Landau Levels, Bardeen Polynomials, and Fermi Arcs in Weyl Semimetals: Lattice-Based Approach to the Chiral Anomaly

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Landau levels, Bardeen polynomials, and Fermi arcs in Weyl semimetals: Lattice-based approach to the chiral anomaly Jan Behrends, Sthitadhi Roy, Michael Kolodrubetz, Jens Bardarson, Adolfo G. Grushin

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Jan Behrends, Sthitadhi Roy, Michael Kolodrubetz, Jens Bardarson, Adolfo G. Grushin. Landau levels, Bardeen polynomials, and Fermi arcs in Weyl semimetals: Lattice-based approach to the chiral anomaly. Physical Review B, American Physical Society, 2019, 99 (14), pp.140201(R). 10.1103/Phys- RevB.99.140201. hal-03276497v2

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Jan Behrends,1 Sthitadhi Roy,2, 3, 1 Michael H. Kolodrubetz,4, 5, 6 Jens H. Bardarson,7, 1 and Adolfo G. Grushin4, 8 1Max-Planck-Institut fürPhysik komplexer Systeme, 01187 Dresden, Germany 2Physical and Theoretical Chemistry, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom 3Rudolf Peierls Centre For Theoretical Physics, Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom 4Department of Physics, University of California, Berkeley, CA 94720, USA 5Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 6Department of Physics, The University of Texas at Dallas, Richardson, TX 75080, USA 7Department of Physics, KTH Royal Institute of Technology, Stockholm, SE-106 91 Sweden 8Institut Néel,CNRS and UniversitéGrenoble Alpes, F-38042 Grenoble, France Condensed matter systems realizing Weyl fermions exhibit striking phenomenology derived from their topologically protected surface states as well as chiral anomalies induced by electromagnetic fields. More recently, inhomogeneous strain or magnetization were predicted to result in chiral electric E5 and magnetic B5 fields, which modify and enrich the chiral anomaly with additional terms. In this work we develop a lattice-based approach to describe the chiral anomaly, which involves Landau and pseudo-Landau levels and treats all anomalous terms on equal footing, while naturally incorporating Fermi arcs. We exemplify its potential by physically interpreting the largely overlooked role of Fermi arcs in the covariant (Fermi level) contribution to the anomaly and revisiting the factor of 1/3 difference between the covariant and consistent (complete band) contributions to the E5 ·B5 term in the anomaly. Our framework provides a versatile tool for the analysis of anomalies in realistic lattice models as well as a source of simple physical intuition for understanding strained and magnetized inhomogeneous Weyl semimetals.

Quantum anomalies describe the breaking of a clas- (Weyl nodes), separated in energy-momentum space by a sical symmetry by quantum fluctuations [1]. The chi- four-vector bµ [9]. A space- and time-dependent bµ, as in ral anomaly, the nonconservation of the chiral charge strained or inhomogenously magnetized Weyl semimet- of three-dimensional Weyl fermions, is relevant to differ- als [15] or Helium-3 [23, 24], generates chiral pseudomag- ent domains in physics since Weyl fermions mediate the netic (B5 = ∇×b) and pseudoelectric (E5 = −∂tb−∇b0) pion-decay into photons [2,3] and are emergent quasipar- fields, which couple with opposite signs to opposite chi- ticles in Weyl semimetals [4–9]. The physics is particu- ralities [16, 17, 25–28]. These pseudofields enhance or larly transparent in the Landau level picture pioneered by generalize phenomena ranging from transport to inter- Nielsen and Ninomiya [10], requiring only basic quantum face physics [16, 17, 29–39]. Unlike B, which gener- mechanics. In a magnetic field B, the conical Weyl dis- ates Landau levels dispersing in opposite directions for persion evolves into Landau levels with a degeneracy pro- opposite chiralities, B5 generates pseudo-Landau lev- portional to |B| [10]. Since momentum along the direc- els that disperse in the same direction for both chiral- tion of the magnetic field remains a good quantum num- ities [16, 17, 25, 26, 40]. ber, the Landau levels disperse in that direction, with Applying the Landau level picture to chiral fields leads the zeroth Landau level having a linear dispersion with to puzzling conclusions: for example, because of the a sign determined by the chirality; all other Landau lev- chirality independent dispersion of the zeroth pseudo- els have a quadratic dispersion. The zeroth Landau level Landau level due to B5, E increases (or decreases) the of the left- and right-handed Weyl fermions furthermore number of fermions for both chiralities at a rate propor- connect at high energy. Consequently, an electric field tional to E · B5, giving an apparent nonconservation of (E) with a component along the dispersion transfers left- the total charge. This is expressed as the so-called co- handed fermions to right-handed fermions (or vice versa) variant anomaly [1, 40] arXiv:1807.06615v2 [cond-mat.mes-hall] 1 Apr 2019 resulting in a nonconservation of left and right particle 1 numbers proportional to E · B [1, 10, 11]. ∂ J µ = (E · B + E · B ) , (1) µ 5,cov 2π2 5 5 Other fields, such as chiral pseudo-electromagnetic 1 ∂ J µ = (E · B + B · E) . (2) fields, torsion or curvature activate the chiral anomaly µ cov 2π2 5 5 beyond E and B [12–22]. Weyl semimetals are ideal to µ µ µ probe the chiral anomaly in the presence of chiral pseudo- Neither the covariant chiral (J5,cov = JL,cov −JR,cov) nor µ µ µ electromagnetic fields. To motivate this, recall that their vector (Jcov = JL,cov +JR,cov) currents are conserved. In low-energy degrees of freedom are pairs of chiral Weyl field theory, to explicitly restore charge conservation, the quasiparticles at topologically protected band touchings covariant currents are supplemented by Bardeen poly- 2

