Physrevlett.124.117002

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Nissinen, Jaakko Emergent Spacetime and Gravitational Nieh-Yan Anomaly in Chiral p plus ip Weyl Superfluids and Superconductors

Published in: Physical Review Letters

DOI: 10.1103/PhysRevLett.124.117002

Published: 19/03/2020

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Please cite the original version: Nissinen, J. (2020). Emergent Spacetime and Gravitational Nieh-Yan Anomaly in Chiral p plus ip Weyl Superfluids and Superconductors. Physical Review Letters, 124(11), [117002]. https://doi.org/10.1103/PhysRevLett.124.117002

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Emergent Spacetime and Gravitational Nieh-Yan Anomaly in Chiral p + ip Weyl Superfluids and Superconductors

Jaakko Nissinen * Low Temperature Laboratory, Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland (Received 27 September 2019; accepted 19 February 2020; published 19 March 2020)

Momentum transport is anomalous in chiral p þ ip superfluids and superconductors in the presence of textures and superflow. Using the gradient expansion of the semiclassical approximation, we show how gauge and Galilean symmetries induce an emergent curved spacetime with torsion and curvature for the quasirelativistic low-energy Majorana-Weyl quasiparticles. We explicitly show the emergence of the spin connection and curvature, in addition to torsion, using the superfluid hydrodynamics. The background constitutes an emergent quasirelativistic Riemann-Cartan spacetime for the Weyl quasiparticles which satisfy the conservation laws associated with local Lorentz symmetry restricted to the plane of uniaxial anisotropy of the superfluid (or superconductor). Moreover, we show that the anomalous Galilean momentum conservation is a consequence of the gravitational Nieh-Yan (NY) chiral anomaly the Weyl fermions experience on the background geometry. Notably, the NY anomaly coefficient features a nonuniversal ultraviolet cutoff scale Λ, with canonical dimensions of momentum. Comparison of the anomaly equation and the hydrodynamic equations suggests that the value of the cutoff parameter Λ is determined by the normal state Fermi liquid and nonrelativistic uniaxial symmetry of the p-wave superfluid or superconductor.

DOI: 10.1103/PhysRevLett.124.117002

— ∂ P ∇ Π −∂ P − ∇ Π ≠ 0 Introduction. Topological phases can be classified in t vac þ · vac ¼ t qp · qp ; ð1Þ terms of quantum anomalies that are robust to interactions and other perturbations [1–3]. Protected emergent quasir- P Π where vac;qp and vac;qp refer, respectively, to the momen- elativistic Fermi excitations coupled to gauge fields and tum and stress-tensor of the order parameter vacuum and geometry arise as dictated by topology and anomaly inflow quasiparticles (qps). Because of textures and superflow, the – [4 6]. In particular, gapless fermions with Weyl spectrum quasiparticles and -holes flow through the gap nodes and chiral anomalies are a recent prominent example transferring net momentum. The total momentum is con- [7–17]. On the other hand, topological phases and their served. The chiral anomaly (1) on vortices [63] has been coupling to geometry (and gravity) is currently a rapidly measured in the early landmark experiment in 3He-A [64]. advancing subject. The well-established results concern the See the Supplemental Material (SM) [65] for a detailed Hall viscosity [18] and chiral central charge [19–21] related review of Eq. (1) in terms of SF hydrodynamics. to gravitational anomalies [22–24] and thermal transport in The anomaly (1) relates to the famous angular momen- quantum Hall systems, topological superfluids (SFs) and tum “paradox” of the chiral superfluid or superconductor superconductors (SCs), as well as semimetals [25–55], the [57,59–61,66,67]: even though each Cooper pair carries net case of SFs and SCs being especially important due to the angular momentum ℏ, the pairs overlap substantially. In a lack of conserved charge. Any purported topological fixed volume, the overlap is of the order ∼a2=ξ2 with mean response of geometrical origin is necessarily more subtle pair separation a and coherence length ξ. This makes the than that based on gauge fields with conserved charges due local angular momentum contribution very small, of the 2 to the inherent dichotomy between topology and geometry. order ∼ðΔ0=EFÞ , instead of (half) the total fermion Topologically protected Weyl quasiparticles also arise in density, where Δ0 is the gap and EF the Fermi energy. three-dimensional chiral p-wave SFs and SCs from the gap At equilibrium, the local variation of angular momentum is nodes [5,56] and lead to nonzero normal density close to still well defined. Similarly, the total linear momentum the Fermi surface zeros, even at zero temperature [57,58]. density is well defined, but it is no longer conserved As any chiral fermion in three dimensions, they suffer from separately between the condensate and normal component, the chiral anomaly in the presence of nontrivial background producing the anomaly [57,58]. This is in contrast to a fields, now as an anomaly where momentum is transferred system of nonoverlapping Bose Cooper pair “molecules” from the order parameter fluctuations to the quasiparticles where the anomaly due to Weyl nodes also naturally [56–62], vanishes [56,57].

