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Torsional Landau levels and geometric anomalies in condensed matter Weyl systems

We consider the role of coordinate dependent tetrads ("Fermi velocities"), momentum space geometry, and torsional Landau levels (LLs) in condensed matter systems with low-energy Weyl quasiparticles. In contrast to their relativistic counterparts, they arise at finite momenta and an explicit cutoff to the linear spectrum. Via the universal coupling of tetrads to momentum, they experience geometric chiral and axial anomalies with gravitational character. We show how torsion and the Nieh-Yan (NY) anomaly appear, with a ultraviolet (UV) parameter with dimensions of momentum, in condensed matter Weyl systems with background fields corresponding to emergent anisotropic Riemann-Cartan and Newton-Cartan spacetimes, depending on the precise form of the low-energy dispersion. The torsional NY anomaly arises in simplest terms from the spectral flow of torsional LLs coupled to the nodes at finite momenta and the linear approximation with a cutoff. We carefully review this for relativistic fermions at zero momentum. We contrast this with the spectral flow and non-zero torsional anomaly and the appearance the dimensionful UV-cutoff parameter in condensed matter systems at finite momentum. We apply this to chiral transport anomalies sensitive to the emergent tetrads in non-homogenous chiral superconductors, superfluids and Weyl semimetals under elastic strain. We also briefly discuss the role torsion in anomalous thermal transport for non-relativistic Weyl fermions, which arises via Luttinger's fictitious gravitational field corresponding to thermal gradients.


I. INTRODUCTION
Gapless quasiparticles with linear spectrum protected by topology arise in many systems in three dimensions [1][2][3]15 . In particular, accidental crossings of two inversion (P ) or time-reversal (T ) breaking bands at the Fermi energy lead to quasirelativistic particles with low-energy dispersion analogous to relativistic Weyl fermions 7,8 . Fourfold degenerate crossings with Diraclike low-energy excitations occur for combined P, T (and/or other similar protecting) symmetries 4 . Similarly, in chiral superconductors and superfluids with gap nodes, Majorana-Weyl excitations arise at low energy 2,11 . By a very general theorem from topology 3 , the lowenergy linear theory near the Fermi point node takes universally the (γ-matrix) form of a quasirelativistic Weyl/Dirac spectrum, with the precise form of the metric and other background fields depending on the microscopic details. It is then of interest to study the detailed form of this emergent Dirac operator with an explicit cutoff and compare to fundamental, Lorentz invariant fermions. Following this logic, the concept of so-called pseudo gauge fields 2,84,86,87 and "emergent" spacetime 2,30,34,66,80,81,91,92,97 in non-relativistic condensed matter systems has emerged, where the lowenergy fermions can experience background fields of various physical origins, similar to what appears for spin-1/2 (or even higher spin) fermions on curved spacetimes in general relativity or its non-relativistic generalizations with non-relativistic coordinate invariance.
Notably the tetrad and its field strength torsion couple to the quasiparticle momentum, as in gravity, making non-relativistic systems at finite density µ F and Fermimomentum p F very different from their relativistic counterparts appearing at p = 0. The system at finite Fermi or crystal momentum is then charged under the field strength these geometric background fields 32,66,67,71,100 . In three spatial dimensions, this corresponds to the anomalous translational symmetry for chiral fermions, leading to axial anomalies in the system 16,30,101 from momentum space. For other relevant condensed matter considerations of the anomaly, see e.g. 2,4,12,14,39,73,74 .
Due to the form of the background fields, non-zero torsion can be expected on general grounds, thus opening the possibility of simulating Riemann-Cartan (or Newton-Cartan) spacetimes for the low-energy fermions. In case of non-trivial background torsion, the so-called gravitational Nieh-Yan anomaly can appear. In contrast to the axial anomaly with gauge fields, this anomaly depends on a non-universal UV cut-off parameter Λ, with canonical dimensions of momentum. While the status of the torsional contribution in relativistic systems has been debated for long [24][25][26][27][28] , the appearance of this term in non-relativistic condensed matter systems with explicit UV cut-offs is a priori plausible. It has been considered in [16][17][18][19][30][31][32][33]93 including Weyl/Dirac superfluids and superconductor, as well as semimetals. This non-universal UV scale is, however, originating from the regime of validity of the quasirelativistic linear spectrum, that is the Taylor expansion close to the node, generically making it (experimentally) a low-energy scale in the system 30 . It seems that the NY anomaly non-universally probes the chiral spectrum and transport, well-defined only at low energies, and conversely, merging in some left-right asymmetric way to other bands as required by global con-sistency and symmetries. At face value the spectrum and spectral flow can be terminated in a multitude of inequivalent ways. If the system is anisotropic, the interplay of different scales in the system becomes essential, as evidenced by the consideration of Newton-Cartan geometry with quadratic spectrum along a preferred direction.
Here we will further argue for this term using the simplest mathematical apparatus for the chiral and axial anomaly: adiabatic spectral flow in the presence of torsional Landau levels. In this context, the torsional LLs appeared implicitly already in Ref. 11,13 and more recently in topological semimetals in 32 within Pauli-Villars regularization of Lorentz invariant fermions. Such a relativistic regularization scheme is at best only an approximation in condensed matter systems, since the linear Weyl regime applies to low-energies with an explicit cutoff scale. This linear regime can be anisotropic and, furthermore, is continuously connected with the non-relativistic regime with quadratic dispersion. It seems that the explicit value of the cutoff parameter has not discussed in detail, except in the recent paper 30 by one of the present authors. Interestingly, the UV-scale can be supplemented by a infrared (IR) temperature scale of thermal fluctuations, in contrast to, say U(1), gauge fields. The contribution to the torsional NY anomaly at finite temperatures was further discussed in 57,58 and 93,98,99 for relativistic fermions at p = 0. The closely related role of torsion and viscoelastic thermal transport has been studied e.g. in 53,59,60 . The dimensional hierarchy and descent relations of the torsional anomaly were recently analyzed in Ref. 94 in a relativistic model.
Here we will mostly focus on the non-universal UV contribution at zero temperature. This is directly related to the momentum dependent charge and LL density of states. This also arises for the momentum space gauge fields, and for the chiral transport close to the Weyl node, is already necessitating the consideration of the cutoff and the appearance of a dimensionful UV paramemeter. It seems that this fact has not been properly taken into account before. Along similar considerations, the universal thermal contribution to the gravitational NY anomaly follows from our arguments by including the thermal fluctuations close to the node. This and the associated currents are considered 57 . For completeness, we comment on thermal effects by non-zero temperature gradients, which point to still new types of torsional anomalies terms not present in systems with Lorentz invariance. This paper is organized as follows. Section II discusses the low-energy Weyl Hamiltonian and the associated geometry in the form of emergent background spacetimes at low energy in condensed matter systems. The following Section III reviews the relativistic torsional anomaly and spectral flow argument, focusing on the finite node momentum and the comparison with the anomaly for U(1) gauge fields presented in Appendix A. Sec. IV discusses the torsional anomaly in chiral superfluids and superconductors, where it can be matched with experiment 30,38,39 . This followed by a model of T -breaking strained semimet-als in Sec. V. We also briefly discuss the role of torsion in the presence of thermal gradients in Sec VI. We conclude with a comparison on previous results Sec. VII and the conclusions and outlook of our results.