µ nomials δj , which act like boundary conditions for the (a) (b) accumulated charge at the cut-off energy [41–44]. This procedure defines the consistent anomaly [1, 40] 1 1 ∂ J µ = E · B + E · B , (3) µ 5 2π2 3 5 5 µ (c) ∂µJ =0. (4) The consistent anomaly conserves charge, and thus de- termines observables [42, 45] by discarding unphysical responses [40–42, 46]. The Bardeen polynomials, however, sacrifice intuition of the covariant picture based on Lan- dau levels and obfuscate the restoration of charge conservation in specific lattice implementations. FIG. 1. Anomalies due to E·B and E·B5. (a) The spectrum of Despite the field theory of the consistent and covariant Hamiltonian (5) for a slab finite along y, showing the coexist- anomalies being well understood for a long time [40], a ing Fermi arcs and Landau levels of B = Bzˆ. The color scale simple physical picture of their origin on a lattice, with denotes the wave function’s position along y. (b), (c) The Landau (pseudo-Landau) levels of B (B ) on the red (blue) guaranteed charge conservation, is still missing. In this 5 triangle plane show the anomaly in the presence of E = Ezzˆ work we provide such a picture using as building blocks (E = Exxˆ), where the spectral flow at the Fermi surface is the Landau and pseudo-Landau levels. It leads to our two shown by green (orange) arrows. We implement E using the main results: first, we identify the Fermi arcs as a source gauge choice k → k − Et. The left panels show the occupa- for the covariant anomaly terms of Eq. (2) and relate tion at t = 0, while the right panels show the dispersion for them to the Bardeen polynomials. Second, we show that t > 0 with Ex,zt = 0.08 and the same momenta occupied as when B > B the Fermi surface twists into a bowtie in the left panels,√ cf. Supplemental Material. Here L = 100 5 and ` = 1/ B = 11.2. shape, a property central to our understanding of how B the term E · B5 redistributes charge within the sample. Our picture allows us to address as well the 1/3 disparity The charge density, ρ(x), is obtained by replacing γ5 with between the second term in Eq. (3) and Eq. (1). We argue the identity in Eq. (6). While our lattice model (5) in- that a necessary condition for its identification is a B5 cludes a τx term that explicitly breaks conservation of profile that spatially separates chiral charge creation and ρ5, we show in the Supplemental Material how this effect annihilation. Similar to how Landau levels simplified our is controlled. understanding of the chiral anomaly [10], we use pseudo- The Hamiltonian (5) (with v = 1) derives from an Landau levels (developed in Ref. 17) to provide a unified effective field theory with action and simple lattice picture of the consistent and covariant Z anomalies with specific implications for experiment. 4 ¯ µ 5 S = d x ψ γ i∂µ − Aµ − bµγ − m ψ, (7) Our starting point is the Weyl semimetal model [46] where ψ¯ = ψ†γ0 and repeated indices are summed. S H =v [sin(ky)σx − sin(kx)σy] τz + v sin(kz)τy + mτx X X yields two species of Weyl fermions of opposite chirali- + t [1 − cos(k )] τ + v uµb , (5) i x µ ties, coupled to an external chiral field bµ and a vector i µ field Aµ, and separated by bµ for m = 0. The lattice reg- ularization is given by the Wilson map ki → sin ki [51], with a = 1 the lattice constant, σi (τi) spin (orbital) Pauli P µ and m → m + t (1 − cos ki)[52]. matrices, and u = (σzτy, −σxτx, −σyτx, σz). For small i 2 2 2 2 Spatial and temporal variations of b generate the chi- bµ = (b0, b) and m < v |b −b0|, the model has one pair ral fields B5 = ∇ × b and E5 = −∂0b. The simplest of Weyl nodes near Γ [46,√ 47]. Unless stated otherwise, we set m = 0 and t = 2v/ 3. When b is oriented along realization of B5 occurs at the boundary of any Weyl a reciprocal lattice vector, this parameter choice gives semimetal with vacuum, where the Weyl node separa- two Weyl nodes located at ±b[1 + O(b2)] + O(b5) and tion bµ goes to zero [17, 26, 40]. For example, for a 0 j slab along y, b(y) = b [Θ(y − L/2) − Θ(y + L/2)]ˆz gives energies ±vb [1+O(b2)]+O(b3). Our results also apply to z 0 j 0 B (y) = b [δ(y − L/2) − δ(y + L/2)]ˆx, localized at the generalizations of Eq. (5) that model Dirac (e.g., Cd As , 5 z 3 2 surface, which generates surface pseudo-Landau levels Na Bi) and Weyl (e.g., TaAs family) materials [48–50]. 