Here, we show that the momentum anomaly (1) is a transformation from the SF comoving frame (cmf) is given x0 x v 0 v manifestation of the so-called chiral gravitational Nieh-Yan as ¼ þ st and t ¼ t. In terms of SF velocity s and μ (NY) anomaly [68–74] on emergent spacetime with torsion chemical potential m, we transform [82,84] (and curvature) coupling to the low-energy Weyl fermions. 1 2 0 0 −i½mvs·xþ2ðmvs −2μmÞt The subtle role of this gravitational (geometric) anomaly ΨðxÞ → Ψ ðx Þ¼e Ψðx þ vst; tÞ; term due to torsion has been debated in the literature since − 2 v x v2−2μ ΔðxÞ → e i½ m s· þðm s mÞtΔðx þ v t; tÞ: ð3Þ its discovery [75]. We show how it arises through the s Galilean symmetries and hydrodynamics of the nonrela- The gauge and Galilean transformations are not tivistic system with an explicit ultraviolet completion. independent for coordinate dependent transformation Therefore, taken at face value, the NY anomaly has been ’ parameters. For infinitesimal constant velocity, the action experimentally verified in the late 90 s, albeit in the context Ψ0 0 δ v μ Rchanges in the cmf with ðx Þ as Scmf½ s; m¼ of emergent condensed matter fermions and spacetime 4 i 2 d xmv J − μ ρ O v ∂μΨ → ∂μ− induced by the SF order parameter. Emergent spacetime in s · m þ ð s Þ. Equivalently, ð Ψ −μ v ˆ two-dimensional topological SCs was recently carefully imvμÞ , where vμ ¼ð m=m; sÞ. Rotations along e3 act i iφ i discussed in Ref. [53]. Related recent work discusses as Δ → e Δ and lead to the combined gauge symmetry emergent tetrads, gauge fields, and anomaly terms in [83]. The SF velocity is determined as Weyl SFs [67,76] and semimetals [54,77–81], without 2 i −ˆ ∂ ˆ ∂ φ 2μ ˆ ∂ ˆ −∂ φ the framework of emergent conservation laws and mvs ¼ e1 · ie2 ¼ i ; m ¼ e1 · te2 ¼ t ; ð4Þ geometry. Model.—We consider an equal spin pairing p-wave SF at where −φ is the rotation angle. In addition, the SF velocity zero temperature on flat Euclidean space in the mean-field satisfies the Mermin-Ho relations [56,85] (MF) approximation [5,82] (see the SM [65] for more κ details). We comment below how to extend our results ∇ v − ϵijklˆ ∇lˆ ∇lˆ × s ¼ 4π i j × k; to SCs. The action for the spinless Grassmann fermion κ fΨðxÞ; Ψ†ðx0Þg ¼ 0 is (ℏ ¼ 1, summation over repeated ∂ v ∇μ − ϵijklˆ ∂ lˆ ∇lˆ t s þ m ¼ 2π i t j k: ð5Þ indices) Z where κ ¼ h=2m is the circulation quantum. The trans- S½Ψ; Ψ†; Δ; Δ†¼ d4xΨ†i∂ Ψ − H ; t MF lation symmetry