Chiral Anomaly in Interacting Condensed Matter Systems
Colin Rylands, Alireza Parhizkar, Anton A. Burkov, Victor Galitski
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Chiral Anomaly in interacting Condensed Matter Systems
Colin Rylands, Alireza Parhizkar, Anton A. Burkov,
2, 3 and Victor Galitski Joint Quantum Institute and Condensed Matter Theory Center,University of Maryland, College Park, MD 20742, USA Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: February 9, 2021)The chiral anomaly is a fundamental quantum mechanical phenomenon which is of great im-portance to both particle physics and condensed matter physics alike. In the context of QED itmanifests as the breaking of chiral symmetry in the presence of electromagnetic fields. It is alsoknown that anomalous chiral symmetry breaking can occur through interactions alone, as is the casefor interacting one dimensional systems. In this paper we investigate the interplay between thesetwo modes of anomalous chiral symmetry breaking in the context of interacting Weyl semimetals.Using Fujikawa’s path integral method we show that the chiral charge continuity equation is mod-ified by the presence of interactions which can be viewed as including the effect of the electric andmagnetic fields generated by the interacting quantum matter. This can be understood further usingdimensional reduction and a Luttinger liquid description of the lowest Landau level. These effectsmanifest themselves in the non-linear response of the system. In particular we find an interactiondependent density response due to a change in the magnetic field as well as a contribution to thenon-equilibrium and inhomogeneous anomalous Hall response while preserving its equilibrium value.
Introduction —Modern condensed matter physics hasbenefited greatly from concepts originally introduced inthe context of high energy physics. One such conceptis the chiral anomaly; the breaking of classical chiralsymmetry in a quantum theory [1, 2]. Within QED itarises through the need to regularize certain loop dia-grams which contain differences of linearly divergent in-tegrals. The appropriate regularization can either pre-serve charge conservation symmetry, chiral symmetry orsome combination of the two but not both. On physicalgrounds, the first of these is chosen, which brings abouta source term for the divergence of the chiral current, j µ ,whenever electric and magnetic fields are not orthogonal, ∂ µ j µ = e π E · B . (1)Here E and B are the electric and magnetic fields andwe have set c = ~ = 1. This expression, although derivedfrom a single triangle diagram in perturbation theory wasshown to obey non-renormalization theorems; higher or-der terms cannot modify the form of this equation andare accounted for by replacing the bare fields and chargewith their renormalized values [3]. Later, this was re-inforced when it was discovered that the chiral anomalymanifests in the path integral formalism through the non-invariance of the measure under a chiral symmetry trans-formation [4–6].The chiral anomaly is present for all odd spatial di-mensions [7–9] but is particularly important in one spa-tial dimension where it is crucial for the proper treat-ment of interacting fermionic theories through bosoniza-tion [10, 11]. A prominent feature therein is that chiralsymmetry breaking can occur due to the presence of in-teractions even when electromagnetic fields are absent.Indeed, it is well known, although perhaps not expressed in this way, that the chiral charge conservation equationfor interacting fermions is [12, 13] ∂ µ j µ = λ π ∂ j , (2)where λ / eE/π term on the righthand side [14].Chiral symmetry, is an emergent low energy propertyin condensed matter systems, appearing due to an evennumber of chiral modes crossing the Fermi surface whichare actually part of the same band. In this respect, theanomaly can be understood in non-interacting systemsvia the pumping of chiral charge through the bottom ofthe band from one node to another [14]. Despite notbeing a fundamental