform, but time-dependent, Eq. (81) becomes

ji = tP32

ijkjAk, i , j, k = x,y, z.

In other words, we have

j = tP32 B. (84)

Since the charge polarization P satisfies j = t P,in a static uniform magnetic field B we have t P =

t

P3 B/2

, so that

P= B

2(P3+ const.) . (85)

Such an equation describes the charge polarization in-duced by a magnetic field, which is a magneto-electriceffect. Compared to similar effects in multiferroicmaterials59,60, the magneto-electric effect obtained hereis of topological origin, and only determined by themagneto-electric polarizationP3.

Similar to the (1 + 1)-d adiabatic pumping effect, theresponse Eq. (84) can be understood in a surface statepicture. For example, consider the lattice Dirac model(73) with periodic boundary conditions in thex, y direc-tions and open boundary conditions in the z-direction.In the presence of a static magnetic field Bz in the z-direction, the single particle energy spectrum En() canbe solved for at a fixed value. As shown in Fig. 11

(b), mid-gap states appear for generic, which are local-ized on the (2+1)-d boundary. It should be noticed thateach state isN-fold degenerate whereN=BzLxLy/2isthe Landau level degeneracy. In the lattice Dirac model,

when4c < m


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20

that any interpolation between two particle-hole sym-metric insulatorsh1(k) andh2(k) carries the same parityof Chern number, so that the relative Chern parity iswell-defined for each two Hamiltonians with particle-holesymmetry. In this section, we will show that the sameapproach can be applied to (3+1)-d insulators, where thetime-reversal symmetry plays the same role as particle-hole symmetry does in (1 + 1)-d.

For a Hamiltonian H =

m,n cmh

mncn, the time-reversal transformation is an anti-unitary operation de-

fined bycm Tcm, where the time-reversalmatrixT satisfies TT = I and TT =I. In k-space time-reversal symmetry requires

Th(k)T=hT(k). (88)The condition TT = I is essential, and leads toKramerss degeneracy. Now we will follow the same ap-proach as Sec. II Dand show how to define a Z2invariantfor the TRI insulators in (3+1)-d. For any two TRI band

insulators h1(k) and h2(k), an interpolation h(k, ) can

be defined, satisfying

h(k, 0) = h1(k), h(k, ) =h2(k)

Th(k, )T=hT(k, ), (89)

and h(k, ) is gapped for any [0, 2]. Since theinterpolation is periodic in , a second Chern numberC2[h(k, )] of the Berry phase gauge field can be defined

in the (k, ) space. In the same way as in Sec. IID,we

will demonstrate below that C2[h(k, )] C2[h(k, )] =

0 mod 2 for any two interpolations h and h. First of all,

two new interpolations g1,2(k, ) can be defined by Eq.(39), which we repeat here for convenience:

g1(k, ) =

h(k, ), [0, ]

h(k, 2 ), [, 2]

g2(k, ) = h(k, 2 ), [0, ]

h(k, ), [, 2] .

By their definition, g1 and g2 satisfy C2[h] C2[h] =C2[g1] + C2[g2] and T

g1(k, )T = gT2(k, ). Todemonstrate C2[g1] = C2[g2], consider an eigenstatek, ;

1ofg1(k, ) with eigenvalue E(k, ). We have

gT2(k, )Tk, ;

1= Tg1(k, )

k, ; 1

= E(k, )Tk, ;

1

g2(

k,

)TTk, ; 1

= E(k, )TTk, ; 1

Thus TT(|k, ; 1) is an eigenstate ofg2(k, ) withthe same eigenvalue E(k, ). Expand over the eigen-

states| k, , 2 ofg2(k, ), we have

TTk, ;

1

=

U(k, )k, ;

2. (90)

Consequently the Berry phase gauge vector of the g1 andg2 systems satisfies

a1j(k, ) = i

k, ;

j k, ; 1

= i,

U

k, ;

j U k, ; 2

=,

Ua2j (k, )(U) i

U(k, )jU(

k, ). (91)

