arXiv:2007.10682v2 [cond-mat.str-el] 25 Aug 2020 · Torsional Landau levels and geometric anomalies in condensed matter Weyl systems Sara Laurila 1,and Jaakko Nissinen y 1Low Temperature

Torsional Landau levels and geometric anomalies in condensed matter Weyl systems

Sara Laurila1, ∗ and Jaakko Nissinen1, †

1Low Temperature Laboratory, Department of Applied Physics,Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland

(Dated: August 26, 2020)

We consider the role of coordinate dependent tetrads (“Fermi velocities”), momentum space ge-ometry, and torsional Landau levels (LLs) in condensed matter systems with low-energy Weyl quasi-particles. In contrast to their relativistic counterparts, they arise at finite momenta and an explicitcutoff to the linear spectrum. Via the universal coupling of tetrads to momentum, they experiencegeometric chiral and axial anomalies with gravitational character. More precisely, at low-energy,the fermions experience background fields corresponding to emergent anisotropic Riemann-Cartanand Newton-Cartan spacetimes, depending on the form of the low-energy dispersion. On thesebackgrounds, we show how torsion and the Nieh-Yan (NY) anomaly appear in condensed matterWeyl systems with a ultraviolet (UV) parameter with dimensions of momentum. The torsional NYanomaly arises in simplest terms from the spectral flow of torsional LLs coupled to the nodes at finitemomenta and the linear approximation with a cutoff. We carefully review the torsional anomalyand spectral flow for relativistic fermions at zero momentum and contrast this with the spectralflow, non-zero torsional anomaly and the appearance the dimensionful UV-cutoff parameter in con-densed matter systems at finite momentum. We apply this to chiral transport anomalies sensitive tothe emergent tetrads in non-homogenous chiral superconductors, superfluids and Weyl semimetalsunder elastic strain. This leads to the suppression of anomalous density at nodes from geometry,as compared to (pseudo)gauge fields. We also briefly discuss the role torsion in anomalous thermaltransport for non-relativistic Weyl fermions, which arises via Luttinger’s fictitious gravitational fieldcorresponding to thermal gradients.

PACS numbers:

I. INTRODUCTION

Gapless fermionic quasiparticles with linear spectrumprotected by topology arise in many condensed mat-ter systems in three dimensions1–5. In particular, ac-cidental crossings of two inversion (P ) or time-reversal(T ) breaking bands at the Fermi energy lead to sta-ble quasirelativistic particles with low-energy dispersionanalogous to relativistic Weyl fermions6,7. Fourfold de-generate crossings with Dirac-like low-energy excitationsoccur for combined P, T (and/or other similar protecting)symmetries8,9. Similarly, in chiral superconductors andsuperfluids with gap nodes, Majorana-Weyl excitationsarise at low energy3,10–14.

By a very general theorem from topology4, the low-energy linear theory near the three-dimensional Fermipoint node takes universally the (γ-matrix) form of aquasirelativistic Weyl/Dirac spectrum, with the preciseform of the metric and other background fields dependingon the microscopic details. It is then of interest to studythe detailed form of this emergent Dirac operator withan explicit cutoff and compare to fundamental, Lorentzinvariant fermions. Following this logic, the conceptof so-called momentum space pseudo gauge fields3,15–24

and “emergent” spacetime3,11–14,25–35 in non-relativisticcondensed matter systems has emerged, where the low-energy fermions can experience background fields of var-ious physical origins, similar to what appears for spin-1/2 (or even higher spin) fermions on curved spacetimesin general relativity or its non-relativistic generalizationswith non-relativistic coordinate invariance.

Notably, in the low-energy quasilinear theory, the localFermi velocities form emergent tetrads which determinethe geometry of the conical dispersion. The tetrads, andits field strength torsion, couple to the quasiparticle mo-mentum effectively as in gravity. The effects of such fieldsin non-relativistic systems appearing at finite density µFand Fermi-momentum pF are expected to be very dif-ferent from their relativistic counterparts appearing atp = 0. Amongst other things, the system at finite Fermior crystal momentum is then charged under the fieldstrength these geometric background fields15,25,36–38. Inthree spatial dimensions, this corresponds to the anoma-lous translational symmetry for chiral fermions, leadingto axial anomalies in the system33,39 from momentumspace translations. For other relevant condensed matterconsiderations of this anomaly, see e.g.3,9,11,40–55. In thispaper we point out that geometric (gravitational) con-tributions in the chiral anomaly, second order in gradi-ents, are expected in generic non-homogenous condensedmatter Weyl systems with momentum space fields (back-ground spacetimes) due to inhomogenous deformationsleading to torsion.

More generally, the appereance of the tetrad back-ground fields in condensed matter Weyl systems is built-in in the low-energy theory, thus opening the possibil-ity of simulating Riemann-Cartan (or Newton-Cartan)spacetimes for the low-energy fermions. In case of non-trivial background torsion, the so-called chiral gravita-tional Nieh-Yan anomaly can appear56,57. In contrastto the axial anomaly with gauge fields, this anomaly de-pends on a non-universal UV cut-off parameter Λ, with

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canonical dimensions of momentum. While the statusof the torsional contribution in relativistic systems hasbeen debated for long58–64, the appearance of this term innon-relativistic condensed matter systems with explicitUV cutoff to the Weyl physics is a priori plausible33,37.Aspects of the gravitational anomaly in condensed mat-ter have been considered in e.g.33,37,50,51,65,67–72 includ-ing Weyl/Dirac fermions in superfluids, superconductorsand semimetals. The dimensional hierarchy and descentrelations of the torsional anomaly were recently analyzedin Ref.72 from a Hamiltonian persperctive in a relativis-tic model. Nevertheless, it seems that any explicit valueof the cutoff parameter has not been discussed in detail,except in the recent paper33 by one of the present au-thors. In the simplest possible terms, the non-universalanomaly UV scale originates from the regime of validityof the quasirelativistic linear spectrum and the associatedanomalous transport. This UV-scale is, in the simplestpossible terms, just the validity of the Taylor expansionclose to the node which is experimentally a low-energyscale in the system33. Generalizing this, it seems thatthe NY anomaly non-universally probes the chiral spec-trum and transport, well-defined only at low energies,and conversely, merging in some left-right asymmetricway to other bands as required by global consistencyand symmetries. Indeed, at face value, the spectrumand spectral flow can be terminated in a multitude ofinequivalent ways. If the system is anisotropic, the inter-play of different scales in the system becomes essential,as evidenced by the consideration of the anomaly in e.g.Newton-Cartan geometry with quadratic spectrum alonga preferred direction or finite temperature (see below).

Here we will further argue for the torsional anomalyterm using the simplest computational apparatus for thechiral and axial anomaly: adiabatic spectral flow in thepresence of torsional Landau levels1,10. In this context,the torsional LLs appeared implicitly already in Refs.10,43

and more recently in topological semimetals in37 in com-parison with Pauli-Villars regularization of Lorentz in-variant fermions. On the other hand, such a relativisticregularization scheme is at best only an approximation incondensed matter systems, since the linear Weyl regimeapplies to low-energies with an explicit cutoff scale. Thislinear regime can be anisotropic and, furthermore, is con-tinuously connected with the non-relativistic regime withquadratic dispersion. Moreover, as discussed in this pa-per, the role of the spectral flow is drastically alteredby the finite node momentum as compared to relativisticfermions.

The role of momentum space pseudo gauge fields, withmomentum dependent axial charge also becomes evidentin the geometric framework for the axial anomaly. Im-portantly, it is incorrect to assume the universal U(1)axial anomaly for such gauge fields, since the effectivemomentum space description has a finite regime of valid-ity. To the best of our knowledge, it seems that this facthas been overlooked thus far. Related to the momentumdependence in the anomaly, the UV-scale can be supple-

mented by a infrared (IR) temperature scale of thermalfluctuations, in contrast to, say U(1), gauge fields. Withsome caveats, this IR anomaly becomes universal dueto universality of thermal fluctuations close to the node.The thermal torsional anomaly and the associated cur-rents were recently considered in Ref.73. Contribution tothe torsional NY anomaly at finite temperatures was fur-ther discussed in74–78 for relativistic fermions at p = 0.The closely related role of torsion and viscoelastic ther-mal transport has been also studied e.g. in79–82. Here wewill mostly focus on the non-universal UV contributionat zero temperature. For completeness, we comment onthermal effects by non-zero temperature gradients, whichpoint to still new types of anisotropic torsional anomaliesterms not present in systems with Lorentz invariance.

This rest of this paper is organized as follows. Sec-tion II discusses the low-energy Weyl Hamiltonian andthe associated geometry in condensed matter systemsfrom the perspective of emergent background spacetimes.The following Section III reviews the relativistic torsionalanomaly and spectral flow argument, focusing on theextension to finite node momentum and the compari-son with the anomaly for U(1) gauge fields presented inAppendix A. Sec. IV discusses the torsional anomalyin chiral superfluids and superconductors, where it canbe matched with experiment33,45,46. This followed by amodel of T -breaking strained semimetals in Sec. V. Wealso briefly discuss the role of torsion in the presence ofthermal gradients in Sec VI. We conclude with a com-parison on previous results Sec. VII and the conclusionsand outlook of our results.

