A note on the parity anomaly from the Hamiltonian point of view
aa r X i v : . [ c ond - m a t . o t h e r] O c t On the parity anomaly from the Hamiltonian point of view
Matthew F. Lapa ∗ Kadanoff Center for Theoretical Physics, University of Chicago, Illinois 60637, USA
We review the parity anomaly of the massless Dirac fermion in dimensions from the Hamiltonian, as op-posed to the path integral, point of view. We have two main goals for this note. First, we hope to make the parityanomaly more accessible to condensed matter physicists, who generally prefer to work within the Hamiltonianformalism. The parity anomaly plays an important role in modern condensed matter physics, as the masslessDirac fermion is the surface theory of the time-reversal invariant topological insulator (TI) in dimensions.Our second goal is to clarify the relation between the time-reversal symmetry of the massless Dirac fermion andthe fractional charge of ± (in units of e ) that appears on the surface of the TI when a magnetic monopole ispresent in the bulk. To accomplish these goals we study the Dirac fermion in the Hamiltonian formalism usingtwo different regularization schemes. One scheme is consistent with the time-reversal symmetry of the mass-less Dirac fermion, but leads to the aforementioned fractional charge. The second scheme does not lead to anyfractionalization, but it does break time-reversal symmetry. For both regularization schemes we also computethe effective action S eff [ A ] that encodes the response of the Dirac fermion to a background electromagnetic field A . We find that the two effective actions differ by a Chern-Simons counterterm with fractional level equal to ,as is expected from path integral treatments of the parity anomaly. Finally, we propose the study of a bosonicanalogue of the parity anomaly as a topic for future work. I. INTRODUCTION
The purpose of this note is to review the parity anomaly ofthe massless Dirac fermion in dimensions [1–4], but fromthe Hamiltonian/Hilbert space point of view. Recall that theparity anomaly is a conflict between the time-reversal sym-metry and large U (1) gauge invariance of the massless Diracfermion. More precisely, the parity anomaly is equivalent tothe statement that it is impossible to regularize the masslessDirac fermion theory, coupled to a background U (1) gaugefield, in a way that preserves both time-reversal symmetry andlarge U (1) gauge invariance. To clarify the meaning of large U (1) gauge invariance here, note that unbroken large U (1) gauge invariance would require all physical states of the the-ory to have integer charge, and so any regularization that leadsto states with fractional charge must violate large U (1) gaugeinvariance.There are two main reasons why we feel that a review ofthe parity anomaly from the Hamiltonian perspective is war-ranted. First, the parity anomaly has been discussed exten-sively in recent years in the context of the time-reversal in-variant topological insulator (TI) [6, 7], which hosts a singlemassless Dirac fermion on its surface. In this context the par-ity anomaly provides one of the classic examples of a theorywith a 't Hooft anomaly [8] appearing at the boundary of asymmetry-protected topological phase [9–12]. However, thediscussion in the recent literature on this topic is almost al-ways from the path integral point of view [5, 14–18]. On theother hand, in condensed matter physics it is more common ∗ email address: [email protected] As emphasized by Witten [5], the word “parity” in “parity anomaly” isa misnomer, and this anomaly is actually an anomaly in time-reversal orreflection symmetry. One exception is the recent mathematical treatment of the parity anomalyin Ref. 13. There the authors studied the projective representation of the U (1) gauge group on the Hilbert space of the massless Dirac fermion. to look at problems from a Hamiltonian point of view. There-fore we believe that there is significant value in explaininghow the parity anomaly works from the point of view of theDirac Hamiltonian on two-dimensional (2D) space . It is alsoworth noting that the TI is one of the few symmetry-protectedtopological phases that have been realized experimentally (see[19] and the review [20]), and so further study and clarifica-tion of the parity anomaly in the context of TI physics seemsjustified.The second reason for our review of the parity anomaly is toexplain the precise connection between the time-reversal sym-metry of the massless Dirac fermion and the half-quantizedelectric charge of ± (in units of e ) that appears on the sur-face of the TI when a magnetic monopole is present in thebulk. In Ref. 1, Niemi and Semenoff studied the massive Dirac fermion in the Hamiltonian formalism using a regular-ization scheme based on the Atiyah-Patodi-Singer (APS) etainvariant [21], or spectral asymmetry of the Dirac Hamilto-nian on 2D space . Within this scheme they found that theground state of the massive Dirac fermion has a charge of ± when the 2D space is pierced by a single unit of magneticflux, and they also found that this charge persists in the limitin which the mass of the fermion is sent to zero. The fractionalcharge of 2D electrons in a magnetic field was also studied byJackiw in Ref. 22.In this note we point out that the regularization scheme usedby Niemi and Semenoff is consistent with the time-reversalsymmetry of the massless Dirac fermion, in a sense that wemake precise below. To the best of our knowledge, the factthat the regularization scheme in [1] is consistent with time-reversal symmetry has not been demonstrated in detail in theexisting literature. The closest discussion that we know ofcan be found in Ref. 23, where it was shown that the resultsobtained by Niemi and Semenoff are identical to the resultsobtained from a point-splitting regularization scheme that pre-serves parity (and time-reversal) symmetry. The fact that theregularization in [1] is consistent with time-reversal shouldnot be unexpected though, as it fits in with the general pictureof the parity anomaly discussed above (i.e., the regularizationof [1] violates large U (1) gauge invariance, so we expect thatit should be consistent with time-reversal symmetry). Thisfact also makes the regularization scheme of [1] the correctscheme to use in the physical situation where the masslessDirac fermion resides on the surface of the TI.In the path integral approach, which was pioneered byRedlich [2, 3], the easiest way to see the parity anomalyis to use Pauli-Villars regularization to compute the parti-tion function of the massless Dirac fermion. This regular-ization scheme preserves large U (1) gauge invariance, butit breaks time-reversal symmetry because of the mass of thePauli-Villars regulator fermion. An alternative regularizationscheme, also considered by Redlich, is to define the partitionfunction of the massless Dirac fermion as the square root ofthe determinant of a Dirac operator for two copies of a mass-less Dirac fermion. This latter determinant can be regularizedin a time-reversal invariant way, which leads to a time-reversalinvariant regularization of the original single massless Diracfermion. Redlich then showed that this second regulariza-tion scheme violates large U (1) gauge invariance. We alsonote here that in a more sophisticated treatment [4, 5, 15, 16],Pauli-Villars regularization leads to an expression for the par-tition function of the massless Dirac fermion in which thephase of the partition function is proportional to the APS etainvariant of the spacetime Dirac operator. The APS eta invari-ant is constructed from the spectrum of the Dirac operator, andso it is manifestly gauge invariant, but this scheme still breakstime-reversal symmetry, again due to the mass of the regulatorfermion.The purpose of this note is to explain how to see the con-flict between time-reversal symmetry and large U(1) gaugeinvariance when the massless Dirac fermion is studied fromthe Hamiltonian point of view. To this end, we study theDirac fermion in the Hamiltonian formalism using two dif-ferent regularization schemes. The first regularization schemeleads to states in the theory with half-integer charge, but weshow that this scheme is consistent with time-reversal sym-metry. The second regularization scheme