II. WEYL FERMIONS IN CONDENSED MATTER AND RELATIVISTIC SYSTEMS
A. Weyl fermions in condensed matter We consider a system with broken time-reversal symmetry (T ) or inversion (P ). Ignoring all other bands, the 2 × 2 Hamiltonian is H = σ a H a in terms of the unit and Pauli matrices σ a , a = 0, 1, 2, 3. In the vicinity of a generic degenerate crossing at p W , it takes the form of an expansion where The expansion is, of course, valid for |p − p W | p W since the remainder is of the order of |p − p W | 2 . This provides an explicit cutoff for the linear Weyl regime that is, nevertheless, continuously connected with the non-relativistic quadratic dispersing spectrum and the other bands. The Weyl node degeneracy is protected since there are three parameters and three constraints 7,8 .
Via rotations and scalings,p a = e i a p i , this becomes the right-or left-handed relativistic Weyl Hamiltonian, at Fermi momentump W , where χ = ±1 = sgn(det e i a ) is the chirality, defined as the direction of (pseudo)spin with respect to the propagation momentum. The band energies are E = The role of the coefficients e µ a is simply to determine the (anisotropic) Fermi velocities of the dispersion where the Einstein summation convention for repeated latin and greek indices will be henceforth assumed. The spatial tetrad e i a is extended to a non-degenerate matrix e µ a by considering the operator σ a e µ a i∂ µ = i∂ t −H(p) with µ = t, x, y, z. In particular, the coefficient e µ 0 = {1, v i } is non-trivial in type-II Weyl semimetals and superfluids and superconductors with superflow. The case with nonzero spatial e t a , a = 1, 2, 3 was considered in 80 . These break different symmetries, while the spacelike tetrads act like gauge potentials corresponding to axial magnetic and electric fields.
While the Hamiltonian (1) is usually analyzed for translationally invariant systems, it remains valid for weak deformations. This can be seen by any consistent gradient expansion scheme, e.g. the semi-classical gradient expansion of the BdG Hamiltonian for superconductors/superfluids, or the Schrieffer-Wolff transformation for Bloch Hamiltonians 91,92 .
In this way, the Hamiltonian (1) has striking similarity, albeit with some important caveats, to relativistic fermions coupled to non-trivial background geometry or gravity. More precisely, if we consider the low-energy Weyl fermion Ψ W in terms of the original excitations Ψ, it is This, however, corresponds to the anomalous (chiral) rotations in the system, thus making the finite node momentum p W very important. In the rest of the paper, we will explicitly consider the anomaly implied by (5) in the presence of non-trivial background fields e µ a (x), after reviewing the necessary geometry in the next section. We will focus here on T -breaking systems, where in the simplest case one finds Weyl nodes of opposite chirality at ±p W , whereas for inversion P breaking systems one has at minimum four Weyl points, which are invariant under T and map non-trivially to themselves under inversion.

B. Quasirelativistic fermions
We briefly summarize quasirelativistic fermions on curved Riemann-Cartan spacetimes here. These spacetimes are defined via an orthonormal frame e a = e a µ dx µ and a (matrix) spin-connectionω µ dx µ , both of which couple to the Dirac (and Weyl) equations. The e a µ is a spacetime "translation gauge field", whileω is the gauge connection corresponding to local (Lorentz) rotations. See e.g. 6 .
As discussed in the Introduction, analogous fields arise in the low-energy Weyl Hamiltonian close to the nodes as well, which are not strictly relativistic however, in the sense that the emergent metric does not follow from locally Lorentz invariant spacetimes implied by general relativity but rather from the microscopic non-relativistic UV theory at low-energy. This what we call quasirelativistic and emergent. Note that the spin-connection is strictly speaking a gauge field entering the Dirac operator, and therefore needs the corresponding local symmetry. Notwithstanding, it arises however, e.g. in chiral superconductors and superfluids due to the local combined U(1) symmetry corresponding to gauge and orbital rotation symmetry 30,42,81 . The tetrad and connection fields give rise to the torsion T a = de a + (ω ∧ e) a and curvaturê R = dω µ −ω∧ω tensors that equivalently characterise the spacetime. From the tetrad one can derive the spacetime metric, which enters as a secondary object, in contrast to usual Riemannian spacetimes where the connection is symmetric and uniquely fixed by the metric.
In terms of equations, the basic quantities are the tetrad e a µ and coordinate connection Γ λ µν . The former gives the metric g µν = e a µ e b ν η ab , e µ a e ν b η ab = g µν (6) by defining a local orthonormal frame (in terms of η ab ). Now tensors X a···µ··· b···ν··· can carry local orthonormal (Lorentz) indices and coordinate indices; the two bases can be transformed by contracting with e a µ or the inverse e µ a . The connection consistent with basis changes has two parts, one for local orthonormal indices and one for coordinate indices, defined as ∇e a µ = 0 and is therefore metric compatible. This determines geometric parallel transport in the system. Without loss of generality it can be written as where Γ λ µν is the coordinate connection with torsion The connection can be decomposed in terms of torsion as whereΓ λ µν = 1 2 g λρ (∂ µ g νρ +∂ ν g µρ −∂ ρ g µν ) is the Christoffel connection fully determined from the metric and C λ µν = 1 2 (T λ µν + T λ µ ν − T λ µν ) is the contorsion tensor. The low-energy quasirelativistic Weyl fermion theory is, in the chiral Dirac fermion basis ψ = ψ L ψ R T , where ψ R,L are Weyl fermions and γ a = σ a ⊕ σ a with σ a = (1, −σ i ), where e ≡ det e a µ and D µ is the covariant derivative corresponding to the canonical momentum where γ ab = i 2 [γ a , γ b ] and A µ is a U(1) gauge potential with charge q. They enter the covariant derivative or canonical momentum due to local Lorentz (rotation) and gauge symmetries. For the emergent spin-connection to exist, the local rotation symmetry has to be dynamically generated. See Sec. IV and 30 . Importantly to our applications, the quantity p W a = (µ W , p W ) is the shift of the of the Weyl (or Dirac) node at chemical potential µ W = e ν 0 p W ν and p W a = e i a p W i 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 is an anisotropic version of Newton-Cartan spacetimes, where we single out a Newtonian time and, in our case, a preferred spatial direction in contrast to the Riemann-Cartan case. In what follows, Sec. IV, this preferred direction is the one with uniaxial symmetry and anisotropic scaling in the chiral superfluid/conductor. Compared to the standard Newton-Cartan case, there is an additional gauge symmetry corresponding to a U(1) number conservation and a local Milne boost symmetry along the anisotropy direction 45,[47][48][49] . These will both be gauge fixed to zero and will be applied in the case of the chiral superconductor/superfluid, where they are absent naturally for Majorana-Weyl fermions. With the time coordinate fixed, the symmetries of the Newton-Cartan spacetime then correspond to the generalized Galilean transforma- 45,46,50,97 .
Then the metric is where n µ is a spacelike vector and a (degenerate) tetrad e a µ and metric h µν restricted to the orthogonal subspace, with e 0 µ = δ 0 µ representing Newtonian time, with inverses The connection and torsion follow as from the condition that L h µν = 0, equivalent to ∇ µ n ν = ∇ λ h µν = 0. The torsion is given as and the standard spin-connection perpendicular to µ , ω µν [h], according to Eq. (7), amounting to local rotation symmetry along µ . The fact that n µ is covariantly constant is natural, since it can be identified with the direction corresponding to non-zero Weyl node separation (in T -breaking Weyl systems). We discuss in Sec. IV the Landau level problem of Majorana-Weyl fermions corresponding to such a spacetime, with the (right-handed Weyl) action where (∂ ) = ∂ 2 /(2m) − µ F in the anisotropic direction with ∂ = µ ∂ µ , corresponding to the non-relativistic dispersion and degenerate metric µ ν = g µν − h µν . In this case the relative anisotropy of the two terms is c ⊥ /c = mc ⊥ /p F , where p F = √ 2mµ F and c = v F the Fermi velocity. Note that a very similar model with Lifshitz anisotropy was considered in 48 , and the ensuing torsional anomalies for momentum transport in 49 .