II. WEYL FERMIONS IN CONDENSEDMATTER AND RELATIVISTIC SYSTEMS

A. Weyl fermions in condensed matter

We consider a fermionic system with broken time-reversal symmetry (T ) or inversion (P ). In the vicinityof a generic degenerate crossing at pW , ignoring all otherbands, the 2 × 2 Hamiltonian is H = σaHa in terms ofthe unit and Pauli matrices σa, a = 0, 1, 2, 3. This leadsto the expansion

H(p) = σaeia(p− pW )i + · · · , (1)

where

eia =∂Ha

∂pi

∣∣∣∣p=pW

. (2)

The expansion is, of course, valid for |p− pW | � pWsince the remainder is of the order of |p− pW |2. Thisprovides an explicit cutoff for the linear Weyl regime thatis, nevertheless, continuously connected with the non-relativistic quadratic dispersing spectrum and the otherbands.

3

The existence of the Weyl node degeneracy is protectedby topologuy in a finite region since there are three pa-rameters and three constraints3,4,6,7. Via rotations andscalings, pa = eiapi, the Hamiltonian becomes the right-or left-handed relativistic Weyl Hamiltonian, at Fermimomentum pW ,

H(p) = χσa(p− pW )a (3)

where χ = ±1 = sgn(det eia) is the chirality, definedas the direction of (pseudo)spin with respect to thepropagation momentum. The band energies are E =

(p− pW )0 ±√|p− pW |2. The role of the coefficients eµa

is simply to determine the (anisotropic) Fermi velocitiesof the conical dispersion ω2 = −gij(p − pW )i(p − pW )jvia the (inverse) metric

gij = −∑

a,b=0,1,2,3

eiaejbδab ≡ −eiae

jbδab (4)

where the Einstein summation convention for repeatedlatin and greek indices will be henceforth assumed. Thespatial tetrad eia is extended to a non-degenerate matrixeµa by considering the operator σaeµai∂µ = i∂t−H(p) withµ = t, x, y, z. In particular, the coefficient eµ0 = {1, vi} isnon-trivial in type-II Weyl semimetals and in superfluidsand superconductors with superflow. The case with non-zero spatial eta, a = 1, 2, 3 was considered in28. Thesebreak different symmetries, while the spacelike tetradstransform like gauge potentials corresponding to axialmagnetic and electric fields. While the Hamiltonian (1)is usually analyzed for translationally invariant systems,it remains valid for weak deformations. This can be seenin any consistent gradient expansion scheme, e.g. thesemi-classical gradient expansion of the BdG Hamilto-nian for superconductors/superfluids, or the Schrieffer-Wolff transformation for Bloch Hamiltonians29,31.

We conclude that the Hamiltonian (1) has striking sim-ilarity to relativistic fermions coupled to non-trivial back-ground geometry or gravity, albeit with some importantcaveats. More precisely, if we consider the low-energyWeyl fermion ΨW in terms of the original excitations Ψ,we see

Ψ(x, t) = eipW ·xΨW (x, t), (5)

which, however, corresponds to the anomalous (chiral)rotations in the system, thus making the finite node mo-mentum pW very important. In the rest of the paper, wewill explicitly consider the anomaly implied by (5) in thepresence of non-trivial background fields eµa(x), from Eq.(2), after reviewing the necessary background geometryin the next section. U(1) gauge fields are assumed to beabsent. We will focus here on T -breaking systems, wherein the simplest case one finds Weyl nodes of opposite chi-rality at ±pW , whereas for inversion P breaking systemsone has at minimum four Weyl points, which are invari-ant under T and map non-trivially to themselves underinversion.

B. Quasirelativistic fermions

We briefly summarize quasirelativistic fermionson curved Riemann-Cartan spacetimes here, seee.g.37,56,59,60. These spacetimes are defined via anorthonormal frame ea = eaµdx

µ, giving rise to metricas in (4), and a (matrix) spin-connection ωµdx

µ, bothof which couple to the Dirac (and Weyl) equations.Informally, the eaµ is a spacetime “translation gaugefield”, while ω is the gauge connection corresponding tolocal (Lorentz) rotations. See e.g.57.

As discussed above and the Introduction, analogousfields arise in the low-energy Weyl Hamiltonian close tothe nodes in condensed matter systems on flat space,giving rise to emergent spacetimes for the low-energyfermions. These are, however, not strictly relativis-tic in the sense that the emergent metric does not fol-low from locally Lorentz invariant spacetimes impliedby general relativity, but rather from the microscopicnon-relativistic UV theory at low energy. This whatwe call quasirelativistic and emergent. Note that thespin-connection is strictly speaking a gauge field of alocal symmetry entering the Dirac operator. Thereforeits emergence needs the corresponding local symmetry.Notwithstanding, it arises however, e.g. in chiral su-perconductors and superfluids due to the local combinedU(1) symmetry corresponding to gauge and orbital rota-tion symmetry30,33,89. The tetrad and connection fieldsgive rise to the torsion T a = dea+(ω∧e)a and curvature

R = dωµ − ω ∧ ω field strength tensors that equivalentlycharacterise the spacetime. From the tetrad one can de-rive the spacetime metric, which enters as a secondary ob-ject, in contrast to usual Riemannian spacetimes wherethe connection is symmetric and uniquely fixed by themetric.

In terms of equations, the basic quantities are thetetrad eaµ and coordinate connection Γλµν . The formeris the metric matrix square-root

gµν = eaµebνηab, eµae

νbηab = gµν (6)

by defining a local orthonormal frame, in terms of ηab =diag(1,−1,−1,−1). Now tensors Xa···µ···

b···ν··· can carry lo-cal orthonormal (Lorentz) indices and coordinate indices;the two bases can be transformed by contracting with eaµor the inverse eµa . The connection consistent with basischanges defined as ∇eaµ = 0, has two parts, one for localorthonormal indices and one for coordinate indices andis metric compatible. The connection determines geo-metric parallel transport in the system. Without loss ofgenerality it can be written as

ωaµb = eaλeνbΓλµν − eaν∂µeνb , (7)

where Γλµν is the coordinate connection with torsion

Tλµν = Γλµν − Γλνµ. (8)

4

The connection can be decomposed in terms of torsionas

Γλµν = Γλµν + Cλµν , (9)

where Γλµν = 12gλρ(∂µgνρ+∂νgµρ−∂ρgµν) is the Christof-

fel connection fully determined from the metric andCλµν = 1

2 (Tλµν + T λµ ν − T λ

µν ) is the contorsion tensor.The low-energy quasirelativistic Weyl fermion theory

is, in the chiral Dirac fermion basis ψ =(ψL ψR

)T,

where ψR,L are Weyl fermions and γa = σa ⊕ σa withσa = (1,−σi),

SD =

∫d4xe

1

2ψγa(eµaiDµ − pWa)ψ + h.c. . (10)

where e ≡ det eaµ and Dµ is the covariant derivative cor-responding to the canonical momentum

Dµ = ∂µ −i

4ωabµ σab − iqAµ (11)

where γab = i2 [γa, γb] and Aµ is a U(1) gauge poten-

tial with charge q. They enter the covariant derivativeor canonical momentum due to local Lorentz (rotation)and gauge symmetries. For the emergent spin-connectionto exist, the local rotation symmetry has to be dynami-cally generated. See Sec. IV and33. Importantly to ourapplications, the quantity pWa = (µW ,pW ) is the shiftof the of the Weyl (or Dirac) node at chemical potentialµW = eν0pWν and pWa = eiapWi in momentum space.The magnitude of latter is a UV-parameter that is fixed(up to small deformations) in the low-energy theory.

C. Anisotropic Newton-Cartan fermions

A related concept to the Riemann-Cartan space-time (10) is an anisotropic version of a non-relativisticNewton-Cartan (NC) spacetime. In the latter, we singleout a Newtonian time and, in our case, a preferred spa-tial direction with quadratic dispersion in contrast to thelinear Riemann-Cartan case. In what follows in Secs. IVand V, this preferred direction is along the Weyl nodeseparation with uniaxial symmetry and anisotropic scal-ing. Compared to the standard NC case, there is an addi-tional gauge symmetry corresponding to a U(1) numberconservation and a local Milne boost symmetry along theanisotropy direction26,68,69,83. These will both be gaugefixed to zero and will be applied mostly in the case of thechiral superconductor/superfluid, where they are absentnaturally for Majorana-Weyl fermions. With the time co-ordinate fixed, the symmetries of the NC spacetime thencorrespond to the generalized Galilean transformationsxi → xi + ξi(x, t)26,27,35,84,85.

The metric is

gµν = nµnν + hµν (12)

where now nµ is a spacelike vector, eaµ a (degenerate)tetrad with metric hµν restricted to the orthogonal sub-space, with e0

µ = δ0µ representing Newtonian time,

hµν = ηabeaµebν , a, b = 0, 1, 2, (13)

with inverses

nµ`µ = 1, eaµ`

µ = 0, eaµeµb = δab , a = 0, 1, 2. (14)

The connection and torsion follow as

Γλµν = Γλµν [h] + `λ∂µnν , (15)

from the condition that L`hµν = 0, equivalent to∇µnν =∇λhµν = 0. The torsion is given as

T 3µν ≡ nλTλµν = −∂µnν + ∂νnµ (16)

and the standard spin-connection perpendicular to `µ,ωµν [h], as in Eq. (7), amounting to local rotation sym-metry along `µ. The fact that nµ is covariantly constantis natural, since it can be identified with the directioncorresponding to non-zero Weyl node separation in e.g.T -breaking Weyl systems.

We discuss in Sec. IV the Landau level problem ofMajorana-Weyl fermions corresponding to such a space-time, with the (right-handed Weyl) action

SW =

∫d4x√gψ†[(τac⊥e

µai∂µ − τ3ε(i∂`)]ψ + h.c.

