Enriched axial anomaly in Weyl materials
EEnriched axial anomaly in Weyl materials
Zachary M. Raines and Victor M. Galitski
Joint Quantum Institute and Department of Physics,University of Maryland, College Park, Maryland 20742-4111, U.S.A.
While quantum anomalies are often associated with the breaking of a classical symmetry in thequantum theory, their anomalous contributions to observables remain distinct and well-defined evenwhen the symmetry is broken from the outset. This paper explores such anomalous contributionsto the current, originating from the axial anomaly in a Weyl semimetal, and in the presence of ageneric Weyl node-mixing term. We find that apart from the familiar anomalous divergence of theaxial current proportional to a product of electric and magnetic fields, there is another anomalousterm proportional to a product of the electric field and the orientation of a spin-dependent node-mixing vector. We obtain this result both by a quantum field-theoretic analysis of an effective Weylaction and solving an explicit lattice model. The extended spin-mixing mass terms, and the enrichedaxial anomaly they entail, could arise as mean-field or proximity-induced order parameters in spin-density-wave phases in Weyl semimetals or be generated dynamically within a Floquet theory.
Quantum anomalies represent a surprising deviationfrom classical intuition, where a global symmetry of theclassical Lagrangian does not necessarily lead to a con-servation law of the corresponding charge. This fact en-tails profound and fundamental consequences in particlephysics such as non-conservation of baryon charge andthe appearance of instantons and θ -vacuum in quantumchromodynamics.Anomalies, including the chiral or axial anomaly [1–4], also appear in the field-theoretic descriptions of con-densed matter models and give rise to “anomalous” con-tributions to observable currents. Most recently, thecondensed-matter chiral anomaly has been discussed ex-tensively in the context of Weyl semimetals, and its signa-tures were experimentally observed in magneto-transportof these materials [5].Arguably, the condensed matter axial anomalies areless impressive than in particle physics as far assymmetry-breaking is concerned. They oftentimes rep-resent properties and “quantum symmetry breaking” ofa low-energy effective theory, while the origin of theanomaly derives from a short-distance regularization,where the low-energy theory is not quantitatively appli-cable and chiral symmetry is poorly defined. Once thefull theory is restored, the symmetry of the low-energymodel is no longer tied to strong conservation laws. Forexample, the non-conservation of charge attached to aparticular Weyl node in a Weyl semimetal is not partic-ularly surprising once we recall that the Weyl nodes areconnected through the bottom of the band in the fulllattice band structure [3, 6]. While the breaking of thechiral symmetry in condensed matter systems is not un-expected, the anomalous contributions predicted withinthe low energy theory remain observable effects, for ex-ample the anomalous Hall and chiral magnetic effects inWeyl semimetals.In fact, these “chiral-anomalous” contributions andfeatures generally survive even if there is no chiral sym-metry to be broken even in the low-energy model. One concrete way to express an anomaly in such a contextis by comparing the divergence of the classical Noethercurrent vs the associated Ward identity obtained fromthe quantum theory. For example, even if Dirac massterms are included