unified topological response theory for gapped and gapless free fermions

Unified Topological Response Theory For Gapped and Gapless Free Fermions

Daniel Bulmash, Pavan Hosur, Shou-Cheng Zhang, and Xiao-Liang QiDepartment of Physics, Stanford University, Stanford, California 94305-4045, USA

(Received 7 November 2014; revised manuscript received 26 February 2015; published 26 May 2015)

We derive a scheme for systematically characterizing the responses of gapped as well as gapless systemsof free fermions to electromagnetic and strain fields starting from a common parent theory. Using the factthat position operators in the lowest Landau level of a quantum Hall state are canonically conjugate, weconsider a massive Dirac fermion in 2n spatial dimensions under nmutually orthogonal magnetic fields andreinterpret physical space in the resulting zeroth Landau level as phase space in n spatial dimensions. Thebulk topological responses of the parent Dirac fermion, given by a Chern-Simons theory, translate intoquantized insulator responses, while its edge anomalies characterize the response of gapless systems.Moreover, various physically different responses are seen to be related by the interchange of position andmomentum variables. We derive many well-known responses and demonstrate the utility of our theory bypredicting spectral flow along dislocations in Weyl semimetals.

DOI: 10.1103/PhysRevX.5.021018 Subject Areas: Condensed Matter Physics

I. INTRODUCTION

Spurred by the discovery of topological insulators,topological phases have become a vital part of con-densed-matter physics over the last decade [14]. Evenin the absence of interactions, a wide variety of gappedtopological phases of fermions are now known, rangingfrom the quantum Hall [5,6] and the quantum spin Hall[713] insulators (among insulators) to the chiral p-wavesuperconductor [14,15] and the B phase of Helium-3[16,17] (among superconducting phases). All these phasesshare some common features: As long as certain symmetryconditions are upheld, they have a bulk band structure thatcannot be deformed into that of an atomic insulatoratrivial insulator, by definitionwithout closing the bandgap along the way. Moreover, they all have robust surfacestates that mediate unusual transport immune to symmetry-respecting disorder. These features lead one to wonderwhether all gapped phases of free fermions can be unifiedwithin a common mathematical framework.Two different approaches have been developed to

provide unified characterization of gapped phases of freefermions. In the topological band theory approach [1820],homotopy theory and K theory are applied to classify free-fermion Hamiltonians in a given spatial dimension andsymmetry class. The topological band theory provides acomplete topological classification of free-fermion gappedstates in all dimensions and all ten Altland-Zirnbauersymmetry classes [21]. However, it does not directly

describe physical properties of the topological states. Incomparison, the topological response theory approach[2226] describes topological phases by topological termsin their response to external gauge fields and gravitationalfields. The advantage of this approach is that the topologi-cal phases are characterized by physically observabletopological effects, so the robustness of the topologicalphase is explicit and more general than in the topologicalband theory. Since it is insensitive to details of the micro-scopic Hamiltonian, a response-theory-based classificationscheme can be further extended to strongly interactingsystems [27].Recently, the advent of Weyl semimetals (WSMs) has

triggered interest in gapless topological phases of freefermions [2834]. These phases are topological in the sensethat they cannot be gapped out perturbatively as long asmomentum and charge are conserved. In this regard,ordinary metals are also topological since their Fermisurfaces are robust in the absence of instabilities towardsdensity waves or superconductivity. Additionally, gaplesstopological phases may have nontrivial surface states suchas Fermi arcs [28,3537] and flat bands [38]. Teo and Kane[39] applied homotopy arguments to classify topologicaldefects such as vortices and dislocations in gapped phases;Matsuura et al. [40] used an analogous prescription toclassify gapless phases by observing that gapless regions inmomentum space such as Fermi surfaces and Dirac nodescan be viewed as topological defects in momentum space ina gapped system. Thus, a common mathematical formalismto describe the Bloch Hamiltonians of gapless phases wasderived.Unlike their gapped counterparts, however, it is not clear

whether the response theories of gapless phases areamenable to a unified description. For gapped systems,the path from the Hamiltonian to the response theory is

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published articles title, journal citation, and DOI.

PHYSICAL REVIEW X 5, 021018 (2015)

2160-3308=15=5(2)=021018(19) 021018-1 Published by the American Physical Society

conceptually straightforward: The fermions are coupled togauge fields and integrated out to get the low-energyeffective field theory, which describes the topologicalresponse properties. In contrast, the low-energy theory ofgapless phases contains fermions as well as gauge fieldsand is distinct from the response theory which containsonly gauge fields. Thus, it is not obvious how thetopological properties of the Hamiltonian affect theresponse. The response depends on system details, ingeneral, and therefore, recognizing its universal featuresand then unifying the responses of various gapless phases isa nontrivial task. A few cases of topological responseproperties of gapless fermions have been studied. Oneexample is the intrinsic anomalous Hall effect of a two-dimensional Fermi gas [4145]. A generalization of thiseffect in three-dimensional doped topological insulators hasbeen discussed [46]. Another example is the topologicalresponse of Weyl semimetals, which has been described inthe form of the axial anomaly [34,4760]. This refers to theapparent charge conservation violation that occurs for eachWeyl fermion branch in the presence of parallel electric andmagnetic fields, although the net charge of the system muststill be conserved. Recently, these ideas were generalized tofind the topological responses of point Fermi surfaces inarbitrary dimensions [61]. The dc conductivity of metalshas also been proposed to be related to a phase-spacetopological quantity [62]. However, a general theory thatdescribes the topological properties of gapless fermions in aunified framework has not been developed yet.In this work, we achieve the above goals for free

fermions: We show that gapless systems have universalfeatures, independent of system details, and derive a unifieddescription of their response. Remarkably, this descriptionalso captures the response of gapped systems. In particular,the response of gapped phases arises from the bulkresponse of a certain parent topological phase, while theuniversal features of the gapless phases correspond to itsedge anomalies. We elucidate this idea below.The backbone of our construction is a mapping from

n-dimensional gapped or gapless systems to a gappedquantum Hall (QH) system that lives in 2n-dimensionalphase space. Such a phase-space system has both bulkresponses, given by a 2n-dimensional Chern-Simons (CS)theory, and boundary (axial) anomalies. We identify thebulk responses with topological responses of insulators.However, the key insight that allows us to include gaplesssystems is to identify a Fermi surface in real space with aphase-space boundary in the momentum directions.Likewise, real-space excitations near the Fermi surfaceare identified with the gapless edge excitations in phasespace. The universal features of the response of gaplesssystems are thus the anomalies associated with thesephase-space edges.There is an important technical point required in order

to bestow the 2n-dimensional QH system with the

interpretation of phase space. Specifically, we must choosethe QH system to consist of a massive Dirac fermion undern uniform magnetic fields of strength B0 in n orthogonalplanes, and then project to the zeroth Landau level (ZLL) ofthe total field. In this case, the projected operators for pairsof dimensions acquire the usual canonical commutationrelations that relate ordinary real and momentum space (upto an overall factor of B0). This result allows us to interpretthe ZLL of the 2n-dimensional QH system as phase spacefor the n-dimensional physical space. We interpret addi-tional perturbations in the phase-space gauge fields asphysical quantities such as the n-dimensional systemselectromagnetic (EM) field, strain field, Berry curvatures,and Hamiltonian. Topological defects, such as monopoles,in the phase-space gauge fields allow us to generalize tosystems with dislocations and with point Fermi surfacessuch as graphene and Weyl semimetals. These ideas aresummarized in Table I.This construction enables us to systematically enumerate

all possible intrinsic, topological responses to electromag-netic and strain fields in the dc limit in any givendimension. One simply has to write down the Chern-Simons action in phase space, vary it with respect to eachgauge field, and consider each boundary to obtain all thebulk, boundary, and gapless responses in real space.Following this procedure, we show carefully that screwdislocations in Weyl semimetals trap chiral modes whichare well localized around the dislocation at momentumvalues away from the Weyl nodes. A related but differenteffect has been studied previously [63]. However, ourframework provides a unified and natural description ofthis effect and other topological effects.It is crucial that the 2n-dimensional system be gapped

even in the absence of the background magnetic fields ofstrength B0. This ensures that its response theory containsterms depending on B0 in addition to fluctuations in thegauge fields. In n dimensions, we will see that theB0-dependent terms translate into quasi-lower-dimensionalresponses, such as the polarization of a system of coupledchains as measured along the chains. If the 2n-dimensional

TABLE I. Dictionary for interpreting phase-space quantities inreal space.

Phase space Real space

Bulk responses Quantized insulator responsesAnomalies from momentumdirection edges

Gapless response

Anomalies from realdirection edges

Real edge anomalies

Gauge-field strength EM-field strength/k-spaceBerry curvature/strain

Monopole in gauge field Magnetic monopole/Weylnode/dislocation

BULMASH et al. PHYS. REV. X 5, 021018 (2015)

021018-2

system is gapless in the absence of the background fields,such responses will be missed by the unified theory.A caveat is that our construction does not capture

responses to spatial, momental, and temporal variationsin the field strengths, such as the gyrotropic effect which isan electric response to a spatially varying electric field.Note that the regular Maxwell response, given byj F, is a response to a variation in the field strength.Another caveat is that in phase-space dimensions equal to4 and above, the Maxwell term in the action is equallyrelevant to or more relevant than the Chern-Simons termand hence will, in general, dominate the dc response.However, our central objective is to demonstrate that thereexists a theory that unifies the responses of gapped andgapless systems, namely, the phase-space Chern-Simonstheory.The rest of this paper is structured as follows. In Sec. II,

we review the key property of the ZLL, which provides thephysical justification for our construction. In Sec. III, weexplain the interpretation of the gauge fields in our mappingand give an example illustrating the validity of the CStheory. In Sec. IV, we write down an explicit model with aCS response theory and show the precise way in which itbehaves as the phase-space response theory of a lower-dimensional model. In Secs. V and VI, we explain theresponses and anomalies (respectively) that come from theCS theory in various dimensions, applying our frameworkto describe spectral flow in Weyl semimetals with dis-locations. Finally, in Sec. VII, we summarize our workand suggest extensions of our theory to more nontrivialsystems.