symmetry, it is intimately relatedto many key concepts including the quantized Hall con-ductance, e.g. through Laughlins’s argument [15], andmore recently the existence of topological metals such asthe Weyl semimetal [16–26]. In this Letter we examinethe interplay between the two modes of chiral symmetrybreaking expressed through (1) and (2) in the context ofinteracting condensed matter systems. Specifically, weshow that for short range interactions the anomaly canbe written as ∂ µ j µ = e π ˜ E · ˜ B , (3)where ˜ E and ˜ B , defined below, contain the effect of boththe electromagnetic fields in a manner similar to (1) andthe interactions through terms like in (2).The effect of interactions in Weyl semimetals has beenconsidered previously using perturbative means [27–31].In contrast, our work takes a non-perturbative approachand considers the interactions from the outset throughthe chiral anomaly itself. By utilizing (3) we predict anumber of new non-perturbative phenomena found be-yond linear response which can be expected in interactingWeyl semimetals and attributed to the chiral anomaly. Model —We consider a model of interacting Diracfermions, ψ , in the presence of a constant backgroundmagnetic field in 3 + 1 dimensions. The action is S = S + S int with S = Z d x ¯ ψ ( x ) (cid:2) i /∂ + e /A (cid:3) ψ ( x ) , (4)where we have employed Dirac slash notation and ¯ ψ = ψ † γ . For later convenience we split the gauge field, A µ = A µ + ˜ A µ , into a part describing the magnetic fieldpointing along the ˆ z direction, A µ = xB z δ µ and a per-turbation around it, ˜ A µ . The magnetic field breaks theLorentz invariance down to rotational invariance in thetransverse plane spanned by the ˆ x and ˆ y directions andreduced (1 + 1)-d Lorentz symmetry in the longitudinaldirections. The general short range current-current in-teraction is of the form S int = − Z d x λ µν j µ ( x ) j ν ( x ) , (5)where j µ ( x ) = ¯ ψ ( x ) γ µ ψ ( x ) is the fermion current with λ µν = λ µα λ αν being the interaction strength. Althoughcertain results presented in this Letter are more gen-eral, we have restricted our focus to special cases of λ µν = λ η µν , which preserves Lorentz symmetry, λ µν = λ η µ η ν + λ η µ η ν which preserves the reduced sym-metries of our system if λ = λ and which gives density-density interaction when λ = 0 [32]. Evidently, depend-ing on the choice of λ µν some of the symmetries of themodel may be broken, e.g. Lorentz invariance, but theydo not break the classical chiral symmetry. These in-teractions are RG irrelevant and typically are not con-sidered, however we will see that in the presence of theconstant magnetic field, they should not be discounted. Chiral Anomaly & Interactions —To study the chiralanomaly in the presence of interactions we proceed usinga generalization of Fujikawa’s path integral method [4, 5].The path integral is I = Z D (cid:2) ¯ ψψa µ (cid:3) exp i n Z d x ¯ ψi /Dψ + 12 a µ a µ o , (6)where we have introduced the Hubbard-Stratonovichfield a µ ( x ) which has been included in the general-ized Dirac operator as D µ = ∂ µ − ieA µ − iλ µν a ν , andwhose equation of motion reads a µ = − λ νµ j ν . Integra-tion over the auxiliary a µ field gives the original action S = S + S int back. We now perform an infinitesimal chi-ral transformation, ψ → e iθ ( x ) γ ψ, ¯ ψ → ¯ ψe iθ ( x ) γ which results in a shift of the action, S → S + Z d x θ ( x ) [ ∂ µ j µ − A ( x )] , (7)where j µ ( x ) = ¯ ψ ( x ) γ µ γ ψ ( x ) is the chiral current. Thefirst term in the brackets arises from the classical shiftof the action itself whereas the second is the anomalousterm which is a result of the non-invariance of the mea-sure. It takes the standard form A ( x ) = 2Tr[ θ ( x ) γ ] ormore explicitly A ( x ) = 2 θ ( x ) X n ϕ † n ( x ) γ ϕ n ( x ) , (8)where ϕ n ( x ) are some orthonormal basis of wavefunc-tions used to expand the Grassmann variables ψ ( x ) = P c n φ n ( x ). In the absence of interactions the natu-ral choice is to take these to be the eigenfunctions of /D = /∂ − ie /A and regularize this divergent sum us-ing the heat kernel method P n → lim M → P n e − /D /M .Such a choice of basis has the crucial benefit of for-mally diagonalizing the action. This results in the famil-iar anomalous term A ( x ) = θ ( x ) e π F µν F ρσ ǫ µνρσ with F µν = ∂ µ A ν − ∂ ν A µ and ǫ µνρσ is the Levi-Cevita sym-bol. The chiral anomaly equation (1) then follows. Notethat owing to the fact that { γ , /D } = 0 it is evidentfrom (8) that anomalous term is generated solely by thezero modes of the Dirac operator.In the presence of interactions we regularize thesum using the generalized Dirac operator, includ-ing the Hubbard-Stratonovich field /D = γ µ ( ∂ µ − ieA µ − iλ µν a ν ) (For similar approaches see [33, 34]).Following the same procedure we find, A ( x ) = θ ( x ) π F µν F ρσ ǫ µνρσ where F µν = ∂ µ ( eA ν + λ να a α ) − ∂ ν ( eA µ + λ µβ a β ) and after integrating over a µ we find ∂ µ j µ = e π F µν F ρσ ǫ µνρσ − e π ǫ µνρσ λ σα ∂ µ A ν ∂ ρ j α + 14 π ǫ µνρσ λ να λ σβ ∂ µ j α ∂ ρ j β . (9)We see that there are terms depending only on the elec-tromagnetic field, only on the presence of interactionsand a mixed term requiring the presence of both. Afterdefining ˜ E i = E i − e (cid:2) λ iβ ∂ − λ β ∂ i (cid:3) j β , (10)˜ B i = B i − e ǫ ijk (cid:2) λ jβ ∂ k − λ kβ ∂ j (cid:3) j β , (11)equation (3) is obtained.We could view this as a screening, by the interac-tions of the electric and magnetic fields which are re-sponsible for the non-conservation of the chiral charge.This can be seen more clearly by allowing the electro-magnetic fields to be dynamical and, for simplicity, con-sidering λ µν = λ η µ η ν , i.e. density-density interac-tions. Upon treating the electromagnetic field in a semi-classical fashion through ej ν = ∂ µ F νµ , we find that˜ E = E − λ e ∇ ( ∇ · E ) and ˜ B = B . Therefore the anoma-lous chiral symmetry breaking is generated not only bythe background fields but also by the fluctuations inducedby the interacting matter. Dimensional reduction to a Luttinger liquid — The chi-ral anomaly, in the free case, can be straightforwardly un-derstood through dimensional reduction of the (3 + 1)-dsystem to the (1 + 1)-d linearly dispersing lowest Landaulevel (LLL) via a magnetic field, B z [14]. We show nowthat one can also arrive at (3) using dimensional reduc-tion provided that the LLL is described by a Luttingerliquid.Let us consider a system that is homogeneous in thetransverse directions along ˆ x and ˆ y . In particular, theonly external fields are in the longitudinal ˆ z directionand there are no currents which vary along ˆ x and ˆ y . Ouranomalous relation then reduces to ∂ µ j µ = e π F µν F ρσ ǫ µνρσ − eB z π λ σα ǫ ρσ ∂ ρ j α . (12)Assuming that the interacting system still forms Lan-dau levels, the zero modes which are responsible for theanomaly are present only on the LLL. As in the freecase, the magnetic field achieves a dimensional reductionfrom the (3 + 1)-d theory to the LLL which is effectively(1 + 1)-d. Within the LLL the following identity is valid ǫ ρσ γ σ = γ γ ρ and after some rearranging we arrive at ∂ µ j µ = 11 + n λ /π e π E z B z − n (cid:0) λ − λ (cid:1) /π n λ /π ∂ j , (13)where n = eB z π . Here we have also specialized to thecase where the interaction tensor is diagonal. In deriv-ing this equation we have assumed that Landau levelsare formed in the interacting system or more preciselythat there is a spin polarized LLL on which the anomalyis generated. We have made no assumptions on the na-ture of Landau levels or how they arise, only that theyexist which seems a physically reasonable proposition es-pecially in the limit of large background field. In theopposite limit of zero background field (13) reduces tothe noninteracting result.The second term in (13) is similar to (2) while themodification of the first has been discovered before inearly studies of interacting (1 + 1)-d fermions [35, 36].To understand their appearance better we introduce thefollowing action consisting of N coupled (1+1)-d bosonicfields S = N