3 dispersing along ±k , with opposite signs at each sur- We further define γµ = (τ , iσ τ , −iσ τ , iτ ) and γ5 = x x y y x y z face [17]. Their Fermi surface traces an arc, establishing iγ0γ1γ2γ3 = σ τ , such that uµ = γ0γµγ5 and the space- z y the correspondence between surface pseudo-Landau lev- dependent chiral charge density is els induced by B5 and topological surface states. Analo- 0 X 5 gously, a uniform external magnetic field B = Bzˆ parallel J5 (x) ≡ ρ5(x) = hψn(x)|γ |ψn(x)i. (6) n∈occ. to the Weyl node separation leads to a spectrum hosting 3 bulk Landau levels dispersing along ±kz, where the sign (a) (b) is set by the Weyl node chirality. When both B and a surface B5 are present, Landau and pseudo-Landau levels coexist and the Fermi surface at the Fermi energy εF (set to εF = 0 hereafter) traces a rectangle [53], see (c) Fig.1(a) [54, 55]. The coexistence of Landau and pseudo-Landau levels provides an ideal platform to discuss the anomalies. Ap- plying E = Ezzˆ pumps charges of one chirality to the other, connecting Landau levels of B through the band bottom and realizing the E · B term in Eq. (3)[10] [see Fig.1(b)]. Similarly, since the pseudo-Landau levels dis- FIG. 2. Anomaly due to E · B with a constant bulk perse along ±k on each surface, applying E = E xˆ de- 5 x x B5 = B5zˆ. (a) Energy spectrum with the same color coding pletes charges from one surface and generate charges on and boundary conditions as Fig.1. (b) The pseudo-Landau the other [Fig.1(c)]. We can interpret [10] this as an levels of B5 show an anomaly when E = Ezˆ with the spectral anomaly of each surface state due to E · B5. In contrast flow indicated by the green arrows. The left panel shows the to the chiral anomaly in the absence of chiral fields, where occupation at kx = 0.2 [gray plane in (a)] at t = 0; in the right the total charge is locally conserved, the spatial separa- panel Et = 0.05. (c) Left: The spatial profile of the charge δρ relative to that at t = 0, normalized by the total charge ρ tion of the two surfaces leads to an apparent violation 0 for different times (colors), showing a surface to bulk charge of local charge conservation, as in Eq. (2). Our pseudo- redistribution. The y axis is chosen such that only the (pos- Landau level picture demonstrates the surface origin of itive) bulk contribution is visible, which is exactly canceled the covariant anomaly, ∼ E · B5. by the (negative) surface contribution. Right: Derivative of Our picture tracks how charge is explicitly conserved. the density with respect to the vector potential A for differ- 2 The spectral flow between the pseudo-Landau levels at ent times, which equals ∂Az ρ = B5/(2π ) in the bulk (dashed each surface happens via the bulk Landau levels connect- line). The derivatives with respect to vector potential and time are related via ∂ ρ = −E · ∂ ρ. In (a) and (b), L = 100 ing them, fixing local charge conservation. The effect of √ t A and `5 = 1/ B5 = 11.1; in (c), L = 200 and `5 = 15.8. E results in an adiabatic shift of kx, which via position- momentum locking [17] generates a Hall current 1 δj = − b E y.ˆ (8) 2π2 z x We can interpret this as the net current flowing along yˆ between surfaces through the bottom of the band. In the bulk, bz is constant leading to ∇ · δj = 0 and no accumulation of charge. At the surface, the Weyl node separation varies, leading to a finite divergence of the spatial current, positive on one surface and negative on the other. Similarly, more general profiles of B and B5 can be FIG. 3. Anomaly due to E5 · B. (a) Occupation at kx = 0.2 understood in terms of Landau and pseudo-Landau lev- and B k E5 k zˆ for different times [different colors, (b)], els. For instance, a uniform bulk B k zˆ arises from where dark (light) colors represent filled (empty) states and 5 the gray shaded region is the low-energy regime, isolating the b = B5yxˆ. Its spectrum [Fig.2(a)] shows a characteris- covariant anomaly. We implement E5 by b → b−E5t and use tic butterfly Fermi surface, obtained from the rectangu- the same boundary conditions as Fig.1 with L = 100 lattice lar Fermi surface Fig.1(a) by noticing that the two bulk sites and `B = 11.2. (b) The charge in the gray low-energy Landau levels have lengths B ± B5. When B5 > B, the regime ρlow linearly increases with E5t. Since the total charge Fermi surface twists, leading to Fig.2(a) (found to lead is conserved, the charge in the band bottom decreases at the to peculiar quantum oscillation signals in an unrelated same rate (not shown). context [56]). An E = Ezˆ parallel to B5 [Figs.2(b) and (c)] makes the bulk gain charge above the Fermi level (upward arrows), while the surface loses charge (down- µ ward arrows). This is consistent with the lattice numerics By construction, the corresponding current Jcov misses [Fig.2(e)] where the spatial profile of the charge relative information from states away from the Fermi level, and to that at t = 0 is shown for different times. thus it is not conserved as dictated by Eq. (2), cf. Supple- Our previous examples (Figs.1 and2) are a conse- mental Material. The consistent current, J µ, is obtained quence of the covariant anomaly which only considers from the covariant current by adding the Bardeen poly- µ µ µ µ the depletion and growth of charges at the Fermi level. nomials δj such that ∂µ(Jcov +δj ) = ∂µJ = 0 [41, 42]. 4