II. REVIEW OF THE ALGEBRA OFTHE ZEROTH LANDAU LEVEL

One of the key features that we use in the intuition forour approach is the fact that projecting position operators tothe ZLL yields nonzero commutators between those oper-ators. We now review this fact, for concreteness as well asfor later convenience, for the case of Dirac electrons in auniform magnetic field in two spatial dimensions in Landaugauge. Although we consider the ZLL of Dirac electronshere, the noncommutativity of position operators is simplya consequence of minimal coupling and Landau quantiza-tion of cyclotron orbits and hence is true for other dis-persions as well as for other Landau levels for a Diracdispersion.Consider a 2D massive Dirac Hamiltonian in a uniform

magnetic field,

H px eByx pyy mz: 1Here, i are the Pauli matrices. We have set the Fermivelocity to unity, written the electron charge as e,and chosen the Landau gauge A Byx^ with B > 0for definiteness. Note that px commutes with the

Hamiltonian, so we may replace it by its eigenvalue. Wecan define an annihilation operator a px eBy ipy=eB

p, which has a; a 1, and the Hamiltonian becomes

H

m2eB

pa

2eBp

a m: 2

It is straightforward to show that the eigenstates are labeledby an eigenvalue n 0 of the number operator aa, withdispersion

2eBnm2

pfor n 0. For n 0, the eigen-

value is m, the spin state is0

1

, and the state is

annihilated by a. As expected, the kinetic energy isquenched and the spectrum becomes discrete, highlydegenerate Landau levels.Let jkxi be the state in the ZLL with px eigenvalue kx.

Then, the projection operator to the ZLL is

P Z

dkxL2

jkxihkxj; 3

with L the system length in the x direction. Writingypx=

eB

p aa= eBp , the projected y operatorbecomes

PyP Z

dkxL2

kxeB

jkxihkxj; 4

where we have used the fact that a and a describe inter-Landau-level processes and thus vanish under projectiononto the ZLL. Next, using the fact that jkxi is an eigenstateof px, we find

PxP Z

dkxL2

ikx jkxihkxj: 5

The commutator can then be easily computed as

PxP;PyP ieB

: 6

Hence, if we absorb the factor of eB into y, then PxP andPyP have the correct commutator structure for us to imbuethem with the interpretation of the position and momentumoperators, respectively, of a 1D system. This interpretationis the primary physical motivation for the construction thatfollows. As mentioned earlier, other dispersions will alsoresult in commutation relations similar to Eq. (6) and thusimbue x and y with interpretations of position andmomentum of a 1D system. However, a massive Diracdispersion is ideal for deriving the unified response theorybecause it does not miss any quasi-lower-dimensionalresponses, as mentioned earlier and detailed later.In higher dimensions, the Dirac model is

H Pipi eAii, where the i are anticommuting

UNIFIED TOPOLOGICAL RESPONSE THEORY FOR PHYS. REV. X 5, 021018 (2015)

021018-3

elements of the Clifford algebra of 2n 2n matrices. If weapply constant magnetic fields Fij for disjoint pairs i; j ofcoordinates, we can form an annihilation operator for eachsuch pair. Annihilation operators from different pairscommute, and the analysis above carries through so thatthe position operators within each pair no longer commuteafter projection.

III. PHASE-SPACE CHERN-SIMONS THEORY

The key idea of our construction is to represent a(possibly gapless) n-dimensional system by a gapped2n-dimensional phase-space system, specifically a massiveDirac model coupled to a gauge field. As we just showed,we can interpret a 2n-dimensional system as living in phasespace by adding background magnetic fields betweendisjoint pairs of spatial directions and projecting to theZLL. Moreover, since the phase-space system is gapped,we can immediately write down a response theory for it, thetopological part of which can be proved to be a CS theory[22,6466]. Note that in a CS theory, real- and momentum-space gauge fields enter the action in similar ways,analogous to our idea of treating position and momentumon the same footing in phase space.Before proceeding, we fix some notation and conven-

tions. We will always use the Einstein summation con-vention where repeated indices are summed. Phase-spacecoordinates will be labeled by x; y; z and x; y; z. Afterprojection, x; y; z will be interpreted as the correspondingreal-space coordinates, while x; y; z will be interpreted asmomentum-space coordinates kx; ky; kz, respectively. Inphase space, we will refer to the U1 background gaugefield that generates the real-space commutator structure asA, with its nonzero field strengths being Fii B0 fori x; y; z. We denote all other contributions to the gaugefield by a and the total gauge field by A A a.Likewise, we write f and F F f for thenonbackground and total field strengths, respectively.The (non-Abelian) field strengths are, as usual, definedby f a a a; a.We will also abbreviate the CS Lagrangian by

aa aa , where is the totally antisymmetricLevi-Civita tensor, with an analogous abbreviation forhigher-dimensional CS terms. Finally, we set e 1and also assume that

B0

p 1=lB is very large compared to

all other wave numbers in the problem.

A. Interpretation of phase-space gauge fields

Our prescription is that the nonbackground contribu-tions a to the phase-space gauge field should be inter-preted as the Berry connection for the lower-dimensionalsystem:

a ihukjjuki; 7

where juki is the (local) Bloch wave function at momen-tum k for the band. Here, x;y;z should be interpreted asB0kx;ky;kz . We can think of the physical EM vectorpotential as a Berry connection, which means that it isincluded in the real-space components of a.As such, we will often use the following heuristic

interpretations in order to more clearly see the physics:at is the lower-dimensional band Hamiltonian plus thephysical EM scalar potential, ax;y;z is the physical EMvector potential, and ax;y;z is the momentum-space Berryconnection. The field strengths that do not mix real andmomentum space then have natural interpretations as thephysical EM field strengths and Berry curvatures.The physical interpretation of mixed field strengths

such as fxy (in four- or higher-dimensional phase space) isless obvious. Here, we present two ways to think aboutthem. First, consider a gauge where yax 0. We find that

Zdyfxy x

Zdyay 2xPy; 8

where Py is the one-dimensional polarization of the system[67]. A spatially varying polarization can be thought of asstrain of the electron wave function, which can come fromeither mechanical strain or some other spatial variation ofthe parameters entering the band structure.To make the connection of fxy to mechanical strain more

explicit, we change the gauge to set xay 0. An intuitiveway to think about a nonzero fxy in this gauge is in terms ofdislocations. In particular, adiabatically moving a particlearound a real-space dislocation leads to a translation, but ifthe particle can locally be treated as a Bloch wave, then thattranslation is equivalent to the accumulation of a phase.This (Berry) phase is equal to k b, with b the Burgersvector of the dislocation. In particular, this is a momentum-dependent Berry phase resulting from adiabatic motion inreal space. Hence, fxy is nonzero. It can be shown explicitly[44] in the perturbative regime that strain typically leads tosuch a Berry phase.

B. Example: 2D phase space

We first consider the case where our real-space theoryconsists of a single filled band living in 1D and that a 2DCS response term in phase space with a background fielddescribes the expected responses. We will, for simplicity,only consider Abelian physics in this example. Considerthe CS action

SCS 1

4

Zdtd2xCx; xAA: 9

(In Sec. IV, we show, in an explicit model, how Eq. (9),with this (quantized) coefficient, appears, but for now wesimply assume that it is the relevant response theory.) Here,


021018-4

Cx; x accounts for the filling at different points; forexample, if the system occupies x > 0, then Cx; x will beproportional tox, with the Heaviside step function, asshown in Fig. 1(a). Likewise, if the system has a Fermimomentum kF, then Cx; x will be proportional tox kF=B0 x kF=B0, as shown in Fig. 1(b).Let us assume that there are no edges so that C 1

everywhere. Then, the responses for this action, given byj S=A, are

j2D 1

4F ; 10

where F A A is the field-strength tensorcorresponding to A. Let us consider each component,assuming for conciseness that the background field is inLandau gauge Ax B0x.First, we examine the real-space response jx2D F xt=2.

This current, in general, depends on x, which we interpretas kx=B0; the observable current should then be given byintegrating the 2D current with respect to x, as the real-space current has contributions from all occupied momenta.The resulting 1D current is

jx1D 1

2

Zdx xat t

Zdxax

: 11

Interpreting the x-dependent part of at as the dispersion, thefirst integral generically gives zero. The second integral is,for a gapped system, exactly the time derivative of thepolarization Px 1=2

Raxdx, which is the expected 1D

real-space current response jx1D tPx. Similarly, thek-space response is jx2D F tx=2. Interpreting jx2D asdk=dt, we recover the real-space semiclassical equationof motion dk=dt E=2, with E the electric field.Finally, the charge response is given by

1D 1

2

ZdxFxx xax xax: 12

In units B0 1, the first term simply gives the total chargein the occupied band, which can be thought of as a quasi-0D response.