(17)

where ε(∂`) = ∂2` /(2m)− µF in the anisotropic direction

with ∂` = `µ∂µ, corresponding to the non-relativisticdispersion and degenerate metric `µ`ν = gµν − hµν .In this case the relative anisotropy of the two terms isc⊥/c‖ = mc⊥/pF , where pF =

√2mµF and c‖ = vF the

Fermi velocity. This NC model can be matched to theresults discussed33. Note that a very similar model withLifshitz anisotropy was considered in68, and the ensuingtorsional anomalies for momentum transport in69. Fora semimetal under strain, the model in Sec. V is cor-respondingly anisotropic but the precise connection to aspecific NC model and symmetries remains to be workedout in full detail.

III. TORSIONAL ANOMALIES AND LANDAULEVELS

A. Torsional Nieh-Yan anomaly

Now consider Weyl fermions coupled to a tetrad withnon-zero torsion and curvature with the U(1) gauge fieldsset to Aµ = A5µ = 0, see however Appendix A. As forthe U(1) gauge fields, or gravitational fields representedby the metric gµν , the Weyl fermions are anomalous inthe presence of non-zero torsion (and curvature).

5

We focus on a pair of complex fermions of oppositechirality with currents jµ±. The (covariant) torsionalanomaly for the axial current jµ5 = jµ+ − j

µ− is58–62

∂µ(ejµ5 ) =Λ2

4π2(T a ∧ Ta − ea ∧ eb ∧Rab) (18)

+1

192π2tr(R ∧R)

=Λ2

4π2εµνλρ(

1

4T aµνTaλρ −

1

2eaµe

bνRabλρ) +O(∂4).

For a discussion of the relativistic torsional anomalyterm, we refer to56,57,59,60,63, and for applications in topo-logical condensed matter systems,33,37,69–71,75,77. For themixed terms between torsion and U(1) gauge potentials,see e.g.64. We focus on the anomaly contribution solelydue to the geometry (tetrads), we will not consider them.Ref.71 also considered novel “axial” tetrads eaµR 6= eaµL at

two Weyl nodes R,L, with (vector like) T 5 appearing asin Eq. (A3). We will require eR = ±eL but this is actu-ally rather strong constraint basically only allowing for(improper) rotations that can be gauged away. In the chi-ral Weyl superfuid/conductor or minimal time-breakingsemimetal, eR = −eL but this just the chirality of thenodes and is built in the axial nature of torsion. Intrigu-ingly the trace part of torsion arises as the gauge fieldof local Weyl scalings but this comes, since non-unitary,with a complex gauge coupling56. The presence of differ-ent (chiral) tetrad couplings and overall symmetry con-siderations would be highly interesting for e.g. paritybreaking and other non-minimal Weyl systems with sev-eral nodes, some of which coincide in momentum space.

To conclude, we note the following salient propertiesrelated to the NY anomaly term: i) Despite appearances,it is given by the difference of topological terms, albeit infive dimensions60. ii) The NY anomaly term is of secondorder in gradients and therefore the leading contributionfrom the background geometry in linear response. iii)The UV-cutoff is isotropic in momentum space by (lo-cal) Lorentz invariance but is multiplied by the geomet-ric term, which can be anisotropic. In condensed mat-ter applications, we do not expect Lorentz invariance soin principle non-isotropic anomaly coefficients can arise(see e.g. Sec. VI). iv) The NY term has contributionsfrom the torsion and curvature, dictated by local exact-ness d(ea ∧ Ta) = T a ∧ Ta − ea ∧ eb ∧ Rab. The twocontributions are a priori independent before the geom-etry (the torsionful connection) is fixed. The anomalyis therefore physical input for the spacetime geometry orconnection33. In more pragmatic terms, the anomaly co-efficient Λ2 can be computed in the case when ωµ = 0,although the constraints of a consistent spacetime geom-etry should be kept in mind.

B. Quasirelativistic fermions and torsional Landaulevels

Now we proceed to compute the torsional NY anomalyin non-relativistic systems utilizing the Landau levelargument. To set the stage and remove confusionsbefore presenting our main results, we briefly review(quasi)relativistic torsional Landau levels with linearspectrum, see e.g.37. The computation of the Landau lev-els is close to and inspired by the spectral flow obtainedin10,43 for momentum space gauge fields at pW 6= 0. Sim-ilar considerations for p = 0 can be found in72,75.

The Weyl particles are governed by the effective Hamil-tonian

HW = σaeia(i∂i − pW,i) + h.c. (19)

where pW is the location of the Weyl point. Due to thelack of protecting symmetries (namely at least broken Por T ) the shift vector

pW,µ = (µW ,pW ) (20)

is necessarily non-zero for the existence of the Weyl point.However, we will focus on the T -breaking case with twonodes of opposite chirality at ±pW and assume that µWis zero unless otherwise specified.

In this section, we assume that the coordinate depen-dence of the Hamiltonian arises solely from the tetradeµa(x), while the location of the node, paW is assumedto be constant. Note that the coordinate momentumpWµ ≡ eaµpWa can still vary and in the case T aµν 6= 0there is non-zero torsion. Torsional LLs arise when, say,12εijkT 3

jk = TB zi is constant with the other torsion com-

ponents and spin connection vanishing. We discuss laterin Secs. IV, V on how to make the identification betweenlow-energy emergent gravitational fields and microscopicbackground fields in specific examples.

1. Torsional Landau levels

Specifically, the assumed (semi-classical) tetrads ea =eaµdx

µ and the inverse ea = eµa∂µ are, following10,37,43,

e0 = dt, e1 = dx, e2 = dy, e3 = dz − T (y)dx

e0 = ∂t, e1 = ∂x + T (y)∂z, e2 = ∂y, e3 = ∂z. (21)

Now we compute the spectrum of the Weyl fermions inthe presence of a constant torsional magnetic field T (y) =T 3By. The corresponding metric is

gµνdxµdxν = ηabe

aeb

= dt2 − (1 + T (y)2)dx2 − dy2 (22)

− 2T (y)dxdz − dz2.

The torsion is given as T 3ij = ∂µe

3ν − ∂νe3

µ or T 3 = de3 =12∂yT (x)dx ∧ dy, i.e. T 3

xy = −∂yT (y) = T 3B . In analogy

6

FIG. 1: Dispersion of left-handed (LLL in blue) and right-handed Weyl fermions (LLL in red) at pW = 0 under a tor-sional magnetic field, respectively.

with the electromagnetic tensor, we will call 12εijkT ajk and

T a0i torsional magnetic and electric fields, respectively.The Weyl hamiltonian couples to the non-trivial vier-

bein as, χ being the chirality,

Hχ =χ

2σaeiapi + h.c.

=χ

[pz px + pzT

3By − ipy

px + pzT3By + ipy −pz

]. (23)

As usual, the energy eigenvalues are obtained from squar-ing the Hamiltonian

H2 = σaeiapiejbσbpj = eiae

jbσaσbpipj + eiaσ

aσb{pi, ejb}pj

= eiaejb(−η

ab + iεabcσc)pipj +iT 3B

2[σ2, σ1]pz

= −gij pj pj − T 3Bσ3pz.

= p2y + p2

z + (px + T 3B ypz)

2 − T 3Bσ3pz.

We see (23) is equivalent to a LL problem in a magneticfield [Eq. (A7) for Bz = T 3

B and e = pz in Appendix A].With those identifications, the spectrum is consequently[from Eq. (A17)]:

E(pz) =

{±√p2z + 2|pzT 3

B |n, n ≥ 1

sgn(T 3B)χ|pz|, n = 0.

(24)

The lowest Landau level (LLL) is chiral and unpairedwith the simple eigenfunctions, σ3 = ±1,

Ψσ3(x, px, pz) ∼ ei(pxx+pzz)e±(pxy−pzTBy2/2) (25)

where the (pseudo)spin or helicity is determined bysgn(pzTB). We stress that the shape of the spectrumis in general also modified due to the momentum replac-ing the electric charge: left-handed states now disperseas E < 0 and right-handed states as E > 0 (or vice versa,depending on the sign of the field), see Fig. 1.

2. Spectral flow and anomaly

Analogously to the Landau level calculation with elec-tromagnetic fields, we may turn on a constant tor-sional electric field parallel to T 3

B by introducing time-dependence to the vierbein as e3

z = 1 + T 3Et where

T 3Et � 1. Then we have ez3 = (1 + T 3

Et)−1 ≈ 1 − T 3

Et.This induces adiabatic time-dependence ∂tpz = (∂te

3z)p3,

analogous to the Lorentz force, which leads to spectralflow of states through the momentum dependent tor-sional electric field. The number currents, in the vicinityof the node pz = e3

zp3 = pWz = 0 are for both chiralities

ej0χ(t) =

T 3B

2π

∫ Λ

−Λ

dp3

2π|pz|

= −Λ2T3B(1 + T 3

Et)

4π2= −Λ2

T 3xye

3z

4π2, (26)

where a cutoff Λ has been introduced to regularize themomentum dependent current density and spectrum. Wesee that for E < 0, particles flow below the cutoff,whereas for E > 0, holes flow above the cutoff, see Fig.2. Then, taking into account the fact that the tensorialcurrent density is modified by the volume element ed4xin the presence of torsion, see e.g.61,80,

˙ej0χ = ∓Λ2

T 3xy∂te

3z

4π2= ∓Λ2T

3BT

3E

4π2

= ∓ Λ2

32π2εµνρσ T 3

µνT3ρσ, (27)

from holes or particles moving above or below the cutoff,respectively, depending on the direction of the torsionalelectric field. This is the vacuum regularization that wasalso used in Ref. 37 in the sense nvac =

∑|En|≤Λ sgn(En),

where an additional factor of one half was present, pre-sumably due to comparison with anomaly inflow fromfive dimensions. Generalizing this to a fully covariantexpression, see the Appendix A, gives

1

e∂µ(ejµ5 ) =

1

e

Λ2

16π2εµνρσ T 3

µνT3ρσ, (28)

and in particular ∂µ(ejµ) = 0 as required. We discussthe relativistic vacuum and the spectral flow leading to(28), as compared to nodes at finite momenta and axialU(1) fields, more in the next section.