explicitly breakstime-reversal symmetry but does not lead to any fractional-ized quantum numbers associated with the U (1) symmetry.Thus, these two regularization schemes serve to demonstratethe parity anomaly in the Hamiltonian/Hilbert space approach.The first regularization scheme that we consider is exactlythe scheme used by Niemi and Semenoff [1]. For this schemewe work on a general curved two-dimensional space M thatis a closed manifold, instead of on flat space R . The spe-cific physical quantity that we calculate within this regulariza-tion scheme is Q A,m , the charge (in units of e ) of the ground We consider closed manifolds (e.g., the two-sphere S ) instead of R tomake the problem mathematically simpler. In particular, on closed mani-folds the Dirac operator has discrete eigenvalues, and in this case we canalso apply the Atiyah-Singer index theorem to answer certain questions re-garding the zero modes of the Dirac operator. See Ref. 22 for a discussionof the difference between the case of the plane R and the case of closedmanifolds. state of the theory in the presence of a time-reversal break-ing mass term (with mass m ), and in the presence of a back-ground time-independent spatial gauge field A = A j dx j (weuse differential form notation and also sum over the spatial in-dex j = 1 , ). We show that the regularization scheme of [1]is consistent with time-reversal symmetry in the sense that itleads to the result Q A,m = Q − A, − m . (1.1)Physically, this result means that in this regularization schemethe charge in the ground state of the theory with mass m andbackground field A is equal to the charge in the ground state ofthe time-reversed theory with mass − m and background field − A (a spatial gauge field is odd under time-reversal). On theother hand, the explicit result for Q A,m (Eq. (3.23) in Sec. III)shows that it can be integer or half-integer valued, Q A,m ∈ Z , (1.2)which shows that this regularization scheme violates large U (1) gauge invariance.The second regularization scheme that we consider is a lat-tice regularization scheme for the massless Dirac fermion on aspatial torus. The lattice model that we use for this regulariza-tion is on the square lattice, but this model is closely relatedto the model on the honeycomb lattice that was introduced inthe seminal work of Haldane [24] on a model for the quan-tum Hall effect without Landau levels. The specific physicalquantity that we calculate in this scheme is σ H,m , the Hallconductivity (in units of e h ) of the Dirac fermion with mass m in the presence of a background time-independent electricfield E . We find that σ H,m is given by σ H,m = sgn ( m ) − ∈ Z . (1.3)This result demonstrates two things. First, the Hall conduc-tivity is an integer for either sign of m , which shows thatlarge U (1) gauge invariance is preserved by this regulariza-tion scheme (there is no fractionalization of quantum num-bers associated with the U (1) symmetry). Second, the Hallconductivity for the theory with mass m is not equal to minusthe Hall conductivity for the time-reversed theory with mass − m , σ H,m = − σ H, − m . (1.4)This shows that this regularization scheme is not consis-tent with the time-reversal symmetry of the original masslessDirac fermion (a regularization scheme consistent with time-reversal symmetry should give σ H,m = − σ H, − m since theHall conductivity is odd under time-reversal).Note that in both cases we never treat the massless theorydirectly — the quantities Q A,m and σ H,m that we study areboth computed for the theory with a non-zero time-reversalbreaking mass m . Instead, we determine whether the resultis consistent with time-reversal symmetry by comparing theanswers for two massive theories that are related to each otherby the time-reversal operation.Finally, for both regularization schemes we also computethe effective action S eff [ A ] that encodes the response of themassive theory to the background gauge field A = A µ dx µ ( µ = 0 , , ). We find that the effective action S (NS)eff [ A ] , com-puted using the regularization scheme of Niemi and Semenoff,is related to the effective action S (lattice)eff [ A ] , computed usingthe lattice regularization, as S (lattice)eff [ A ] = S (NS)eff [ A ] −
12 14 π Z A ∧ dA . (1.5)The last term on the right-hand side is a Chern-Simons term(written in differential form notation), but with a fractionallevel equal to − . Thus, the two effective actions differ bya Chern-Simons counterterm with fractional level − , whichis exactly the result that we expect based on the original pathintegral treatment of the parity anomaly [2, 3].This note is organized as follows. In Sec. II we reviewthe form of the Hamiltonian for the Dirac fermion on flat andcurved 2D space, and we also review the time-reversal sym-metry of the massless Dirac fermion. In Sec. III we studythe Dirac fermion on a closed spatial manifold M using theregularization scheme of Niemi and Semenoff [1], and wecompute the charge Q A,m of the ground state for the mas-sive Dirac fermion in the presence of a background time-independent spatial gauge field A . In Sec. IV we study theDirac fermion using a lattice regularization scheme on a spa-tial torus, and we compute the Hall conductivity σ H,m for themassive Dirac fermion in the presence of a time-independentelectric field E . In Sec. V we compute the effective action S eff [ A ] for both regularization schemes, and we show that thetwo effective actions are related as shown in Eq. (1.5). InSec. VI we present concluding remarks and propose the studyof a similar anomaly in bosonic systems for future work. Fi-nally, Appendix A contains important background material onDirac fermions on curved space and on the notation used in thepaper. Note:
Throughout the paper we work in system of unitswhere the Dirac fermion has charge e = 1 , and where ~ = 1 (so h = 2 π ~ → π ) and c = 1 . Here c would be the speedof light in a high-energy context or the Fermi velocity in acondensed matter context. We use a summation conventionin which we sum over any index that appears once as a sub-script and once as a superscript in any expression, and we useLatin indices j, k, . . . taking values { , } to label spatial di-rections and Greek indices µ, ν, . . . taking values { , , } tolabel spacetime directions. We also use Latin indices a, b, . . . near the beginning of the alphabet for frame indices on curvedspace (see Appendix A). In general, we recommend that read-ers glance at Appendix A before reading the paper, to makesure that they are familiar with our notation and conventionsfor the Dirac operator on curved space, and also to review therelation between the U (1) gauge field A = A µ dx µ and theordinary electric and magnetic fields E and B on flat space. II. DIRAC HAMILTONIAN AND TIME-REVERSALSYMMETRY
In this section we introduce the Dirac fermion on flat andcurved two-dimensional space . We also discuss the time-reversal symmetry of the massless Dirac fermion, and wediscuss the effect of time-reversal on the Dirac fermion withnon-zero mass m and in the presence of a background time-independent spatial U (1) gauge field A = A j dx j . A. Flat space