The second term of Eq. (12), in a gauge where ax 0,becomes xPx for a gapped system. This is again intuitive;if, say, the system is strained, then the polarization andhence the charge density will change accordingly.If we now impose a pair of edges at x kF, two things

happen. First, the 1D system lies between a pair ofmomentum points kF, so the integrals in Eqs. (11)and (12) run from kF to kF instead of the full Brillouinzone. Consequently, the background charge becomesbg1D 1=2

R kFkF Fxx kF=, as expected, while theterms proportional to fxx cease to have a simple interpre-tation as the polarization but can be nonzero nonetheless.Second, the 2D system develops a chiral anomaly at theedge, given by

t2D xjx2D 12F tx xfx E x

2 xCx; 13

where Cx x kF x kF. Integrating over1D real space under a constant electric field and transla-tional invariance yields

tZ

dx2D L2

x kF x kFE; 14

where L is the length of the system and is the Dirac deltafunction. This is precisely the chiral anomaly in the 1Dsystem: The electric field tilts the 1D Fermi surface,effectively converting right-moving charge in the vicinityof one Fermi point into left-moving charge near the other.Thus, we have derived a property of a gapless 1D bandstructure from the edge anomaly of the parent 2D QHsystem.Notice also that integration of (13) over momentum

space leads to

t1D xjx1D 0; 15which correctly tells us that there is no anomaly in the totalcharge. The precise value of 1D and jx1D depends on systemdetails; therefore, calculating them in our formalism wouldrequire knowledge of nonuniversal properties of the 2D QHedge, such as the velocity of the chiral modes. However, wehave shown here that they still have universal propertiesthat reflect the universal properties of a higher-dimensionaltopological state.A different type of anomaly occurs when the system has

real-space edges and a filled band. In this case, the anomalyequation in 2D is

t2D xjx2D 12F txxCx: 16

Integrating the above in x yields

t1d x x0 x x0tPx: 17FIG. 1. Phase-space realization of (a) real-space edges and(b) Fermi points for a 1D real-space system. Arrows indicate thedirection of the edge modes.


021018-5

This is the known result [68] that charge can be adiabati-cally pumped from one edge of the system to the other via atime-dependent local polarization.We thus see that the standard responses, including

anomalies, that we expect in a 1D theory are retrievedfrom the 2D CS theory. However, detail-dependent edgeresponses are described in our theory only by the anomaly(or lack thereof) that they create. We expect the sameprocedure to generalize to higher dimensions, and we willshow that the expected topological responses appearin Sec. V.

IV. EXPLICIT MODEL

In this section, we elucidate the precise way in which aZLL behaves as the phase space of a system in half thenumber of spatial dimensions. In particular, we explainwhy the response theory of the lower-dimensional systemshould be of CS form in phase space and describe thephysical meaning of projecting onto the ZLL. We alsoanswer the question of when a CS theory in 2nD can beinterpreted as a phase-space response theory in nD.To begin, consider a massive Dirac Hamiltonian in 2n

dimensions coupled to the gauge fieldA defined in Sec. III:

H2nD Xni1

ipi Ai ipi Ai 0M: 18

A corresponds to large constant background fields B0 in northogonal planes plus small fluctuations; thus, Fii B0 fij; fij; fi j for all i; j 0;; n. The s are2n 2n anticommuting matrices with eigenvalues 1,and they satisfy 0

Qni1 ii. To zeroth order in f,

the spectrum of H2nD can be easily derived by generalizingthe calculation of Sec. II; it consists of Landau levels withenergies

2kBM2

pfor positive integers k together with

a ZLL state that has energy M and a spinor wave functionthat has a 0 eigenvalue of 1.We have two tasks. First, we must isolate the topological

response theory of the ZLL of this system, which we expectto be a Chern-Simons theory. Second, we must relate thisHamiltonian, projected onto the ZLL of the total field, tothe Hamiltonian of the real-space system.For the first task, note that the ZLL is occupied

(unoccupied) in the ground state if M > 0 (M < 0), whilethe occupation of all the other Landau levels is independentof the sign of M. This should hold for nonzero f as well ifM

B0

p. As a result, the response of the ZLL to A is

given by the terms in the total response that are odd in M.Moreover, it is known that the two signs ofM correspond toa topological and a trivial insulator (which sign correspondsto which phase is determined by the regularization far awayfrom the Dirac point). Therefore, the difference betweentheir response theories, which by definition is the topo-logical part of the effective action, equals the response of

the ZLL. In the absence of vertex corrections, this is knownto be the 2n-dimensional CS action with coefficient 1 tolowest order in the coupling constant e. In short, theresponse of the ZLL is precisely the CS action withcoefficient 1 in appropriate dimensions under suitablewell-controlled perturbative approximations. We empha-size that this statement is true even if H2nD is modified athigh energies to change the total (nth) Chern numbers ofthe occupied and unoccupied bands. The only requirementis that the Chern numbers of the M > 0 and M < 0 casesdiffer by unity; their actual values are irrelevant fordetermining the ZLL response.Next, we recall that xi; xi i in the ZLL as shown in

Sec. II, so xi and xi can be thought of as a pair ofcanonically conjugate position and momentum variables.Therefore, projecting H2nD onto the ZLL gives ann-dimensional system whose response theory is guaranteedto be of CS form in phase space. In this response theory, thegauge fields in the momentum directions are to bereinterpreted as momentum-space Berry connections.This flow of logic is depicted in Fig. 2 (where we haverenamed H2nD as HMDirac to make the figure self-contained).Having shown that the response of the n-dimensional

system is given by the phase-space CS theory, we turn toour second task and show in detail how the Hamiltonianin n dimensions is related to H2nD. For clarity, wechoose n 1; i.e., we demonstrate this in 1D real spacewith a U1 gauge field. The procedure generalizes

FIG. 2. Logical flow of the derivation in Sec. IV. An nDHamiltonian hnD can be obtained by projecting a 2nD massiveDirac Hamiltonian HMDirac in a magnetic field onto the ZLL,denoted by the thick red bar. For a Fermi level in the Dirac massgap, the M > 0 and M < 0 ground states differ only in theoccupation of the ZLL, while their response theories SMDirac differby the CS term in 2nD. Thus, the response theory of the ZLLSZLL2nD , which is the real-space response theory S

realnD of hnD, is the

phase-space CS theory SphaseCS .


021018-6

straightforwardly to more dimensions and to larger gaugegroups. To construct the 2D phase-space model, let x, x,and 0 be the Pauli matrices x, y, and z. [The notationis for consistency with the higher-dimensional generaliza-tion in Eq. (18).] The appropriate 2D Hamiltonian is

H2D px Axx px Axx M0 A0: 19We are projecting onto the ZLL of the total fieldA, so we

need to make some approximations to make progress. Weassume that the field fluctuations f are much smaller thanB0. In other words, we identify 1=

B0

pwith some micro-

scopic length scale like a lattice constant for the underlyingreal-space system, and assume that all the gauge-fieldfluctuations are small over that length scale. If this is true,then we can make the gauge choice that a B0 for all; . In this case, the Hamiltonian of the ZLL of A can becomputed by considering a to be a perturbation on theHamiltonian with A A. We implement degenerateperturbation theory as follows.Let us write

H2D H0 H0 20with

H0 px Axx px Axx M0 A0; 21

H0 axx axx a0: 22

Since A is a constant background field of strength B0, weknow how to diagonalize H0; let jn; ki be the eigenstates,where n labels the LL and k labels a momentum (in Landaugauge). Using hxj0; ki eB0xk=B02=21; 0, where thespinor indicates that the ZLL states are polarized in thebasis of 0 eigenstates, and denoting k k q=2, first-order degenerate perturbation theory gives an effective 1DHamiltonian as

h0;kjH0j0;kiZdxdxeiqxa0x;xeB0xk=B02eq2=4B0

a0i0q;k=B0h1dka0ik;k=B0: 23

Thus, the desired 1D Hamiltonian h1dx; k can easily beobtained by choosing a0x; x h1dx; B0x. Since theZLL is spin polarized, the dependence on ax and axdisappears from Eq. (23); these fields only appear atsecond order in perturbation theory. Degenerate perturba-tion theory tells us that, if P is the projector onto thedegenerate subspace, then the second-order correction tothe energy is given by the eigenvalues of

h ijH2Dj ji h0; kijH0 H0 H0 PH0PH0 E01H0 PH0Pj0; kji h0; kijH2D H2j0; kji; 24where

j ii j0; nii Xn>0;l

jn; klihn; kljH0 E01H0 PHPjn 0; kii 25

is a basis for the perturbed ZLL wave functions up to firstorder in H0. In particular, the unitary transformation U thattakes H2D to H2D H2, to second order in H0, is the onethat takes j0; kii to a state living in the ZLL of the fullHamiltonian, to first order in H0.Therefore, if we find this unitary transformation and then

perform the projection in the ZLL of the backgroundfield, we still get our desired projected Hamiltonian.We write U expiS, with S Hermitian, and expandS S1 S2 . where the subscripts indicate an expan-sion in orders ofH0 (by inspection, S can be chosen to haveno zeroth-order term). Then, we can match, order by order,terms in eiSH2DeiS with those in H2D H2 to find theconditions

H0; S1 0; 26

H0; S2 iH2: 27

We do not claim to be able to demonstrate explicitly aunitary transformation that obeys the second of theseconditions, as computing H2 is highly nontrivial. How-ever, we will proceed first by exhibiting an ansatz for U,then showing that the projection onto the ZLL of A yieldsthe correct Hamiltonian in real space, and finally giving thephysical motivation for the ansatz.Let us start in the gauge Ax B0x, Ax At 0.