A. Torsional Nieh-Yan anomaly
Now consider Weyl fermions coupled to a tetrad with non-zero torsion and curvature with the U(1) gauge fields set to A µ = A 5µ = 0. As for the U(1) gauge fields or gravitational fields represented by the metric g µν , the Weyl fermions are anomalous in the presence of non-zero torsion and curvature.
We focus on a pair of complex fermions of opposite chirality with currents j µ ± . The (covariant) torsional anomaly for the axial current j µ For a discussion of the relativistic anomaly term, we refer to 5,6,24,25,28 , and for applications in topological condensed matter systems, [30][31][32][33]49,93,96 . For the mixed terms between torsion and U(1) gauge potentials, see e.g. 29 .
Since they are higher order than the NY contribution and we focus on the anomaly contribution solely due to the geometry (tetrads), we will not consider them. Ref. 33 also considered novel "axial" tetrads e a µR = e a µL at two Weyl nodes R, L, with (vector like) T 5 appearing as in Eq. (A3). We will require e R = ±e L but this is actually rather strong constraint basically only allowing for (improper) rotations that can be gauged away. In the chiral Weyl superfuid/conductor or minimal time-breaking semimetal, e R = −e L but this just the chirality of the nodes and is built in the axial nature of torsion. Intriguingly the trace part of torsion arises as the gauge field of local Weyl scalings but this comes, since non-unitary, with a complex gauge coupling 5 . For parity breaking and other non-minimal Weyl systems with several nodes, the presence of different tetrads and overall symmetry considerations would be highly interesting.
We note the following salient properties related to the NY anomaly term: i) Despite appearances, it is given by the difference of topological terms, albeit in five dimensions 25 . ii) The NY anomaly term is of second order in gradients and therefore the leading contribution from the background geometry in linear response.
iii) The UV-cutoff is isotropic in momentum space by (local) Lorentz invariance but is multiplied by the geometric term which can be anisotropic in condensed matter applications where we do not expect Lorentz invariance. iv) The NY term has contributions from the torsion and curvature, dictated by local exactness d(e a ∧ T a ) = T a ∧ T a − e a ∧ e b ∧ R ab . The two contributions are a priori independent before the geometry (the torsionful connection) is fixed. The anomaly is therefore physical input for the spacetime geometry or connection 30 . In more pragmatic terms, the anomaly coefficient Λ 2 can be computed in the case whenω µ = 0,s although the constraints of a consistent spacetime geometry should be kept in mind.

B. Quasirelativistic fermions and torsional Landau levels
Now we proceed to compute the torsional NY anomaly in non-relativistic systems utilizing the Landau level argument. To set the stage and remove confusions before presenting our main results, we briefly review (quasi)relativistic torsional Landau levels, see e.g. 32 . The computation of the Landau levels is close to and inspired by the spectral flow obtained in 11,13 for momentum space gauge fields at p W = 0. Similar calculations for p = 0 can be found in 93,94 .
The Weyl particles are governed by the effective Hamiltonian where p W is the location of the Weyl point. Due to the lack of protecting symmetries (namely at least broken P or T ) the shift vector is necessarily non-zero for the existence of the Weyl point. However, we will focus on the T -breaking case with two nodes of opposite chirality at ±p W and assume that µ W is zero unless otherwise specified.
In this section, we assume that the coordinate dependence of the Hamiltonian arises solely from the tetrad e µ a (x), while the location of the node, p aW is assumed to be constant with p W µ = e a µ p W a . Note that the coordinate momentum p W µ can still vary and in the case T a µν = 0 there is non-zero torsion. Torsional LLs arise when, say, 1 2 ijk T 3 jk = T Bẑ i is constant with the other torsion components and spin connection vanishing. We discuss later in Secs. IV, V on how to make the identification between low-energy emergent gravitational fields and microscopic background fields in specific examples.

Torsional Landau levels
Specifically, the (semi-classical) tetrads e a = e a µ dx µ and the inverse e a = e µ a ∂ µ are, following 11,13,32 , Now we compute the spectrum of the Weyl fermions in the presence of a constant torsional magnetic field T (y) = T 3 B y. The corresponding metric is The torsion is given as In analogy with the electromagnetic tensor, we will call 1 2 ε ijk T a jk and T a 0i torsional magnetic and electric fields, respectively. The Weyl hamiltonian couples to the non-trivial vierbein as, χ being the chirality, As usual, the energy eigenvalues are obtained from squaring the Hamiltonian We see (23) is equivalent to a LL problem in a magnetic field [Eq. (A7) for B z = T 3 B and e = p z in Appendix A]. With those identifications, the spectrum is consequently [from Eq. (A17)]: The lowest Landau level (LLL) is chiral and unpaired with the simple eigenfunctions, σ 3 = ±1, where the (pseudo)spin or helicity is determined by sgn(p z T B ). We stress that the shape of the spectrum is in general also modified due to the momentum replacing the electric charge: left-handed states now disperse as E < 0 and right-handed states as E > 0 (or vice versa, depending on the sign of the field), see Fig. 1.