Torsional anomaly for pW 6= 0

If we now displace the Weyl nodes in the relativisticcase (23) by pz = ±pW in momentum space, correspond-ing to a T -breaking Weyl system, the spectrum (24) takesthe form

E(pz) =

{±√

(pz ± pW )2 + 2|pzT 3B |n, n ≥ 1

sgn(χpzT3B)(pz ± pW ), n = 0.

(29)

7

FIG. 2: Relativistic spectral flow at k = 0 in the presenceof torsion, with the adibatic transfer of states. Dashed lineindicates the location of the cutoff Λ.

The lowest, chiral Landau level looks exactly like thatof a Weyl fermion in an axial magnetic field, Eq. (A27).Higher levels are distorted due to the effective chargecarried by the particles being their momentum. See Fig.3.

FIG. 3: Left-handed Weyl particles at kz = k0 (LLL in red)and right-handed Weyl holes at kz = −k0 (LLL in blue) undera torsional magnetic field. Spectral flow is indicated with thearrows.

Since the node is at finite momentum pW 6= 0, alsothe spectral flow summation is centered around pW ±Λ′,where Λ′ is a cutoff from e.g. the validity of the lin-ear spectrum. For notational convenience and compari-

son to Eq. (28), we introduce the momentum cutoff as

Λ′ =Λ2

rel

2 pW , where we expectΛ2

rel

2 � 1, this being thedimensionless ratio of the cutoff of the linear spectrumto pW . The spectral flow results in the expression, whereparticles and holes simply add at the two nodes,

1

e∂µ(ejµ5 ) =

1

e

p2WΛ2

rel

16π2εµνρσ T 3

µνT3ρσ (30)

which shows that the NY anomaly cutoff is proportionalto the node momentum pW , and is small by a factorΛ2

rel � 1 corresponding to the validity of the linear Weylapproximation.

3. Comparison of torsion to U(1) fields

From Figs. 1 and 3, we see that the spectrum of tor-sional LLs resemble the LL spectrum of charged particlesin U(1) axial and vector fields, with the momentum de-pendent charge to torsion kept in mind. See appendixA for a complete review of the U(1) case for compari-son. It is well-known that the contribution of torsion forcomplex chiral Weyl fermions can be equivalently castin terms of the axial gauge field γ5Sµ ≡ γ5εµνλρTνλρcorresponding to the totally antisymmetric torsion, seee.g.60,61. We stress that while the spectral equivalence oftorsional and U(1) LLs is of course expected, the phys-ical appearance of the anomaly is drastically different:the density of states of the LLs depend on momentumand thus the dimensional coefficient Λ2 and the need foran explicit UV-cutoff appears. Similarly, the physics ofFigs. 2 and 3 is completely different, although both arisefrom spectral flow in momentum space under torsion.

On this note, although the relativistic result in (27) isfamiliar, there seems to be still confusion in the literatureabout the role of torsional Landau levels in momentumspace and the validity of the NY anomaly due to theexplicit UV cutoff. For relativistic Weyl fermions withLorentz invariance up to arbitrary scales, the spectralflow is symmetric around p = 0, leading to the conclu-sion that the anomaly indeed can cancel. This is simplyby the observation that, in the absence of Lorentz sym-metry breaking at high energy, no net transfer of occu-pied and empty states in the vacuum takes place duringthe adiabatic spectral flow, cf. Fig. 2. The net transferof j5 requires left-right asymmetric regularization at thescale of Λ with chirality disappearing above that scale,maintaining ∂µj

µ = 037. Alternatively, at the very least,there is a divergence as Λ→∞. In contrast, for quasirel-ativistic Weyl fermions at finite node momentum and anexplicit cutoff to the Weyl spectrum, the spectral flowcan terminate due to the non-relativistic corrections atthe cutoff scale of Λ2

rel, also implying that chirality isno longer well-defined, leading to net transport of statesand momenta relative to the vacuum (and other quan-tum numbers of the Weyl fermions if present). A relatedfact is that the momentum that plays the role of chirality,which remains physically well-defined irrespective of the

8

scale. We also note that the flow is composed of particlesand antiparticles (holes) at the different nodes. It wouldbe interesting to study the detailed role of the breakdownof relativistic spectrum and spectral flow numerically, fol-lowing Ref. 22. There only the charge density at finitechemical potential from the node is analyzed, correspond-ing to Fig. 6 and the expected deterioration away fromthe Weyl node is verified.

IV. CHIRAL WEYL SUPERFLUIDS ANDSUPERCONDUCTORS

Now we discuss the role of the torsionalanomaly in p-wave superfluids and superconduc-tors with gap nodes and associated Weyl-Majoranaquasiparticles13,14,41,65,67,86. Close to the nodes, theFermi energy is tuned to the Weyl point due to the exis-tence of the p+ip pairing amplitude. The chiral anomalyis related to the non-conservation of momentum in thecondensate and normal state quasiparticles46. The rela-tion of this to the torsional gravitational anomaly andthe LL spectral flow was briefly pointed out in Ref.33.Earlier related work can be found in10,12,13,42,43,87,88.

The spinless gap amplitude, with equal spin pairingunderstood, takes the form

∆(p) =∆0

pF(m + in), (31)

where c⊥ = ∆0/pF has units of velocity. The direction

l = m × n is a low-energy Goldstone variable for the

condensate. At low-energy, the direction of l can fluc-tuate and there is combined U(1) gauge symmetry89 inthe m− n plane, leading to the Mermin-Ho relations be-

tween l and vs3,86,90. In the following, we focus on the

Landau levels and torsion, keeping the magnitudes of pFand ∆0 fixed. Related to this, for the superconductors,the end results apply the case where the EM potentialAµ = 0 which amounts to the case where we work in thegauge where vs − A → vs. In the following computa-tions we will set vs = 0 as well, since this corresponds tothe case where one has only torsion, see Ref. 33 for thegeneral case with superfluid velocity. The orientation of

the orthonormal triad l can still rotate for the torsionaltextures.

Considering first the simple homogenous case, the lin-earization of the BdG Hamiltonian takes the form of aWeyl Hamiltonian close to the nodes of E(p) at p =

∓pF l,

HBdG(p) =

(ε(p) 1

2{p,∆(p)}12{p,∆

†(p)} −ε(−p)

)(32)

≈ ±τaeia(pi ∓ pF,i).

Note that the BdG excitations are Majorana, Φ†(p) =τ1Φ(−p), as expected in a BCS paired system. Here we

have taken the normal state dispersion ε(p) =p2−p2

F

2m ,

where m is the 3He atom mass. The tetrads are

ei1 = c⊥m, ei2 = −c⊥n, ei3 = −c‖ l, (33)

where c‖ ≡ pFm = vF . Henceforth, to conform with

relativistic notation, we will work with dimensionlesstetrads in units of c‖ = 1. The quasiparticle dispersion

is E(p) = ±√ε(p)2 + |∆(p)|2 ≈ ±

√c‖q

2‖ + c2⊥q

2⊥, with

q = p − pF for the Weyl quasiparticles. The linear ex-pansion is valid when |p− pF | � pF which provides anexplicit cut-off for the Weyl description, requiring thatthe remainder

1

2

∂ε(k)

∂ki∂kj(p− pF )i(p− pF )j =

1

2m(p− pF )2 (34)

� eia(p− pF )i.

This leads to the condition, in addition to the trivial|p− pF | � pF from the Taylor expansion of ε(p), that

EWeyl � mc2⊥ =

(c⊥c‖

)2

EF . (35)

which will prove important later. In particular, the en-ergy cutoff for the Weyl quasiparticles is anisotropic inmomenta q = p− pF around the Weyl point,

q⊥ �(c⊥c‖

)pF , q‖ �

(c⊥c‖

)2

pF , (36)

if we consider the Weyl fermion system in the case wherethe background fields couple parallel and perpendiculardirections33. This happens in the chiral system since the

three direction are coupled by l = m× n and the corre-sponding Mermin-Ho relations.

A. Landau levels in linear approximation

To compute the LL levels in the order parameter tex-ture corresponding to a torsional magnetic field, we cantake the ”weak-twist” texture m + in = x + iy − iTBxzwith |Bx| � 1, which corresponds to l = z+TBxy

10,42,43.The BdG Hamiltonian then takes the form

HBdG =

[ε(p) 1

2{∆i, pi}

12{∆

† i, pi} −ε(−p)

](37)

=

[ε(px, py, pz)

∆0

pF[px + i(py − TBpzx)]

∆0

pF[px − i(py − TBpzx)] −ε(−px,−py,−pz)

].

Near the gap node p = −pF l we may linearize the oper-

ator ε(p) as εp ≈ −vF l · (p+ pF l) ≈ −vF (pz + pF ). Thisleads to

H+ = eiaτa(pi − pF e3

i ) = τa(eiapi − pF δ3a) (38)

9

FIG. 4: The torsional LL spectrum for the anisotropicNewton-Cartan model in chiral superfluids/conductors withthe spectral flow indicated. Note that we have inverted thehole-like right-handed Landau level at −pF and the spectrumis particle-hole doubled. Overall there is a corresponding fac-tor of 2 from spin-degeneracy.

with

eia = (c⊥δi1,−c⊥[δi2 − TBxδi3],−c‖δi3), (39)

where we remind that c‖ ≡ vF and c⊥ ≡ ∆0

pF. This corre-

sponds, up to the sign of the field TB and the tetrad, tothe case (21) after a rotation in the x− y plane.