correspondsR to the energy-momentum 4 1 tensor conservation law, S ¼ d xL, H ϵ Ψ ∂ Ψ ΔiΨ†∂ Ψ† ΔiΨ∂ Ψ MF ¼ ð ; i Þþ ð i þ i Þ: ð2Þ 2i μ ∂ Πν ∂ L † μ ¼ ν jexplicit; ð6Þ The normal state energy is ϵðΨ; ∂iΨÞ¼ð∂iΨ ∂iΨ=2mÞ− μ Ψ†Ψ, where m is the constituent mass, μ p2 =2m is μ μ F F ¼ F where Πν ¼ −ð∂L=∂ð∂μΨÞÞ∂νΨ þ H:c: þ Lδν. In particu- the normal state Fermi level. The spinless MF gap ampli- lar, this conservation law is broken by the anomaly (1) for Δi ∝ 1 2 Ψ∂ Ψ Δi Δ mˆ − nˆ i≡ tude ðxÞ ð = iÞh i i is ¼ð 0=pFÞð i Þ ν lˆ Πi i ˆ ¼ i along . Moreover, j is not symmetric in the plane c⊥ðeˆ1 − ieˆ2Þ , the unit vector eˆ3 ¼ mˆ × nˆ ¼ l being the determined by Δi, as is well known in 3He-A [53,56,82]. axis of orbital angular momentum of the Cooper pairs. The See the SM [65] for more details on the symmetries of dynamics of the SF free energy is ignored, and ΔðxÞ; Δ†ðxÞ the model. represent given background fields. We ignore Fermi liquid Linearized comoving quasiparticle action.P—We define corrections [82] for simplicity as we expect that these will Ψ the Bogoliubov transformation ðxÞ¼ s usðxÞasþ not affect the arguments which are based on the sym- † metries, hydrodynamics, and anomalies of the system. For vsðxÞas, where s is a generalized index in quasi- † 0 particle (particle-hole) space with fas ;a g¼δ 0 . The SCs, we perform minimal substitution in ϵðΨ; ∂iΨÞ. s ss Symmetries and Galilean transformations.—The unbro- Bogoliubov-de Gennes (BDG) quasiparticles form a spin- ken continuous symmetries of the normal Fermi liquid state Pless two-component Grassman Nambu spinor ΦðxÞ∼ 1 R3 3 ð u ðxÞv ðxÞÞTa , with the Lagrangian are Uð ÞN × ×SOð ÞL, where the translations and s s s s 3 rotations R ×SOð3Þ form a subgroup of the Galilean L ∂ − ϵ − ∂ 1 Δi ∂ group. The gauge and rotational symmetries are broken to i t ð i iÞ 2 f ;i ig L ¼ Φ†ðxÞ ΦðxÞ: ð7Þ the combined gauge symmetry U 1 [83]. In addi- BDG 1 i ð ÞN=2þL3 2 fΔ ;i∂ig i∂t þ ϵði∂iÞ tion, time-reversal symmetry is broken allowing for the emergence of Weyl quasiparticles. If ðusvsÞ is a solution to the equation of motion with energy 1 In the SF, the global Uð ÞN gauge symmetry leads to the εs, ðv−su−sÞ satisfies the particle-hole conjugate solution ∂ μ −ΔiΨ†∂ Ψ† ΔiΨ∂ Ψ 1 † conservation law μJ ¼ i þ i , where ε−s ¼ −εs. This implies the Majorana relation τ Φ ∼ Φ. Jμ ¼ðρ;JiÞ is the normal state fermion current. A Galilean For a homogenous state, the dispersion vanishes on the