in a description of a Weyl semimetal(which physically implies scattering between the Weylnodes that breaks chiral symmetry already at the classi-cal level), one can still identify a well-defined anomalouscontribution, as was considered for example by Zyuzinand Burkov in Ref. [7]. In this case, one finds classicallythat ∂ µ j µ = − im ¯ ψγ ψ , whereas in the quantum theorythis is modified to become ∂ µ (cid:104) j µ (cid:105) = − im (cid:104) ¯ ψγ ψ (cid:105) + A ( x ).The presence of the anomaly function A ( x ) = ( e / π ) E · B is a hallmark of a quantum anomaly, which persistseven though the symmetry corresponding to j µ is alreadybroken at the classical level by the presence of a Diracmass.Motivated by these considerations, we explore theanomalous divergence of the Noether current in the pres-ence of a generic node-mixing term in a Weyl semimetal.Our main result, derived and discussed below, is thatthe spin-dependent node-mixing terms do not affect the“conventional” chiral anomaly, but give rise to anotheranomalous term — A spin ( x ) = − eπ E · Re ( g m ∗ ), wherethe vector g defined in Eq. (12) below, determines spin-mixing between the nodes and m is the complex Diracmass. As is the case with the conventional anomaly, thepresence of such an anomaly term can lead to anomaloustransport, which in this case is reminiscent of the chiralmagnetic effect.We will begin by examining a low-energy theory ofDirac fermions within the functional integral technique.Our aim is to understand what electromagnetic responsecan arise from the addition of new terms to the La-grangian. To do so we employ the chiral rotation tech-nique as first illustrated by Fujikawa [4]. Our startingpoint is the Dirac Lagrangian in imaginary time L = ¯ ψ (cid:104) i /D − /bγ − | m | e iαγ − ∆ µν σ µν (cid:105) ψ ≡ i ¯ ψ D ψ, (1) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r where ¯ ψ = ψ † γ denotes Dirac conjugation and we areusing Euclidean gamma matrices { γ µ , γ ν } = 2 g µν = − δ µν , with µ, ν = 1 , , ,
4. We use the Feynmanslash notation /D = D µ γ µ where D µ = ∂ µ + ieA µ isthe gauge covariant derivative. We also define the fifthgamma matrix γ = γ γ γ γ and the sigma matrices σ µν = ( i/ γ µ , γ ν ]. With these definitions the matrices γ µ are anti-Hermitian, while γ and σ µν are Hermitian.We defer a discussion of the specific origin of m and ∆until later in the paper.We now perform a chiral gauge transformation to re-move the axial vector b µ from the fermionic action. Sub-sequently, we make the following claims in the absence ofan external field. Firstly, the fermionic sector exhibits noaxial-vector dependent currents. Secondly, the Jacobianintroduced via the anomaly in the path integral measureproduces a current in response to the ∆ µν term.As pointed out by Fujikawa, when considering the chi-ral anomaly it is important to specify which basis one isusing to define the functional integral [4]. In his origi-nal work, Fujikawa used the basis states of the Euclideanoperator /D which is Hermitian. Here, the operator D isnot Hermitian, and so we follow the approach of Refs. [7and 8]. We define the eigenfunctions and eigenvalues ofthe operators D † D and DD † by D † D φ n ( x ) = λ n φ n ( x ) and ˜ φ † n ( x ) DD † = ˜ φ † n ( x ) λ n . (2)These operators are manifestly Hermitian and thus theireigenvectors form a basis. In defining the eigenvaluesas λ n we have made use of the fact that the operators D † D and DD † are positive semi-definite. Note that thereis a one-to-one correspondence