Spectral flow and anomaly
Analogously to the Landau level calculation with electromagnetic fields, we may turn on a constant torsional electric field parallel to T 3 B by introducing timedependence to the vierbein as e 3 z )p 3 , analogous to the Lorentz force, which leads to spectral flow of states through the momentum dependent torsional electric field. The number currents, in the vicinity of the node p z = e 3 z p 3 = p W z = 0 are for both chiralities where a cutoff Λ has been introduced to regularize the momentum dependent current density. We see 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 tensorial current density is modified by the volume element ed 4 x in the presence of torsion, see e.g. 26,59 , from holes or particles moving above or below the cutoff, respectively, depending on the direction of the torsional electric field. This is the vacuum regularization that was used in Ref. 32 in the sense N vac = |En|≤Λ sgn(E n ), where an additional factor of one half was present, presumably due to comparison with anomaly inflow from five dimensions. Generalizing this to a fully covariant expression, see the Appendix A, gives and in particular ∂ µ (ej µ ) = 0 as required. We discuss the regularization of the relativistic vacuum leading to this and the anomaly spectral flow, as compared to nodes at finite momenta and axial U(1) fields, more in the next section. Torsional anomaly for pW = 0 If we now displace the Weyl nodes in the relativistic case (23) by p z = ±p W in momentum space, corresponding to a T -breaking Weyl system, the spectrum (24) takes the form The lowest, chiral Landau level looks exactly like that of a Weyl fermion in an axial magnetic field, Eq. (A27). Higher levels are distorted due to the effective charge carried by the particles being their momentum. See Fig.  3.
Since the node is at finite momentum p W = 0, also the spectral flow summation is centered around p W ± Λ . For notational convenience and comparison to Eq. (28), we introduce the momentum cutoff as Λ = 1, this being the dimensionless ratio of the cutoff of the linear spectrum to p W . The spectral flow results in the expression, where particles and holes simply add at the two nodes, which shows that the NY anomaly cutoff is proportional to the node momentum p W , and is small by a factor Λ 2 1 corresponding to the validity of the linear Weyl approximation.

Comparison of torsion to U(1) fields
From Figs. 1 and 3, we see that the spectrum of torsional LLs resemble the LL spectrum of charged particles in U(1) axial and vector fields, with the momentum dependent charge to torsion kept in mind. It is well-known that the contribution of torsion for complex chiral Weyl fermions can be equivalently cast in terms of the axial gauge field γ 5 T µ ≡ γ 5 ε µνλρ T νλρ corresponding to the totally antisymmetric torsion, see e.g. 25,26 . See appendix A for a complete review of the U(1) case for comparison. We stress that while the spectral equivalence of torsional and U(1) LLs is of course expected, the physical appearance of the anomaly is drastically different: the density of states of the LLs depend on momentum and thus the dimensional coefficient Λ 2 and the need for an explicit UV-cutoff appears.
On this note, although the relativistic result in (27) is familiar, there seems to be still confusion in the literature about the role of torsional Landau levels in momentum space and the validity of the NY anomaly due to the explicit UV cutoff. For relativistic Weyl fermions with Lorentz invariance up to arbitrary scales, the spectral flow is symmetric around p = 0, leading to the conclusion that the anomaly indeed can cancel. This is simply by the observation that, in the absence of Lorentz symmetry breaking at high energy, no net transfer of occupied and empty states in the vacuum takes place during the adiabatic spectral flow, cf. Fig. 2. The net transfer of j 5 requires left-right asymmetric regularization at the scale of Λ, maintaining ∂ µ j µ = 0, as was utilized above. Alternatively, at the very least, there is a divergence as Λ → ∞. In contrast, for quasirelativistic Weyl fermions at finite node momentum and an explicit cutoff to the Weyl spectrum, the spectral flow can terminate due to the non-relativistic corrections at the cutoff scale of Λ 2 , leading to net transport of states and momenta relative to the vacuum (and other quantum numbers of the Weyl fermions if present). We note however that the flow is composed of particles and antiparticles (holes) at the different nodes in general. It would be interesting to study the role of the breakdown of relativistic spectrum and spectral flow numerically 89 , where only the charge density at finite chemical potential from the node is analyzed, Fig. 6, with the expected deterioration away from the Weyl node.

IV. CHIRAL WEYL SUPERFLUIDS AND SUPERCONDUCTORS
Now we discuss the role of the torsional anomaly in p-wave superfluids and superconductors with gap nodes and the associated Weyl-Majorana quasiparticles 21,22,36,44,72,76 .
Close to the nodes, the Fermi energy is tuned to the Weyl point due to the existence of the p + ip pairing amplitude. The chiral NY anomaly is related to the non-conservation of momentum in the condensate and normal state quasiparticles. The relation of the torsional gravitational anomaly and the LL spectral flow was briefly pointed out in Ref. 30 . Earlier related work can be found in 11,13,20,21,37,82,83 .
The spinless gap amplitude, with equal spin pairing understood, takes the form where c ⊥ = ∆ 0 /p F has units of velocity. The direction l =m ×n is a low-energy Goldstone variable for the condensate. At low-energy, the direction ofl can fluctuate and there is combined U(1) gauge symmetry in them−n plane, leading to the Mermin-Ho relations betweenl and v s . In the following, we focus on the Landau levels and torsion, keeping the magnitudes of p F and ∆ 0 fixed, as well as ignoring the contributions from v s , see Ref. 30 where c ≡ p F m = v F . Henceforth, to conform with relativistic notation, we will work with dimensionless tetrads in units of c = 1. The quasiparticle dispersion is E(p) = ± (p) 2 + |∆(p)| 2 ≈ ± c q 2 + c 2 ⊥ q 2 ⊥ , with q = p − p F for the Weyl quasiparticles. The linear expansion is valid when |p − p F | p F which provides an explicit cut-off for the Weyl description, requiring that the remainder This leads to the condition, in addition to the trivial |p − p F | p F from the Taylor expansion of (p), that which will prove important later. In particular, the energy cutoff for the Weyl quasiparticles is anisotropic in momenta q = p − p F around the Weyl point, if we consider the Weyl fermion system in the case where the background fields couple parallel and perpendicular directions 30 .