Then, let

U eiaxpxds=B0NeiB0xxeiaxpxds=B0NeiB0xx; 28where ds is an infinitesimal parameter and N suchthat Nds 1. This transformation is a gauge transforma-tion, followed by a translation of x by ax, followed by thereverse gauge transformation, followed by a translation of xby ax.By inspection, U commutes with H0, so Eq. (26) is

satisfied. To second order in H0, we now have


021018-7

UH2DU H0 axx ax

B0; x ax

B0

x ax

x ax

B0; x ax

B0

x a0

x ax

B0; x ax

B0

; 29

where ai appearing without explicit functional dependencemeans aix; x. The terms that we have neglected aredouble-nestings of a=B0; our aforementioned approxi-mation that a is slowly varying (which was a gauge choicepossible when the corresponding field strengths were weak)allows us to write

a0

x ax

B0; x axx

axB0; x

B0

a0

x ax

B0; x ax

B0

:

30

Let us now perform the projection on the ZLL of thebackground field. As before, everything projects to zeroexcept for the a0 term and the mass term of H0. The latterjust projects to a constant, which we can absorb by a shift ofa0. However, we now obtain a different 1D Hamiltonianh01dx; x a0x ax; x ax, which is simply h1dx; xwith minimal coupling to the gauge fields ax and ax,respectively (B0 is set to unity for convenience). Wetherefore have correctly retrieved the full 1DHamiltonian from a projection to the ZLL, as the functionalform of the projected Hamiltonian is correct if we imbue xand x with the interpretations of a parameter trackinga locally periodic Hamiltonian in space and Blochmomentum, respectively.A major question remains: Why, physically, should this

choice of U be the correct one? First of all, the projectedHamiltonian, if it is to describe a real system, must be gaugeinvariant. Hence, the gauge fields should be minimallycoupled, and U indeed accomplishes this goal.A more fundamental reason, though, is the following.

Consider H2D in some local region over which a isapproximately constant, and for convenience, choose agauge in which ax is zero. In this region, ax functions as aconstant shift of the momentum px which dictates, in theZLL of A, the wave-function center in x. Hence, we should,roughly speaking, identify the (local) eigenvalue of px axwith x. In the original basis, then, the variable canonicallyconjugate to x is identified in the ZLL with x ax. If weare to interpret the commutator of the projected x and xoperators in phase space as being the canonical commu-tation relation of x and p in real space, then we need to shiftx by ax in order to do so. By a similar argument in thegauge where ax 0, we should shift x by ax to identify xwith px in the ZLL.Having derived the real-space Hamiltonian from an

ansatz for the solution to the phase-space one, we nowcomment on a few details.

First, notice that this derivation generalizes easily tohigher dimensions, as the background field only couples xto x, y to y, etc. The primary difference is that in2n-dimensional phase space, the matrices must beanticommuting elements of the Clifford algebra of2n 2n matrices with diagi 0 for i 0.We next comment on gauge invariance. It may appear

that there is extra gauge invariance in the phase-spacetheory; in particular, it may seem strange that the Berryconnections ax can be gauge transformed into real-spacegauge fields ax and vice versa. We claim that this is simplya reflection of the usual gauge invariance in the lower-dimensional Hamiltonian. To see this, consider a unitaryoperator U expifx; x which implements the gaugetransformation a a f, and let the ZLL wavefunctions be jni for some set of labels n. Since U is agauge transformation in the phase-space system, it mustcommute with the projection operator P (as U must takestates in a given LL to the same LL). Hence, we can projectU to get its action on the projected Hamiltonian; by thesame argument we used for projecting the Hamiltonian, wemust have PUP expifx; k. To understand the mean-ing of this operator, recall that, locally, any state can belabeled as a Bloch wave function jk; xi at momentum k fora local Hamiltonian at x. Therefore, a gauge transformationin the higher-dimensional system is equivalent to a spatiallydependent U1 gauge transformation on the eigenstatesjk; xi of the local Hamiltonian parametrized by x.Finally, after seeing the derivation, we may answer the

following question: When can a CS theory in 2nD beinterpreted as the phase-space response theory of a systemin nD? The key physical requirement in our derivation wasthat the total field in the 2nD system could be separatedinto two parts: a uniform background field, which setssome length scale, and another portion which varies slowlyon that length scale. When this condition holds, the CStheory may be interpreted as a phase-space theory for somelower-dimensional system.

V. ENUMERATION OF BULK RESPONSES

Having shown that the phase-space CS theory is thecorrect unified theory, we now systematically enumerate allthe bulk responses of the CS theory for each possibledimensionality of phase space, and interpret them in realspace. To avoid cluttering the notation, we set B0 1.The 2D responses were discussed in Sec. III B. There, we

showed that the real-space current density is the rate ofchange of polarization, while the k-space current densityreflects the expected relation dk=dt E, with E the electric


021018-8

field. The charge-density response is just the band fillingcorrected for strain-induced changes in the lattice constant.We summarize the 2D responses in Table II.

A. 4D phase space

The action is given by

S 1242

Zdtd4xtrCx; xAAA ; 31

where indicates the non-Abelian terms. We set C 1uniformly to look at bulk responses.Spatial components.Without further information

about the system under consideration there is no differencebetween the spatial directions x and y, so we focus on the xresponses,

jx2D 1

42

Zd2xtrF yyF tx F x yF ty F yxF yt: 32

The first term includes the background field;setting F yy Fyy B0 turns this into jx2D 1=2 R dxtrF tx tPx, with Px the polarization inthe x direction. This is the same response that appears in1D; and is illustrated in Fig. 3(a). The second term is theanomalous Hall response; in the simple case where F ty issimply an electric field, this term gives a currentjx Ey

Rd2xtrF x y=42 EyC1=2, with C1 the first

Chern number of the occupied bands. This formula also

applies to systems with open boundaries in the x; ydirections, in which case C1 is not quantized but stilldetermines the intrinsic Hall conductivity of the two-dimensional Fermi liquid [4145].The third term, illustrated in Fig. 3(b), says the follow-

ing. Suppose that there is a change in time of thepolarization in the y direction, i.e. F yt 0, without anystrain in the system. If we now add shear in the system, i.e.,have F yx 0, then some of that polarization changebecomes a current along the x direction as defined beforeadding strain.k space components.The x-direction responses are

jx4D 1

42trF txF yy F tyF yx F tyF xy: 33

The first term is quasi-1D, which means that dkx=dt isproportional to the electric field Ex. The second term saysthat an electric field Ey leads to a change in kx if there isshear in the system. The third term is, semiclassically, theLorentz forcechanging the polarization in the y direction(F ty 0) leads to a 1D current in the y direction, whichthen feels the Lorentz force of the magnetic field (F xy 0),causing kx to change.Charge component.The charge responses are

jt2D 1

42

Zd2xtrF xyF y x F xxF yy F xyF yx: 34

The first term is the Hall response for a Chern insulator.Specifically, if Fxy is just the magnetic field, this termbecomes jt Bz=42

Rd2xFy x C1Bz=2, where C1 is

the first Chern number.Consider the remaining terms

jt2D 1

42

Zd2xtrF xxF yy F xyF yx: 35

In the simplest case of a single featureless, flat band,fij iuj, where u is the displacement vector. Thisproportionality occurs because infinitesimal motion dxiin the i direction leads to a translation dxj iujdxi in thej direction, which is, for Bloch wave functions, the same asaccumulating a j-dependent phase B0jdxj. Hence, ai jiuj with our convention of B0 1.In this simple case, then, Eq. (35) becomes

jt2D 1

42

Zd2x1 xux1 yuy xuyyux:

36

The expression inside the parentheses is the determinant ofthe deformation gradient, that is, the area of the strainedunit cell in units of the original unit-cell area. Hence, thenonbackground terms are just due to the change in the area

TABLE II. Summary of 2D phase-space responses.

Current component Response

Real space Change in polarizationk space Electric forceCharge density Band filling

FIG. 3. Cartoon of the polarization response of an (a) unstrainedsquare lattice and (b) sheared square lattice. In (b), Fyx 0because motion along the y lattice vector leads to translation inthe unstrained x direction. The directed lines are the flow ofcharge due to a positive tPy; the polarization is measured alongthe lattice vector, which is deformed in (b) because of strain. Onlycase (b) has a nonzero current in the x direction, which is due tothis deformation.


021018-9

of the unit cell. Adding features to the bands will lead tocorrections due to, for example, strain changing the localdensity of states.We summarize the 4D responses in Table III.

B. 6D phase space

The action is given by

S 11923

Zdtd6xtrCx; xAAAA 37

with again representing the terms for a non-Abeliangauge field. For simplicity of exposition and interpretation,we assume that the momentum-space and time componentsof A are UN and the real-space components are U1;that is, the latter couple to all the bands in the same way.This assumption is not necessary for our theory, however.We again assume that C 1 uniformly to look at bulkresponses.There are 15 different responses in each component. If

we separate F xx into its background and nonbackgroundcomponents, for the spatial and momentum components weget an extra 7 terms for a total of 22. For the chargecomponent, there are 28. We sort them, neglecting relativeminus signs between the groups.Spatial components.Quasi-1D response (5 terms):

jx3D 1

83

Zd3xtrF txFyyfyyFyy fyyF yzF zy:

38By the same computation that was done for the chargeresponse in 4D, F yyF zz F yzF zy is the change in areaperpendicular to the current. This response thus has theform of the 1D real-space response (time-varying polari-zation) times the change in area perpendicular to thecurrent.Layered Chern insulator response (2 terms):

jx3D 1

83

Zd3xtrF tyF x yFzz y z; 39

where y z means to switch y and z as well as y and z.This is the Hall response corresponding to thinking of the

3D system as 2D systems layered in momentum space.Note that this includes the Hall response of a Weylsemimetal (WSM) [34,52,53,55,69] appearing from itsmonopoles of F x y. This can be seen by thinking of the(two-node) WSM as stacks of 2D insulators parametrizedby the momentum direction along which the Weyl nodesare split; as shown in Fig. 4, each insulator lying betweenthe nodes is a Chern insulator and thus contributes to F x y(for kz-direction Weyl node splitting). In this special case,integration yields

jx3D 1

42Eykz Ezky; 40

where ki is the splitting of the Weyl points in the kidirection.Topological magnetoelectric effect (3 terms):

jx3D 1

83

Zd3xtrF xtF y z F x yF tz F x zF tyF yz:

41Assuming that F yz does not depend on momentum forsimplicity, this term is an x-direction current proportional toBx. Indeed, if we assume that the real-space system isgapped so that there are no monopoles of Berry curvature,simple but tedious manipulations (see the Appendix) turnEq. (41) into

jx3D 1163 tZ

d3xIJKtr

aIJaK 2

3aIaJaK

Bx

12

tP3Bx; 42

TABLE III. Summary of responses from 4D phase space.