X j =1 Z d x π ( [ ∂ t φ j ] + [ ∂ x φ j ] − e [ ǫ mn A m ∂ n ] φ j + X j ≤ k λ π [ ∂ x φ j ][ ∂ x φ k ] + λ π [ ∂ t φ j ][ ∂ t φ k ] ) , (14)with ǫ mn the 2-d Levi-Cevita symbol. This is equivalent,through bosonization, to a system of N flavors of inter- acting chiral fermions, χ †± ,j = √ ρ e i [ ± φ j − R t dt∂ x φ j ] where ρ is the background density [12, 13]. The bosons are re-lated to the fermionic charge and chiral charge density via P σ = ± : χ † σ,j χ σ,j := − ∂ x φ j /π and P σ = ± σ : χ † σ,j χ σ,j := ∂ t φ j /π with : : indicating normal ordering.The model is flavor symmetric and accordingly boththe interactions and the gauge field affect only the sym-metric combination, φ S = √ N P j φ j . After a canonicaltransformation and retaining only the symmetric termswe arrive at the following action S S = Z d x π (cid:0) λ N/π (cid:1) [ ∂ x φ S ] + (cid:0) λ N/π (cid:1) [ ∂ t φ S ] − √ N eA ∂ x φ S + 2 √ N eA ∂ t φ S . (15)Note that here the gauge field couples to the fermionicdensity rather than through minimal coupling with thesymmetric boson, an important distinction which wecomment on further below. The chiral anomaly is nowmanifest in the Euler-Lagrange equation for φ S . Cal-culating this we find agreement with (13) provided oneidentifies the number of flavors with the Landau level de-generacy, N = n = eB z / π as well as j = P ∂ t φ j /π and j = P ∂ x φ j /π which follows from the properties of γ µ in (1 + 1) − d.Our path integral calculation is therefore consistentwith a description of the LLL as a Luttinger liquid. ALuttinger liquid approach has also been adopted in [37]to investigate the effect of disorder which we shall notconsider here. The Luttinger liquid consists of a pair ofinteracting chiral fermions χ †± ,S = √ ρ e i [ ± φ S − R t dt∂ x φ S ]formed from the symmetric boson which couple to thegauge field and the decoupled non-symmetric fields whichplay no role. The excitations of the LLL are still chiralbut are distinct from these bare fermions and are cre-ated by Ψ †± = √ ρ e i h ± √ λ N/πφ S − √ λ N/π R t dt∂ x φ S i which coincide with χ †± ,S only when interactions are ab-sent. In general these excitations carry different electricand chiral charges from χ †± ,s which can be seen throughthe coefficients of φ S and R t dt∂ x φ S in the exponential.Had our gauge field coupled to these instead then wewould find that the chiral anomaly equation was unmod-ified. A similar situation also arises when comparing con-ductances in one dimensional systems [38].As mentioned in the introduction, the chiral anomalyis related to Laughlin’s argument for quantized Hall con-ductance [15]. Therein one can argue that the invari-ance of the Hall conductance to local interactions im-plies invariance of the chiral anomaly for the edge modesof Laughlin’s cylinder and vice versa. We remark thatour results are not in contradiction to this as our (1 + 1)-d chiral modes are not spatially separated as they arein Laughlin’s argument. In order to see similar interac-tion effects as ours one would need to include non-localinteractions between the edges. Consequences for Weyl Semimetals —We now turn ourattention to the consequences of (3) for interacting in-teracting condensed matter systems, in particular Weylsemimetals. These are a recently discovered type of gap-less topological matter possessing a number of distinctivefeatures which arise due to the chiral anomaly includinga large negative magnetoresistance [14, 39–41] and ananomalous Hall response [34, 42]. The low energy de-scription of such systems is given by S = S + S b + S int with S b = R d x b µ j µ , where b µ separates the Weyl nodesin momentum and energy space. The effect of this termis most conveniently seen by performing a chiral rotation ψ → e ib µ x µ γ ψ, ¯ ψ → ¯ ψe ib µ x µ γ which removes S b at thecost of generating a Chern-Simons term, S CS due to thechiral anomaly. In terms of the Hubbard-Stratonovichfield this is S CS = Z d x π ǫ νµρσ b µ [ eA ν + λ να a α ] ∂ ρ (cid:2) eA σ + λ σβ a β (cid:3) . (16)Then, following [34] we vary S + S CS with respect to A to obtain the anomalous Hall current. Specializing to thecase b µ = b z δ µ , λ µν = λη µν and after integrating over a µ we find j x = eb z π ˜ E y or more expicitly j x = eb z π E y − λ b z π [ ∂ t j y − ∂ y ρ ] , (17)with E y being the electric field along ˆ y and ρ ( x ) = j ( x ).The first term here gives the quantum anomalous Hallcurrent while the interaction dependent contribution van-ishes in equilibrium. Thus, the interactions do not affectthe equilibrium Hall current however they may contributeto the non-equilibrium or inhomogeneous response. Com-bining (17) with the corresponding expression for j y andswitching to Fourier space we obtain the homogeneousfinite frequency Hall conductivity expected from S CS , σ xy ( ω ) = " (cid:18) λ b z π ω (cid:19) − e b z π . (18)The effect of interactions can also be seen in the equilib-rium density response to a change in the magnetic field, B z → B z + δB z . In the absence of any fields along thetransverse components we may use (13) as our anoma-lous relation. After subtracting the background density,the leading order density response is δj = 11 + λ eB z π eb z π δB z . (19)Due to the dimensional reduction, the density is equiv-alent to a chiral current in the longitudinal direction, (cid:10) j (cid:11) = (cid:10) j (cid:11) and so (19) can be viewed as the generationof a chiral current in response to a change in the mag-netic field which is known as the chiral separation effect(CSE) [43–45]. Photon Action —As was pointed out in [42] the Chern-Simons term obtained via chiral transformation requiressome subtle interpretation if it is to describe a Weylsemimetal. The appropriate understanding comes fromintegrating out the fermionic degrees of freedom to de-termine the linear response. We adopt this approach toconfirm the equilibrium response of the system expectedfrom S CS . To O ( e ), after integrating out the fermions, S = − e Z d qdω (2 π ) Tr [ G λ ( q , ω ) γ µ ] ˜ A ∗ µ ( q , ω ) − e Z d qdω (2 π ) ˜ A µ ( q , ω )Π µνλ ( q , ω ) ˜ A ∗ ν ( q , ω ) , (20)where G λ ( q , ω ) is the single particle, interacting, Green’sfunction in the presence of B z and b z and Π µνλ ( q , ω ) = R d q ′ dω ′ (2 π ) Tr [ γ µ G λ ( q ′ , ω ′ ) γ ν G λ ( q ′ − q , ω ′ − ω )]. Theanomalous terms we are interested in can then be iso-lated by considering the leading q , ω → G λ ( q , ω ) cannot be carried out ex-actly however we are only interested in computing thedensity response and the form of (19) suggestive of anRPA approximation. Indeed, the low energy responsein the longitudinal directions is determined solely by theLLL whose current and density responses are completelycaptured by an RPA summation owing to its reduced di-mensionality. Using the non-interacting Green’s functionin the Landau level basis derived in [30] we obtainlim q → ω → Π µν RPA ( q , ω ) = "
11 + λ eB z π P k + P ⊥ µρ lim q → ω → Π ρν ( q , ω ) , (21)where, for λ µν = λ η µν , P k = (1 − γ ) / P ⊥ = 1 − P k projectsonto the transverse components. When λ µν = λ η µ η ν we use instead P k = [(1 − γ ) / − γ ) /
2] which projectsonly onto the temporal components. We see here ascreening of the density response due to the interactionswhile the transverse components are unaffected. Theequilibrium Hall response is therefore the same as the freecase, in agreement with (17). The linear density responseis then found after computing lim q → lim ω → Π ( q , ω ) /iq x . Sur-prisingly however, this vanishes. Thus the anomalousdensity response comes from the first term in (20) andcan be attributed to the change in degeneracy of the LLL.The same RPA screening occurs for this term also andwe find agreement with (19).In the absence of B z , the density response depends onall filled bands [42]. When it is present however, this isnot the case and the density response is determined onlyby the LLL. Therefore we can understand this by return-ing to our description of the LLL given in (15). The S b term can be accounted for by the inclusion of