We start with the action for the massless Dirac fermion onflat Minkowski spacetime, S [Ψ , Ψ] = Z d x Ψ i ˜ γ µ ∂ µ Ψ . (2.1)The quantities appearing here are as follows. First, x =( x , x , x ) is the spacetime coordinate, ∂ µ ≡ ∂∂x µ for µ =0 , , , and Ψ = Ψ( x ) is a two-component Dirac spinorfield on spacetime. Next, ˜ γ µ are a set of gamma matricesthat satisfy the Clifford algebra { ˜ γ µ , ˜ γ ν } = 2 η µν , where η = diag (1 , − , − is the Minkowski metric in “mostly mi-nus” convention. Finally, Ψ = Ψ † ˜ γ is the Dirac adjoint of Ψ .Next, we discuss the coupling to a background U (1) gaugefield (electromagnetic field) represented by the vector poten-tial one-form A = A µ dx µ . Our convention is that the Diracfermion has charge . The correct action for Ψ coupled to A is then S [Ψ , Ψ , A ] = Z d x Ψ i ˜ γ µ ( ∂ µ + iA µ )Ψ . (2.2)To see that the sign of the coupling to A µ is correct forfermions with charge , note that the term with A is − Z d x Ψ † Ψ A , (2.3)and this is the correct action for a distribution of charge withcharge density Ψ † Ψ in the presence of a scalar electromag-netic potential A (for charge e the correct covariant derivativeis ∂ µ + ieA µ ). We refer the reader to the end of Appendix Afor more details on how the components of the one-form A are related to the usual electric and magnetic fields E and B in the case of flat Minkowski spacetime.Finally, the mass term for the Dirac fermion takes the sim-ple form S m [Ψ , Ψ] = − m Z d x ΨΨ , (2.4)where the mass m is a real parameter that can be positive ornegative.We now pass to the Hamiltonian formulation of the mass-less Dirac fermion on flat space. The momentum canonicallyconjugate to Ψ is i Ψ † . As a result, the Dirac Hamiltonian onflat 2D space takes the form ˆ H = − Z d x ˆΨ † i ˜ γ ˜ γ j ∂ j ˆΨ , (2.5)where x = ( x , x ) is the spatial coordinate, j = 1 , , and ˆΨ = ˆΨ( x ) is the operator-valued Dirac spinor on 2D space. To proceed, it is convenient to define a new set of spatialgamma matrices by γ j = − ˜ γ ˜ γ j . These new gamma ma-trices obey the Clifford algebra { γ j , γ k } = 2 δ jk . In addition,we define the Dirac (differential) operator H on 2D space by H = iγ j ∂ j . (2.6)In terms of these new quantities the massless Dirac Hamilto-nian on flat space takes the form ˆ H = Z d x ˆΨ † H ˆΨ . (2.7)The Hamiltonian ˆ H for the massless Dirac fermion com-mutes with a time-reversal operator ˆ T that is defined as fol-lows. First, we define a third gamma matrix γ = i ǫ jk γ j γ k = iγ γ , which satisfies { γ, γ j } = 0 and γ = 1 . The matrix γ is sometimes referred to as the chirality matrix. Next, wechoose a concrete realization for the three gamma matrices γ j ( j = 1 , ) and γ such that the γ j have real matrix elementsand γ has imaginary matrix elements. For example we couldchoose γ = σ x , γ = σ z , and then γ = σ y , where σ x,y,z arethe Pauli matrices.With these conventions in place, the action of the time-reversal operator ˆ T on ˆΨ is defined to be ˆ T ˆΨ α ˆ T − = γ αβ ˆΨ β (2.8a) ˆ T ˆΨ † ,α ˆ T − = ˆΨ † ,β γ βα , (2.8b)where ˆΨ α , α = 1 , , are the two components of the spinor-valued field ˆΨ , ˆΨ † ,α are the two components of ˆΨ † , and γ αβ are the matrix elements of γ . As usual, ˆ T is an anti-unitary operator, so it will complex-conjugate any c-numbersthat it passes through. With this definition of ˆ T we find that ˆ T ˆΨ α ˆ T − = − ˆΨ α and likewise for ˆΨ † (this property is usu-ally summarized by the equation ˆ T = ( − ˆ N , where ˆ N is thefermion number operator). In addition, one can show that themassless Dirac Hamiltonian above commutes with this time-reversal operator ˆ T ˆ H ˆ T − = ˆ H , (2.9)and to show this it is necessary to use the fact that ˆ T is anti-unitary. We emphasize here that in the definition of ˆ T it was Later on we define the operator ˆΨ( x ) more precisely using a mode expan-sion in terms of eigenfunctions of the appropriate Dirac differential opera-tor on flat or curved space — see Eq. (3.5). crucial that we chose the gamma matrices so that γ has imag-inary matrix elements and the γ j have real matrix elements.In Sec. III we will be interested in coupling this theory to a time-independent background electromagnetic field which isspecified by the spatial vector potential A = A j dx j (we donot turn on a time-component A for this discussion). We willalso be interested in adding a mass term to ˆ H . Starting fromthe Dirac action coupled to A and with a non-zero mass term,it is straightforward to see that the resulting Hamiltonian takesthe form ˆ H A,m = Z d x ˆΨ † H A,m ˆΨ . (2.10)where H A,m is the massive Dirac operator coupled to A on2D space, H A,m = iγ j ( ∂ j + iA j ) + mγ . (2.11)To arrive at this form of H A,m we have also chosen ourgamma matrices so that ˜ γ = γ , where ˜ γ was the originalgamma matrix associated with the time direction. Since A isa background field (as opposed to a quantum operator), andsince it is real-valued, it commutes with ˆ T . Then we find thatunder time-reversal the Hamiltonian for the massive theorycoupled to A transforms as ˆ T ˆ H A,m ˆ T − = ˆ H − A, − m . (2.12)In other words, the theories with ( A, m ) and ( − A, − m ) aretime-reverses of each other, and only the theory with A = 0 and m = 0 is invariant under the action of ˆ T . B. Generalization to curved space
We now discuss the form of the Dirac Hamiltonian oncurved space. In this case the flat two-dimensional plane R (the spatial part of Minkowski spacetime) is replaced by acurved manifold M . We assume that M is a 2D orientableRiemannian manifold. We also assume that M is closed (i.e., compact and without boundary) and connected. In acoordinate patch on M with coordinates x = ( x , x ) , thecomponents of the metric g will be denoted by g jk ( x ) , anddet [ g ( x )] > is the determinant of g at the point x . Since M is 2D it is also a spin manifold, and so we do not need to worryabout the issue of whether or not fermions can be consistentlyplaced on M .The Hamiltonian for the massless Dirac fermion on M takes the form ˆ H = Z d x p det [ g ( x )] ˆΨ † H ˆΨ , (2.13)where H = i / ∇ (2.14)is the Dirac operator on M . In Appendix A we review theform of the Dirac operator on a general spin manifold M ,including our conventions for gamma matrices and so on, andwe suggest that readers take a look at that appendix beforereading the rest of this note.In 2D the Dirac operator simplifies greatly and we have / ∇ = e ja γ a (cid:18) ∂ j − i ω j γ (cid:19) , (2.15)where γ a , a = 1 , , are gamma matrices with frame indices, e ja are the components of the frame vector field e a = e ja ∂ j on M , ω j are the components of the spin connection one-form ω = ω j dx j on M , and the matrix γ is now defined using thegamma matrices with frame indices as γ = i ǫ ab γ a γ b (it stillsatisfies { γ, γ a } = 0 and γ = 1 ). We can write the Dirac op-erator in this simplified form in 2D because in this dimensionthe only non-zero components of the spin connection ω jab on M are ω j = − ω j , and so we can write everything interms of the single quantity ω j := ω j .The massless Dirac Hamiltonian on the curved space M has the same time-reversal symmetry as on flat space. If wechoose the gamma matrices with frame indices so that the γ a are real, then we again find that γ is imaginary, and the time-reversal operation for the case of curved space can be definedusing γ just as in Eq. (2.8) on flat space. With that definitionwe again find that ˆ T ˆ H ˆ T − = ˆ H , so that the massless Diracfermion is still time-reversal invariant even on curved space.Finally, on curved space the Hamiltonian for the massiveDirac fermion coupled to the time-independent spatial gaugefield A = A j dx j takes the form ˆ H A,m = Z d x p det [ g ( x )] ˆΨ † H A,m ˆΨ , (2.16)where H A,m = i / ∇ A + mγ (2.17)and / ∇ A = e ja γ a (cid:18) ∂ j + iA j − i ω j γ (cid:19) (2.18)is the massless Dirac operator on curved space and coupled to A . We again find that ˆ H A,m transforms under time-reversalas ˆ T ˆ H A,m ˆ T − = ˆ H − A, − m . III. REGULARIZATION SCHEME 1
In this section we study the Dirac fermion using our firstregularization scheme, which is the scheme used by Niemiand Semenoff in Ref. 1. In this regularization scheme wecompute the charge Q A,m in the ground state of the massiveDirac fermion theory on the curved space M and in the pres-ence of the time-independent background spatial gauge field A = A j dx j . We then explain that this regularization schemeis consistent with the time-reversal symmetry of the mass-less Dirac fermion, in the sense that Eq. (1.1) holds, i.e., inthe sense that this regularization leads to equal ground state charges for the theory with ( A, m ) and the time-reversed the-ory with ( − A, − m ) .We start by introducing the normal-ordered charge operator ˆ Q for the Dirac fermion ˆ Q = 12 Z d x p det [ g ( x )] h ˆΨ α, † ( x ) , ˆΨ α ( x ) i . (3.1)For comparison, the non-normal-ordered version ofthis operator would just be the familiar expression R d x p det [ g ( x )] ˆΨ † ( x ) ˆΨ( x ) . The reason that we usethe normal-ordered charge operator is that the expectationvalue of this operator is zero in the ground state of thetheory with the background field A set to zero. Using thetime-reversal operation defined above, it is simple to showthat this operator is time-reversal invariant, ˆ T ˆ Q ˆ T − = ˆ Q , (3.2)which is exactly what we expect for the physical electriccharge.