Current density component Response

Real space Change in polarizationHall response

Change in polarization with straink space Electric force

Sheared response to electric fieldLorentz force

Charge density Hall responseChange in unit-cell area due to strain

FIG. 4. Slab of WSM with Weyl nodes separated along x. Eachslice in momentum space with fixed x can be characterized by aChern number C1x, which changes by unity across the Weylnodes. Thus, the region between the nodes in the above figure is aseries of Chern insulators. The edge states of these Cherninsulators constitute the Fermi arcs, marked as thick red lineswith an irregular shape. Note that the cones are only present as acartoon to depict the position of the Weyl nodes; the verticaldirection in the figure is z and should not be confused withenergy. The figure is adapted from Ref. [34].


021018-10

where I; J; K run over x; y; z and P3 is the three-dimensional analog of charge polarization. Equation (42)is precisely the contribution of the topological magneto-electric effect to jx [22].Topological insulator (TI)-type anomalous Hall response

(6 terms):

jx3D 1

83

Zd3xtrF tzF x yF yz F y zF xy

F tyF x yfzz y z: 43

If we choose a gauge such that the real-space Berryconnections do not depend on momentum, then by similarlogic to the topological magnetoelectric effect terms (andwith similar assumptions), we can manipulate this contri-bution into the form

jx3D 12 Eyy EzzP3: 44

Here, E is the real-space electric field. This is theanomalous Hall effect that appears in a 3D TI [22]. Itdiffers from the Hall effect that appears as a quasi-2Dresponse in that it does not originate from having a nonzerototal Chern number at each 2D slice of momentum space.Sheared polarization responses (6 terms):

jx3D 1

83

Zd3xtrF tyF xzF zy F xyFzz fzz

y z: 45

The first term here corresponds to a current flowing in y dueto a changing polarization, but that current is redirected intothe z and then the x direction by shear. The second term isthe same current in y being redirected into the x directiontogether with a change in the perpendicular area due touniaxial strain. These are the 3D real-space analogs of the2D real-space response illustrated in Fig. 3(b).k-space components.Quasi-1D response (5 terms):

jx 183

trF txFyy fyyFzz fzz F yzF zy: 46

As in the real-space response, this is the 1D responseaccounting for changes in the area perpendicular to thecurrent.Strained electric forces (6 terms):

jx 183

trF tyF xzF yz F xyFzz fzz y z:47

These terms correspond to a typical electrical force in they direction, which is then redirected to the x direction by

shears and correcting for change in the area perpendicularto the current.Strained polarization or Lorentz-force responses (8

terms):

jx 183

trF tyF zxF yz F xzF yz F xyFzz fzz y z: 48

The first two terms say that if the polarization changes inthe y direction, then either shear or the Lorentz force canchange this into a current in the z direction. That current canthen be redirected by the Lorentz force or shear (respec-tively) to the x direction. The third term is the same currentdue to polarization, leading to a current in the x direction bythe Lorentz force, correcting for change in the areaperpendicular to the new current.WSM-type E B charge pumping (3 terms):

jx 183

trF txF yz F tyF zx F tzF xyF y z: 49

Assuming the real-space field strengths are k indepen-dent, this term is E B times the Berry curvature. If weintegrate over y and z, we find

Zdydzjx 1

42C1xE B; 50

where C1x is the Chern number of the slice of theBrillouin zone at fixed x. For the case of a WSM with Weylpoints split along the x direction, C1x is nonzero betweenthe Weyl points (see Fig. 4), so this response is a currentfrom one Weyl point to the other. This result is exactly thechiral anomaly [34,5153,55,69,70], which says that anE B field in a WSM pumps charge from oneWeyl point tothe other.Charge component.All 16 terms which only contain

mixed field strengths F ij combine to form the unit-cell volume, corrected for strain. The other two types ofresponse are as follows.Layered Chern insulator Hall response (3 terms):

j0 183

Zd3xtrF xzF z xFyy perm; 51

where perm indicates terms created by cyclically permut-ing x; y; z and x; y; z. This is the charge densitycounterpart to the spatial layered Chern insulator Hallresponse; it analogously comes from adding the Hallresponse of each subsystem at fixed ki to a magneticfield Bi.


021018-11

TI-type Hall response (9 terms):

j0 183

Zd3xtrF xzF y zF yx F x yF yz F z xfyy

perm: 52

By the same methods used to derive Eq. (42), these termscan be manipulated into the form

j0 12B P3: 53

This is exactly the charge component of the Hall responsethat appears in a TI [22].We summarize the 6D responses in Table IV.

VI. ANOMALIES

In the previous section, we enumerated the phase-spacebulk responses, which, as we have seen, correspond to thetopological responses of filled states in real space. Thisincludes the responses of insulators and semimetals as wellas the responses of metals that involve all the occupiedstates, such as the anomalous Hall effect. We now wish todescribe the topological features of Fermi surfaces and real-space system edges. We also approach the response ofsemimetals from another perspective. All these featurestake the form of anomalies in phase space.Given the CS theory for a phase-space system, such as

that which appears in Eq. (9) or its higher-dimensional and/or non-Abelian generalization, suppose that iC 0 forsome coordinate i. This means that the phase-space systemcontains an edge; that is, the real-space system has a Fermisurface or an edge. Then, in general, the responses of thesystem will depend on details; for example, edge currentsof quantum Hall systems depend on the nonuniversal edgemode velocity. However, there will be a universal anomaly

(or lack thereof) along such edges. We see this from theanomaly resulting from the phase-space CS term in 2nD:

Xij 1n!22nn iC

i122n trF 12F 2n12n :

54From here, integration over the appropriate phase-spacedirections determines the anomalies in the real system. Welist some physically interesting effects below.

A. Fermi surface anomalies

In Sec. III B, we computed the chiral anomaly in 1Dmomentum space by imposing edges at x kF in 2Dphase space. Integrating over x yielded the fact that anelectric field pumps electrons in states near kF into statesnear kF, or vice versa. At a down-to-earth level, thissimply corresponds to tilting of the 1D Fermi surface in anelectric field due to semiclassical motion of the electrons.This idea is straightforward to generalize to higher

dimensions. For instance, an open Fermi surface in 2Dis obtained by confining the 4D phase-space systembetween x kF while leaving the other three directionsinfinite, while a 3D spherical Fermi surface results frommaking the x, y, and z directions in 6D phase space finiteunder the constraint x2 y2 z2 k2F and leaving the x, y,and z directions unconstrained. For each phase-spacegeometry, the corresponding anomaly characterizes proper-ties of the resultant Fermi surface. Importantly, if a specialobject such as a Dirac or a Weyl point is buried under theFermi surface, its observable effects in local transportphenomena should emerge from the anomaly equation.We demonstrate this first for a 3D spherical Fermi

surface, which carries a Chern number, in general, andexhibits a chiral anomaly proportional to the Chern numberand the electromagnetic field E B. The most well-knownoccurrence of this phenomenon is in Weyl semimetals.We begin with the anomaly equation in 6D phase space.In the absence of any strains and ignoring quasi-lower-dimensional terms (i.e., terms such as Fxx that contain thebackground field), it readsX

j rjr

183

r kFF F txF yz F tyF zx F tzF xy;55

where r; ; are the spherical coordinates correspondingto x; y; z. Integrating over the barred coordinates immedi-ately yields the chiral anomaly in Weyl semimetals:

Xt;x;y;z

j3D 142 CFSE B; 56

TABLE IV. Summary of 6D phase-space responses.

Current component Response

Real space Quasi-1D responsesQuasi-2D (layered Chern insulator) response

Topological magnetoelectric effectTI-like anomalous Hall responseChange in polarization with strain

k space Quasi-1D responseElectric fields with strain

Change in polarization plus Lorentzforce with strain

WSM-like E B charge pumpingCharge density Density response to change

in unit-cell volumeLayered Chern insulator Hall response

TI-like anomalous Hall response


021018-12

where CFS 12Hdd sin F Z is the Chern number

of the Fermi surface which equals the total chirality of allWeyl points enclosed by it.Unlike in 3D, in 2D systems the one-dimensional Fermi

surface carries a nonquantized Berry phase instead of aChern number. An analogous analysis, i.e., starting with theanomaly equation in 4D phase space with a x; y boundarythat satisfies x2 y2 k2F and integrating over x; y, gives

Xt;x;y

j2D 142 Bzt; 57

ignoring strain and quasi-lower-dimensional terms, where HFS ar dr is the Berry phase on the Fermi surface.Equation (57) is the statement that adiabatically changingthe Hall conductivity of an anomalous Hall metal in amagnetic field creates charged excitations bound tothe field.In the presence of strains, both Eqs. (56) and (57) contain

more terms on their right-hand sides. We encounter theseterms in the next subsection when we discuss the effects ofdislocations. Before moving on, however, we wish to stressthat the physical anomaly in a given dimension is inde-pendent of the topology of the Fermi surface. However,certain topologies are more convenient for studying a givenphysical anomaly. For instance, the chiral anomaly in Weylmetals is easier to see for a spherical Fermi surface, but itcan equally well be derived for open Fermi surfaces thatspan one or two directions in the Brillouin zone.