After moving to scaled coordinates c−1⊥ x ≡ x, c−1

⊥ y ≡y, c−1

‖ z ≡ z, corresponding to dimensionless and scaled

momenta pa ≡ eiapi, we can define the annihilation oper-ator a ≡ 1√

2|TBpz|[(|TBpz|x− py) + ipx] to arrive at the

Hamiltonian

Hpz<0 =

[p3 + pF

√2|TBpz|ia†

−√

2|TBpz|ia −(p3 + pF )

], (40)

which is (A7) after a Galilean boost p3 → p3 + pF . Theeigenstates are then

Ψn,pz<0 =

(unφnvnφn−1

)ei(pzz+pyy). (41)

where φn ≡ φn(x), for n ≥ 0, are harmonic oscillatoreigenstates and vanish otherwise. The condition for nor-malization is |un|2 + |vn|2 = 1, corresponding to the BdGparticle and hole amplitudes. Carrying out a correspond-

ing calculation at the Weyl point p = pF l, we have theHamiltonian

Hpz>0 =

[p3 − pF −

√2|TBpz|ia√

2|TBpz|ia† −(p3 − pF )

], (42)

which can be identified as the left-handed HamiltonianH− = −eiaτapi after a rotation about l such that m →−m and n→ −n. Its eigenstates are

Ψn,pz>0 =

(unφn−1

vnφn

)ei(pzz+pyy). (43)

Depending on the chirality, i.e. sign of momentum atthe node, the LLL is either particle- or holelike as in Eq.(25). The conclusion is that the spectrum looks like therelativistic spectrum in Fig. 3, when the linear approxi-mation for ε(p) ≈ ±c⊥(pz − pF ) is valid, Eq. (36). Thiscorresponds to the spectrum of axial U(1) fields with mo-mentum dependent charge and density of states per LL.The density of states is (A19) in the scaled coordinates,which gives, with e0

µ = δ0µ,

j0dV = ej0dV =|pzTB |

4π2dV . (44)

B. Anisotropic Newton-Cartan model

We just showed that the simple order parameter tex-ture in chiral superfluid or superconductor gives rise tothe torsional LLs for the low-energy Weyl quasiparticles,in the linear regime close to nodes. We can howeverconsider quadratic dispersion beyond the linear approxi-mation

ε(p) =p2

2m− µF →

p2z

2m− µF , (45)

which corresponds to the anisotropic Newton-Cartan(Majorana-Weyl) fermion model in Sec. II C.

The above model has the same regime of validity in thechiral superfluid or superconductor as the linear approxi-mation in Eq. (36), since it also neglects the rotationallyinvariant dispersion ε(p) of the normal state, see alsoRef. 33. The chiral p-wave BCS state has the uniaxialanisotropy of Eq. 45, however, and this carries to thelow-energy Weyl description in the form of the emergentspacetime. The other benefit of the anisotropic model(45) is that the LL spectrum can be computed for mo-menta far from pF , up till p = 0, corresponding to thefilled levels of the non-relativistic Fermi system, whichare absent in the relativistic linear model. This is impor-tant for the global properties of the chiral spectrum andanomaly. In this way the contribution to the anomalouscurrent from the superfluid vacuum can be analyzed, seeSec. IV C.

The spectrum follows simply from Eqs. (40), (42) bythe substitution ∓(p3 ± pF )→ ±ε(±pz). From squaringthe Hamiltonian, the corresponding eigenvalues are atboth nodes

En = ±√ε(pz)2 + c2⊥|TBpz|2n,

E0 = ±sgn(pzTB)ε(pz). (46)

10

for n ≥ 1. The LLL state retains the gaussian form (25).The condition for normalization is |un|2 + |vn|2 = 1, andconsequently the particle and hole amplitudes are in bothcases

un =

√En + ε(pz)

2En, vn = i

√En − ε(pz)

2En. (47)

With E0 = ε(pz) we have v0 = 0, meaning that the lowestlevel particles appear only for pz < 0. For pz > 0 u0 = 0when E0 = −ε(pz), so for positive momenta only holesappear at the lowest level, as we found for the linearmodel. In this case we must, however, remember thatthe hole spectrum arises due to the Majorana doublingof the BdG spectrum and is not physical. This cancelswith a corresponding factor of two from spin-degeneracyin the Fermi system. This leads to the LL spectrum inFig. 4.

C. Spectral flow, axial density and consistentanomalous vacuum current

Now we are equipped to compute the spectral flow re-sulting from torsional Landau levels, corresponding tothe covariant torsional NY anomaly. For the anisotropicNewton-Cartan model we can also compute the consis-tent vacuum current of the condensate, since the disper-sion takes into account the filled states below the Fermi-level which is not the case for the linear approximationclose to the Weyl nodes. For the chiral superfluid (or-conductor) we have to take into account that the parti-cles are Majorana-Weyl but a factor of two results fromthe spin-degeneracy.

1. Axial density

The torsional spectral flow leads to the anomalous den-sity as

ej0± =

∫ ∓pF+pFΛ2

2

∓pF−pF Λ2

2

dp3NLL(pz) = ±p2F ( c⊥c‖ )2

4π2TBe

3z.

(48)

where the cutoff for the Weyl spectrum is taken at

Λ2 =(c⊥c‖

)2

, corresponding to Eq. (36) with 12 � 1.

Remarkably the LL results matches the more generaltorsional contribution for the NY anomaly including cur-vature, as implied by the anomalous momentum non-conservation in the system as found in Ref. 33. Thisresult was found by matching the anomaly on emergentspacetime of background the chiral p-wave system to thecorresponding BCS hydrodynamic result of the super-fluid. In particular, including the effects of superflowleads to a spin-connection and curvature perpendicular

to l, as required by the Mermin-Ho relations90.

In the chiral superfluid (or superconductor) the aboveresult holds for both the linear quasirelativistic and theanisotropic Newton-Cartan spacetime, as defined by thetetrad (33). This simply follows from the fact that thecutoff for the validity of both models coincides with (36).In this case, therefore, the anisotropic model NC is ex-pected to require the same cutoff as the linear model sincethe system is probed also in the perpendicular direction.

This morally happens since l = m× n, making the triaddependent33,89,90. Strictly speaking in the LL-model we

approximated l ≈ z which for the general non-trivial tex-tures is given higher order corrections42.

2. Axial current

On the other hand, for the non-relativistic anisotropicNC model, however, we can also compute the anomalousvacuum current, corresponding to the anomalous super-fluid momentum from the filled states below pF

10. Theglobal spectrum has correct form, valid also outside thevicinity of the Weyl points. The anomalous momentumcurrent is given by

janom,‖ = −2

∫ pF

0

dp3NLL(pz)p3 = − p3F

6π2l(l · ∇ × l)

(49)

and even extending to pz = 0, there is no need for acutoff. See Fig. 4.

This is actually the correct hydrodynamic result forthe (weak-coupling) BCS system10,40,42 to lowest order ingradients, since the final answer for the anomalous vac-

uum current is sensitive only to the e3 = l direction, evenin the presence of vs (corresponding to curvature in theperpendicular plane). Upon taking the time-derivative ofthis momentum, the hydrodynamics of the system pro-duce the covariant current implied by the Weyl anomaly.If we assume, without any supporting arguments, thatthe curvature and torsion contribute to the current (49)as they enter the anomaly Eq. (18), we get the sameresult if we apply the cutoff (36) as above, even in thelinear model. We note that these findings are corrobo-rated by the thermal contribution to the NY anomaly, asfound in Ref.73. The proper inclusion of curvature alsoensures that states far away from the Fermi surface donot contribute to the currents.

These considerations beyond the LL spectral flowaside, what we want to here emphasize is that the(49) current corresponds to the consistent anomaly,and can be derived from a corresponding Wess-Zumino terms that should be generalized for torsionalspacetimes18,62,69,72,91–93. See especially69, where theconsistent and covariant anomalies are discussed in ananisotropic Lifshitz model, closely related to Eq. (17).We leave the study of the consistent vacuum current fromthe perspective of gravitational anomalies with torsionfor the future.

11

V. STRAINED WEYL SEMIMETALS

Semimetals with Weyl fermions arise in solid-state sys-tems where the Fermi energy is tuned to a band-crossingin the Brillouin zone1,5. The tetrads arise universallyvia the coefficients of the linear expansion. In this case,the fermions are also charged leading to the possibilityof the U(1) anomaly with electric fields1. In additionto the tetrads, related effective background (axial) fieldscan be considered with similar origin as in the chiralsuperconductor3 – the (constant) shift of the Weyl nodein momentum space that leads to the existence of theprotected Fermi arc states18,19,94. Here we would like toclarify the related but physically distinct torsional con-tribution to anomalous transport from the tetrads in thepresence of elastic strains. In fact, due to the universalcoupling of the tetrads to momentum15,37, as in gravity,one expects that deformations of the (lattice) geometrywould lead to effects that probe the Weyl fermions viathe background tetrads. This framework correctly takesinto account the anomalous physics of the momentumdependent fields, see nevertheless17,19,20,51,70,71,75,95,96.

We start in a roundabout way, first discussing the low-energy Weyl Hamiltonian and then considering a latticemodel for a realistic T -breaking material.

A. Bloch-Weyl fermions in crystals

The low-energy Bloch-Weyl Hamiltonian is of theform1,5,9

h±(k) = ±σa(ka ∓ kF,a) + h.c.