A. Landau levels in linear approximation
To compute the LL levels in the order parameter texture corresponding to a torsional magnetic field, we can take the "weak-twist" texturem + in =x + iŷ − iT B xẑ with |Bx| 1, which corresponds to l =ẑ + T B xŷ. The BdG Hamiltonian then takes the form Near the gap node p = −p F l we may linearize the operator (p) as ε p ≈ −v F l · (p + p F l) ≈ −v F (p z + p F ). This leads to with where we remind that c ≡ v F and c ⊥ ≡ ∆0 p F . This corresponds, up to the sign of the field T B and the tetrad, to the 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 p a ≡ e i a p i , we can define the annihilation operatorâ ≡ which is (A7) after a Galilean boost p 3 → p 3 + p F . The eigenstates are then where φ n ≡ φ n (x), for n ≥ 0, are harmonic oscillator eigenstates and vanish otherwise. The condition for normalization is |u n | 2 + |v n | 2 = 1, corresponding to the BdG particle and hole amplitude. Carrying out a corresponding calculation at the Weyl point p = p F l, we have the Hamiltonian which can be identified as the left-handed Hamiltonian H − = −e i a τ a p i after a rotation about l such thatm → −m andn → −n. Its eigenstates are Ψ n,pz>0 = u n φ n−1 v n φ n e i(pzz+pyy) .
Depending on the chirality, i.e. sign of momentum at the node, the LLL is either particle-or holelike. The conclusion is that the spectrum looks like the relativistic spectrum in Fig. 3, when the linear approximation for (p) ≈ ±c ⊥ (p z − p F ) is valid, Eq. (36). This corresponds to the spectrum of axial U(1) fields with momentum dependent charge and density of states per LL. The density of states is (A19) in the scaled coordinates, which gives, with e 0 µ = δ 0 µ ,

B. Anisotropic Newton-Cartan model
We just showed that the simple order parameter texture in chiral superfluid or superconductor gives rise to the torsional LLs for the low-energy Weyl quasiparticles, in the linear regime close to nodes. We can however consider quadratic dispersion beyond the linear approximation 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 the chiral superfluid or superconductor as the linear approximation in Eq. (36), since it also neglects the rotationally invariant dispersion (p) of the normal state. The chiral p-wave BCS state has the same uniaxial anisotropy, however, and this carries to the low-energy Weyl description in the form of the emergent spacetime. The benefit of the anisotropic model (45) is that the LL spectrum can be computed for momenta far from p F , up till p = 0, corresponding to the filled levels of the non-relativistic Fermi system, which are absent in the relativistic linear model. In this way the contribution to the anomalous current from the superfluid vacuum can be analyzed, see Sec. IV C.
The spectrum follows simply from Eqs. (40), (42) by the substitution ∓(p 3 ± p F ) → ± (±p z ). From squaring the Hamiltonian, the corresponding eigenvalues are at both nodes for n ≥ 1. The LLL state retains the gaussian form (25). The condition for normalization is |u n | 2 + |v n | 2 = 1, and consequently the particle and hole amplitudes are in both cases With E 0 = (p z ) we have v 0 = 0, meaning that the lowest level particles appear only for p z < 0. For p z > 0 u 0 = 0 when E 0 = − (p z ), so for positive momenta only holes appear at the lowest level, as we found for the linear model. In this case we must, however, remember that the hole spectrum arises due to the Majorana doubling of the BdG spectrum and is not physical. This cancels with a corresponding factor of two from spin-degeneracy. This leads to the LL spectrum in Fig. 4.

C. Spectral flow and consistent anomalous vacuum current
Now we are equipped to compute the spectral flow resulting from torsional Landau levels, corresponding to the covariant torsional NY anomaly. For the anisotropic Newton-Cartan model we can also compute the consistent vacuum current of the condensate, since the dispersion takes into account the filled states below the Fermilevel which is not the case for the linear approximation close to the Weyl nodes.
For the (spinless) chiral superfluid (or -conductor) we have to take into account that the particles are Majorana-Weyl but a factor of two results from taking spin-degeneracy into account. The spectral flow arises as where the cutoff for the Weyl spectrum is taken at , corresponding to Eq. (36) with 1 2 1. Remarkably this reproduces the more general torsion contribution result for the NY anomaly including curvature, as implied by the anomalous momentum non-conservation in the system as found in Ref. 30. This result was found by matching the anomaly on emergent spacetime of background the chiral p-wave system to the corresponding BCS hydrodynamic result of the superfluid. In particular, including the effects of superflow leads to a spinconnection and curvature perpendicular tol, as required by the Mermin-Ho relations 43 .
In the chiral superfluid (or superconductor) the above result holds for both the linear quasirelativistic and the anisotropic Newton-Cartan spacetime, as defined by the tetrad (33). This simply follows from the fact that the cutoff for the validity of the both dispersion coincides with (36). In this case, therefore, the anisotropic model is expected to require the same cutoff since the system is probed also in the perpendicular direction. This morally happens sincel =m ×n, making the triad dependent 30,43 . Strictly speaking in the LL-model we approximatedl ≈ẑ for the non-trivial texture.
For the non-relativistic model, however, we can also compute the anomalous vacuum current, corresponding to the anomalous superfluid momentum from the filled states below p F 2 . In the anisotropic Newton-Cartan model, it is given by and extending to p z = 0, there is no need for a cutoff. See Fig. 4. This is actually the correct hydrodynamic result for the (weak-coupling) BCS system 10,11,37 to lowest order in gradients, since the final answer for the anomalous vacuum current is sensitive only to the e 3 =l direction, even in the presence of v s (corresponding to curvature in the perpendicular plane). Upon taking the time-derivative of this momentum, the hydrodynamics of the system produce the covariant current implied by the Weyl anomaly. If we assume, without any supporting arguments, that the curvature and torsion contribute to the current (49) as they enter the anomaly Eq. (18), we get the same result if we apply the cutoff (36) as above. We note that these findings are corroborated by the thermal contribution to the NY anomaly, as found in Ref. 57 .
These considerations beyond the LL spectral flow method aside, 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 term 35,40 . See also 49 where the consistent and covariant anomalies are discussed in the closely anisotropic Lifshitz model as what we consider in Eq. (17). We leave the study of the consistent vacuum current from the perspective of gravitational anomalies with torsion for the future.

V. STRAINED WEYL SEMIMETALS
Semimetals with Weyl fermions arise in solid-state systems where the Fermi energy is tuned to a band-crossing in the Brillouin zone. The tetrads arise universally via the linear expansion. In this case, the fermions are also charged leading to the possibility of the U(1) anomaly with electric fields. In addition to the tetrads, effective background (axial) fields still arise with similar origin as in the chiral superconductor -the (constant) shift of the Weyl node in momentum space. These considerations lead to the existence of the protected Fermi arc states. Here we would like to clarify the contribution from the tetrads in the presence of elastic strains. In fact, due to the universal coupling of the tetrads to momentum, as in gravity, one expects that deformations of the (lattice) geometry would lead to effects that probe the Weyl fermions via the background tetrads.
We start in a roundabout way, first discussing the lowenergy Weyl Hamiltonian and then considering a lattice model for a realistic T -breaking material.