B. Anomalies in real space

1. Real-space edge

The simplest example of a real-space edge anomaly wasderived in Sec. III B, where we imposed x-direction edgesin 2D phase space and showed that a time-dependentpolarization in 1D can be used to pump charge across thelength of the chain. As a more nontrivial example, considera real, single-band 2D Chern insulator that occupies x > 0.Then, in 4D phase space, the anomaly equation (54) withCx x reads

t yjy xjx yjy

142

xF tyF x y F txF yy F tyF yx: 58

Integrating xjx yjy over momentum space gives zerosince there is no boundary in those directions. Hence,integrating the previous equation over momentum spacegives

t2D yjy2D 142 xZ

d2kF x yEy 1

2xC1Ey;

59

with C1 the first Chern number of the occupied band of the2D Hamiltonian. We have ignored quasi-1D terms andterms containing strain. Equation (59) is recognizable asthe usual anomaly for a 2D Chern insulator where anelectric field parallel to the edge builds up a charge densityalong that edge.

2. Dislocations

4D phase space.The simplest example of a dislocationis an edge dislocation in 2D real space. The key feature ofthe dislocation, as we discussed in Sec. III A, is that, farfrom the dislocation line itself, electrons accumulate aBerrys phase of k b upon encircling the dislocation. Wecan thus model the dislocation by a Berry connectionar; a 0;b k=2, leading to a k-independent Berrycurvature F i bi=2r. Our theory breaks down at thedislocation itself because the system changes quickly onthe scale of a lattice constant. We can avoid this problem bykeeping the Berry connection but surrounding the dislo-cation by a finite-size puncture in the system of radius r0,i.e., choose C r r0, with r the radial coordinate inthe xy plane. The resulting anomaly equation reads

X

j rjr 183

F txby F tybx; 60

plus quasi-1D terms on the right-hand side, which weignore. Integrating over x; y and gives the chargeradiating from the core of an edge dislocation in thepresence of a time-dependent polarization:

t2D z b tP: 61This result is shown in Fig. 5, which makes the physicalpicture of the anomaly clear in the limit of weakly coupled

P

b

FIG. 5. An edge dislocation in 2D with the Burgers vector b. Inthe presence of a polarization Pb, charge gets accumulated atthe core of the dislocation, shown by the red dot.


021018-13

chains perpendicular to b; the core of the dislocation is theend of such a chain, so polarizing that chain adds charge toits end. The nontrivial result is that the extra charge remainsbound to the dislocation core and does not leak into otherchains even when they are strongly coupled.6D phase space.A similar analysis for a dislocation in

3D real space running along z and with the Burgers vectorb gives

t 1r j zjz

r r083

Zd3xF x yrF z F y zrF x

F z xrF yF tz r r0

83Ez

Zd3x b; 62

where i 12 ijkF j k is the Berry curvature of the bands inthe plane perpendicular to i.To understand Eq. (62), let us first consider a layered

Chern insulator, that is, a system composed of layers ofChern insulators stacked along a certain direction. Theintegral in Eq. (62) then gives the Chern number of thelayers in each direction, so

t 1r j zjz r r0 EzC b; 63

where Ci 1=82Rd3xi. Now, add a dislocation run-

ning along z with the Burgers vector b. Edge dislocationsare defined by bz, whereas screw dislocations have bz.The two scenarios are shown in Fig. 6. Equation (63) showsthat in either case, there exists a chiral mode along thedislocation that participates in an anomaly in response toEz. We can understand this as follows.

An edge dislocation can be thought of as a semi-infinitesheet perpendicular to b and unbounded along z insertedinto the 3D lattice. If the sheet has a Chern number, weexpect it to have a chiral mode along z. For weakly coupledsheets, this is precisely the chiral mode along the edgedislocation. For a screw dislocation, the existence of achiral dislocation mode follows from an argument adaptedfrom one that predicts helical dislocation modes in weaktopological insulators [71,72]. Suppose that our system isof finite size in the z direction. Then, on each surface, thereis a semi-infinite edge emerging from the dislocation.However, this edge must carry a chiral mode since thesurface layer is a Chern insulator. By charge conservation,this chiral mode cannot terminate at the dislocation, so thechiral mode must proceed along the dislocation to the othersurface. Moreover, in each case, the chiral mode is expectedto survive for strongly coupled layers as well, where thesystem is better thought of as stacked sheets in momentumspace and is typically termed an axion insulator. Thisrobustness occurs because layered Chern insulators andaxion insulators are actually the same phasethere isno phase transition as the interlayer coupling isstrengthenedso their topological defects such as dislo-cations have qualitatively similar behavior. Indeed, thepresence of a chiral mode was shown explicitly for an axioninsulator created from a charge-density wave instability of aWSM in Ref. [73].Spectral flow due to dislocations in Weyl semimetals.

Having seen examples of anomalies being the universalfeature of gapless systems, we use our theorys anomalymachinery to derive a new result: prediction of an anomalyat dislocation lines in a WSM. This result is closely relatedto the case of a layered Chern insulator (axion insulator)just discussed.Consider a WSM with two nodes split by K (and thus

having broken time-reversal symmetry) with a dislocationalong z. In contrast to the layered Chern insulators or axioninsulators, WSMs have a gapless bulk and thus cannotsupport localized modes the same way that the former do.However, our theory allows us to confirm that there isindeed an anomaly at the dislocation in a WSM. Theanomaly calculation is identical to the axion insulator case,except that 1=4 R d3xi Ki instead of 2. The resultis that the anomaly is r r0EzK b=2, which reflectsthe fact that chiral modes appear only in the region ofmomentum space between the Weyl nodes, where theChern number of the layers is 1. From now on, weassume K Kz for concreteness.The physical interpretation of this anomaly is more

subtle for the WSM than the axion insulator. In the lattercase, because of the bulk gap, the anomaly means that thereis a chiral zero mode on the dislocation. In the WSM case,there is no bulk gap. Furthermore, if the region carrying anonzero Chern number is near kz 0, then that region seesonly small perturbations from the dislocation because the

FIG. 6. Screw (left diagram) and edge (right diagram) dis-locations in 3D. Dislocations in an axion insulator harbor chiralmodes, denoted by red lines in both figures. In the screwdislocation, thick black lines represent the standard chiral edgemode. The screw dislocation geometry with Weyl nodes splitalong the screw axis was used for the numerical results presentedin Fig. 7.


021018-14

dislocation acts like a flux proportional to kz. Hence, weshould not necessarily expect a zero mode on the dis-location. On the other hand, if this region is located nearkz , then there may be such a zero mode. In general,however, the existence of a localized zero mode is notguaranteed.Since the anomaly need not imply a localized zero mode,

we numerically solved a simple k p model for a WSM inthe presence of a dislocation in order to directly verify theanomaly. The Hamiltonian we used is

H M0 M1k2z M2k2x k2y5 L1kz4 L2ky1 kx2 U012: 64

Here, the anticommuting matrices are defined by1;2;3 x;y;zx, 4 y, 5 z, and ij i;j=2i,where is a spin index and is an orbital index. This modelleads to Weyl points at k jU0j=L1z^ when the quadraticterm is neglected. It has been previously investigated in aradial geometry [55] with no dislocation. The only effect ofa screw dislocation at r 0 with the Burgers vector bz^ isthat the dependence of the components of the wavefunction on the in-plane angle changes from ein toeilbkz=2, where the half-integer l is the eigenvalue of Lzin the absence of the dislocation.We solved the discretized version of this model at fixed

angular momentum for a cylinder of size R 120 sites atfixed angular momentum l 1=2 kz=2. For compari-son, we show the band structure in the WSM phase with nodislocation in Fig. 7(a). The mode localized near r 0 (inblue) is always at higher energy than the Fermi arc, and it isnot topological. Adding the dislocation, we see in Fig. 7(b)that now the r 0 mode changes from unoccupied tooccupied after crossing the Weyl points; an electron hasbeen pumped from the Fermi arc (in red) to the dislocation.This is the anomaly that we discussed above, even thoughthere is no zero-energy mode localized on the dislocation.This system can smoothly evolve, by bringing the Weylpoints together and annihilating them, into the axioninsulator in Fig. 7(c). That state has a single chiral modelocalized on the dislocation which crosses the band gapwithout mixing with the outer edge mode, as expected. InFig. 7(d), we have a WSM with a topologically nontrivialregion centered about kz ; here, there is a zero modelocalized on the dislocation, and the charge pumping ismore obvious than in Fig. 7(b).The result of charge pumping due to disclinations has

been previously predicted [63]. However, our picture isdifferent from the one considered there. The claim inRef. [63] is that a chiral magnetic field, which in our caseis created by the dislocation, causes a net spontaneouscurrent to flow. As can be seen from our picture, this is nottrue; an electric field is necessary to have an anomaly andthus a net current. Fundamentally, the total current mustvanish in the absence of an electric field. If the current did

not vanish, adding an electric field parallel to the currentwould cause dissipation and lower the system energy, butthis is impossible for a system already in its ground state.The difference in Ref. [63] stems from neglectingmomentum-space regions away from the Weyl nodesand the real-space boundary in determining the totalcurrent. Thus, while the general expression for the currentdensity derived by Ref. [63] is correct, the total currentvanishes when these contributions are included. For thecase that we show in Fig. 7(b), the net current due to thedislocation is canceled by the current along the Fermi arcs.In the case of Fig. 7(d), the dislocation mode near one Weylpoint connects directly to the mode on the other sidethrough a zero mode which cancels the net current.To summarize, dislocations in a WSM indeed cause

pumping of charge to (or from) the dislocation line when anelectric field is applied along the dislocation. Such a chargepumping is smoothly connected to that which occurs in theaxion insulator, but it may or may not, depending ondetails, result in a zero mode localized on the dislocation.Although we presented numerics for a screw dislocationthat runs along the same direction as the Weyl node

FIG. 7. Band structure of the lattice regularized version ofEq. (64) in a cylindrical geometry. Color corresponds to hri withthe dislocation at r 0; red indicates localization on thedislocation, and blue is localization on the outer boundary. Para-meters are M0 0, M1 0.342 eV2, M2 18.25 eV2,L1 1.33 eV, L2 2.82 eV, R 120 radial sites, and l 1=2 angular momentum unless otherwise stated. Note thatbecause of the dislocation, the system is not periodic in kz atfixed angular momentum. (a) WSM phase (U0 1.3 eV), nodislocation. (b) Same as (a), but with dislocation. (c) Axioninsulator phase (U0 1.7 eV) with dislocation. (d) WSM phase(U0 1.3 eV, M1 0.342 eV, M0 1.4 eV) withtopologically nontrivial BZ slices centered at kz and adislocation.