= ±σa

2eia(ki ∓ kF,i) + h.c.. (50)

where now

eia =∂HTB(k)

∂ka

∣∣∣∣kF

(51)

are simply the linear coefficients of the expansion of theunderlying (tight-binding) Bloch Hamiltonian HTB(k)near the Weyl nodes. Before we consider lattice defor-mations in this model, we remark on the interplay of thetetrads and momentum. The lattice momentum is15

pa =i

2a

∑x

c†xcx+a − c†x+acx =∑k

sin(kaa)c†kck. (52)

Under non-trivial background fields, the Weyl system it-self is anomalous under the lattice translation symmetry,T3 = Tz, corresponding to the conservation of the latticemomentum p3,

T †z c±kTz = e±iakFc±kF (53)

which corresponds to an anomalous chiral rotation of thelow-energy Weyl fermions at the T -breaking nodes ±kF .

Here c†k creates the state corresponding to the latticeperiodic Bloch state |vk〉 = |vk+K〉, with wave function

ψk(x) = eik·xvk(x). (54)

In the presence of elastic deformations correspondingto torsion, i.e. phonons, the anomalous chiral symme-try corresponding to translations is manifested as thenon-conservation of (lattice) momenta between the Weylfermions and the background phonons33,39, as found insuperfluid 3He-A for the p+ip-wave paired Fermi-liquid3.See also16,69,71,97.

B. Elastic deformations

Now consider general lattice deformations. The origi-nal unstrained lattice momenta entering the Weyl Hamil-tonian are represented as ka and the deformed lattice isgiven as ki = e ai ka in the coordinate system of the labo-ratory, where e ai 6= δai to first order in the strains. Thesewill couple as expected in the continuum model, as longas we take into account the lattice model properly, as wenow recall following15. See also71. We have the contin-uum linear strain tensor,

e ai = δai + w ai = δai + ∂iu

a

eia = δia − wia = δia − ∂jubδabδij (55)

where ua/a � 1, in terms of the lattice constant. Thismeans that kF,a is held fixed, whereas kF,i with δkF,i =w ai kF,a is deformed (in the laboratory coordinates). This

becomes on the lattice

ka → ka − wiasin kia

a≈ eiaki,

ki → ki + w ai

sin kaa

a≈ e ai ka. (56)

where wia = ∂jubδabδ

ij is defined above and in the lastapproximation, the linear approximation for strain aswell as kia � 1, close to the Γ-point, are used. Inaddition we assume that we work with low-frequenciescorresponding to the acoustic phonons, below the Debyeenergy15.

C. Lattice model

In general, a model for a T -breaking Weyl semimetalconsist of layered 2D Wilson fermions tuned to a zeroenergy crossing in three dimensions3,22. For a model ofthis kind pertaining to a real material, Ref.20 considereda time-reversal invariant k · p close to the Γ-point, wherethe the Weyl node itself will be at finite momentum cor-responding to four momenta in the Brillouin zone, theminimum for P -breaking system. While the k · p modelis realistic, it is more convenient to work with an ex-plicit model with a lattice regularization that produces

12

the same results. In terms of a tight-binding model, theyconsidered

Hlat(k) = ε(k) +

(hlat(k)

−hlat(k)

), (57)

where we focus on the time-reversal odd block hlatt(k) ofthe T -invariant model3,20,22,

hlat(k) = tz(M −∑

i=x,y,z

ci cos kia)σ3 (58)

+(tx sin kxa)σ1 + (ty sin kya)σ2.

For −1 <M−cx−cy

cz< 1 the model hlat(k) has Weyl

points at

±akF = (0, 0,± arccosM − cx − cy

cz), (59)

otherwise it is gapped. The dimensionful tetrads are

eia(±kF ) = a(tx, ty,±tzcz sin akF,z)δia. (60)

Inversion symmetry P acts as hlat(k) → σzhlatt(−k)σz.For simplicity we set cz = 1, cx,y = c⊥, tx,y = t⊥ andassume uniaxial symmetry along z in the following. Weexpect (56) to hold for the Weyl semimetal model Eq.(57), originating from the k ·p model close to the Γ-point.

For this tetrad we can moreover ignore the differenceof lattice and coordinate indices, with uij = 1

2 (∂iuj +

∂jui) + O(u2) the symmetric lattice strain. The straininduces the deformation considered in Ref.16 and19,20

δhlat(k) =− tzβeluzzσ3 cos akz

+ t⊥βel(uxzσ1 + uyzσ

2) sin akz (61)

which gives

δeia = atzβeluiiδia sin(kFa) + at⊥βel

∑i′ 6=i

uii′δi′

a cos(kFa)

(62)

where βel is the Grunesein parameter. Restricting to auniaxial strain corresponding to the axis of the Weyl nodeorientation, with the approximation that akF � 1,

eza → atz(1 + βeluzz)δa3 + at⊥∑i=x,y

βeluzjδja,

δez3 = atzuzz, δez1 = at⊥uzx, δez2 = at⊥uyz. (63)

This has the (dimensionless) inverse tetrad, up to theneglected terms O(u2) in strains,

e1i = x, e2

i = y,

e3i = z− βel

(uzx,

(tzt⊥

)uzy,

(tzt⊥

)uzz

). (64)

This is what we expected, based on the correspondinguniversal continuum limit (55) and the lattice substi-tution (56) coupling to geometry, apart from the (non-

universal) couplings βel,(tzt⊥

)between the phonons and

electrons of the lattice model15. Now in the presence ofnon-homogenous strain vector e3

z depending coordinatesand time, torsion T 3

µν and spectral flow will arise. TheLandau level arguments of Sec. III and IV apply fora torsional magnetic field from uzx,zy(x, y) (in the “sym-metric gauge”) and an adiabatic electric field from uzz(t),as in19,20.

D. Torsional density of states in anomaloustransport

Armed with the geometric background fields corre-sponding to torsional (magnetic field), we can considerthe anomaly resulting from the chiral rotation (53). Thelinear Weyl model is valid up to the approximation

tz(M −∑

i=x,y,z

ci cos kia) (65)

≈ tza2

2

[c⊥(k2

x + k2y) + (kz ∓ kF )2

]≈ tzaei3(ki−kF,i) = (tza sin kFa)qz (66)

which is simply restricted by the ignored terms of theremainder in the expansion. Apart from the trivial qz �kF � 1/a, also

cx cos qxa+ cy cos qya ≈c⊥a

2

2(q2x + q2

y) =c⊥a

2

2q2⊥

� txtzaqx +

tytzaqy =

t⊥tzaq⊥ (67)

leading to the constraint q⊥ � 2t⊥c⊥atz

, meaning

EWeyl �t2⊥c⊥tz

, (68)

for the perpendicular direction. We are working in theunits where −1 < M−2c⊥ < 1 and cos kFa = M−2c⊥ ≈1. For the effects of any torsional anomaly from magneticstrain, we can just evaluate the chiral densities at thenodes,

n±(Λ) = ej0± =

∫ ∓kF (1+ Λ2

2 )

±kF (1−Λ2

2 )

dk3NLL(kz)

= ∓k2FΛ2

4π2βel

(tzt⊥

)TBe

3z. (69)

It is interesting to recall that for the chiral superfluid,while strictly it must be that Λ2 � 1 since qz � kF , wefound that the cutoff was parametrically high “ 1

2 � 1”in terms of the validity of the Weyl description. Therehowever, due to the orthonormal triad, also the perpen-dicular direction couples to the transport, with the cutoffEq. (36) which in real 3He-A is actually ∼ 10−6pF .

For the semimetal, the case where qz ∼ t⊥tz sin kF a

q⊥ �kF arises when assuming that we isotropically couple to

13

the perpendicular directions for general strain field con-figurations. Plugging in real parameters, we expect thatfor e.g. Cd3As2, t⊥ ∼ tz sin kFa

20. Another option wouldbe to consider the Newton-Cartan model with quadraticspectrumM−2c⊥−cos kza along the Weyl node directionwith uniaxial strain only, with the constraint qz � kF .The same model with different parameter also applies forthe Dirac semimetal Na3Bi20 and references therein.

Independent of whether one has a torsional electricfield ∂te

3z 6= 0 or an electric field Ez driving the spectral

flow, as in Fig. 6 and 8, this will lead to the suppressionof the density proportional to Λ2, corresponding to thevalidity of the linear Weyl approximation, in the anoma-lous transport, as compared to the Fermi wavevector kFand the pseudo gauge field in momentum space19,20. Wenote that this reduction of anomalous axial density is sim-ply due to the momentum dependent density of states.This, as we have explained, naturally follows from thetetrads and torsion coupling to momenta and should becontrasted with a U(1) gauge field and constant densityof states, as dictated by the universal minimal couplingand the topology of U(1) gauge fields.

VI. THERMAL EFFECTS

Finally we briefly recall and discuss thermal contribu-tions to the torsional anomaly. There are two possibleeffects: i) the small but finite temperature enters the NYanomaly as the scale of thermal fluctuations in momen-tum space. These are analyzed in73,75,76 ii) There is a re-lated finite thermal gradient in the system and one com-putes the thermal response via Luttinger’s fictitious grav-itational field98. We note that non-zero time-like torsionfor the Luttinger spacetime implies the non-single valuedtime coordinate in the fictitious gravitational field80. Seealso81,82,87,99–101.