between their non-zeroeigenvalues.We then define the path-integral by expressing ψ and¯ ψ in terms of the eigenfunctions of D † D and DD † respec-tively as ψ ( x ) = (cid:88) n φ n ( x ) a n , ¯ ψ ( x ) = (cid:88) n ¯ a n ˜ φ † n ( x ) . (3)With our basis states defined we now consider an in-finitesimal chiral gauge transformation of the form. ψ ( x ) = e − i d s b · xγ ψ (cid:48) ( x ) and ¯ ψ ( x ) = ¯ ψ (cid:48) ( x ) e − i d s b · xγ . (4)Under such a transformation the Lagrangian becomes L (cid:48) = ¯ ψ (cid:48) (cid:104) i /D − (1 − d s ) /bγ − | m | e i ( α − s b · x ) γ − ∆ µν e − i d s b · xγ σ µν (cid:105) ψ (cid:48) . (5)We must also include the contribution from the Jacobianfactor in the path integral measure which is introduced bythis transformation. To that end, let us consider the par-tition function, which under the chiral rotation Eq. (4) transforms as Z = (cid:90) D [¯ a, a ] e − S [¯ a,a ] = (cid:90) D [¯ a (cid:48) , a (cid:48) ](det J ) − e − S (cid:48) [¯ a (cid:48) ,a (cid:48) ] (6)where S is the action corresponding to the Dirac La-grangian Eq. (1), expressed in terms of the basis states a n , and S (cid:48) is the action corresponding to the LagrangianEq. (5). The new fields of integration a (cid:48) n and ¯ a (cid:48) n areimplicitly defined in terms of the old via the relations a n = (cid:88) m U nm a (cid:48) m , ¯ a n = (cid:88) m ¯ a (cid:48) m ˜ U mn (7)with U nm = (cid:82) dxφ † n ( x ) e − i d s b · xγ φ m ( x ) and (cid:101) U nm = (cid:82) dx ˜ φ † n ( x ) e − i d s b · xγ ˜ φ m ( x ), being the matrix elements ofthe chiral rotation operator in the a -basis. We can nowsimply express the Jacobian determinant in terms of thematrices U and (cid:101) U as det J = det (cid:101) U U .We now reinterpret the Jacobian as a term in the ac-tion via the relation det J = e tr ln J . Thus, our partitionfunction becomes Z = (cid:90) D [¯ a (cid:48) , a (cid:48) ] e − S (cid:48) [¯ a (cid:48) ,a (cid:48) ] − S J , where S J = tr ln J = tr ln ˜ U + tr ln U . Using the fact thatd s is infinitesimal we rewrite the above as S J = − i d s (cid:90) d x ( b · x ) (cid:104) I ( x ) + ˜ I ( x ) (cid:105) , (8)where I ( x ) = (cid:88) n φ † n ( x ) γ φ n ( x ) and ˜ I ( x ) = (cid:88) n ˜ φ † n ( x ) γ ˜ φ n ( x ) . (9)This is where one encounters an anomaly. Let us firstnote that I ( x )+ ˜ I ( x ) is exactly analogous to the anomalyfunction one encounters in computing the divergence ofthe axial current. Now, the expressions in Eq. (9) canreadily be seen to be indeterminate. Naively, from thecompleteness of the eigenstates φ and ˜ φ we have I ( x ) = ˜ I ( x ) = δ (4) (0) (cid:124) (cid:123)(cid:122) (cid:125) ∞ × tr γ (cid:124)(cid:123)(cid:122)(cid:125) . This ambiguity is due to the continuum representation ofthe path integral. In order to resolve it we must introducea proper regulator. Following Refs. [4 and 7], we evaluate I and ˜ I by heat kernel regularization as follows I ( x ) = lim M →∞ lim y → x (cid:88) n φ † n ( y ) γ e − λ n /M φ n ( x )= lim M →∞ lim y → x (cid:90) k tr e ik · y γ e −D † D /M e − ik · x , (10)where we have used the completeness of the eigenfunc-tions φ n ( x ) and e − ik · x . The analogous expression holdsfor ˜ I ( x ). This has the benefit of regulating the expres-sion in a gauge invariant manner. The calculation of theanomaly functions I and ˜ I is presented in more detail inthe appendix. If we define the Hodge dual of the Maxwelltensor ∗ F µν = (1 / (cid:15) µναβ F αβ and the real and imaginarytensors ∆ R(I) µν = ∆ µν ± ∆ ∗ µν we can express the result ofthe limit of I + ˜ I in Eq. (10) as − e π ∗ F µν F µν + e π Re (cid:104) ∗ F µν ∆ R µν + F µν ∆ I µν (cid:105) , (11)where we have assumed b µ to be constant.