A. Ground state charge and the eta invariant of the spatialDirac operator
We now calculate the charge of the ground state of the mas-sive Dirac fermion theory in the presence of the backgroundfield A . Our discussion is similar to the original derivationin [1], but adapted to the case of curved space. The key ideaof the calculation is to define the regularized charge of theground state using the Atiyah-Patodi-Singer (APS) eta invari-ant [21] of the spatial Dirac operator H A,m . Note that in [1]the APS eta invariant was also referred to as the spectral asym-metry , since the the APS eta invariant of a differential operatoris a regularized version of the difference between the numbersof positive and negative eigenvalues of that operator. We alsonote that the calculation of the ground state charge in this sub-section is quite general, and would also apply to the massiveDirac fermion on a general D -dimensional space. Thus, al-though our notation is specialized to the case of D = 2 , thefinal result of Eq. (3.13) is also valid for spatial dimensions D = 2 as well.The operator H A,m has discrete eigenvalues E n with cor-responding eigenfunctions Φ n ( x ) , where n is an index la-beling the different eigenfunctions. The differential operator H A,m is self-adjoint with respect to the inner product ( φ, ψ ) = Z d x p det [ g ( x )] φ † ( x ) ψ ( x ) , (3.3)and we assume that the eigenfunctions Φ n are orthonormalwith respect to this inner product, (Φ n , Φ n ′ ) = δ nn ′ . (3.4) The eigenvalues are discrete because M is a closed manifold. Then the fermion operators can be defined by the mode ex-pansion ˆΨ( x ) = X n ˆ b n Φ n ( x ) , (3.5)where ˆ b n are fermionic annihilation operators with the stan-dard anticommutation relations { ˆ b n , ˆ b † n ′ } = δ nn ′ . Using thismode expansion, we find that the Hamiltonian operator takesthe diagonal form ˆ H A,m = X n E n ˆ b † n ˆ b n . (3.6)We now define the ground state | i A,m for this system, cor-responding to a Fermi (or Dirac) sea filled up to the energy E = 0 . In the case that H A,m has zero modes, we havea choice about whether to keep those states empty or filledwhen we define the ground state | i A,m . For mathematicalreasons that we discuss below, we choose to leave the zeroenergy states empty in the state | i A,m . Note also that oncewe have computed the regularized charge of the ground state | i A,m , the charges of all other states will be well-defined andwill differ from the charge of | i A,m by integer amounts. Thisfollows from the fact that we can obtain all of the other statesby acting on | i A,m with the ˆ b n and ˆ b † n operators, which addor remove charge from the state | i A,m .With these considerations in mind, we now define theground state | i A,m by the conditions ˆ b n | i A,m = 0 , E n ≥ , (3.7a) ˆ b † n | i A,m = 0 , E n < , (3.7b)i.e., | i A,m has all states with E n < occupied. The chargein the ground state | i A,m is then given by Q A,m = A,m h | ˆ Q | i A,m , (3.8)where ˆ Q is the normal-ordered charge operator from Eq. (3.1).If we plug in the mode expansion for ˆΨ( x ) into this expres-sion for Q A,m , then after some algebra we find the ill-definedexpression Q A,m = − X n ; E n > − X n ; E n < + h , (3.9)where h := dim [ Ker [ H A,m ]] (3.10)is the number of zero modes of H A,m .As discussed by Paranjape and Semenoff [25] (and thenused later by Niemi and Semenoff in [1]), it is possible tomake sense of this expression by defining a regularized ver-sion of it using the APS eta invariant of the spatial Dirac This is not the same as the eta invariant of the spacetime
Dirac operatorthat appears in path integral treatments of the parity anomaly [4, 5, 15, 16]. operator H A,m . Recall that the eta function η ( s ) associatedwith H A,m is [21] η ( s ) = X n ; E n =0 sgn ( E n ) | E n | − s , (3.11)which is an analytic function of s ∈ C when the real part of s is sufficiently large. It is a nontrivial fact that η ( s ) possessesa well-defined analytic continuation to s = 0 . This analyticcontinuation is known as the APS eta invariant and it is de-noted by η (0) . Following [1, 25], we can now use η (0) todefine the regularized difference of the numbers of positiveand negative eigenvalues of H A,m as X n ; E n > − X n ; E n < reg. = η (0) . (3.12)Using this regularization scheme, we find that the charge ofthe ground state | i A,m is given by Q A,m = −
12 ( η (0) + h ) . (3.13)Note that if we had instead decided to define the ground state | i A,m as having the zero modes all filled, then this would bemodified to Q A,m = −
12 ( η (0) − h ) . (3.14)The mathematical reason for choosing the ground state | i A,m to have all zero modes empty is that the combination η (0) + h is exactly the combination that appears in the APS index the-orem (Theorem 3.10 of Ref. 21). This means that if if wechoose to define | i A,m in this way, then the APS index theo-rem can be applied to compute the ground state charge Q A,m in various systems that we might want to study. As we re-marked above, once we have computed an appropriate reg-ularized charge for the state | i A,m , the charges of all otherstates in the Hilbert space are well-defined and differ from Q A,m by integer amounts.