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ˆ nˆ lˆ Fermi surface at k ¼pFl. The same BDG action applies ðc⊥=ckÞ ; g with uniaxial symmetry. Note that, after 3 ϵ − ∂ for SCs with the replacements ∂t → ∂t − iA0τ , linearization of ð i Þ, we perform in (8) the transforma- 3 Φ → χ˜ i − ∂ − e3 → i ∂ τ3ϵð∓ i∂iÞ → τ ϵ½∓ ði∂i þ AiÞ, where the signs are tion , i.e., fe3; ð i i pFÞig fe3; ig corre- 3 according to eigenstates of τ and vμ → vμ − Aμ, where sponding to a chiral rotation in momentum space due to the Aμ is the electromagnetic gauge potential. node. This is precisely the anomalous chiral symmetry in Now, we consider the low-energy fermions in the the system and produces the quasirelativistic anomaly (18) Λ ∝ presence of slowly varying order parameter texture and below, proportional to pF in the path-integral repre- Galilean transformation parameters vsðxÞ; μmðxÞ in the sentation [72,73,87]. semiclassical approximation, see, e.g., [86]. We assume Further, in the presence of SF velocity vμ ¼ ∂μφ ¼ 2 2 μ −v that the BDG fermions with gap ≳mc⊥ have been mð m=m; sÞ, the fermions transform in the comoving −iφτ3=2 i i −iφ i integrated out and restrict ourselves to the linear expansion frame as χ˜ → e χ˜, e1 − ie2 → e ðe1 − ie2Þ where ˆ φ close to the Weyl nodes pFl. The dispersion is ðxÞ is slowly varying compared to pF. The derivative a μ a μ i 3 operator transforms to τ ie ∂μ → τ ie ð∂μ − ∂μφτ Þ. X a a 2 † v These coincide with local spin-1=2 Lorentz transformations Φ†ðxÞϵð−i∂ÞΦðxÞ≈ χ˜ ðxÞ F lˆiðxÞ;−i∂ χ˜ ðxÞ; ð8Þ 2 i in the 12 plane, and we can attempt to associate the linearized action to a nontrivial spacetime. Denoting the a a i 2 Pauli matrices τ ; τ¯ ¼ð1; τ Þ in Nambu space, velocities and we neglectR terms of order Oð∂ χ˜Þ, where ΦðxÞ¼ x c ¼ v ¼ p =m and c⊥ ¼ðΔ0=p Þ, and ∂μ ¼ð∂ ;c ∂ Þ, P lˆ x0 x0 k F F F t k i eipF ð Þ·d χ˜ ðxÞ and χ˜ ðxÞ is slowly varying ˆ the linearized action close to the node þpFl, written in compared to pF. The order parameter vectors form the explicitly Hermitian form, becomes i mˆ spatial part of an inverse tetrad ea ¼fðc⊥=ckÞ ;