a chemicalpotential term S S,b = − R d x √ N b z ∂ x φ S /π . Recallingthat N = eB z / π is identified with the degeneracy of theLLL we compute the density response to N → N + δN and once again find agreement with (19). Furthermore,the modification of the anomalous terms is natural fromthis viewpoint as we can identify (1 + λ eB z / π ) − asbeing the charge susceptibility or the chiral charge stiff-ness of the LLL [12, 13]. This is in agreement with (19)being viewed either as the density response or the CSE. Conclusions — In this Letter we have explored the in-terplay between anomalous chiral symmetry breaking viaelectromagnetic fields and interactions. We have shown,using Fujikawa’s path integral method, that the chiralcharge continuity equation contains new interaction de-pendent terms which can be absorbed into effective elec-tromagnetic fields which are responsible for the breakingof chiral symmetry. Furthermore this result was shown tobe consistent with the lowest Landau level being a Lut-tinger liquid. We investigated the consequences of thisresult for interacting Weyl semimetals and found thatinteraction effects will be present in the non-equilibriumHall response as well as the density response to a changein the magnetic field. These results were then reproducedvia direct perturbative calculation.Recently, it was discovered that the circular photogal-vanic effect [46], originally thought to be quantized asa result of the chiral anomaly, is actually renormalizeddue to the presence of interactions [47]. It would be de-sirable to understand our results in the context of thisobservable also. Lastly, we note that other anomalousWard identities, including the gravitational anomaly canbe derived using Fujikawa’s method and our analysis canlikewise be applied in those situations with the possibilityof additional observable interaction effects [48].We acknowledge useful discussions with Natan An-drei. This work was supported by the U.S. Departmentof Energy, Office of Science, Basic Energy Sciences un-der Award No. DE-SC0001911, the Simons Foundation(A.P., C.R., and V.G.) and Natural Sciences and Engi-neering Research Council (NSERC) of Canada (AAB).Research at Perimeter Institute is supported in part bythe Government of Canada through the Department ofInnovation, Science and Economic Development and bythe Province of Ontario through the Ministry of Eco-nomic Development, Job Creation and Trade. [1] Stephen L. Adler, “Axial-vector vertex in spinor electro-dynamics,” Phys. Rev. , 2426–2438 (1969).[2] J. S. Bell and R. Jackiw, “A PCAC puzzle: π → γγ inthe σ -model,” Nuovo Cimento A Serie , 47–61 (1969).[3] Stephen L. Adler and William A. Bardeen, “Absence ofhigher-order corrections in the anomalous axial-vector di-vergence equation,” Phys. Rev. , 1517–1536 (1969).[4] Kazuo Fujikawa, “Path-integral mea-sure for gauge-invariant fermion theories,”Phys. Rev. Lett. , 1195–1198 (1979). [5] Kazuo Fujikawa, “Erratum: Path integral for gauge theo-ries with fermions,” Phys. Rev. D , 1499–1499 (1980).[6] Kazuo Fujikawa and Hiroshi Suzuki, Path integrals andquantum anomalies , 122 (Oxford University Press on De-mand, 2004).[7] Paul H. Frampton and Thomas W. Kephart, “Ex-plicit evaluation of anomalies in higher dimensions,”Phys. Rev. Lett. , 1343–1346 (1983).[8] Paul H. Frampton and Thomas W. Kephart, “Ex-plicit evaluation of anomalies in higher dimensions,”Phys. Rev. Lett. , 232–232 (1983).[9] Bruno Zumino, Wu Yong-Shi, and A. Zee, “Chiralanomalies, higher dimensions, and differential geometry,”Nuclear Physics B , 477–507 (1984).[10] C. M. Na´on, “Abelian and non-Abelianbosonization in the path-integral framework,”Phys. Rev. D , 2035–2044 (1985).[11] D.K.K. Lee and Y. Chen, “Functional Bosonization ofthe Tomonaga-Luttinger Model,” J. Phys. A , 4155(1988).[12] T. Giamarchi, Quantum Physics in One Dimension , In-ternational Series of Monographs on Physics (ClarendonPress, 2003).[13] A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik,
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