Here we focus on the effects of a thermal gradient, thecurrents induced can be computed by coupling the sys-tem to fictitious spacetime metric, following Luttinger98.Specifically, we assume a thermal gradient

∇σ = − 1

T∇T (70)

which is equivalent to a weak gravitational potentialg00 = 1 + 2σ in the system. The perturbation δg00 cou-ples to the Hamiltonian (energy current) T 00. In unitswhere the velocity of propagation is v = 1, the metric is

ds2 = e+2σdt− δijdxidxj (71)

≈ (1 + 2σ)dt2 − δijdxidxj (72)

from which the linear response to the thermal gradient σcan be calculated98. This can be generalized to a metric

ds2 = e2σ(dt+ e−σNidxi)2 − δijdxidxj (73)

= e0µe

0νdx

µdxν − δijdxidxj , (74)

now with a small gravimagnetic potential3,100

Agµ = (eσ, Ni) ≈ (1 + σ,Ni) ≡ e0

µ, (75)

where Ni describes a velocity field in the units wherev = 1. The gravitational thermal potential3,66,100

− 1

T∇T = ∇σ − ∂tNi. (76)

whence

e0µ = (eσ, Ni), eaµ = δaµ, a = 1, 2, 3 (77)

eµ0 = (e−σ, 0), eµa = (e−σNi, δia), a = 1, 2, 3. (78)

In this case Eq. (76) becomes

− 1

T∇T = ∇σ − ∂tNi = ∂ie

0t − ∂te0

i = T 0it (79)

where T 0µν = ∂µe

0ν − ∂νe

0µ is the temporal torsion, as-

suming zero temporal spin-connection ω0µb ≡ 0. It is ex-

pected then, that one would have possibility for anoma-lous transport in terms of the combination of thermalgradient and vorticity T 0

ij = ∂iNj − ∂jNj in the veloc-ity field Ni(x), as in the chiral vortical (and magnetic)effect66,74.

Now similarly as we expect momentum density at theWeyl node (Pµ)node = Πtµ = pF e

i3δµi ej

05

33 for the Weylsystems at finite pWa = pF δ3a, or since T 0µ = eeµaT

ta,

eΠt3 =p3FΛ2

16π2e3µei3δµi ε

0νλρe3νT

3λρ (80)

we expect an energy density of the form

J tε = eT t0 = pF ej05 =

pFT2

12v2εtijke0

iT0jk (81)

where Tµa ≡ 1eδSδeaµ

. The anomaly of this current would

be proportional to T∇T , and is indeed reminiscent ofthe chiral vortical effect66,81. We can also expect mixedterms, in the sense that there should be a correspond-ing energy current from both the momentum density andthermal current, ∂te

i3 6= 0, at the node

J iε = eT i0 =pFT

2

6v2ε0ijke3

j × T 00k +

pFT2

12v2ε0ijke0

tT3jk,

(82)

these “mixed” contributions to the currents were identi-fied and discussed in Ref.77.

The message we want to convey here is that one canindeed expect anisotropic and “mixed” contributions tothe torsional anomalies, in the sense that the Lorentzinvariant Λ2ηab → ΛaΛb a generalized anisotropic ten-sor, in various condensed matter systems depending onthe symmetries, perturbations and cutoffs. We leave thedetailed discussion of such thermal gravitational contri-butions for the future, see however75,77 and the generaldiscussion in73.

14

VII. ON THE RELATION OF EMERGENTTORSION AND PSEUDO GAUGE FIELDS

Here we summarize our findings in relation to earlierliterature, where the momentum space field correspond-ing to the shift of the node is often considered as an axialgauge field3,10,16,17,19–21,24,71,95. We note that torsion canbe shown to enter as an axial gauge field constructed fromthe totally antisymmetric torsion γ5Sµ = εµνλρTνλρ

60,61

coupling to the momentum. This is essentially what wefound in Secs. III and IV with the momentum space de-pendent LL density of states. The LL calculation andanomaly itself should be performed by taking this mo-mentum dependence into account, as we have done here.

How are tetrads with torsion otherwise different fromthe momentum gauge field? The symmetries correspond-ing to the tetrads are translations which for finite nodemomenta, requisite for condensed matter Weyl fermions,corresponds to the anomalous chiral symmetry. There isno local gauge symmetry corresponding to the Berry cur-vature in momentum space. On the other hand, the geo-metric formulation is suited for such translation symme-tries and reveals the background geometry of the space-time emerging from the node4. The overall geometrycan made consistent with the non-relativistic symmetriesaway from the Weyl node for a finite momentum range.For the anomalous axial density and anomaly, this leadsto the parametric suppression compared to U(1) anomalyand the UV-scale pW . The phenomenological implica-tions of this are significant, even without the theoreticalrecourse to the emergent geometry.

We also note that Ref.71 discusses torsion (and the con-servation of momentum) in strained semimetals in termsof a model with both the axial gauge field from the nodeand the tetrad with elastic deformations. While sucha “splitting” between low-energy and high-energy mo-menta is in principle allowed, it makes the considerationof the momentum dependent anomalies more involved,with the danger of double counting. The momentumanomaly (without EM gauge fields) should be propor-tional kW∂µ(ejµ5 ), as found in33.

The original paper15 for elastic deformations takes anexplicitly geometrical view point which nicely connectswith the strain induced tetrad formalism proposed here.In the simplest possible terms, we start with the Weyl(or Dirac) Hamiltonian in flat space with the small de-formation eia = δia + δeia,

H+ = σa(ka − kWa)→ σa

2eia(ki − kWi) + h.c.

=σa

2(eiaki − kWa) + h.c.. (83)

≈ σa

2([δia + δeia]qi + kW δe

ia) + h.c.

where now kW δeia = −kW δeai is the momentum space

gauge field in the Hamiltonian with (almost) constanttetrads10,15,19,20,43,71. The right-hand side is the Hamil-tonian in coordinate (or laboratory) space, which is the

one we have experimental access to, and is deformed withrespect to the orthogonal frame of ka. We see that the

momentum ki couples to eia, as expected, and the shiftis essentially constant in the Hamiltonian, in the sensethat kFa is constant corresponding to the undeformedcase, irrespective of the deformation. At the same time,the laboratory value changes though as kFi = eai kFa. Inthe examples we considered, in the chiral superfluid andsuperconductor we explicitly have that kF,i = pF e

3i , giv-

ing kFa = pF δ3a. Similarly, for the strained semimetal we

consider the originally unstrained lattice Fermi wave vec-tor kFa(x)→ k′Fa(x+u) ≈ kFa(x)+∂iu

akFa(x) ≡ eai kFaunder strain x′ = x+ u, giving Eq. (55) as expected.

What this means more generally is that ∇kFa = 0,in terms of the connection corresponding to the emer-gent spacetime, as discussed in Sec. II. In fact this isone of the requirements for the consistent assignment ofthe low-energy geometry. On the other hand, all the tor-sional spacetimes we considered are in some sense abelian(or gravitoelectromagnetic) since the relevant fields canbe identified as an abelian gauge fields in momentumspace, amounting to what was called “minimal coupling”trick in15,37. In this case however, the gravitational char-acter comes still evident in the momentum dependentcharge and density of LLs, as expected for gravitationalresponse, coupling to momenta and energy densities in-cluding thermal effects.

VIII. CONCLUSIONS AND OUTLOOK

In this paper, we have argued for the emergence ofnon-zero torsional anomalies in Weyl (and Dirac) sys-tems with simple Landau level arguments. In particular,we were motivated by the possibility of non-zero torsionalNieh-Yan anomalies in condensed matter systems with anexplicit cutoff and the lack of relativistic Lorentz symme-tries. For the anomaly, the spectral flow in the presenceof torsion clearly renders non-zero results for Weyl nodesat finite momentum. Although obtained with specificsimple field configurations corresponding to the torsionwith Landau level spectra, they are expected to general-ize covariantly in terms of the relevant spatial symmetriesof the system. We discussed two idealized spacetimes re-lated to the symmetries, the linear Riemann-Cartan andthe anisotropic Newton-Cartan spacetime with quadraticdispersion.

We also briefly discussed the thermal torsion via Lut-tinger’s fictitious spacetime, since we can expect mixedanomalies already from the inclusion of thermal gra-dients. This connects to gravitational anomalies andtransport in general73. The recent results on universalanomaly coefficients in linear response thermal transportrelated to gravitational anomalies53,54,102–105 are related.From the non-universal torsional anomaly, via e.g. themomentum dependent LL density of states, the expectedgravitational anomaly polynomials at finite temperaturearise already at the level of linear response from the uni-

15

versality of IR thermal fluctuations73. Moreover, we ex-pect that the emergent tetrads with coordinate depen-dence arise rather generally in any Weyl system, makingsense of evaluating the linear response to these, even inflat space.

We clarified the relation between momentum spacepseudo gauge fields and the emergent tetrads. It is im-portant to realize that the spectral or Hamiltonian cor-respondence between torsion and U(1) magnetic fields,e.g. in a Landau level problem, is not yet enough for theanomalies to match in general. The simple LL spectralflow argument is enough to identify the non-universalcutoff appearing in the NY anomaly term. The mes-sage is that low-energy tetrads and geometry couple tothe momentum in a universal way, even in lattice modelswith some caveats15,16, due to the non-universal couplingof the lattice phonons and fermions as compared to purecontinuum. The UV scales appearing in the termina-tion of anomalous chiral transport from such emergentfields, related to the Fermi-point momentum pW and theregime of validity of the effective Weyl/Dirac descrip-tion, are naturally understood from the geometric per-spective. In the presence of both independent U(1) fieldsand momentum space tetrads we should also expect manymixed terms, as studied e.g. in37,64. The mixed torsionalanomalies should also be carefully reconsidered with re-gards to finite node momentum, where we again expectdifferences to relativistic fermions. On this note our re-sults for the anomaly at finite momentum are in contrastto96, where a model with torsion is compared to a rela-tivistic model at p = 0 with pseudo gauge fields withoutconsideration of node momentum coupling to the torsionor the cutoff of the quasirelativistic dispersion.