Z 1 lî χ˜† χ˜ Δ μ v 4 χ˜† ∂ χ˜ − ∂ χ˜†χ˜ μ − μ χ˜†τ3χ˜ − pF χ˜† lîτ3χ˜ S½ ; ; ; m; s¼ d x2ð i t i t Þþð F mÞ 2 vF v lî 1 vi F χ˜†τ3 ∂ χ˜ − ∂ χ˜†τ3χ˜ − χ˜† i τ1 ∂ i τ2 ∂ χ˜ lˆ v χ˜†χ˜ − s χ˜† ∂ χ˜ − ∂ χ˜†χ˜ þ 2 ð i i i i Þ 2½ ðe1 i i þ e2 i iÞ þ H:c:þpF · s 2 ð i i i i Þ: ð9Þ

— 2 lˆ2 2 χτ˜ 3χ˜ Emergent Riemann-Cartan spacetime. The action (9) density pF = m in the action, which cancels with τa 3 is (after a rotation of in the 12 plane) equivalent to a the Fermi level μFχτ˜ χ˜, since this term represents nonzero relativistic chiral (right-handed) Majorana-Weyl fermion on physical Galilean momentum present at the Weyl node, see Riemann-Cartan spacetime [69,88,89], see the SM [65] for Eq. (16) below. 12 a review of Rieman-Cartan spacetimes and our conventions Equations (10), (11), and the spin-connection ωμ define on relativistic fermions, a Weyl fermion on an emergent Riemann-Cartan spacetime μν μ ν ηab a ν δμ Z [69,88], with the uniaxial metric g ¼ eaeb , eμea ¼ ν, 1 μ a a a a λ a † 4 † a ⃗ and ∇μeν ≡∂μeμ þ ωμ eν − Γμνeλ ¼ 0. The nonzero tor- S ½χ; χ ;e;ω¼ ed xχ τ eaiDμχ þ H:c:; ð10Þ b Weyl 2 a a sion and curvature tensors are given by Tμν ¼ ∂μeν − a a b a b 12 12 12 ∂νeμ þ ω eν − ω eμ and Rμν ¼ ∂μων − ∂νωμ ¼ −1=2 a 2 2 μb νb with the identifications χ ¼ e χ˜, e ¼ det eμ ¼ðc =c⊥Þ, k 2mð∂μvsν − ∂νvsμÞ. Although seemingly pure gauge, the latter is, in general, nonzero by the Mermin-Ho relations μ μ c⊥ ˆ e0 ¼ð1; −vsÞ;e1 ¼ ð0; mÞ; Eq. (5). This correspondence is the first major result of the ck Letter. See the SM [65] for a glossary of Riemann-Cartan μ c⊥ μ ˆ spacetimes [88,89]. e2 ¼ ð0; nˆ Þ;e3 ¼ð0; lÞ; ð11Þ ck Conservation laws on curved spacetime.—Now, we formulate the SF conservation laws in terms of the Weyl ω12 2 2 −μ vi ∂ φ and μ ¼ mvsμ ¼ mð m=m; sÞ¼ μ . The covariant fermions coupled to emergent spacetime [38,53,69]. The ⃗ ab ab a b ρ i derivate is Dμ ¼ ∂μ − ði=4Þωμ σab, σ ¼ði=2Þ½τ¯ ; τ . current ð ;J Þ in terms of the Weyl quasiparticles is † 3 ˆi † 1 † † Note that the Galilean invariance leads to both the shift ðχ˜ τ χ˜;pFl χ˜ χ˜ − 2 ½χ˜ i∂iχ˜ − i∂iχ˜ χ˜Þ. To first order in i − i ∂2 μ 1 † i 1 i 2 e0 ¼ vs, although it is Oð Þ in the action, and the spin gradients, ∂μJ is equal to 2 ðχ˜ ½e1τ i∂i þ e2τ i∂iχ˜þ i ω12χ˜†χ˜ ∝ lˆ v connection term e3 i pF · s in Eq. (9). We also H:c:Þ. On the other hand, this becomes (assuming only 12 emphasize that we explicitly retained the momentum ωμ ≠ 0 and e ¼ const)