In other words, a1j(k, ) is equal to a2j(k, ) up to

a gauge transformation. Consequently, the Berry phase

curvature satisfies f1ij (k, ) = Uf

2ij (k, )(U) ,

which thus leads to C2[g1(k, )] = C2[g2(

k, )]. Insummary, we have proved C2[h(k, )] C2[h(k, )] =

2C2[g(k, )] = 0 mod 2 for any two symmetric interpola-tionsh and h. Thus the relative second Chern parity

N3[h1(k), h2(k)] = (1)C2[h(k,)]

is well-defined for any two time-reversal invariant (3+1)-d insulators, independent on the choice of interpolation.In the same way as in (1 + 1)-d, a vacuum Hamiltonian

h0(k) h0, k can be defined as a reference. All the

Hamiltonians withN3[h0, h] = 1 are calledZ2 nontriv-ial, while those with N3[h0, h] = 1 are trivial.

Similar to the (1 + 1)-d case, there is a more intu-itive, but less rigorous, way to define the Z2invariantN3.

Through the derivation of Eq. (91) one can see that for aTRI Hamiltonian satisfying Eq. (88), the Berrys phase

gauge potential satisfies ai(k) = U ai(k)U iU iU,so that the magneto-electric polarization P3 satisfies

2P3 = i

242

d3kijkTr

U iU

U jU

U kU Z.

Consequently, there are only two inequivalent, TRI val-ues of P3, which are P3 = 0 a nd P3 = 1/2. Fortwo Hamiltonians h1 and h2, the second Chern number


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21

C2[h(k, )] = 2 (P3[h2] P3[h1]) mod 2, so the differenceofP3 determines the relative Chern parity N3[h1, h2] byN3[h1, h2] = (1)2(P3[h1]P3[h2]). Since the trivial Hamil-tonian h0 obviously has P3 = 0, we know that all theHamiltonians withP3 = 1/2 are topologically non-trivial,while those with P3 = 0 are trivial.

Once theZ2classification is obtained, the physical con-sequences of this topological quantum number can be

studied by the effective theory (80), as has been done inthe last subsection. In the (1 + 1)-d case, we have shownthat a zero-energy localized state exists at each openboundary of aZ2nontrivial particle-hole symmetric insu-lator, which leads to a half charge Qd= e/2( mode) con-fined on the boundary. Similarly, the nontrivial (3 + 1)-d insulators also have topologically protected surfacestates. The easiest way to study the surface physics of the(3 + 1)-d insulator is again by dimensional reduction. Asdiscussed above, for any three-dimensional Hamiltonianh1(k), an interpolation h(k, ) can be defined betweenh1 and the vacuum Hamiltonian h0. If we interpret

as the fourth momentum, h(k, ) defines a (4 + 1)-

d band insulator. Moreover, the constraint Eq. (89)onh(k, ) requires time-reversal symmetry for the corre-

sponding (4 + 1)-d system. The Hamiltonianh(k, ) canbe written in a real space form and then defined on a four-dimensional lattice with open boundary conditions in thez-direction and periodic boundary conditions for all theother directions. As discussed in Sec. IIIB,there will be|C2[h]| flavors of (3 + 1)-d chiral fermions on the surfacewhen the second Chern number C2[h] is nonzero. In otherwords, in the 3D BZ of the surface states there are |C2[h]|nodal points (kxn, kyn, n) , n = 1,.., |C2[h]| where theenergy spectrum En(kx, ky, ) is gapless and disperseslinearly as a Dirac cone. From time-reversal symmetry

it is easy to prove that the energy spectrum is iden-tical for (kx, ky, ) and (kx, ky, ). Consequently,if (kx, ky, ) is a nodal point, so is (kx, ky, ). Inother words, time-reversal symmetry requires the chiralfermions to appear in pairs, except for the ones at time-reversal symmetric points, as shown in Fig. 12(a). Thus,when the second Chern numberC2[h] is odd, there mustbe an odd number of Dirac cones at the 8 symmetricpoints in the 3D BZ. Actually, the (4 + 1)-d lattice Diracmodel (62)provides an example of TRI insulators withnontrivial second Chern number, since one can define 0

to be time-reversal even and 1,2,3,4 to be odd, as in con-ventional relativistic quantum mechanics62. As shown

in Fig. 9, all the nodal points of the surface states arelocated at the symmetric points , M, R orX.