(a) (c) E5 k zˆ.A B 6= 0 activates the second term of (2), sug- gesting that charge is created at the Fermi surface at a 2 rate E5 · B/(2π ). For our lattice model (5), E5 shifts the band bottom, pushing charge above a fixed energy (d) [Fig.3(a)]. Rigid shifts of the band conserve total charge, giving the consistent picture of the anomaly. However, if one insists on only considering the low-energy gray region in Fig.3(a), the charge appears to emerge from the (b) vacuum, as expected from the covariant anomaly. The connection between these pictures is shown in Fig.3(b), (e) where charge growth near the Fermi surface equals charge loss near the band bottom, which in turn equals the Bardeen polynomial δj0, Eq. (9). We end by addressing the factor of 1/3 disparity between the prefactors of the E5 · B5 anomalies. This difference implies that the band bottom current must add −2/3 to the Fermi surface contribution, irrespective of the precise pseudo-field profile. One may argue that this factor arises from the topological nature of the Bardeen FIG. 4. Anomaly due to E5 · B5. (a) The Weyl node separa- polynomials [44], yet we find that topology alone does tion (10) and its corresponding B5. (b) Energy spectrum with a real space color encoding that reflects the periodic bound- not explain the conditions which give 1/3 for a generic ary conditions. (c), (d) Where B5 > 0 [red in (d)], a parallel lattice model. E5 pushes left- and right-handed chiral charges above and To illustrate the conditions for isolating the 1/3, con- below the Fermi level respectively, annihilating chiral charge. sider [36] Where B5 < 0 (blue), chiral charge is created. (e) The spatial distribution of chiral charge creation and annihilation follows " L ! L ! # FS c y − y + and equals B5 (red solid line) for charges ρ5 traversing the B 4 4 b = erf √ − erf √ − 1 xˆ + bzz,ˆ original Fermi surface (blue dashed line). With all bands, 2 2ξ 2ξ the total chiral charge creation is ∼ 1/3 of that at the Fermi (10) surface, as predicted by the consistent anomaly (black dot- with cB a constant and erf(x) the error function ted line). We use L = 360, a B5 broadening ξ = 6 and peak −6 [Fig.4(a)]. In a periodic system of length L in the height cB = 1.25, and a delta function broadening η = 10 v, cf. Supplemental Material. y direction, this profile realizes interfaces at ±L/4 between two Weyl semimetals with different node separations, connected by regions where b smoothly changes to c −(y− L )2/(2ξ2) give B = B (y)ˆz with B (y) = √ B [e 4 − Using Eq. (2) and the definition of the pseudo-fields, 5 5 5 2πξ −(y+ L )2/(2ξ2) e 4 ]. This profile maintains the useful prop- 1 1 δj0 = b · B, δj = (b B − b × E) . (9) erty that the two chiralities (eigenstates of γ5) are 2π2 2π2 0 well separated in momentum space [Fig.4(c)]. Adding Comparing (9) to (8) of our first example (Fig.1), we E5 k B5 to produce a chiral anomaly, chiral charge (6) identify the latter as a part of the Bardeen polynomi- is created/annihilated in spatially separated regions als [40]. The benefit of the Landau level approach is [Fig.4(d)], at a rate that closely follows the spatial pro- its intuitive interpretation: in the first example, E5 = 0 file of B5 as expected from Eq. (3) [Fig.4(e)]. The Fermi and the finite E · B5 pumps charge from one surface to surface contribution to the chiral charge is calculated as another via the anomalous Hall effect [Eq. (8)] through FS 0 2 dρ5 (y) X 5 ∂tJcov = E · B5/(2π ). Our second example, Fig.2, can = hψ |γ Π |ψ ihψ |∂ H|ψ iδ(µ − ε ). db n y n n b n n be interpreted similarly. In the bulk charge grows as n 0 2 ∂tJcov = E · B5/(2π ) locally while at the surface charge (11) is depleted since B5 has the opposite sign. The corre- with Πy the projector to the position y, cf. Supplemen- sponding current (9) pumps charge from the surface to tal Material. Comparing the chiral charge pumped at the bulk. Since b = B5yxˆ, there are local currents in the Fermi surface and that in the full spectrum to the 2 the bulk δj ∼ B5Eyyˆ and ∇ · δj = B5 · E/(2π ) is pre- expected value proportional to B5, we see that they are cisely the growth rate of local charge [Fig.2(c)], reconcil- close, but neither perfectly reproduces the field theory ing the Fermi surface (covariant) picture with the charge prediction. conserving (consistent) picture on the lattice [57]. To determine the factors behind this mismatch we note Moving on to spatio-temporally varying Weyl node that the erf profile minimizes the spatial overlap be- separations, consider first b = −E5tzˆ, yielding uniform tween regions of chiral charge creation and annihilation 5