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μ e∂μS12 ¼ −eT12 þ eT21; ð12Þ nonconservation Eq. (1) in the system. The gravitational NY anomaly is [34,68,70,72], for a chiral pair of Weyl a a μ in terms of the Weyl fermions [53]. Here, the currents fermions, with e ¼ eμdx ,

1 δS 1 1 Λ2 a W χ†τa ⃗ χ − χ† ⃖ τ¯aχ ∂ μ 4 a ∧ − a ∧ b ∧ T μ ¼ μ ¼ iDμ i Dμ μðej5d xÞ¼ 2 ðT Ta e e RabÞ; ð18Þ e δea 2 2 4π 1 1 χ†τa ∂ χ − ∂ χ†τaχ ωbcχ† τa σ χ 2 ¼ 2 ð i μ i μ Þþ8 μ f ; bcg ; ð13Þ where the higher order term OðR Þ is neglected. For the p þ ip SF, the anomalous chiral Weyl action is (10) with μ 1 δS 1 μ spacetime defined by Eq. (11). Note that, although the 2 W χ† τc σab χ Sab ¼ ab ¼ ec f ; g ; ð14Þ χ˜ e δωμ 4 quasiparticles are Majorana-Weyl contributing one-half of Eq. (18) per node, a factor of 2 comes from accounting 0 are derived from the relativistic Weyl action Eq. (10).In for spin degeneracy. The temporal torsion T ¼ 0, and we μ 1 χ† μ τa τ3 χ a μ a compute the spatial contribution, particular S12 ¼ 4 eaf ; g and T b ¼ ebTμ. The relativistic conservation law Eq. (12) follows from the local T1 ∧ T1 þ T2 ∧ T2 Lorentz symmetries 2 ck 0ijk ˆ ˆ ˆ ˆ 4 ¼ 2 ϵ ½ðl · vsÞ∂ila − ∂ivsalj∂klad x i ab a a b δχ − Λ σ χ; δeμ Λ eμ; c⊥ ¼ 4 ab ¼ b c 2 ab a cb ac b ab 2 k ϵ0ijklˆ ∂ v ∂ lˆ 4 ≈ 0 ∂3 δωμ ¼ Λ cωμ þ ωμ Λ c − ∂μΛ ; ð15Þ ¼ i j s · k d x þ Oð Þ; c⊥ which, when restricted to the 12 plane, coincide with the 0 where ϵ xyz ¼ −1 and Galilean transformations. Indeed, the energy momentum μ tensor Πν was not symmetric either in this plane, the 3 3 0 4 T ∧ T ¼f2ϵ ijk½∂ lˆ − ∂ ðv · lˆÞ∂ lˆ gd x linearization of which is equal to, see the SM [65], t i i s j k 0ijk ˆ ˆ ˆ ˆ ˆ 4 ¼ 2ϵ f½∂tli − ðvs · ∇Þli∂jlk − la∂ivsa∂jlkgd x ν ð1Þ ð2Þ ν ˆi ν ν a 12 ν Πμ ¼ðΠ þ Π Þμ ¼ pFl ej δiμ − eeaTμ þ eωμ S12: 0ijk ˆ ˆ ˆ 4 3 ≈ 2ϵ f½∂tli − ðvs · ∇Þli∂jlkgd x þ Oð∂ Þ:ð19Þ ð16Þ ab The curvature term −ea ∧ eb ∧ R is The Galilean term Πð1Þ proportional to ejν ¼ χ˜†eν τaχ˜ a 4π c 2 arises due to the finite momentum density þp lˆ at the − k ϵ0ijk mˆ nˆ ∂ − ∂ F κ f i jð 0vsk kvs0Þ node and, therefore, contributes to energy momentum. The c⊥ μ 4 corresponding relativistic conservation law related to Πν of þ½ðmˆ · v Þnˆ − ðnˆ · v Þmˆ ð∂ v Þgd x s i s i j sk the linearized Weyl action follows from spacetime diffeo- c 2 k 0ijk ˆ ˆ ˆ ˆ ˆ morphisms and leads to ¼ ϵ ½2minjðl · ∂tl × ∂klÞ c⊥ 1 1 ˆ ˆ ˆ ˆ 4 μ a ab μ μ a ab μ þðl × vsÞ ðl · ∂jl × ∂klÞd x: ð20Þ ∂μ −e Tν ων S ∂νe Tμ ∂νωμ S ; i a þ 2 ab ¼ a þ 2 ab ð17Þ To lowest order in gradients, we arrive to ð2Þμ ∂ Πν ∂ L i.e., μ ¼ ν . The field theory conservation equation 2 2 μ Λ c for the energy-momentum Eq. (6) is, then, equivalent to ∂ 1 − ⊥ ϵ0ijk ∂ lˆ − v ∇ lˆ ∂ lˆ e μj5 ¼ 2π2 e 2 ½ t i ð s · Þ i j k; ð21Þ Eq. (17) and the conservation of the quasiparticle current ck density ejμ ¼ðχ˜†χ˜; χ˜†½−vi þ v lˆiτ3χ˜Þ at the node [up to s F 2 subleading terms Oð∂=pFÞ, see, e.g., [58,60] ]. Although where e ¼ðck=c⊥Þ . Matching the expression with the the Weyl action (10) implies the classical conservation law hydrodynamic anomaly [56,57,62] in the SM [65], the μ Λ ∂μj ¼ 0, this suffers from the chiral anomaly at the anisotropic cutoff is ¼ðc⊥=ckÞpF and applies in a Weyl quantum level due to the emergent spacetime (11). SF with the nodes at p lˆ, such as 3He-A, or in a Weyl SC ˆ F Nieh-Yan anomaly.—Adding both chiralities pFl, the after minimal substitution [53,62]. The expression is a conservation law for momentum is broken since, in spite of Galilean invariant and the coefficient is proportional to the ∂ μ ∂ μ − μ ≠ 0 Eq. (17), μj5 ¼ μðjþ j−Þ at the quantum level, i.e., weak-coupling normal state density [without the logarithm the conservation law suffers from the axial anomaly lnðEF=Δ0Þ due to the neglected gapped fermions [57] ]. (however,P the Weyl qp number is conserved This the central result of the Letter. The NY anomaly ∂ μ 0 μ j ¼ ) and leads to the observed momentum equation can also be derived with simple arguments using