∆ µν in Eq. (1) can be conveniently parametrized bythe complex vector g i = (cid:15) ijk ∆ jk + i (cid:16) ∆ i − ∆ i (cid:17) , (12)where i, j, k are spatial indices and the first and secondterms are, respectively, purely real and imaginary due toHermiticity. This allows Eq. (11) to be written in termsof vector quantities as I ( x ) + ˜ I ( x ) = − e π E · B + e π E · ( g m ∗ + g ∗ m ) . (13)In the above we have assumed g to have plane waveform. The first term in Eq. (13) is the conventional chi-ral anomaly, while the second term describes a new effectdue to the added terms in the Lagrangian.Note that the action in Eq. (8) is linear in d s . We canthus perform a series of chiral gauge transformations toremove the axial vector b µ from the electronic Lagrangianso that it becomes L = ¯ ψ (cid:48) (cid:104) i /D − | m | e i ( α − b · x ) γ − ∆ µν e − ib · xγ (cid:105) ψ (cid:48) . (14)This corresponds to integrating the Jacobian in s from 0to 1. The Lagrangian arising from the Jacobian is L J = ib · x (cid:34) e π E · B − eπ E · Re [ g m ∗ ] (cid:35) . (15)Note that this result is unchanged if m and g , insteadof being constant, are taken to be plane waves with thesame four-momentum, i.e. | m | → | m | e − iQ · xγ , ∆ µν → ∆ µν e − iQ · xγ . (16)We may then, for example, take Q = − b . In this case,the fermionic sector reduces to L = ¯ ψ (cid:48) (cid:104) i /D − | m | e iαγ − ∆ µν (cid:105) ψ (cid:48) . (17)Since this has no dependence on b µ we have isolated theaxial vector dependent part of the current into the Jaco-bian term L J . Alternatively, consider the case where m Eµ b b m, g k z b z − b z FIG. 1. Two-band lattice model with a pair of Weyl nodes.The nodes are located at (0 , , ± b z ) in the Brillouin zone andseparated by an energy of 2 b . and ∆ arise from the decoupling of short range interac-tions in the appropriate channels. In that case they willhave Hubbard-Stratonovich Lagrangians L HS = 1 λ m | m ( x ) | + 1 λ ∆ ∆ µν ( x )∆ µν ( x ) . (18)We can then absorb the chiral phase into the defini-tions of the parameters without affecting the Hubbard-Stratonovich term.Now to illustrate the effects of Eq. (15) let us considerthe case of g = g z ˆz . After analytic continuation back toreal time, the new term in the action reduces to S g = eπ (cid:90) d x b · x | m | g z E z cos α. (19)This term has a form similar to the typical anomalywhere in place of the magnetic field B , we have2Re ( g m ∗ ). As such, we expect this term should also becapable of producing currents. As there are no fermionoperators in this term, we can obtain the induced cur-rents by simply differentiating the action with respect tothe vector potential. Doing so we obtain a contribution j J = δSδ A = eπ b | m | g z cos α ˆz . (20)We thus arrive at the prediction of a current without anexternal field. This is best understood in a way akin tothe chiral magnetic effect. As has been discussed in anumber of works, one can find zero or non-zero valuesfor the chiral magnetic effect depending on the order oflimits one uses in evaluating the result [9–11]. Taking thefrequency to zero before momentum corresponds to theequilibrium case and as one would expect due to generalarguments one finds no current in the absence of an elec-tric field. However, in the opposite limit, correspondingto a near equilibrium DC