(∂ρ5/∂bz > 0 and < 0 at y ≈ +L/4 and y ≈ −L/4 Supplemental Material respectively). However, for both chiralities, there ex- ist regions in momentum space [light/dark regions in Summary of consistent and covariant anomalies Fig.4(b)] where the states are not well-localized in real space. These regions spread out with increasing ξ, such For completeness, we provide a summary of the rele- that the erf profile evolves into a sinusoid-like function, vant facts of consistent and covariant anomalies derived resulting in a poorer match between the B5(y) profile from quantum field theory. As discussed in the main text and the chiral charge response, cf. Supplemental Mate- electric and magnetic fields that satisfy E·B 6= 0 generate rial. The sensitivity of the result to these regions, high- an anomalous imbalance between left and right movers. lighted by the failure of the sinusoid-like profile where Additionally pseudo-magnetic (B5) and pseudo-electric no clear spatial separation exists, implies that the value (E5) field—the fields that couple with opposite signs to 1/3 is modified by such effects for generic profiles of B5. opposite chiralities—will result in the same chiral imbal- This detrimental overlap is minimized if L ξ a is ance, since the contributions of the two chiralities to the satisfied, allowing the intriguing possibility that an exact µ µ µ chiral current J5,cov = JL,cov − JR,cov (µ = 0, 1, 2, 3) will 1/3 may be recovered in this limit. Finally, all anomaly add up. However, these considerations imply also that terms present finite size and quadratic corrections to the µ µ µ the vector current Jcov = JL,cov +JR,cov is not conserved. low-energy field theory (7), which we discuss in the Sup- Mathematically plemental Material. 1 ∂ J µ = (E · B + E · B ) , (12a) In summary, we have provided an intuitive lattice pic- µ 5,cov 2π2 5 5 ture that is based on Landau and pseudo-Landau lev- 1 ∂ J µ = (E · B + E · B) , (12b) els and connects the covariant and consistent anoma- µ cov 2π2 5 5 lies. We explicitly identified the Hall current as the Bardeen polynomial that connects the covariant anoma- a result which can be obtained either diagrammati- lies of the Fermi arcs and restores charge conservation, cally [1, 41] or semiclassically [16, 17, 25, 42]. Eqs. (12) define the covariant chiral anomaly [40, 41] of the covari- most notably when the Fermi surface knots into a bowtie. µ µ We expect that the bowtie Fermi surface and its re- ant vector and axial currents Jcov and J5,cov respectively. sponse to external fields will be important to understand This result is troublesome: a theory which does not strained Weyl semimetals experimentally, in particular conserve charge is unphysical, yet the fields B5 and E5 their transport properties. can be physically realized in the solid state [16, 17, 25]. At the quantum field theory level curing the nonconser- Our work highlights that measuring the consistent or vation of charge amounts to acknowledging that Eqs. (12) covariant anomaly (e.g., the factor of 1/3) depends on are determined solely by the Fermi surface, but that whether the experimental probe is sensible to only the counteracting currents are necessary at energies com- Fermi surface or rather the entire Fermi sea. Addition- parable to the cut-off energy [41]. To ensure this con- ally, perturbations such as strain allow other model pa- straint is satisfied, two additional Chern-Simons currents, µ µ rameters to depend on position, e.g, the Fermi velocity, δj and δj5 , known as Bardeen polynomials, are added, as well as additional terms in Eq. (7)[58]. Our work mo- which transfer the necessary charge and guarantee charge tivates the study of these questions to interpret incipient conservation [43]. They relate the covariant currents µ µ µ µ experiments in strained Weyl semimetals. (Jcov,J5,cov) to the physical consistent currents (J ,J5 ) µ µ µ µ µ µ through J5 = J5,cov + δj5 and J = Jcov + δj , respec- Acknowledgements.- We thank B. Bradlyn, Y. Fer- tively, and take the form reiros, R. Ilan, F. de Juan, K. Landsteiner and D. Pesin for insightful discussions, and especially N.G. Goldman 0 1 1 0 δj = A5 · B; δj = A B − A5 × E ; (13) for discussions that initiated this work and comments 2π2 2π2 5 on the manuscript. This research was funded by the 0 1 1 0 δj = A5 · B5; δj5 = A B5 − A5 × E5 , Marie Curie Program under EC Grant Agreement No. 5 6π2 6π2 5 653846 (A.G.G.), the ERC Starting Grant No. 679722 (14) and the Knut and Alice Wallenberg Foundation 2013- where the chiral gauge field Aµ defines the pseudo-fields 0093 (J.H.B.), the EPSRC Grant No. EP/N01930X/1 5 B = ∇ × A and E = −∇A0 − ∂ A . Together with (S.R.), and supported by the Laboratory Directed Re- 5 5 5 5 0 5 Eqs. (12) results in the consistent anomaly [41, 42] search and Development (LDRD) funding from Lawrence Berkeley National Laboratory, provided by the Director, 1 1 ∂ J µ = E · B + E · B ; ∂ J µ = 0. (15) Office of Science, of the U.S. Department of Energy un- µ 5 2π2 3 5 5 µ der Contract No. DEAC02-05CH11231, and the DOE Basic Energy Sciences (BES) TIMES initiative as well as Eqs. (12) and (15) summarize the two versions of the UTD Research Enhancement Funds (M.H.K.). quantum anomalies. 6