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i i Landau levels and spectral flow in the case of a torsional tetrad e3 ¼ zˆ þ δe3, formally equivalent to gauge field 3 ˆ magnetic field Tμν [9,11,34,58]. In general, the dimensional ∼pFδlμ in the Hamiltonian [34,54,90]. In contrast, coefficient Λ is seen simply to follow from the fact that Refs. [67,76] inconsistently consider both contributions torsion couples to momentum and that the density of states independently. Moreover, the emergent gauge field does of the anomalous chiral lowest Landau level branches is not correspond to physical symmetry in the system and pF momentum dependent. Lorentz invariance would require is an explicit UV scale. This is distinct from the emergent that the Weyl nodes are symmetrically at pμ ¼ 0 which spacetime (11), which, in addition, is valid for arbitrary leads to Λ ¼ 0 at the node. For chiral Weyl nodes with a (semiclassical) textures and superflow, or considerations of nonzero separation 2pμ in momentum space, the coefficient other Weyl systems, where the emergent tetrads or gauge of the torsion anomaly is Λ ∝ jpj according to the spectral fields (e.g., elastic deformations, Fermi velocities, node flow calculation. separation) are independent [54,77,78]. On the other hand, In condensed matter systems, however, the Weyl descrip- Ref. [54] sets the NY cutoff to the lattice scale and neglects tion of the quasiparticles and the chiral anomaly breaks the breakdown of the linear Weyl spectrum. down at some cutoff scale. This is in contrast to funda- Interestingly, for the emergent spacetime in Eq. (11), the mental Weyl fermions, where the conventional chiral anomaly coefficient seems to vanish in the relativistic case anomalies satisfy IR-UV independence: the anomaly is c⊥ ¼ ck, but this is probably an artefact of the breakdown the same at each energy scale since it can be computed by of the (weak-coupling) BCS model. Our findings corrobo- comparing to a theory with no anomaly simply by adding a rate the subtle interplay of broken Lorentz invariance, high-energy chiral fermion that cancels the anomaly of the anisotropic dispersion, renormalization, and the NY- 3 original theory [23]. On the other hand, for He-A the UV anomaly coefficient Λ and should be verified by detailed completion is fully known in terms of the Fermi-liquid field theory computations [34,53,55,72,91,92]. Similarly, theory and the anomalous SF hydrodynamics of 3He-A [5]. the relation of the emergent quasirelativistic (or uniaxial) In the idealized p-wave BCS pairing model (2), the cutoff spacetime to Newton-Cartan geometries should be clarified Δ energy scale EIR ¼ for the SF is determined from the MF [84,93–96]. We did not consider the dynamics of the SF ∼ −m3n=g ∼ Δ gap equation c⊥ ðEUV=pFÞe ð =pFÞ, where g is order parameter or Goldstone modes, orbital nonanalycity δ ∼ the -function interaction coupling constant and EUV [5,56,97], nor derive the Wess-Zumino consistency equation and action for the chiral NY anomaly. This will be a vFpF ¼ ckpF the normal state Fermi energy. The linear ≪ gravitational Chern-Simons term for the tetrad and spin quasirelativistic Weyl regime emerges when E EW ¼ 2 connection [16,98–100]. Likewise, we did not consider mc⊥ ¼ðc⊥=c ÞΔ. Therefore, the uniaxial anisotropy is k singular vortices, which will lead to additional curvature simply the relative scale ðc⊥=c Þ¼ðE =E Þ, while the k IR UV and zero modes, as well as the Iordanskii force and linear Weyl regime is suppressed by an additional factor of gravitational Aharonov-Bohm phase [5] for the quasipar- c⊥=ck compared to EIR leading to the value of ticles. The connection of emergent spacetime and thermal Λ 3 10−3 ¼ðc⊥=ckÞpF.In He-A, c⊥=ck is of the order transport should be explored [40,76]. In particular, it is [82]. Remarkably, the hydrodynamic anomaly (1) is the possible that the UV scale Λ is supplemented by the IR same as in Eq. (18) when all states beyond the linear temperature scale in the anomaly, which can be universal quasirelativistic Weyl approximation are taken into [101]. These, and other considerations extending previous account. results in the literature, will be left for the future. Outlook.—We have revisited the anomalous momentum transport in chiral p-wave SFs and SCs in terms of a I wish to thank S. Moroz, C. Copetti, and S. Fujimoto for consistent hydrodynamic and low-energy effective theory useful discussions, partly during the NORDITA program “ ” description. Using the gauge and Galilean symmetries of Effective Theories of Quantum Phases of Matter. I want the system, we have shown how the quasirelativistic Weyl to especially thank G. E. Volovik for invaluable feedback approximation, emergent spacetime, and symmetries and constructive criticism during the course of this work. appear in the semiclassical derivative expansion. The This work has been supported by the European Research ’ anomalous transport is a consequence of the axial gravi- Council (ERC) under the European Union s Horizon 2020 tational NY anomaly due to the chiral Weyl fermions on an research and innovation programme (Grant Agreement emergent Riemann-Cartan spacetime with torsion. No. 694248). Here, we have shown that the emergent spacetime formulation satisfies all the symmetries and conservation laws of the effective field theory required for the gravita- *[email protected] tional NYanomaly and saturates the nonzero value from SF – [1] S. Ryu, J. E. Moore, and A. W. W. Ludwig, Electromag- hydrodynamics [64]. The early papers [9 12] treat the netic and gravitational responses and anomalies in topo- anomaly in terms of a momentum space axial gauge field; logical insulators and superconductors, Phys. Rev. B 85, this follows from our formalism via the substitution of the 045104 (2012).