B. Ground state charge of the 2D Dirac fermion
We now compute η (0) + h for the Dirac fermion in 2D withHamiltonian ˆ H A,m . This will give us the ground state charge Q A,m within the regularization scheme of [1]. To compute η (0) + h , first note that since { / ∇ A , γ } = 0 , we have H A,m = ( i / ∇ A ) + m , (3.15)which means that H A,m has no zero modes, and so h = 0 .Next, we consider the calculation of η (0) for H A,m . Let ǫ n and φ n ( x ) be the eigenvalues and eigenfunctions of the mass-less spatial Dirac operator i / ∇ A , i / ∇ A φ n ( x ) = ǫ n φ n ( x ) . (3.16)Then for eigenfunctions φ n ( x ) with non-zero eigenvalue ǫ n ,we have H A,m φ n ( x ) = ǫ n φ n ( x ) + mγφ n ( x ) (3.17a) H A,m γφ n ( x ) = mφ n ( x ) − ǫ n γφ n ( x ) . (3.17b)By diagonalizing the × matrix (cid:18) ǫ n mm − ǫ n (cid:19) , (3.18)we see that for any non-zero ǫ n , the massive Dirac Hamilto-nian H A,m has eigenvalues ± p ǫ n + m . These cancel eachother in the computation of the eta invariant η (0) of H A,m ,so this means that only zero modes of i / ∇ A will contribute to η (0) . We consider these zero modes next.As is well known, zero modes of i / ∇ A can be chosen to beeigenvectors of the chirality matrix γ with eigenvalue ( chiral-ity ) equal to ± . It is now easy to see that zero modes of i / ∇ A that have chirality ± are also eigenfunctions of H A,m witheigenvalue ± m . Let us assume for the moment that m > .Then we find that the eta invariant for H A,m reduces in thiscase to η (0) = ( of positive chirality zero modes of i / ∇ A ) − ( of negative chirality zero modes of i / ∇ A )= Index [ i / ∇ A ] , (3.19)where the last line follows from the definition of Index [ i / ∇ A ] ,the index of the massless Dirac operator i / ∇ A . An applicationof the Atiyah-Singer index theorem [26] (a useful referencefor physicists is Ref. 27) for the Dirac operator i / ∇ A on theclosed spatial manifold M then givesIndex [ i / ∇ A ] = 12 π Z M F , (3.20)where F = F ij dx i ∧ dx j = dA is the field strength for thespatial gauge field A = A j dx j .It is important to note here that we have assumed that thebackground field F obeys a Dirac quantization condition ,which states that the flux of F through M must be an inte-ger multiple of π , π Z M F ∈ Z . (3.21)Since the index of i / ∇ A is integer-valued by definition, it isclear that Eq. (3.20) would not make sense without this condi-tion. Mathematically, this condition is equivalent to the state-ment that A is a connection on a complex line bundle over M ,and the integer (2 π ) − R M F is the first Chern number of thisline bundle (see, for example, Sec. 6 of [27]).As a final comment on the Atiyah-Singer index theorem, wenote that the sign on the right-hand side of Eq. (3.20) can beseen to be correct by considering a simple example with M = S , the unit two-sphere. In this case we have R M dω = 4 π bythe Gauss-Bonnet theorem (the Euler characteristic of S is ). If we consider the field configuration A = ω , then we have (2 π ) − R M F = 1 , and we can also see from Eq. (2.18)that for this choice of A the operator i / ∇ A has a zero modeequal to a constant function on M times the eigenvector of γ with eigenvalue +1 . This confirms that the sign in Eq. (3.20)is correct.Using the result from the Atiyah-Singer index theorem(3.20), we find that within this regularization scheme theground state charge for this system, for m > , is Q A,m> = − π Z M F = −
12 12 π Z M F . (3.22)This is in agreement with the result of Niemi and Semenoffwho considered the case of flat space [1]. We see that curv-ing the space does not change the result. This is true becausethe Atiyah-Singer index theorem for the Dirac operator in 2Dshows that the index of the operator i / ∇ A does not receiveany gravitational contribution (this is not true in higher di-mensions). The connection of the expression for Q A,m to theAtiyah-Singer index theorem was pointed out by Jackiw inRef. 22.The above result was derived under the assumption that m > . If we instead chose m < , then our expressionfor the ground state charge would change sign because wewould instead find that η (0) = − Index [ i / ∇ A ] . Therefore, inthe general case we find that Q A,m = − sgn ( m )2 12 π Z M F . (3.23)An important property of this formula is that when F is anodd multiple of π , we find a half-integer charge in the groundstate. This can occur, for example, if the Dirac fermion theoryis located on the surface of the TI (i.e., M is the surface ofthe TI) and if there is a magnetic monopole of the backgroundelectromagnetic field present in the bulk of the TI. In this casethere would be a flux of π passing through M , and our resultfor Q A,m shows that the ground state of the surface theorywith the mass term mγ would have a charge of ± dependingon the sign of m . C. Discussion on symmetries
We now explain that the regularization scheme of [1],which we have been studying in this section, violates large U (1) gauge invariance, but is consistent with the time-reversalsymmetry of the massless Dirac fermion. The violation oflarge U (1) gauge invariance is easy to see from the fact thatthe charge Q A,m from Eq. (3.23) can take on half-integer val-ues. We now explain the sense in which this regularizationscheme is consistent with time-reversal symmetry.Recall that time-reversal acts on the Hamiltonian ˆ H A,m as ˆ T ˆ H A,m ˆ T − = ˆ H − A, − m , i.e., the effect of time-reversalis to negate A and m . Within the regularization scheme of[1], which uses the eta invariant of the spatial Dirac operator H A,m to define Q A,m , we find using Eq. (3.23) that Q A,m satisfies the relation Q A,m = Q − A, − m . (3.24)This means that in this regularization scheme the ground statecharge for the theory with Hamiltonian ˆ H A,m is equal to theground state charge of the time-reversed theory with Hamil-tonian ˆ H − A, − m . This is the precise sense in which the etainvariant regularization scheme of [1] is consistent with thetime-reversal symmetry of the massless Dirac fermion.One way to understand why Eq. (3.24) holds within thisregularization is to note that the eta invariant is built from thespectrum of the massive Dirac operator H A,m , and H A,m and H − A, − m have the same spectrum. To see this, observe thatif Φ( x ) is an eigenfunction of H A,m with eigenvalue E , then γ Φ ∗ ( x ) is an eigenfunction of H − A, − m with the same eigen-value (the star ∗ denotes complex conjugation). IV. REGULARIZATION SCHEME 2
In this section we study the Dirac fermion using our sec-ond regularization scheme, which is a lattice regularizationscheme for the Dirac fermion on a spatial torus. In this reg-ularization scheme we compute the Hall conductivity σ H,m in the ground state of the Dirac fermion with mass m andin the presence of a background time-independent electricfield E . We find that σ H,m is always an integer (in units of e h ), which implies that this regularization scheme preservesthe large U (1) gauge invariance of the Dirac fermion (thereare no fractionalized quantum numbers found in the Hall re-sponse of the system to the background electric field). On theother hand, we show that this regularization scheme explic-itly breaks the time-reversal symmetry of the massless Diracfermion in the continuum. This fact is also reflected in theresult of the Hall conductivity calculation, where we find that σ H,m = − σ H, − m .As we mentioned in the Introduction, the calculation in thissection is closely related to the calculation of Haldane [24]on a lattice model on the honeycomb lattice that displays anon-zero Hall conductivity in the absence of any net exter-nal magnetic field (i.e., zero total magnetic flux through eachunit cell). Our results here are consistent with the findings inRef. 24. Just as in Haldane’s model on the honeycomb lat-tice, the model that we consider also features a single mass-less Dirac fermion at low energies, but at the cost of breakingtime-reversal symmetry. In fact, it was emphasized in Ref. 24that the honeycomb model considered there should be thoughtof as a condensed matter realization of the parity anomaly. A. Lattice regularization and Hall conductivity