021018-15

splitting, Eq. (62), and hence the qualitative result, is validfor edge dislocations as well as for other directions of theWeyl node splitting.

VII. DISCUSSION AND CONCLUSIONS

We have shown that the responses and anomalies of agapped or gapless system living in n spatial dimensions canbe described by a single response theory of a gappedsystem living in 2n spatial dimensions. Conceptually, this isbecause adding magnetic fields in the 2n-dimensionalsystem and projecting onto the zeroth Landau level allowsus to interpret that system as living in phase space. We haveused this theory to reproduce well-understood responsesand anomalies in systems with noninteracting electrons andAbelian real-space gauge fields, as well as to demonstratethe existence of spectral flow due to dislocations in Weylsemimetals.There are several interesting fundamental questions

about our theory which are at present open. It would beinteresting to see how our theory connects to the use ofphase space in statistical mechanics; how might theLandau-Boltzmann transport equation, which describestransport in Fermi liquids via Wigner functions, orLiouvilles theorem, which describes the time evolutionof general classical systems in phase space via a densitymatrix, arise in our context? Both the Wigner function andthe phase-space density matrix treat real and momentumspace on an equal footing; thus, our theory holds promise incapturing these phenomena.In addition to these fundamental questions, we envision a

number of extensions of our theory to more complicatedsystems. In particular, the responses that we have explicitlydiscussed have so far been only those of noninteractingsystems, which only feel a U1 real-space gauge field,though the k-space Berry connection has been allowed to benon-Abelian. The latter constraint is not an inherent limi-tation of the theory; perhaps there are interesting responsesto a larger real-space gauge group. SU2 groups in 4D and3D have been studied and shown to give topologicalinsulator- and WSM-like responses, respectively [74]. It isthus conceivable that general gauge groups can lead to othertopological responses, possibly of phases with emergentfermions such as partons [75] or composite fermions [76].As for interactions, it is not immediately clear if there are

sensible real-space systems that are well described by aphase-space theory with only local interactions. However,if there are such real-space systems, then working in phasespace could be very useful because, for example, in theabsence of a magnetic field, mean-field theory is moreaccurate because of the higher dimensionality. This advan-tage may be mitigated by the fact that our constructionrequires gauge fields, however. Alternatively, it is possiblethat there is a simple way to directly incorporate theinteractions of the real-space system into the phase-spacetheory.

Another interesting question is if there is an extension ofour theory that describes nodal superconductors. Ourtheory as written requires U1 charge conservation;perhaps there is some way to incorporate spontaneousbreaking of this symmetry. Finally, it could also beinteresting to explicitly incorporate other symmetries ofthe lower-dimensional system; this could allow a betterunderstanding of gapless symmetry-protected phases likeDirac semimetals.

ACKNOWLEDGMENTS

D. B. is supported by the National Science Foundationunder Grant No. DGE-114747. P. H. is supported by theDavid and Lucile Packard Foundation and the U.S. DOE,Office of Basic Energy Sciences, Contract No. DEAC02-76SF00515. S. C. Z. is supported by the National ScienceFoundation under Grant No. DMR-1305677. X. L. Q. issupported by the National Science Foundation throughGrant No. DMR-1151786.

APPENDIX: TOPOLOGICALMAGNETOELECTRIC EFFECT

Here, we derive explicitly the topological magnetoelec-tric effect from our response theory. The topologicalmagnetoelectric effect is only quantized in gapped systems,so we assume the system is gapped.The relevant terms are in Eq. (41), which we rewrite

here as

jx3D 1163

Zd3xtrIJKF tIF JKF yz; A1

where I; J; K run over x; y; z. We assume that F yz is thereal-space magnetic field Bx, so we can pull it out. Forsimplicity, we choose a gauge such that Ai 0for i x; y; z.Expanding Eq. (A1) and manipulating some indices

yields

jx3D Bx83Z

d3xIJKtrtaI Iat at; aIJaK aJaK: A2

We first show that several sets of terms in this expansionare zero. First, notice that

Zd3xIJKIatJaK

Zd3xIJKIJataK A3

Z

d3xIJKJatIaK A4

Zd3xIJKIatJaK; A5


021018-16

where we integrated by parts twice and then switched the indices I and J. Hence,

Zd3xIJKIatJaK 0: A6

Next, consider the terms

Zd3xIJKtrat; aIJaK IataJaK

Zd3xIJKtrJataI aIataK IataJaK A7

Z

d3xIJKtrIataJ aJataK IataJaK A8

Z

d3xIJKtratIaJ IaJat aJIataK A9

Z

d3xIJKtrat; aIJaK IataJaK: A10

We have integrated by parts, manipulated indices, and used the cyclic property of the trace. Hence, the left-hand side here isalso zero.Finally, trivial manipulations show that

Rd3xIJKtrat; aIaJaK 0 as well.

The remaining terms in the expansion are those that do not involve at:

jx3D Bx83

Zd3xIJKtrtaIJaK aJaK A11

Bx83

Zd3xIJKtrtaIJaK aIaJaK aItJaK 2taJaK A12

Bx83

Zd3xIJKtrtaIJaK aIaJaK JaItaK 2taIaJaK A13

Bx83

Zd3xIJKtr

taIJaK aIaJaK taIJaK taIaJaK 1

3taIaJaK

A14

Bx83

Zd3xIJKtr

taIJaK 2

3aIaJaK

jx3D: A15

This immediately gives the desired relation (42).

[1] X.-L. Qi and S.-C. Zhang, The Quantum Spin Hall Effectand Topological Insulators, Phys. Today 63, 33 (2010).

[2] M. Z. Hasan and C. L. Kane, Colloquium: TopologicalInsulators, Rev. Mod. Phys. 82, 3045 (2010).

[3] X.-L. Qi and S.-C. Zhang, Topological Insulators andSuperconductors, Rev. Mod. Phys. 83, 1057 (2011).

[4] M. Z. Hasan and J. E. Moore, Three-Dimensional Topo-logical Insulators, Annu. Rev. Condens. Matter Phys. 2, 55(2011).

[5] F. D. M. Haldane, Model for a Quantum Hall Effectwithout Landau Levels: Condensed-Matter Realizationof the Parity Anomaly, Phys. Rev. Lett. 61, 2015(1988).

[6] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. denNijs, Quantized Hall Conductance in a Two-DimensionalPeriodic Potential, Phys. Rev. Lett. 49, 405 (1982).

[7] C. L. Kane and E. J. Mele, Quantum Spin Hall Effect inGraphene, Phys. Rev. Lett. 95, 226801 (2005).

[8] C. L. Kane and E. J. Mele, Z2 Topological Order and theQuantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802(2005).

[9] R Roy, Topological Phases and the Quantum Spin HallEffect in Three Dimensions, Phys. Rev. B 79, 195322(2009).

[10] L. Fu, C. L. Kane, and E. J. Mele, Topological Insulators inThree Dimensions, Phys. Rev. Lett. 98, 106803 (2007).


021018-17

[11] B. A. Bernevig and S.-C. Zhang, Quantum Spin Hall Effect,Phys. Rev. Lett. 96, 106802 (2006).

[12] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, QuantumSpin Hall Effect and Topological Phase Transition in HgTeQuantum Wells, Science 314, 1757 (2006).

[13] R. Roy, Z2 Classification of Quantum Spin Hall Systems: AnApproach Using Time-Reversal Invariance, Phys. Rev. B79, 195321 (2009).

[14] N. Read and D. Green, Paired States of Fermions in TwoDimensions with Breaking of Parity and Time-ReversalSymmetries and the Fractional Quantum Hall Effect, Phys.Rev. B 61, 10267 (2000).

[15] C. Kallin and A. J. Berlinsky, Is Sr2RuO4 a Chiral p-WaveSuperconductor?, J. Phys. Condens. Matter 21, 164210(2009).

[16] D. D. Osheroff, W. J. Gully, R. C. Richardson, and D. M.Lee, New Magnetic Phenomena in Liquid He3 below 3 mK,Phys. Rev. Lett. 29, 920 (1972).

[17] A. J. Leggett, Interpretation of Recent Results on He3 below3 mK: A New Liquid Phase?, Phys. Rev. Lett. 29, 1227(1972).

[18] A. Kitaev, Periodic Table for Topological Insulators andSuperconductors, L. D. Landau Memorial ConferenceAdvances in Theoretical Physics (AIP, New York,2009), pp. 2230.

[19] S. Ryu, A. P. Schnyder, A. Furusaki, and A.W.W. Ludwig,Topological Insulators and Superconductors: Tenfold Wayand Dimensional Hierarchy, New J. Phys. 12, 065010(2010).

[20] A. P. Schnyder, S. Ryu, A. Furusaki, and A.W.W. Ludwig,Classification of Topological Insulators and Superconduc-tors in Three Spatial Dimensions, Phys. Rev. B 78, 195125(2008).

[21] A. Altland and M. R. Zirnbauer, Nonstandard SymmetryClasses in Mesoscopic Normal-Superconducting HybridStructures, Phys. Rev. B 55, 1142 (1997).

[22] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological FieldTheory of Time-Reversal Invariant Insulators, Phys. Rev. B78, 195424 (2008).

[23] A. M. Essin, J. E. Moore, and D. Vanderbilt, Magnetoelec-tric Polarizability and Axion Electrodynamics in CrystallineInsulators, Phys. Rev. Lett. 102, 146805 (2009).

