-
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
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PHYSICAL REVIEW X 5, 021018 (2015)
2160-3308=15=5(2)=021018(19) 021018-1 Published by the American
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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
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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
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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,
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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.
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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 .
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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
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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
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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.
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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].
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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.
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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
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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.
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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.
BULMASH et al. PHYS. REV. X 5, 021018 (2015)
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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.
UNIFIED TOPOLOGICAL RESPONSE THEORY FOR PHYS. REV. X 5, 021018
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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
BULMASH et al. PHYS. REV. X 5, 021018 (2015)
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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).
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