For the lattice regularization we consider a set of two-component fermions on the square lattice and with periodicboundary conditions, and we set the lattice spacing equal to .The Fourier transform of the two-component lattice fermionoperator will be denoted by ˆΨ( k ) , with components ˆΨ α ( k ) , α = 1 , . Here k = ( k , k ) is a wave vector in the firstBrillouin zone of the square lattice, k ∈ ( − π, π ] × ( − π, π ] .The Hermitian conjugate of ˆΨ( k ) is ˆΨ † ( k ) with components ˆΨ † ,α ( k ) , α = 1 , . We take the Hamiltonian for the latticeregularization of the Dirac fermion to be ˆ H lattice = X k ˆΨ † ( k ) H ( k ) ˆΨ( k ) , (4.1)where the Bloch Hamiltonian H ( k ) is given by H ( k ) = sin( k ) σ x +sin( k ) σ z +( ˜ m +2 − cos( k ) − cos( k )) σ y , (4.2)Here ˜ m is a tunable parameter that, in a certain parameterregime, can be identified with the mass m of the continuumDirac fermion. This model features two bands, labeled “ + ”and “ − ”, with energies given by E ± ( k ) = ± λ ( k ) with λ ( k ) = q sin ( k ) + sin ( k ) + ( ˜ m + 2 − cos( k ) − cos( k )) . (4.3)In what follows we will be interested in the case in which thelower band is completely filled and the upper band is com-pletely empty. We also note here that essentially the samemodel was studied in Sec. II.B of Ref. 7.Consider the parameter regime | ˜ m | ≪ . In this regimethe upper and lower bands of the model come closest to eachother at the origin k = (0 , of the Brillouin zone, and thetwo bands actually touch at k = (0 , when ˜ m = 0 . If weTaylor expand the Bloch Hamiltonian H ( k ) near k = (0 , ,then we find that it takes the approximate form H ( k ) ≈ k σ x + k σ z + ˜ mσ y . (4.4)To make contact with our previous discussion of the Diracoperator in the continuum, recall that we worked in a basisin which the gamma matrices γ a , a = 1 , , were both real,and so the third matrix γ was imaginary. One concrete choicefor these matrices is γ = σ x , γ = σ z , which gives γ = σ y . With this choice, we see that the Fourier transform of themassive Dirac operator i / ∇ + mγ on flat space has exactly theform of Eq. (4.4) with m = ˜ m . (4.5)The discussion in the previous paragraph shows that in theregime | ˜ m | ≪ , the low energy description of this latticemodel consists of a single continuum Dirac fermion with mass m = ˜ m and located at the point k = (0 , in the Brillouinzone of the square lattice. In addition, the full lattice modeldoes not have any additional phase transitions for any ˜ m > ,while the next transition for ˜ m < occurs at ˜ m = − . At ˜ m = − the upper and lower bands touch at the two points k = ( π, and k = (0 , π ) . This means that this latticemodel is a sensible regularization for a single continuum Diracfermion as long as we keep the parameter ˜ m in a region near ˜ m = 0 and far away from the next transition at ˜ m = − .We now turn to the calculation of the Hall conductivity forthe Dirac fermion in this lattice regularization. We first com-pute the Hall conductivity σ lattice H, ˜ m for the lattice model, which is By a phase transition we mean a value of the parameter ˜ m at which theupper and lower bands touch. well-defined for any value of the parameter ˜ m for which thereis a gap between the upper and lower bands of the model.We then identify the Hall conductivity σ H,m of the contin-uum Dirac fermion with the lattice Hall conductivity σ lattice H, ˜ m in the appropriate parameter regime where the lattice modelis a sensible regularization of the continuum Dirac fermion.Specifically, we have the following identifications, σ H,m> = σ lattice H, ˜ m> (4.6a) σ H,m< = σ lattice H, − < ˜ m< . (4.6b)The Hall conductivity σ lattice H, ˜ m for the lattice model is definedprecisely as follows. We first place the system in a static elec-tric field E that points in the x direction, so that E = (0 , E ) .We then compute the current j that flows in the x direction.Then σ lattice H, ˜ m is defined as the constant that relates j to E , j = σ lattice H, ˜ m π E . (4.7)More precisely, σ lattice H, ˜ m encodes the spatially uniform (i.e., zerowave vector) part of the linear response of j to the appliedfield E . Note also that the factor of (2 π ) − appearing here isactually e h in our units where e = ~ = 1 .As discussed above, we consider the case where the lowerband is completely filled and the upper band is completelyempty, as this filling corresponds to the continuum groundstate in which the Dirac sea of negative energy states is com-pletely filled. In this case we can compute σ lattice H, ˜ m using variousmethods including a direct linear response calculation usingthe Kubo formula [28], or the semi-classical theory of wavepacket dynamics in solids [29]. Both methods lead to the re-sult that σ lattice H, ˜ m = − Z d k π Ω − ( k ) , (4.8)where Ω jℓ − ( k ) (with j, ℓ = 1 , ) are the components of the Berry curvature of the filled lower band (the “ − ” band) of thelattice model, and where the integral is taken over the Bril-louin zone of the square lattice. The Berry curvatures Ω jℓ ± ( k ) for the “ + ” and “ − ” bands of the model are defined preciselyas follows. Let | u k , ± i be the eigenvector of the Bloch Hamil-tonian corresponding to the ± band of the model, H ( k ) | u k , ± i = E ± ( k ) | u k , ± i . (4.9)If we define the Berry connection for the ± band as A j ± ( k ) = i D u k , ± (cid:12)(cid:12)(cid:12) ∂u k , ± ∂k j E , then the Berry curvature for the ± band isgiven by Ω jℓ ± ( k ) = ∂ A ℓ ± ( k ) ∂k j − ∂ A j ± ( k ) ∂k ℓ . (4.10)We now provide some details of the Berry curvature calcu-lation. For this calculation it is convenient to introduce spher-ical coordinate variables Θ( k ) and Φ( k ) and to rewrite theBloch Hamiltonian in terms of these variables as H ( k ) = λ (cid:16) sin(Θ) cos(Φ) σ x + sin(Θ) sin(Φ) σ y + cos(Θ) σ z (cid:17) , (4.11) where λ ( k ) was defined in Eq. (4.3), and where we have sup-pressed the dependence of λ ( k ) , Θ( k ) , and Φ( k ) on k forbrevity. This type of parametrization for a two-band Hamilto-nian has been used, for example, in Sec. I.C.3 of Ref. 29. Interms of these variables the eigenvector | u k , − i for the lowerband of the model takes the form | u k , − i = (cid:18) e − i Φ sin (cid:0) Θ2 (cid:1) − cos (cid:0) Θ2 (cid:1) (cid:19) . (4.12)A straightforward calculation then shows that the Berry cur-vature for the lower band is given by Ω − ( k ) = − ǫ jℓ ∂ Φ( k ) ∂k j ∂ Θ( k ) ∂k ℓ sin(Θ( k )) , (4.13)and so we have σ lattice H, ˜ m = 14 π Z d k ǫ jℓ ∂ Φ( k ) ∂k j ∂ Θ( k ) ∂k ℓ sin(Θ( k )) . (4.14)This expression shows that σ lattice H, ˜ m is an integer and is equalto the number of times that the unit vector specified by Θ( k ) and Φ( k ) covers the unit two-sphere S as k varies over theBrillouin zone of the square lattice. This can be seen from thefact that sin(Θ) d Θ d Φ is the area element on S , and from thefact that ǫ jℓ ∂ Φ( k ) ∂k j ∂ Θ( k ) ∂k ℓ is the Jacobian of the map from theBrillouin zone to S (the normalizing factor of π is also thetotal area of S ).One way to proceed with the calculation of σ lattice H, ˜ m would beto work out explicit expressions for Θ( k ) and Φ( k ) in termsof k and ˜ m and then evaluate the integral in Eq. (4.14). Asa practical matter, however, the easiest way to compute σ lattice H, ˜ m is to evaluate the integral numerically for a particular value of ˜ m in each parameter range where the Hamiltonian has a gapbetween the upper and lower bands. We can use this methodbecause we already know that σ lattice H, ˜ m is an integer-valued topo-logical invariant that takes a constant value in each parameterrange where the Hamiltonian has a gap. We find that the Hallconductivity for the lattice model is given by σ lattice H, ˜ m = , ˜ m > − , − < ˜ m < , − < ˜ m < − , ˜ m < − . (4.15)Only the first two cases ˜ m > and − < ˜ m < are relevantfor our original goal of studying a regularization of the con-tinuum Dirac fermion. Using Eq. (4.6) to compute σ H,m from σ lattice H, ˜ m , we find that the result for σ H,m , for either sign of themass m , can be written in the compact form σ H,m = sgn ( m ) − . (4.16) B. Discussion on symmetries
We now point out that this lattice regularization preservesthe large U (1) gauge invariance of the original massless Dirac0fermion, but breaks the time-reversal symmetry. To see thatlarge U (1) gauge invariance is preserved, it is sufficient tonote that this regularization yields an integer value for the Hallconductivity σ H,m . In other words, we do not find any frac-tionalization of quantum numbers associated with the U (1) symmetry of charge conservation. This result makes sensesince in this regularization scheme we are dealing with a well-defined lattice model with charge conservation symmetry.We now show that the lattice regularization that we havebeen discussing does not possess the time-reversal symmetryof the continuum massless Dirac fermion, even when the massparameter ˜ m in the lattice model is set to zero. To see this,note that for the choice γ = σ x , γ = σ z , and γ = σ y , thetime-reversal operator defined in Eq. (2.8) would act on thelattice fermions ˆΨ( k ) as ˆ T ˆΨ α ( k ) ˆ T − = ( σ y ) αβ ˆΨ ,β ( − k ) (4.17a) ˆ T ˆΨ † ,α ( k ) ˆ T − = ˆΨ † ,β ( − k )( σ y ) βα , (4.17b)where we note that k is negated by the time-reversal oper-ation. Then the condition of time-reversal invariance of theHamiltonian, ˆ T ˆ H lattice ˆ T − = ˆ H lattice , is equivalent to the ma-trix equation σ y H ∗ ( k ) σ y = H ( − k ) . (4.18)However, it is easy to check that this condition is not satisfiedby H ( k ) , even when ˜ m = 0 . It follows that this lattice regu-larization explicitly breaks the time-reversal symmetry of thecontinuum massless Dirac fermion.Finally, we note that the breaking of time-reversal symme-try in this regularization scheme can also be seen from the factthat σ H,m = − σ H, − m , (4.19)i.e., time-reversed theories do not have opposite values of theHall conductivity within this regularization scheme. V. EFFECTIVE ACTIONS FOR THE TWOREGULARIZATIONS