transport one finds that thereis a current. In the same way, one can interpret Eq. (20)as the response to a slow but non-zero frequency pertur-bation by the ∆ µν term in the action.The “enriched” chiral anomaly as derived above issomething only sharply defined for unbounded linearlydispersing particles [12]. In reality the Dirac theory ofthe previous section is only the low-energy description ofsome bounded dispersion in the Brillouin zone. As such, ǫ n , k − ζ b ǫ n k + ζ b ǫ n − Ω m k + ζ b v i ( k + ζ b ) σ z e − iζα ǫ n , k + ζ b ǫ n k − ζ b ǫ n − Ω m k − ζ b v i ( k − ζ b ) σ z e − iζα FIG. 2. Diagrams contributing to the susceptibility χ i . Thesymbol ⊗ indicates the current vertex v i ( k ), while the circleand square represent the vertices for m and g respectively. (cid:15) n is the internal fermionic Matsubara frequency while Ω m isthe external bosonic Matsubara frequency to be analyticallycontinued to obtain the retarded correlator. Internal loopmomenta k and (cid:15) n are summed over as well the index ζ = ± we need to establish that the predicted effects can beobserved within a lattice regulated model.In order to verify the validity of the above conclusionsindependent of the subtleties of the low energy theory,we study the current response of a lattice model of Weyl-fermions. Our purpose is to show that the current re-sponse of the lattice system is in agreement with the pre-diction of the low energy theory, Eq. (20).In particular, we use the following inversion and time-reversal symmetry breaking two-band lattice model [13] H = (cid:88) k c † k (cid:2) (cid:15) ( k ) + d ( k ) · σ (cid:3) c k (21)with (cid:15) ( k ) = t sin k z and d † ( k ) = (cid:0) sin k x , sin k y , b z − (cid:80) i cos k i (cid:1) , which is hostto a pair of Weyl fermions as depicted in Fig. 1. Themomentum-space separation of the nodes is given by2 b = 2 b z ˆz and the energy separation by b = 2 t sin b z .To this bare Hamiltonian we add the perturbations H m = m (cid:88) k c † k + b e − iα σ z c k − b + h . c ., (22)which corresponds to the mass term in the low-energytheory of Eq. (1). The ∆ µν term can be modeled as H g = (cid:88) k c † k + b σ z g ( τ ) · σ c k − b + h . c .. (23)We wish to establish the existence of a DC currentin response to the combined terms m and g . In par-ticular, we calculate the retarded susceptibility of thecurrent to m and g in the uniform limit χ Ri ( ω →
0) =lim ω → lim q → χ R ( ω, q ). χ R is obtained as the analyticcontinuation from Matsubara frequency of the object χ i ( i Ω m , q ) = δj i ( i Ω m , q ) δmδg ( − i Ω m , − q )= δF [ A , g, m ] δmδg ( − i Ω m , q ) δA i ( i Ω m , q ) (cid:12)(cid:12)(cid:12)(cid:12) g =0 m =0 A =0 , (24) t cos( α ) χ ( ω → , q = ) × − T = 10 − T = 10 − T = 10 − T = 10 − FIG. 3. The susceptibility χ in the uniform limit versus theparameter t of Eq. (21). The induced current grows linearlywith t cos α = b sin b z cos α as predicted by the low-energy the-ory, and vanishes in the absence of a nodal energy separation. where F [ A , g, m ] is the free energy in the presence ofan external vector potential A and perturbations m , g .Eq. (24) corresponds to the diagrams in Fig. 2 and de-scribes the lowest order contribution of the m and g fieldsto the current in the spirit of linear response theory.As shown in Fig. 3, the induced current grows linearlywith the nodal energy separation and vanishes, as ex-pected, when t = 0. We have also verified that in theopposite order of limits (with ω → L = L − λ m ¯ ψe iαγ ψ ¯ ψe − iαγ ψ − λ ∆ ¯ ψσ µν ψ ¯ ψσ µν ψ. (25)A Hubbard-Stratonovich decoupling, then leads to aspacetime-dependent analog of Eq. (1). In a condensedmatter context we can write Eq. (25) in terms of spin, σ ,and valley, τ i , degrees of freedom as L = c † ( ∂ τ + k · σ τ z − b · σ − b τ z ) c − λ m e iα (¯ cτ + c ) (¯ cτ − c ) − λ ∆ (¯ c g · σ τ + c ) (¯ c g ∗ · σ τ − c ) . (26)We therefore interpret m as a charge-density wave, while g describes a spin-density wave, since the valley degree offreedom denotes a separation in momentum space. Thissuggests that our model may potentially be realized in asystem of interacting Weyl electrons, where interactionsgive rise to such density-wave orders. Alternatively, suchperturbations may be externally induced, e.g. in Floquet-driven Weyl materials.This work was supported by U.S. Department of En-ergy BES-DESC0001911 and Simons Foundation. [1] S. L. Adler, Phys. Rev. , 2426 (1969).[2] J. S. Bell and R. Jackiw, Nuovo Cim. A , 47 (1969).[3] H. B. Nielsen and N. Masao, Phys. Lett. , 389 (1983).[4] K. Fujikawa, Phys. Rev. D , 285 (1984).[5] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang,H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, andG. Chen, Phys. Rev. X , 031023 (2015).[6] D. T. Son and B. Z. Spivak, Phys. Rev. B , 1 (2013).[7] A. A. Zyuzin and A. A. Burkov, Phys. Rev. B , 1(2012).[8] H. Kikuchi, Phys. Rev. D , 4704 (1992).[9] K. Landsteiner, Phys. Rev. B , 1 (2014).[10] M.-c. Chang and M.-f. Yang, Phys. Rev. B , 115203(2015).[11] Y. Alavirad and J. D. Sau, Phys. Rev. B , 115160(2016).[12] A. A. Burkov, J. Phys. Condens. Matter , 113201(2015).[13] K. Y. Yang, Y. M. Lu, and Y. Ran, Phys. Rev. B ,12 (2011).[14] D. Bohm, Phys. Rev. , 502 (1949).[15] Y. Ohashi and T. Momoi, J. Phys. Soc. Japan , 3254(1996).[16] N. Yamamoto, Phys. Rev. D , 085011 (2015). Supplemental information: Heat kernelregularization of the anomaly functions
We wish to evaluate the heat kernel regulated expres-sions I ( x ) = lim M →∞ lim y → x (cid:90) k tr e ik · y γ e −D † D /M e − ik · x . (A.27)and˜ I ( x ) = lim M →∞ lim y → x (cid:90) k tr e ik · y γ e −DD † /M e − ik · x . (A.28)To do so, let us first define the operator O = i | m | e iαγ + i ∆ µν σ µν . (A.29) We then expand D † D = − D + e F µν σ µν + i [ /D, /b ] γ + /DO + O † /D + i/bγ O + iO † /bγ + O † O − b (A.30)and similarly DD † = − D + e F µν σ µν + i [ /D, /b ] γ + O /D + /DO † + Oi/bγ + i/bγ O † + OO † − b . (A.31)where X denotes (cid:80) i X i . Here, we have used the iden-tity [ D µ , D ν ] = ieF µν , where F µν is the electromagnetic field tensor. The pres-ence of the plane waves in Eqs. (A.27) and (A.28), onlyleads to the replacement D µ → D µ − ik µ . We thus canrewrite Eq. (A.27) as I ( x ) = lim M →∞ M (cid:90) p e − p tr γ exp (cid:34) − D † D M + M − (cid:16) − ip µ D µ + [ /p, /b ] γ − i/pO − iO † /p (cid:17)(cid:21) , (A.32)where we have also made the change of variables p µ = M k µ . Writing the exponential term asexp (cid:32) − D † D M + KM (cid:33) , we can expand in powers of MI ( x ) = lim M →∞ M (cid:90) p e − p × tr γ (cid:34) KM + K − D † D M + K / − { K, D † D} M + K /
4! + D † D / M (cid:35) . (A.33)As in the conventional case, the only terms which sur-vive are at order M . All lower-order or K -dependentterms either vanish due to the matrix traces or cancelbetween I and ˜ I . We are then left with I ( x ) + ˜ I ( x ) = 12 tr (cid:104) ( D † D ) + ( DD † ) (cid:105) (cid:90) d p (2 π ) e − p . (A.34)We can easily perform the Gaussian integral to obtain afactor of π2