[24] Z. Wang, X.-L. Qi, and S.-C. Zhang, TopologicalField Theory and Thermal Responses of InteractingTopological Superconductors, Phys. Rev. B 84, 014527(2011).

[25] S. Ryu, J. E. Moore, and A.W.W. Ludwig, Electromagneticand Gravitational Responses and Anomalies in TopologicalInsulators and Superconductors, Phys. Rev. B 85, 045104(2012).

[26] X.-L. Qi, E. Witten, and S.-C. Zhang, Axion TopologicalField Theory of Topological Superconductors, Phys. Rev. B87, 134519 (2013).

[27] C. Wang, A. C. Potter, and T. Senthil, Classification ofInteracting Electronic Topological Insulators in ThreeDimensions, Science 343, 629 (2014).

[28] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov,Topological Semimetal and Fermi-Arc Surface States in theElectronic Structure of Pyrochlore Iridates, Phys. Rev. B83, 205101 (2011).

[29] A. A. Burkov and L. Balents, Weyl Semimetal in a Topo-logical Insulator Multilayer, Phys. Rev. Lett. 107, 127205(2011).

[30] A. A. Burkov, M. D. Hook, and L. Balents, TopologicalNodal Semimetals, Phys. Rev. B 84, 235126 (2011).

[31] A. M. Turner and A. Vishwanath, Beyond Band Insulators:Topology of Semi-metals and Interacting Phases,arXiv:1301.0330.

[32] O. Vafek and A. Vishwanath, Dirac Fermions in SolidsFrom High Tc Cuprates and Graphene to TopologicalInsulators and Weyl Semimetals, Annu. Rev. Condens.Matter Phys. 5, 83 (2014).

[33] W. Witczak-Krempa and Y.-B. Kim, Topological andMagnetic Phases of Interacting Electrons in the PyrochloreIridates, Phys. Rev. B 85, 045124 (2012).

[34] P. Hosur and X. Qi, Recent Developments in TransportPhenomena in Weyl Semimetals, Topological Insulators/Isolants Topologiques, C.R. Phys. 14, 857 (2013).

[35] P. Hosur, Friedel Oscillations Due to Fermi Arcs in WeylSemimetals, Phys. Rev. B 86, 195102 (2012).

[36] A. C. Potter, I. Kimchi, and A. Vishwanath, QuantumOscillations from Surface Fermi-Arcs in Weyl and DiracSemi-Metals, Nat. Comm. 5, 5161 (2014).

[37] F. D. M. Haldane, Attachment of Surface Fermi Arcs tothe Bulk Fermi Surface: Fermi-Level Plumbing inTopological Metals, arXiv:1401.0529.

[38] T. T. Heikkila, N. B. Kopnin, and G. E. Volovik, Flat Bandsin Topological Media, JETP Lett. 94, 233 (2011).

[39] J. C. Y. Teo and C. L. Kane, Topological Defects andGapless Modes in Insulators and Superconductors, Phys.Rev. B 82, 115120 (2010).

[40] S. Matsuura, P.-Y. Chang, A. P. Schnyder, and S. Ryu,Protected Boundary States in Gapless Topological Phases,New J. Phys. 15, 065001 (2013).

[41] R. Karplus and J. M. Luttinger, Hall Effect in Ferromag-netics, Phys. Rev. 95, 1154 (1954).

[42] F. D. M. Haldane, Berry Curvature on the Fermi Surface:Anomalous Hall Effect as a Topological Fermi-LiquidProperty, Phys. Rev. Lett. 93, 206602 (2004).

[43] T. Jungwirth, Q. Niu, and A. H. MacDonald, AnomalousHall Effect in Ferromagnetic Semiconductors, Phys. Rev.Lett. 88, 207208 (2002).

[44] D. Xiao, M.-C. Chang, and Q. Niu, Berry Phase Effectson Electronic Properties, Rev. Mod. Phys. 82, 1959 (2010).

[45] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, andN. P. Ong, Anomalous Hall Effect, Rev. Mod. Phys. 82,1539 (2010).

[46] M. Barkeshli and X.-L. Qi, Topological Response Theory ofDoped Topological Insulators, Phys. Rev. Lett. 107, 206602(2011).

[47] S. L. Adler, Axial-Vector Vertex in Spinor Electrodynamics,Phys. Rev. 177, 2426 (1969).

[48] J. S. Bell and R. Jackiw, A PCAC Puzzle: 0 in the-Model, Il Nuovo Cimento A 60, 47 (1969).

[49] H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on aLattice: (I). Proof by Homotopy Theory, Nucl. Phys. B185,20 (1981).

[50] H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on aLattice: (II). Intuitive Topological Proof, Nucl. Phys. B193,173 (1981).


021018-18

[51] H. B. Nielsen and M. Ninomiya, The Adler-Bell-JackiwAnomaly and Weyl Fermions in a Crystal, Phys. Lett. B 130,389 (1983).

[52] A. A. Zyuzin and A. A. Burkov, Topological Response inWeyl Semimetals and the Chiral Anomaly, Phys. Rev. B 86,115133 (2012).

[53] Y. Chen, S. Wu, and A. A. Burkov, Axion Response in WeylSemimetals, Phys. Rev. B 88, 125105 (2013).

[54] V. Aji, Adler-Bell-Jackiw Anomaly in Weyl Semimetals:Application to Pyrochlore Iridates, Phys. Rev. B 85, 241101(2012).

[55] C.-X. Liu, P. Ye, and X.-L. Qi, Chiral Gauge Field andAxial Anomaly in a Weyl Semimetal, Phys. Rev. B 87,235306 (2013).

[56] G. Basar, D. E. Kharzeev, and H.-U. Yee, Triangle Anomalyin Weyl Semi-metals, Phys. Rev. B 89, 035142 (2014).

[57] D. T. Son and B. Z. Spivak, Chiral Anomaly and ClassicalNegative Magnetoresistance of Weyl Metals, Phys. Rev. B88, 104412 (2013).

[58] K. Landsteiner, Anomaly Related Transport of Weyl Fermionsfor Weyl Semi-Metals, Phys. Rev. B 89, 075124 (2014).

[59] P. Goswami and S. Tewari, Axionic field theory of 3 1-dimensional Weyl semimetals, Phys. Rev. B 88, 245107(2013).

[60] A. G. Grushin, Consequences of a Condensed MatterRealization of Lorentz-Violating QED in Weyl Semi-Metals,Phys. Rev. D 86, 045001 (2012).

[61] S. T. Ramamurthy and T. L. Hughes, Patterns ofElectro-Magnetic Response in Topological Semi-Metals,arXiv:1405.7377v1.

[62] B. Hetnyi, DC Conductivity as a Geometric Phase, Phys.Rev. B 87, 235123 (2013).

[63] Z. Jian-Hui, J. Hua, N. Qian, and S. Jun-Ren, TopologicalInvariants of Metals and the Related Physical Effects, Chin.Phys. Lett. 30, 027101 (2013).

[64] H. Yao and D.-H. Lee, Topological Insulators andTopological Nonlinear Models, Phys. Rev. B 82,245117 (2010).

[65] S.-C. Zhang and J. Hu, A Four-Dimensional Generalizationof the Quantum Hall Effect, Science 294, 823 (2001).

[66] B. A. Bernevig, C.-H. Chern, J.-P. Hu, N. Toumbas, andS.-C. Zhang, Effective Field Theory Description of theHigher Dimensional Quantum Hall Liquid, Ann. Phys.(Amsterdam) 300, 185 (2002).

[67] R. D. King-Smith and D. Vanderbilt, Theory of Polarizationof Crystalline Solids, Phys. Rev. B 47, 1651 (1993).

[68] D. J. Thouless, Quantization of Particle Transport, Phys.Rev. B 27, 6083 (1983).

[69] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Quantum Hall Effectsin a Weyl Semimetal: Possible Application in PyrochloreIridates, Phys. Rev. B 84, 075129 (2011).

[70] G. E. Volovik, The Universe in a Helium Droplet,International Series of Monographs on Physics (OxfordUniversity Press, Oxford, 2009).

[71] Y. Ran, Y. Zhang, and A. Vishwanath, One-DimensionalTopologically Protected Modes in Topological Insulatorswith Lattice Dislocations, Nat. Phys. 5, 298 (2009).

[72] T. L. Hughes, H. Yao, and X.-L. Qi, Majorana Zero Modesin Dislocations of Sr2RuO4, Phys. Rev. B 90, 235123(2014).

[73] Z. Wang and S.-C. Zhang, Chiral Anomaly, Charge DensityWaves, and Axion Strings fromWeyl Semimetals, Phys. Rev.B 87, 161107 (2013).

[74] Y. Li, S.-C. Zhang, and C. Wu, Topological Insulatorswith SU(2) Landau Levels, Phys. Rev. Lett. 111, 186803(2013).

[75] X.-G. Wen, Projective Construction of Non-Abelian Quan-tum Hall Liquids, Phys. Rev. B 60, 8827 (1999); B.Swingle, M. Barkeshli, J. McGreevy, and T. Senthil,Correlated Topological Insulators and the FractionalMagnetoelectric Effect, Phys. Rev. B 83, 195139 (2011);J. Maciejko, X.-L. Qi, A. Karch, and S.-C. Zhang, Frac-tional Topological Insulators in Three Dimensions, Phys.Rev. Lett. 105, 246809 (2010).

[76] J. K. Jain, Composite-Fermion Approach for the FractionalQuantum Hall Effect, Phys. Rev. Lett. 63, 199 (1989).


021018-19

unified topological response theory for gapped and gapless free fermions

Documents

topological termsin

topological states

thebulk topological

homotopy theory

gapped phases of free

common parent theory

superconducting phases

chernsimons theory