In this final section we compute, for each regularizationscheme, the effective action S eff [ A ] that encodes the responseof the system to the background electromagnetic field A = A µ dx µ . On curved space the physical three-current j µ ( x ) thatarises as a response to the background field A is obtained from S eff [ A ] by functional differentiation as j µ ( x ) = − p det [ g ( x )] δS eff [ A ] A µ ( x ) , (5.1) The electric field E is invariant under time-reversal, so the time-reversalpartner of the theory with mass m in the presence of E is the theory withmass − m in the presence of the same field E . where we remind the reader that x = ( x , x , x ) is the space-time coordinate and x = ( x , x ) is the spatial coordinate.The overall minus sign appearing here is a matter of conven-tion. We chose it because with this sign the j ( x ) term in theeffective action has the form − Z d x p det [ g ( x )] j ( x ) A ( x ) , which has the correct sign for the action arising from the po-tential energy of the charge density j in the presence of thescalar electromagnetic potential A .For both regularizations considered in this note, we find thatthe effective action that encodes the response has the Chern-Simons form S CS [ A ] = k π Z A ∧ dA = k π Z d x ǫ µνλ A µ ∂ ν A λ , (5.2)for an appropriate choice of the level k . To find the correctvalue of k in each case, we compute the response that followsfrom S CS [ A ] in the two situations considered in this note.We start by computing the charge that follows from S CS [ A ] for a system on a closed spatial manifold M and in the pres-ence of a time-independent spatial gauge field. This is exactlythe physical quantity that we computed using the regulariza-tion scheme of Niemi and Semenoff in Sec. III. This charge isgiven by Q = Z d x p det [ g ( x )] j ( x )= − k π Z M F , (5.3)where in the second line we plugged in the result for j ( x ) thatfollows from functional differentiation of S CS [ A ] . To matchwith our answer for Q A,m from Eq. (3.23), we find that wemust choose the level k to be k NS = sgn ( m )2 . (5.4)Then for the regularization scheme of Niemi and Semenoff wefind the effective action S ( NS ) eff [ A ] = sgn ( m )2 14 π Z A ∧ dA . (5.5)Next, we compute the Hall conductivity for the case wherespace is a flat torus, which is exactly the physical quantitythat we computed using the lattice regularization scheme inSec. IV. For this calculation we study the current that flowsin the x direction in response to a static electric field E =(0 , E ) pointing in the x direction. For the effective actionof the Chern-Simons form we find that j = k π E , (5.6)where we needed to use the equation F = ∂ A − ∂ A = − E , which relates the physical electric field to the compo-nents of A (see the end of Appendix A for a review of this re-lation). This equation implies a Hall conductivity of σ H = k .1In this case, to match our answer for σ H,m from Eq. (4.16),we must choose the level k as k lattice = sgn ( m ) − . (5.7)Then for the lattice regularization scheme on a spatial toruswe find the effective action S ( lattice ) eff [ A ] = (cid:18) sgn ( m ) − (cid:19) π Z A ∧ dA . (5.8)It is now clear that the effective actions computed using thetwo different regularization schemes differ by a Chern-Simonscounterterm as S (lattice)eff [ A ] = S (NS)eff [ A ] −
12 14 π Z A ∧ dA , (5.9)where we see that the Chern-Simons counterterm has a fractionally-quantized level equal to − . This differencebetween the effective actions for these two regularizationschemes exactly matches the expectation from the originalpath integral treatment of the parity anomaly [2, 3]. VI. CONCLUSION
In this note we reviewed the parity anomaly of the masslessDirac fermion in dimensions in the context of the Hamil-tonian formalism, as opposed to the more conventional dis-cussion within the path integral formalism. Our first goal withthis presentation was to explain the parity anomaly in a waythat would be more approachable for condensed matter physi-cists. To this end, we have tried to show how the anomaly ismanifested in the calculation of concrete physical quantitiessuch as the charge of the ground state in a background spa-tial gauge field A (Sec. III) and the Hall conductivity of theground state in a background electric field E (Sec. IV).Our second goal was to understand the precise relation be-tween time-reversal symmetry and the charge of ± that ap-pears on the surface of the TI (whose surface theory is themassless Dirac fermion) when a magnetic monopole is presentin the bulk of the TI. The regularization scheme that leads tothis half-quantized charge is known and was originally con-sidered by Niemi and Semenoff in Ref. 1. In this note weexplained that this regularization scheme is consistent withthe time-reversal symmetry of the massless Dirac fermion, inthe precise sense of Eq. (1.1). To the best of our knowledge,the consistency of the regularization scheme of [1] with time-reversal symmetry has not been discussed in detail in the ex-isting literature (see, however, the comparison with a parity-preserving point-splitting regularization scheme in Ref. 23).This observation is important because it fits in with the gen-eral picture of the parity anomaly, which states that a givenregularization scheme can preserve either the time-reversalsymmetry, or the large U (1) gauge invariance of the masslessDirac fermion, but not both.An interesting direction for future work on this topic wouldbe to investigate a bosonic analogue of the parity anomaly in quantum field theories with U (1) and time-reversal symmetrythat can appear on the surface of the bosonic topological insu-lator [30, 31]. The bosonic topological insulator is the closestanalogue, in a bosonic system with U (1) and time-reversalsymmetry, of the more familiar fermion TI state. Some ideasabout the form of this bosonic anomaly have already been pre-sented in Ref. 32. The key physical property of the anomalythat was discussed there was the fact that a bosonic theory pos-sessing this anomaly can be driven into a time-reversal break-ing state with a Hall conductivity of (in units of e h ), whichis exactly half of the allowed Hall conductivity that can beachieved in a (non-fractionalized) phase of bosons that canexist intrinsically in dimensions [33, 34].An additional reason to look for such an anomaly in dimensions is the demonstration in Ref. 35 that in dimen-sions there is a bosonic anomaly that is an exact analogue ofa well-known fermionic anomaly in the same dimension [36].The action for a massless Dirac fermion in dimensionscoupled to a background gauge field A = A dx has bothlarge U (1) gauge invariance and a unitary charge conjugation (or particle-hole) symmetry. However, it was shown in Ref. 36that it is impossible to regularize this theory in a way that pre-serves both of these symmetries. This anomaly is clearly anal-ogous to the parity anomaly of the massless Dirac fermion in dimensions, but with charge conjugation instead of time-reversal as the relevant discrete symmetry. It was recentlyshown in Ref. 35 that an exact analogue of this anomaly existsin a bosonic theory in dimensions with the same sym-metries . In addition, the calculation of the bosonic anomalyin Ref. 35 employed the equivariant localization technique,which revealed that the bosonic and fermionic anomalies in dimensions have the same mathematical origin. Indeed,the derivation showed that both anomalies follow from theform of the APS eta invariant for the Dirac operator in dimensions. Based on this simple example, we expect that itwould be interesting to search for a bosonic analogue of theparity anomaly in dimensions. ACKNOWLEDGMENTS
We thank M. Levin for helpful discussions on this topic andfor a collaboration on a related project. M.F.L. acknowledgesthe support of the Kadanoff Center for Theoretical Physics atthe University of Chicago.
Appendix A: Conventions
In this appendix we review our conventions and notationfor the Dirac operator on curved space and our conventionsfor the electromagnetic field. The information contained inthis appendix is used in Sec. II of the main text of this note,where we review the Hamiltonian for the Dirac fermion on flatand curved space.We consider a Dirac fermion on a spacetime of the form M× R , where M represents D -dimensional space and R rep-resents time. We assume that M is an orientable Riemannian2manifold. We also need to assume that M is a spin manifold so that we can consistently place fermions on M× R . We alsoassume that M is closed (i.e., compact and without bound-ary) and connected. Coordinates on the full spacetime will bedenoted by x µ where the (Greek) spacetime indices µ, ν, . . . take the values { , , . . . , D } . The spatial coordinates on M are x j where the (Latin) spatial indices j, k, . . . take the val-ues { , , . . . , D } . We denote by x = ( x , . . . , x D ) the fullvector of spacetime coordinates and by x = ( x , . . . , x D ) thevector of spatial coordinates. We also use the standard sum-mation convention in which we sum over any index (Latin orGreek) that is repeated once as a subscript and once as a su-perscript in any expression.We denote by G µν the components of the spacetime metric G , which we choose to have signature (1 , − , . . . , − (i.e., a“mostly minus” signature). Since our spacetime is a productof a curved space M and flat time direction R , the spacetimemetric G has the form G = dx ⊗ dx − g jk ( x ) dx j ⊗ dx k , (A1)where g jk ( x ) are the components of an ordinary Rieman-nian metric g on M . Note that with this definition we havedet [ g ( x )] > for all x ∈ M , where det [ g ( x )] is the determi-nant of g jk ( x ) .We now discuss the construction of the spatial Dirac opera-tor on M . The first step is to define the the coframe one-forms e a = e aj dx j and frame vector fields e a = e ja ∂ j ( ∂ j ≡ ∂∂x j ),where frame indices a, b, . . . take the values { , . . . , D } . Thecomponents of these objects are defined in terms of the metric g jk by e aj δ ab e bk = g jk (A2a) e ja g jk e kb = δ ab . (A2b)The frame and coframe components are inverses of each otherwhen considered as matrices, e aj e jb = δ ab and e ak e ja = δ jk . Inaddition, we have the relation det [ e ( x )] = p det [ g ( x )] , wheredet [ e ( x )] is the determinant of the coframe e aj viewed as amatrix with row index a and column index j .To construct a Dirac operator on M we also need a setof gamma matrices γ a with frame indices a ∈ { , . . . , D } .These satisfy the Clifford algebra { γ a , γ b } = 2 δ ab , andin terms of them we define the rotation generators γ ab := [ γ a , γ b ] (these are generators of the group Spin ( D ) ). Notethat these gamma matrices all square to the identity and so wecan choose them to be Hermitian.The next ingredient we need is the spin connection on M .The spin connection one-form ω ab = ω jab dx j on M is de-fined by the relation ∇ j e b = ω jab e a , (A3)where ∇ j ≡ ∇ ∂ j denotes the connection on the tangent bun-dle of M . Under a local rotation of the coframes e a → Λ ab e b ,the spin connection transforms as ω jab → (Λ ω j Λ − ) ab − ( ∂ j ΛΛ − ) ab . (A4) If we assume metric compatibility of the spin connection, thenwe have ω ab = − ω ba , i.e., ω ab is a one-form that takes valuesin the Lie algebra of the group SO ( D ) .The curvature and torsion two-forms on M are R ab = dω ab + ω ac ∧ ω cb (A5a) T a = de a + ω ab ∧ e b . (A5b)If we assume that the torsion vanishes, T ajk = 0 for all a, j, k , where T ajk are the components of the two-form T a = T ajk dx j ∧ dx k , then the metric-compatible spin connectiontakes the explicit form ω jab = e kb Γ ℓjk e aℓ − e kb ∂ j e ak , (A6)where Γ ℓjk is the Levi-Civita connection, Γ ℓjk = 12 g ℓm ( ∂ k g mj + ∂ j g mk − ∂ m g jk ) . (A7)With all of these conventions in place, we can now constructthe Dirac operator on the D -dimensional space M . The Diracoperator D is given by D = i / ∇ , (A8)where / ∇ = e ja γ a ( ∂ j + 14 ω jbc γ bc ) . (A9)One can check that D is Hermitian with respect to the innerproduct ( ψ, φ ) = Z d D x det [ e ( x )] ψ † ( x ) φ ( x )= Z d D x p det [ g ( x )] ψ † ( x ) φ ( x ) , (A10)which is the appropriate inner product for spinors φ and ψ onthe Riemannian manifold M . To verify that D is Hermitianone needs to use the fact that the torsion two-form vanishes, T ajk = 0 for all a, j, k , as well as the fact that M is closed(no boundary terms arise in integration by parts because M does not have a boundary).To close this appendix we also discuss our conventions forthe electromagnetic field. We specialize the discussion to thecase of D = 2 , which is the case that we consider in themain text of this note. Let A = A µ dx µ be the one-form for aconfiguration of a background electromagnetic field. Since weassume a spacetime metric with signature (1 , − , − , whenthe space M is flat (i.e., M = R ) the components F µν ofthe field strength two-form F = dA = F µν dx µ ∧ dx ν arerelated to the usual electric and magnetic fields as F = − B (A11a) F = − E (A11b) F = E , (A11c)where B is the magnetic field perpendicular to the plane and E = ( E , E ) is the usual electric field in the plane. 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