A. Bloch-Weyl fermions in crystals
The low-energy Bloch-Weyl Hamiltonian is of the form 4,9,15 where now are simply the linear coefficients of the expansion of the underlying (tight-binding) Bloch Hamiltonian H TB (k) near the Weyl nodes. Before we consider lattice deformations in this model, we remark on the interplay of the tetrads and momentum. The lattice momentum is 71 Under non-trivial background fields, the Weyl system itself is anomalous under the lattice translation symmetry, T 3 = Tẑ, corresponding to the conservation of the lattice momentump 3 , which corresponds to an anomalous chiral rotation of the low-energy Weyl fermions at the T -breaking nodes ±k F . Here c † k creates the state corresponding to the lattice periodic Bloch state |v k = |v k+K , with wave function In the presence of elastic deformations corresponding to torsion, i.e. phonons, the anomalous chiral symmetry corresponding to translations is manifested as the non-conservation of momenta between the Weyl fermions and the background phonons 30,101 , as found in superfluid 3 He-A for the p + ip-wave paired Fermi-liquid 2 . See also 33,49,78,84 .

B. Elastic deformations
Now consider general lattice deformations. The original unstrained lattice momenta entering the Weyl description are represented as k a and the deformed lattice is given as k i = e a i k a in the coordinate system of the laboratory, where e a i = δ a i to first order in the strains. These will couple as expected in the continuum model, as long as we take into account the lattice model properly, as we now recall following 71 . See also 33 . We have the continuum linear strain tensor, where u a /a 1, the lattice constant. This means that k F,a is held fixed, whereas k F,i with δk F,i = w a i k F,a is deformed. This becomes on the lattice where w i a = ∂ j u b δ ab δ ij is defined above and in the last approximation, the linear approximation for strain as well as k i a 1, close to the Γ-point, are used. In addition we assume that we work with low-frequencies corresponding to the acoustic phonons, below the Debye energy 71 .

C. Lattice model
In general, a model for a T -breaking Weyl semimetal consist of layered 2D Wilson fermions tuned to a zero energy crossing in three dimensions 2,89 .
where the the Weyl node itself will be at finite momentum corresponding to four momenta in the Brillouin zone, the minimum for P -breaking system. While the k·p model is realistic, it is more convenient to work with an explicit model with a lattice regularization that produces the same results. In terms of a tight-binding model, they considered where we focus on the time-reversal odd block h latt (k) of the T -invariant model 2,87,89 , otherwise it is gapped. The dimensionful tetrads are e i a (±k F ) = a(t x , t y , ±t z c z sin ak F,z )δ i a .
Inversion symmetry P acts as h lat (k) → σ z h latt (−k)σ z . For simplicity we set c z = 1, c x,y = c ⊥ , t x,y = t ⊥ and assume uniaxial symmetry alongẑ in the following. We expect (56) to hold for the Weyl semimetal model Eq. (57), originating from the k · p model For this tetrad we can ignore the difference of lattice and coordinate indices, with u ij = 1 2 (∂ i u j +∂ j u i )+O(u 2 ) the symmetric lattice strain. The strain induces the deformation considered in Ref. 84 which gives where β el is the Grünesein parameter. Restricting to a uniaxial strain corresponding to the axis of the Weyl node orientation, with the approximation that ak F 1, This has the (dimensionless) inverse tetrad, up to the neglected terms O(u 2 ) in strains, This is what we expected, based on the corresponding universal continuum limit (55) and the lattice substitution (56) coupling to geometry, apart from the (nonuniversal) couplings β el , tz t ⊥ between the phonons and electrons of the lattice model 71 . Now in the presence of non-homogenous strain vector e 3 z depending coordinates and time, torsion T 3 µν will arise. The Landau level arguments of Sec. III and IV apply for a torsional magnetic field from u zx,zy (x, y) ("symmetric gauge") and an adiabatic electric field from u zz (t), as in 86,87 .

D. Torsional density of states in anomalous transport
Armed with the geometric background fields corresponding to torsional (magnetic field), we can consider the anomaly resulting from the chiral rotation (53). The linear Weyl model is valid up to the approximation which is simply restricted by the ignored terms of the remainder in the expansion. Apart from the trivial q z k F 1/a, also c x cos q x a + c y cos q y a ≈ c ⊥ a 2 2 (q 2 for the perpendicular direction. We are working in the units where −1 < M −2c ⊥ < 1 and cos k F a = M −2c ⊥ ≈ 1. For the effects of any torsional anomaly, we can just evaluate the chiral densities at the nodes, It is interesting to recall that for the chiral superfluid, while strictly it must be that Λ 2 1 since q z k F , we found that the cutoff was parametrically high " 1 2 1". There however, due to the orthonormal triad, also the perpendicular direction couples to the transport, with the cutoff Eq. (36) which in real 3 He-A is actually ∼ 10 −6 p F . It is interesting to consider also the case where q z ∼ t ⊥ tz sin k F a q ⊥ k F assuming that we isotropically couple to the perpendicular directions for general field configurations. Another option would be to consider the Newton-Cartan model with quadratic spectrum M − 2c ⊥ − cos k z a taken into account along the uniaxial Weyl node direction In any case, plugging in real parameters, we expect that for e.g. Cd 3 As 2 , t ⊥ ∼ t z sin k F a 87 . The same model with different parameter also applies for the Dirac semimetal Na 3 Bi 87 and references therein.
Independent of whether one has a torsional electric field ∂ t e 3 z = 0 or an electric field E z driving the spectral flow, as in Fig. 6 and 8, this will lead to the suppression of the density proportional to Λ 2 , corresponding to the validity of the linear Weyl approximation, in the anomalous transport, as compared to the Fermi wavevector k F . We note that this is simply due to the momentum dependent density of states from the tetrads and torsion coupling to momenta, as compared to a U(1) gauge field with constant density of states, as dictated by minimum coupling and the topology of U(1) gauge fields.