More formally, what we did amounts to applying theK-theory theorem of Horava4 to the geometry of specificWeyl nodes in three dimensions, by keeping track of theUV symmetries and scales in the problem for the precisethe form of the emergent geometry and fields couplingto the quasiparticles. The topology only guarantees theeffectively Dirac like spectrum, with everything else de-pending on the microscopics.

Many interesting avenues remain in the geometric de-scription of topological condensed matter systems withgapless fermions, including also nodal line systems32,106.It would be extremely interesting to study the grav-itational anomalies in Weyl and Dirac systems fromthe global symmetry perspective with many nodesWeyl, taking into account the relevant space groupsymmetries16,36,107–109. More generally, the appearanceof low-energy quasirelativistic fermions with exotic geo-metric backgrounds within feasible experimental reach isexpected to give more insight also to the physics of rela-tivistic gravitational anomalies with torsion60, althoughthe symmetries and status of the background fields aredramatically different.

Acknowledgements. — We thank Z.-M. Huang for cor-respondence on his work, T. Ojanen and P.O. Sukhachovfor discussions. Finally we especially thank G.E. Volovik

for discussions, support and collaborations on relatedsubjects. This work has been supported by the EuropeanResearch Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (GrantAgreement no. 694248).

Appendix A: Review of the chiral anomaly and thespectral flow argument

1. Weyl fermions in vector and axial U(1) fields

The simplest way to argue for the axial anomaly incondensed matter systems is the spectral flow argument,utilizing Landau level spectrum in 3+1d1,10. To that end,one envisages a Hamiltonian of the form

HR,L = ±σi(i∂i − qAi;R,L) (A1)

where Ai is some U(1) gauge field with charge q, i.e. thecharges in the system are quantized in terms of q. Notethat we still assume that the vector and axial combina-tions could still be both non-zero,

Aµ =1

2(AR +AL)µ, A5,µ =

1

2(AR −AL)µ. (A2)

Under these gauge fields, the chiral fermions always ex-perience the chiral anomaly, where the classical conser-vation laws corresponding to these fields are broken.

The are summarized by the anomaly equations18,21–24

∂µjµ =

q2

8π2εµνλρFµνF5λρ (A3)

∂µjµ5 =

q2

16π2εµνλρ(FµνFλρ + F5µνF5λρ).

Naturally, if the particles couple to both axial and vec-tor gauge fields, countercurrents must be introduced tothe system to conserve the total particle number, as re-quired by conservation of charge. This is done by addingBardeen-Zumino counterterms to the effective action18,

Γ[A,A5]→ Γ[A,A5]−∫

d4x

12π2εµνρσ FµνAρA

5σ. (A4)

This is the Bardeen counter term, and it introduces thecountercurrents

δjµ =1

12π2εµνρσ(2FρνA

5σ + F 5

ρσAν)

δjµ5 =1

12π2εµνρσ FνρAσ. (A5)

These modify the anomaly in Eq. (A3) so that jµ isconserved:

∂µjµ = 0

∂µjµ5 =

1

16π2εµνρσ

(3FµνFρσ + F 5

µνF5ρσ

). (A6)

Please see Refs.18 and22,23 for more discussion.

16

a. Landau levels

We consider the minimally coupled Weyl Hamiltonianwith vector potential A = (−By, 0, 0),

Hχ = χσi(pi − qAi) (A7)

= χ

[pz px + qBy − ipy

px + qBy + ipy −pz

], (A8)

where χ = ±1 denotes the chirality fo the fermion. Withan eigenstate ansatz ψ = ei(pzz+pxx)φ the eigenvalueproblem becomes

Hχψ = χ

[pz px + qBy − ipy

px + qBy + ipy −pz

]ψ. (A9)

For qB > 0 the off-diagonals can be identified as raisingand lowering operators for a harmonic oscillator in they-direction (displaced by px),{

a =√

2qB−1

[qBy + px + ipy]

a† =√

2qB−1

[qBy + px − ipy] ,(A10)

which satisfy the properties {a, a†} = 1, aφn =√nφn−1,

and a†φn =√n+ 1φn+1 for eigenstates of the harmonic

oscillator φn. The eigenvalue equation becomes

Hχψ = χ

[pz

√2qBa†√

2qBa −pz

]ψ. (A11)

The energy eigenvalues are obtained from considering thesquared Hamiltonian operator:

H2χ = (p− qA)2 − qσ ·B

= p2y + p2

z + (px + qBy)2 − qBσ3

(A12)

whence

E2 = p2z + 2|qB|(n+ 1)− qBσ3,

E = ±√p2z + 2|qB|n, n ≥ 0. (A13)

Looking now at the action of the ladder operators oncomponents of the eigenstates ψ, they must of the form

ψ = ei(pzz+pxx)

[φn

Cnφn−1

](A14)

where φn are eigenstates of the harmonic oscillator,φn−1 = 0 and Cn is a factor determined from the eigen-

value equation to be Cn =

√2qBn

±E + pzfor n 6= 0. The

n = 0 state is ”half” occupied, since

ψ = ei(pzz+pxx)φ0

[10

]with chiral dispersion relation E = pz for H+ andE = −pz for H−, after the elimination of the trivial zeromodes HχΨ.

For qB < 0 the spectrum is the same but the eigen-states are now

ψn = ei(pzz+pxx)

[Dφn−1

φn

], n ≥ 1, (A15)

ψ0 = ei(xpx+zpz)φ0

[01

](A16)

where D =

√pz ∓ E

2|qB|n. The zeroeth Landau level disper-

sion relation is E = −pz for H+ and E = +pz for H−.In summary:

E =

{±√p2z + 2|qB|n, n ≥ 1

sgn(qBχ)pz, n = 0.(A17)

The degeneracy of each state can be determined fromcontaining the system within a finite volume LxLyLz andrequiring the center of the harmonic oscillator be withinit:

0 ≤ px|qB|

≤ Ly. (A18)

The x-direction is free and is therefore quantized as px =n 2πLx

with n ∈ N. The z-direction is similarly quantized

in units of ∆pz = 2πLz

, so the number of states in thexy-plane per ∆pz is

n =|qB|4π2

LxLyLz. (A19)

b. Spectral flow

When an electric field parallel to B is turned on adi-abatically, for example as A = (−By, 0,−Ezt), thestates flow in the spectrum according to Lorentz’s law askz = qEz. The unpaired LLL chiral modes flow to spe-cific direction, whereas the higher LLs cancel. The statesconsequently move in or out of the vacuum depending ontheir chirality as

∂tj0χ = sgn(qχ)

q2EB

4π2

= −sgn(qχ)q2

32π2εµνρσ FµνFρσ. (A20)

We need to generalize (A20) from Minkowski space-time (with metric signature +−−−) to a general space-time with or without torsion. The Landau level calcu-lation genralizes to a non-trivial metric and coordinatedependent tetrads, when we work in the coordinate spacexi ≡ eiaxa, edV = dV , where e = det eai , compared to thelocal Minkowski space. The invariant density of statesand fields to be (for e0

µ = δ0µ)

dN

dVdV =

|qB|4π2

dV (A21)

=dV

dV

dN

dVedV =

|qB|4π2

edV , (A22)

17

which we need to use when we do not want the (scalingof the) tetrads to affect the physical density or flux we

are interested in. With kz = −Ez the anomaly becomes

1

e∂t(ej

χ0 ) =

1

e

χ

32π2εµνρσ FµνFρσ, (A23)

which matches (A3) after covariantly generalized to anon-trivial metric.

Inclusion of axial fields

Left and right chiral fermions may also couple indepen-dently to different gauge fields A+ and A− depending onthe chirality:

Hχ = χσi(pi − qA±i ) (A24)

This is often the case in condensed matter systems withpseudo gauge fields. We then define the axial vector po-tential A5 = 1

2 (A+ − A−) with corresponding axial elec-

tric and magnetic fields B5 and E5, while the total vectorpotential is A = 1

2 (A+ + A−). The corresponding cur-rents are from (A20)

j0 = ∂tj0+ + ∂tj

0− =

1

2π2(EzB

5z + E5

zBz)

=−1

8π2εµνρσ FµνF

5ρσ (A25)

j05 = ∂tj

0+ − ∂tj0

− =1

2π2(EzBz + E5

zB5z )

=−1

16π2εµνρσ

(FµνFρσ + F 5

µνF5ρσ

). (A26)

This is the covariant chiral anomaly (A3), represented asspectral flow under parallel electric and magnetic fields.The pictorial version for these equations in form of theLL the spectral flow can be found in Figs. 5 to 8.

Weyl node at finite momentum

In condensed matter systems the Weyl nodes are dis-placed from p = 0. Let us fix the symmetry by settingthe node at pz = pW . It is straight-forward to see thatshifting the momentum as pi = pi ± δ3

i pW in the Hamil-tonian

H± = ±σi(pi − qA±i ), (A27)

simply shifts the spectrum by ±pW .An axial anomaly can then arise from the momentum

space structure itself, corresponding to e.g. the anoma-lous quantum Hall effect39,48, since one has to shift Ψby Ψ → e∓ipW ·xΨ to obtain the low-energy Weyl exci-tation Ψ = e±ipW ·xΨW . While this is not important forthe spectrum of U(1) states, and the axial anomaly withparallel field, it is crucial in the case of the protectedFermi arcs, anomalous quantum Hall effect, as well asthe torsional anomaly with the tetrads.

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FIG. 5: Dispersion and spectral flow of right-handed (red) and left-handed (blue) particles (q > 0) under parallel B,E.

FIG. 6: Spectral flow of parallel B5, E with the same conventions.

FIG. 7: Spectral flow of parallel B, E5 with the same conventions.

FIG. 8: Spectral flow of parallel B5, E5 with the same conventions.

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