We note that the consistent and covariant currents or, expressed in terms of Heaviside functions, have been discussed in the context of the two-dimensional quantum Hall effect [59]. In this case the edge current is ± X 5 dρ5 (y) = ± hψn|γ Πy|ψniΘ(±dεn) the sum of the edge consistent current plus the anomaly n inflow from the 2+1D bulk Chern Simons term, which × [Θ(µ − εn) − Θ(µ − (εn + dεn))] . (22) combined form the covariant edge current [60]. In a similar way, consistent and covariant currents can be defined To first order in perturbation theory, the infinitesimal for a 3+1D Weyl fermion living at the edge of a 4+1D change in the energies is quantum Hall effect.

dεn = db · hψn|∂bH|ψni (23) Details of the lattice calculations and the sum of Heaviside functions reduces to We are interested in the response of a Weyl semimetal Θ(µ − ε ) − Θ(µ − (ε + dε )) = dε δ(µ − ε ), (24) to external fields E and E5. A time-dependent vector po- n n n n n tential A = −Et gives an electric field, and analogously a time-dependent node separation b = −E5t. We express which finally gives the time-derivative of any expectation value f in terms of the response to changes in A and b, dρFS(y) X 5 = hψ |γ5Π |ψ ihψ |∂ H|ψ iδ(µ − ε ) db n y n n b n n df n = −E · ∂ f − E · ∂ f, (16) (25) dt A 5 b as used in the main text. Numerically, we use a which is especially handy when evaluating time- Lorentz approximation to the delta function, δ(x) = µ 2 2 derivatives proportional to E and E5, such as ∂µj and lim η/(π(η + x )). µ η→0 ∂µj5 where f corresponds to the densities ρ or ρ5. The response of the density to external fields E and E5 is the response to a shift A → A + dA and b → b + db respectively. Explicitly, the chiral density at site y reads The quest for 1/3

X 5 ρ5(y) = hψn|γ Πy|ψni, (17) Two effects challenge the calculation of the chiral n∈occ. anomaly due to E5 · B5 on a lattice. First, B5 always where Πy projects onto site y. When performing an in- averages to zero over the whole sample [17], which im- finitesimal shift of the Weyl node separation b → b+db, plies that a length scale ξ exists that characterizes the the single-particle states change as typical size of the variations in B5. Second, a mass term µ m changes the response of the chiral charge ∂µj5 . X |ψmihψn|∂bH|ψmi |ψn(b + db)i = |ψni + db · (18) εn − εm m6=n Effect of ξ resulting in a change of the chiral density