VI. THERMAL EFFECTS
Finally we briefly recall and discuss thermal contributions to the torsional anomaly. There are two possible effects: i) the small but finite temperature enters the NY anomaly as the scale of thermal fluctuations in momentum space. These are analyzed in 57,58,93 ii) There is a finite thermal gradient in the system and one computes the thermal response via Luttinger's fictitious gravitational field 51 . We note that non-zero time-like torsion for the Luttinger spacetime implies the non-single valued time coordinate in the fictitious gravitational field 59 . See also [54][55][56]60,82 Here we focus on the effects of a thermal gradient, the currents induced can be computed by coupling the system to fictitious spacetime metric, following Luttinger 51 . Specifically, we assume a thermal gradient which is equivalent to a weak gravitational potential g 00 = 1 + 2σ in the system. The perturbation δg 00 couples to the Hamiltonian (energy current) T 00 . In units where the velocity of propagation is v = 1, the metric is from which the linear response to the thermal gradient σ can be calculated 51 . This can be generalized to a metric now with a small gravimagnetic potential 2,55 where N i describes a velocity field in the units where v = 1. The gravitational thermal potential 2,55,75 whence e 0 µ = (e σ , N i ), e a µ = δ a µ , a = 1, 2, 3 e µ 0 = (e −σ , 0), e µ a = (e −σ N i , δ i a ), a = 1, 2, 3. (78) In this case Eq. (76) becomes where T 0 µν = ∂ µ e 0 ν − ∂ ν e 0 µ is the temporal torsion, assuming zero temporal spin-connection ω 0 µb ≡ 0. It is expected then, that one would have possibility for anomalous transport in terms of the combination of thermal gradient and vorticity T 0 ij = ∂ i N j − ∂ j N j in the velocity field N i (x), as in the chiral vortical effect 75,98 .
Now similarly as we expect momentum density at the Weyl node (P µ ) node = Π tµ = p F e i 3 δ µ i ej 0 5 30 for the Weyl systems at finite p W a = p F δ 3a , or since T 0µ = ee µ a T ta , we expect an energy density of the form The anomaly of this current would be proportional to T ∇T , and is reminiscent of the chiral vortical effect 60,75 . We can also expect mixed terms, in the sense that there should be a corresponding energy current from both the momentum density and thermal current, ∂ t e i 3 = 0, at the node these "mixed" contributions to the currents were identified and discussed in Ref. 96 . The message we want to convey here is that one can indeed expect anisotropic and "mixed" contributions to the torsional anomalies, in the sense that the Lorentz invariant Λ 2 η ab → Λ a Λ b a generalized anisotropic tensor, in various condensed matter systems depending on the symmetries, perturbations and cutoffs. We leave the detailed discussion of such gravitational contributions for the future, see however 93,96 .

VII. ON THE RELATION OF EMERGENT TORSION AND PSEUDO GAUGE FIELDS
Here we summarize our findings in relation to earlier literature, where the momentum space field corresponding to the shift of the node is often considered as an axial gauge field 2,11,33,41,77,79,84,[86][87][88] . We note that torsion can be shown to enter as an axial gauge field constructed from the totally antisymmetric torsion γ 5 S µ = µνλ T λµν 25,26 coupling to the momentum. This is essentially what we found in Secs. III and IV with the momentum space dependent LL density of states. How are tetrads torsion otherwise different from the momentum gauge field? The symmetries corresponding to the tetrads are translations which correspond to the anomalous chiral symmetry that is requisite for the condensed matter Weyl description. There is however no local gauge symmetry corresponding to this field translational with Berry curvature in momentum space. On the other hand, the geometric formulation is suited for such translation symmetries and reveals the background geometry of spacetime emerging from the node. The overall geometry can made consistent with the non-relativistic symmetries away from the Weyl node for a finite momentum range. For the anomalous axial density and anomaly, this leads to the parametric suppression compared to U(1) anomaly and the UV-scale p W . The phenomenological implications of this are significant even without the theoretical recourse to the emergent geometry.
We also note that Ref. 33 discusses torsion (and the conservation of momentum) in strained semimetals in terms of a model with both the axial gauge field from the node and the tetrad with elastic deformations. While such a "splitting" between low-energy and high-energy momenta is in principle allowed, it makes the consideration of the momentum dependent anomalies more involved, with the danger of double counting. The momentum anomaly (without EM gauge fields) should be proportional k W ∂ µ (ej µ 5 ), as found in 30 . Nevertheless, the original paper 71 takes an explicitly geometrical view point which nicely connects with 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 deformation e i a = δ i a + δe i a , where now p W δe i a = −p W δe a i is the momentum space gauge field in the Hamiltonian with (almost) constant tetrads 11,13,33,71,86,87 . The right-hand side is the Hamiltonian in coordinate (or laboratory) space, which is the one we have experimental access to, and is deformed with respect to the orthogonal frame of k a . We see that the momentumk i couples to e i a , as expected, and the shift is essentially constant in the Hamiltonian, in the sense that k F a is constant corresponding to the undeformed case, irrespective of the deformation. At the same time, the laboratory value changes though as k F i = e a i k F a . In the examples we considered, in the chiral superfluid and superconductor we explicitly have that k F,i = p F e 3 i , giving k F a = p F δ 3 a . Similarly, for the strained semimetal we consider the originally unstrained lattice Fermi wave vector k F a (x) → k F a (x+u) ≈ k F a (x)+∂ i u a k F a (x) ≡ e a i k F a under strain x = x + u, giving Eq. (55) as expected.
What this means more generally is that ∇k F a = 0, in terms of the connection corresponding to the emergent spacetime, as discussed in Sec. II. In fact this is one of the requirements for the consistent assignment of the low-energy geometry. On the other hand, all the torsional spacetimes we considered are in some sense abelian (or gravitoelectromagnetic) in the sense that the relevant fields can be identified as an abelian gauge fields in momentum space, amounting to what was called "minimal coupling" trick in 32,71 . In this case however, the gravitational character comes still evident in the momentum dependent charge and density of LLs, as expected for gravitational response, coupling to momenta and energy densities.