Now we return to the surface of (3 + 1)-d insulator.

Since h1(k) = h(k, 0), h0 = h(k, ) by definition of theinterpolation, the surface energy spectra of h1 and h0are given by the = 0 and = slices of the 3D surfacespectrum. Since all 8 time-reversal symmetric points (,X and M) are at = 0 or = , we know that the netnumber of Dirac cones on the surface energy spectrum of

h1 and h0 is odd (even) when C2[h(k, )] is odd (even).

M

A

A

B

0

+B

kx

ky

X

(3+1)d surface of

(4+1)d system

X

M

B

B

(2+1)-d surface of(3+1)d system

kx

ky

(a) (b)

FIG. 12: Illustration of the nodal points in the surface stateenergy spectrum of (4+1)-d and (3+1)-d insulators. (a) Thenodal points in the (kx, ky , ) BZ, for a z= const. surface ofthe (4 + 1)-d system. The red (blue) points stand for nodalpoints at time-reversal symmetric (asymmetric) wavevectors.The dashed lines are a guide for the eyes. There are twopairs of asymmetric nodal points and three symmetric pointsin this example, which correspond to a bulk second Chern

number C2 = 7. (b) The nodal points in the (kx, ky) BZ,for az= const.surface of the (3 + 1)-d system. According tothe dimensional reduction procedure (see text) the 2D surfaceenergy spectrum is given by the = 0 slice of the 3D surfacespectrum in (a). Since = corresponds to a vacuum Hamil-tonianh0, no nodal points exist in that plane. Consequently,the number of nodal points in the 2D BZ (5 in this example)has the same parity as C2 in the (4 + 1)-d system.

However,h0is defined as the vacuum Hamiltonian, whichis totally local without any hopping between differentsites. Thus, there cannot be any mid-gap surface states

for h0. Consequently, the number of 2D Dirac cones inthe surface state spectrum ofh1must be odd (even) whenC2[h] is odd (even). Since the parity ofC2[h] determinesthe Z2 invariant N3[h1], we finally reach the conclusionthat there must be an odd (even) number of (2 + 1)-d gapless Dirac fermions confined on the surface of a(3 + 1)-d nontrivial (trivial) topological insulator.

Compared to earlier works onZ2invariants and surfacestates in (3+1)-d, one can see that the Z2nontrivial topo-logical insulator defined here corresponds to the strongtopological insulator of Ref. 22. The present approachhas the advantage of (i) demonstrating the bulk-edge re-lationship more explicitly, (ii) clarifying the connectionbetween the second Chern number and theZ2topologicalnumber and (iii) naturally providing the effective theorythat describes the physically measurable topological re-sponse properties of the system. The weak topologicalinsulators defined in Ref. 22 are not included in thepresent approach, since these Z2 invariants actually cor-respond to topological properties of (2 + 1)-d insulators(QSH insulators, as will be discussed in next section), justas the QH effect in (3 + 1)-d systems63 still correspondsto a first Chern number, but defined in a 2D projectionof the 3D BZ.


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D. Physical properties ofZ2-nontrivial insulators

In the last subsection we have defined theZ2 topologi-cal quantum number for the (3+1)-d TRI insulators, anddiscussed the gapless Dirac fermions on the surface of anon-trivial insulator. Now we will study the physical re-sponse properties of the non-trivial insulators. Since thenon-trivial insulator has a magneto-electric polarization

P3 = 1/2 mod 1, according to Eq. (80) the effective ac-tion of the bulk system should be

S3D =2n + 1

8

d3xdtAA. (92)

in whichn = P3 1/2 Zis the integer part ofP3. Un-der time-reversal symmetry, the term AA =2E B is odd, so that for general P3, the effective ac-tion (80) breaks time-reversal symmetry. However, whenthe space-time manifold is closed (i.e., with periodicboundary conditions in the spatial and temporal dimen-sions), the term

d3xdtAA is quantized to

be 82m, m Z. Consequently, S3D = (2n+ 1)m sothat the action e

iS3D= e

im

= (1)m

is time-reversalinvariant and is independent of n, the integer part ofP3. This time-reversal property of the effective action i

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Topological Field Theory of Time-Reversal Invariant Insulators