5 dρ5(y) X X hψn|γ Πy|ψmihψn|∂bH|ψmi We demonstrate the lattice effect set by the length ξ = + h.c. db εn − εm explicitly by using the same profile of the node separa- n∈occ. m6=n (19) tion b as in the main text (10). In Fig.5 (a) and (b), we show how the responses of the total chiral charge ρ5 and Note that the sum goes over the states n that are initially FS occupied at a node separation b. the charge around the Fermi surface ρ5 are affected by The Fermi surface contribution, on the other hand, just different sizes of ξ respectively. The two conditions for ξ counts the subset of states that are lifted above the Fermi discussed in the main text that allow for the identifica- level when shifting b → b+db, minus the states that are tion of 1/3 are both demonstrated explicitly. When ξ ≈ a pushed below the Fermi level, dρFS = dρ+ − dρ− with the variation of B5 occurs on the scale of the lattice con- 5 5 5 stant forcing the density variations Eqs. (19) and (25) to + X 5 interpolate smoothly between lattice sites such that the dρ (y) = hψn|γ Πy|ψni (20) 5 quantum field theory predictions breaks down. When εn<µ, εn+dεn>µ ξ L is not satisfied, tails of the regions with positive − X 5 and negative B5(y) (positive and negative chiral charge dρ5 (y) = hψn|γ Πy|ψni (21) εn>µ, creation) overlap, leading to notable finite size effect and εn+dεn<µ again conflicting with the field theory predictions. 7

(a)

(b)

FS FIG. 5. (a) Responses ∂ρ5/∂bz and (b) ∂ρ5 /∂bz for our lattice model [Eq. (5)] with periodic boundary conditions in FIG. 6. (a) Convergence of the response ∂ρ5/∂Az for a tight- y-directions of length L = 360 and cB = 1.25, cf. Eq. (10). binding system governed by the Hamiltonian (5) with periodic We show results for different widths ξ of the regions with non- boundary conditions in y-directions of length L and a node zero B5. The best matching of the field theory predictions are separation b = bzˆ. A magnetic field B = Bzˆ that satisfies found for ξ = 6, which mostly satisfies the condition a ξ periodic boundary√ conditions is included, with a magnetic L. For ξ = 2, the condition a ξ is not satisfied, making length `B = 2πL. The crosses denote the numerical results lattice scale effects extremely relevant and deviating from the from tight-binding calculations , the solid line the results of field theory limit. For ξ = 30, tails of the two chiral charge a fit with f(L) = c0 + c1 exp(−L/ξ), and the dashed line the pumping regions centered at ±L/4 overlap, leading to notable coefficient c0, i.e., the result in the limit L → ∞. The color finite size effects. denotes different values ofm ¯ , as shown in panel (b), and we further choose b = 0.4. (b) The response ∂ρ5/∂Az in the limit L → ∞ as a function ofm ¯ 2/b2. The different symbols denote Effect of a finite mass different combinations of b and t and the gray line shows the expectation from the lattice model (29). (c) Response of the FS The response of the chiral and total charge to fields E chiral charge around the Fermi surface to E ∂ρ5 /∂A, as a function of the level broadening η for b = 0.4. The dashed and E5 computed on a lattice can deviate from the ex- lines denote the result from panel (a), lim ∂ρ /∂A. (d) pectation based on the simplest quantum field theory due L→∞ 5 Response of the total charge around the Fermi surface to an to corrections set by the mass term, as mentioned in the FS axial field, ∂ρ /∂A5, as a function of the level broadening main text. In this section, we carefully investigate one η. The dashed line denotes the field theory result ∂ρ/∂A5 = example, the electromagnetic contribution to the chiral B/(2π2) as a guide for the eyes. µ 2 anomaly on a lattice, ∂µj5 = E · B/(2π ) and argue how a mass term influences the chiral anomaly. We further show that the mass term does not play the same role for that couples both chiralities and changes the response µ 2 the covariant anomaly ∂µJcov = E5 · B/(2π ). to electric and magnetic fields [62–64]. Such a term re- The main source of lattice corrections to the anomaly sults in an additional classical contribution to the chiral µ ¯ 5 is that the Wilson fermion Hamiltonian [61] used for all anomaly, ∂µj5,class = 2miψγ ψ [1], that is zero in all tight-binding calculations, Eq. (5), has a momentum- equilibrium situations we consider, which we confirm nu- dependent mass term merically. Instead, another consequence of m affects the chiral anomaly: when m 6= 0 in the field theory, the X Mk = m + t (1 − cos ki), (26) eigenfunctions of the corresponding Hamiltonian are no i longer eigenfunctions of the chiral matrix γ5. Instead, that ensures the absence of copies of the Weyl nodes, or close to the Weyl node of chirality χ, the overlap with 5 doublers, on the lattice [61]. When t > 0, the minimum γ for the low-energy eigenstates with a linear dispersion can be obtained analytically to be of Mk is at the Γ point and its maximum at (π, π, π); for m = 0, as set in the main text, the term is zero at s m2 k = 0 and increases away from Γ. The term Mk sets the hψχ|γ5|ψχi = χ 1 − , (27) momentum-independent mass term m in the action (7) |b2| 8