VIII. CONCLUSIONS AND OUTLOOK
In this paper, we have argued for the emergence of non-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 torsional Nieh-Yan anomalies in condensed matter systems with an explicit cutoff and the lack of relativistic Lorentz symmetries. For the anomaly, the spectral flow in the presence of torsion clearly renders non-zero results for Weyl nodes at finite momentum. Although obtained with specific simple field configurations corresponding to the torsion with Landau level spectra, they are expected to generalize covariantly in terms of the relevant spatial symmetries of the system. We discussed two idealized spacetimes related to this, the linear Riemann-Cartan and the anisotropic Newton-Cartan spacetime with quadratic dispersion. We also briefly discussed the thermal torsion via Luttinger's fictitious spacetime, since we can expect mixed anomalies already from the inclusion of thermal gradients.
This connects to gravitational anomalies and transport in general 57 . The universal anomaly coefficients related to gravitational anomalies and thermal transport are related to the non-universal torsional anomaly in the sense that from the momentum dependent density of states, the expected gravitational anomaly polynomials at finite temperature arise at the level of linear response 57,[61][62][63][64][65]70 . Moreover, we expect that the emergent tetrads with coordinate dependence arise rather generally in any Weyl system, making sense of evaluating the linear response to these, even in flat space.
We also discussed the relation between momentum space pseudo gauge fields and the emergent tetrads. It is important to note that the spectral correspondence between torsion and U(1) magnetic fields, e.g. in a Landau level problem, is not yet enough for the anomalies to match in general. The simple LL spectral flow argument is enough to identify the non-universal cutoff appearing in the NY anomaly term. The message is that momentum space fields are very naturally described as tetrads that couple to the momentum in a universal way, even in lattice models with caveats 71,84 , regarding nonuniversal parameters due to the coupling of the lattice phonons and fermions. The UV scales appearing in the termination of anomalous transport from such emergent fields, related to the Fermi-point momentum p W and the regime of validity of the effective Weyl/Dirac description, are naturally understood from this perspective. In the presence of both independent U(1) fields and momentum space tetrads we should also expect many mixed terms, as studied e.g. in 29,32 . The mixed anomalies should also be clarified with regards to the momentum space pseudo gauge fields. Our results for the anomaly at finite momentum are in contrast to 95 , where a model with torsion is compared to a relativistic model at p = 0 with pseudo gauge fields without consideration of node momentum coupling to the torsion or the cutoff of the quasirelativistic dispersion.
Formally, what we did amounts to applying the Ktheory theorem of Horava 3 to specific Weyl nodes in three dimensions, by keeping track of the UV symmetries and scales in the problem for the precise the form of the emergent geometry and fields coupling to the quasiparticles. The topology only guarantees the effectively Dirac like spectrum, with everything else depending on the microscopics.
Many interesting avenues remain in the geometric description of topological condensed matter systems with gapless fermions. It would be extremely interesting to study the gravitational anomalies in Weyl and Dirac systems from the global symmetry perspective with many nodes Weyl 67-69,84 taking into account the spatial space group symmetries. More generally, the appearance of low-energy quasirelativistic fermions gives insight also to the physics of relativistic gravitational anomalies with torsion, although the symmetries and status of the background fields is dramatically different.
The simplest way to argue for the axial anomaly in condensed matter systems is the spectral flow argument, utilizing Landau level spectrum in 3+1d 9,11 . To that end, one envisages a Hamiltonian of the form where A i is some U(1) gauge field with charge q, i.e. the charges in the system are quantized in terms of q. Note that we still assume that the vector and axial combinations could still be both non-zero, Under these gauge fields, the chiral fermions always experience the chiral anomaly, where the classical conservation laws corresponding to these fields are broken.
Naturally, if the particles couple to both axial and vector gauge fields, countercurrents must be introduced to the system to conserve the total particle number, as required by conservation of charge. This is done by adding Bardeen-Zumino counterterms to the effective action, This is the Bardeen counter term, and it introduces the countercurrents These modify the anomaly in Eq. (A3) so that j µ is conserved:

a. Landau levels
We consider the minimally coupled Weyl Hamiltonian with vector potential A = (−By, 0, 0), where χ = ±1 denotes the chirality fo the fermion. With an eigenstate ansatz ψ = e i(pzz+pxx) φ the eigenvalue problem becomes H χ ψ = χ p z p x + qBy − ip y p x + qBy + ip y −p z ψ. (A9) For qB > 0 the off-diagonals can be identified as raising and lowering operators for a harmonic oscillator in the y-direction (displaced by p x ), which satisfy the properties {â,â † } = 1,âφ n = √ nφ n−1 , andâ † φ n = √ n + 1φ n+1 for eigenstates of the harmonic oscillator φ n . The eigenvalue equation becomes The energy eigenvalues are obtained from considering the squared Hamiltonian operator: whence E 2 = p 2 z + 2|qB|(n + 1) − qBσ 3 , E = ± p 2 z + 2|qB|n, n ≥ 0. (A13) Looking now at the action of the ladder operators on components of the eigenstates ψ, they must of the form ψ = e i(pzz+pxx) φ n C n φ n−1 (A14) where φ n are eigenstates of the harmonic oscillator, φ n−1 = 0 and C n is a factor determined from the eigenvalue equation to be C n = √ 2qBn ±E + p z for n = 0. The n = 0 state is "half" occupied, since ψ = e i(pzz+pxx) φ 0 1 0 with chiral dispersion relation E = p z for H + and E = −p z for H − , after the elimination of the trivial zero modes H χ Ψ. For qB < 0 the spectrum is the same but the eigenstates are now ψ n = e i(pzz+pxx) Dφ n−1 φ n , n ≥ 1, (A15) . The zeroeth Landau level dispersion relation is E = −p z for H + and E = +p z for H − . In summary: E = ± p 2 z + 2|qB|n, n ≥ 1 sgn(qBχ)p z , n = 0. (A17) The degeneracy of each state can be determined from containing the system within a finite volume L x L y L z and requiring the center of the harmonic oscillator be within it: The x-direction is free and is therefore quantized as p x = n 2π Lx with n ∈ N. The z-direction is similarly quantized in units of ∆p z = 2π Lz , so the number of states in the xy-plane per ∆p z is n = |qB| 4π 2 L x L y L z . When an electric field parallel to B is turned on adiabatically, for example as A = (−By, 0, −E z t), the states flow in the spectrum according to Lorentz's law aṡ k z = qE z . The unpaired LLL chiral modes flow to specific direction, whereas the higher LLs cancel. The states consequently move in or out of the vacuum depending on their chirality as ∂ t j 0 χ = sgn(qχ) q 2 EB 4π 2 = −sgn(qχ) q 2 32π 2 ε µνρσ F µν F ρσ . (A20) We need to generalize (A20) from Minkowski spacetime (with metric signature + − −−) to a general spacetime with or without torsion. The Landau level calculation genralizes to a non-trivial metric and coordinate dependent tetrads, when we work in the coordinate spacẽ x i ≡ e i a x a , edṼ = dV , where e = det e a i , compared to the local Minkowski space. The invariant density of states and fields to be (for e 0 µ = δ 0 µ ) which we need to use when we do not want the (scaling of the) tetrads to affect the physical density or flux we are interested in. Withk z = −E z the anomaly becomes which matches (A3) after covariantly generalized to a non-trivial metric.

Inclusion of axial fields
Left and right chiral fermions may also couple independently to different gauge fields A + and A − depending on the chirality: This is often the case in condensed matter systems with pseudo gauge fields. We then define the axial vector potential A 5 = 1 2 (A + − A − ) with corresponding axial electric and magnetic fields B 5 and E 5 , while the total vector potential is A = 1 2 (A + + A − ). The corresponding currents are from (A20) This is the covariant chiral anomaly (A3), represented as spectral flow under parallel electric and magnetic fields. The pictorial version for these equations in form of the LL the spectral flow can be found in Figs. 5 to 8.

Weyl node at finite momentum
In condensed matter systems the Weyl nodes are displaced from p = 0. Let us fix the symmetry by setting the node at p z = p W . It is straight-forward to see that shifting the momentum asp i = p i ± δ 3 i p W in the Hamiltonian simply shifts the spectrum by ±p W . An axial anomaly can then arise from the momentum space structure itself, corresponding to e.g. the anomalous quantum Hall effect 16,101 , since one has to shift Ψ by Ψ → e ∓ip W ·x Ψ to obtain the low-energy Weyl excitation Ψ = e ±ip W ·x Ψ W . While this is not important for the spectrum of U(1) states, and the axial anomaly with parallel field, it is crucial in the case of the protected Fermi arcs, anomalous quantum Hall effect, as well as the torsional anomaly with the tetrads.