(a) (b) i.e., it scales with the value of Mk at the Weyl nodes,

m¯ ≡ MkW . In the presence ofm ¯ 6= 0 the eigenstates are not strictly eigenstates of γ5 yet in order to define the chiral anomaly using Eq. (6) we effectively assume that they are. An alternative definition of chirality that does not possess this problem is to partition the Brillouin zone into two and declare left and right chiral charges as done in Ref. 65. While it has the advantage of having a clear definition of chirality, it does not reduce to any representative matrix at the field theory level and so we (c) (d) retain the first definition. With these definitions it is tempting to upgrade the chiral anomaly to r 1 m¯ 2 ∂ jµ = 1 − E · B. (29) µ 5 2π2 b2 It is interesting to note that a similar equation would (e) be obtained if one interprets the prefactor as an effective renormalization of the chiral electric charge p 2 2 e5 = e 1 − m¯ /b that would enter the chiral anomaly throughout the coupling to an external chiral gauge field e5Aµ,5 in analogy with ordinary charge [66]. We have un- successfully attempted to derive Eq. (29) from the effective field theory responses for the action Eq. (7) discussed in [62–64]. However, our numerical results on the corrections presented below suggest a richer structure when the mass is non-zero, which has motivated us to leave the FIG. 7. (a) Response ∂ρ5/∂bz for a tight-binding system governed by the Hamiltonian (5) with periodic boundary con- precise connection between the quantum field theory dis- ditions in y-directions of length L = 360. An axial field cussed in these works and our results for a separate study. B5 = B5(y)ˆz generated by b(y), Eq. (10), with ξ = 6 and At the end of the section we specify the conditions where cB = 1 that satisfies periodic boundary conditions is included. our numerics coincide with the massless field theory ex- The different colors denote different values ofm ¯ (cf. panel (b)) pectation validating our results in the main text, up to a and the bold red line shows the profile of B5(y). (b) The in- given order in momentum. tegral over y around the peak is computed numerically for different values ofm ¯ . The dashed line denotes the value of To gain insight on the above expectation numerically, the integral that is expected from quantum field theory. In we investigate how the chiral anomaly changes withm ¯ . (c) and (d), we show the response of the chiral charge around Differentiating between the total charge and the charge FS the Fermi surface ∂ρ5 /∂bz as a function of the level broad- around the Fermi surface, we compute the response to ening η with the different colors denoting different values of external fields E and E5 via Eqs. (19) and (25) on a η. The bold red line shows the profile of B5(y), the filled re- FS lattice with periodic boundary conditions and a constant gion is minimal difference between B5 and ∂ρ5 /∂bz. In (e), B. we show the numerically computed integral over y around the We show the response of the chiral charge ρ5 in peak in B5 for different values ofm ¯ . The dashed line denotes the value of the integral that is expected from quantum field Fig.6(a) and (b). In panel (a), we show the conver- theory. gence with system size, whereas in panel (b), we show that, in the limit of L → ∞, the response of the chiral charge scales with p1 − m¯ 2/b2 as dictated by Eq. (29). 2 µ with b = bµb , i.e., the chiral density around the Weyl is Taking into account just states around the Fermi sur- reduced by a nonzero mass term. We note that a similar FS face via Eq. (25), the chiral charge ρ5 responds approx- factor appears in the effective action for this model and imately in the same way as shown in Fig.6(b), cf. panel is related to the Weyl node separation [63, 64]. (c). Thus, the consistent and covariant anomalies are the On a lattice, the eigenstates of the Hamiltonian (5) same for the term E · B, even in the presence of a mass 5 are only eigenstates of γ for those momenta k where m. In panel (d), we show the response of the total charge 5 Mk = 0. In particular, the overlap with γ of states of around the Fermi surface to a field E5: due to the covari- chirality χ = ± close to the Weyl nodes at χk is FS FS W ant anomaly, ∂ρ /∂A5 6= 0. In contrast to ∂ρ5 /∂A, the FS contribution ∂ρ /∂A5 is not much affected bym ¯ 6= 0. s 2 χ χ Mk The chiral-symmetry breaking term M has more in- hψ |γ5|ψ i = χ 1 − W , (28) k χkW χkW |b2| tricate consequences for the anomaly in the chiral charge 9 due to E5 · B5. We find numerically that the dependence 130, 389–396 (1983). FS [11] Barry R. Holstein, “Anomalies for pedestrians,” Am. J. of ∂ρ5/∂bz and ∂ρ5 /∂bz onm ¯ are different. 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