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[2] X.-G. Wen, Classifying gauge anomalies through sym- [19] G. E. Volovik, Analog of gravity in superfluid 3He-A, metry-protected trivial orders and classifying gravitational Pis’ma Zh. Eksp. Teor. Fiz. 44, 388 (1986) [JETP Lett. 44, anomalies through topological orders, Phys. Rev. D 88, 498 (1986)], http://jetpletters.ac.ru/ps/1393/article_21138 045013 (2013). .shtml. [3] A. Kapustin, Symmetry protected topological phases, [20] G. E. Volovik, The gravitational topological Chern-Simons anomalies, and cobordisms: Beyond group cohomology, term in a film of superfluid 3He-A, Pis’ma Zh. Eksp. Teor. arXiv:1403.1467. Fiz. 51, 111 (1990) [JETP Lett. 51, 125 (1990)], http:// [4] C. G. Callan and J. A. Harvey, Anomalies and fermion zero www.jetpletters.ac.ru/ps/1136/article_17200.shtml. modes on strings and domain walls, Nucl. Phys. B250, 427 [21] N. Read and D. Green, Paired states of fermions in two (1985). dimensions with breaking of parity and time reversal [5] G. E. Volovik, The Universe in a Helium Droplet (Oxford symmetries and the fractional quantum Hall effect, Phys. University Press, Oxford, 2003). Rev. B 61, 10267 (2000). [6] P. Horava, Stability of Fermi Surfaces and K Theory, Phys. [22] T. Eguchi and P. Freund, Quantum Gravity and World Rev. Lett. 95, 016405 (2005). Topology, Phys. Rev. Lett. 37, 1251 (1976). [7] S. L. Adler, Axial-vector vertex in spinor electrodynamics, [23] L. Alvarez-Gaum´e and E. Witten, Gravitational anomalies, Phys. Rev. 177, 2426 (1969); J. S. Bell and R. Jackiw, A Nucl. Phys. B234, 269 (1984). PCAC puzzle: π0 → γγ in the σ-model, Nuovo Cimento A [24] L. Alvarez-Gaum´e and P. Ginsparg, The structure of gauge 60, 47 (1969). and gravitational anomalies, Ann. Phys. (N.Y.) 161, 423 [8] H. B. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw (1985). anomaly and Weyl fermions in a crystal, Phys. Lett. 130B, [25] N. Read, Non-Abelian adiabatic statistics and Hall vis- 389 (1983). cosity in quantum Hall states and px þ ipy paired super- [9] G. E. Volovik, Normal Fermi liquid in a superfluid 3He-A fluids, Phys. Rev. B 79, 045308 (2009). at T ¼ 0 and the anomalous current, Pis’ma Zh. Eksp. [26] K. Landsteiner, E. Megías, and F. Pena-Benitez, Gravita- Teor. Fiz. 42, 294 (1985) [JETP Lett. 42, 363 (1985)], tional Anomaly and Transport Phenomena, Phys. Rev. http://www.jetpletters.ac.ru/ps/1435/article_21833.shtml. Lett. 107, 021601 (2011). [10] G. E. Volovik, Chiral anomaly and the law of conservation [27] M. Stone, Gravitational anomalies and thermal Hall effect of momentum in 3He-A, Pis’ma Zh. Eksp. Teor. Fiz. 43, in topological insulators, Phys. Rev. B 85, 184503 (2012). 428 (1986) [JETP Lett. 43, 551 (1986)], http://195.178.214 [28] B. Bradlyn, M. Goldstein, and N. Read, Kubo formulas for .34/ps/1407/article_21374.shtml. viscosity: Hall viscosity, Ward identities, and the relation [11] A. V. Balatskii, G. E. Volovik, and A. V. Konyshev, On the with conductivity, Phys. Rev. B 86, 245309 (2012). chiral anomaly in superfluid 3He-A, Zh. Eksp. Teor. Fiz. [29] C. Hoyos and D. T. Son, Hall Viscosity and Electromag- 90, 2038 (1986) [Sov. Phys. JETP 63, 1194 (1986)], http:// netic Response, Phys. Rev. Lett. 108, 066805 (2012). jetp.ac.ru/cgi-bin/e/index/e/63/6/p1194?a=list. [30] G. Basar, D. E. Kharzeev, and I. Zahed, Chiral and [12] G. E. Volovik, Peculiarities in the dynamics of superfluid Gravitational Anomalies on Fermi Surfaces, Phys. Rev. 3He-A: Analog of chiral anomaly and of zero-charge, Zh. Lett. 111, 161601 (2013). Eksp. Teor. Fiz. 92, 2116 (1987) [Sov. Phys. JETP 65, [31] T. L. Hughes, R. G. Leigh, and O. Parrikar, Torsional 1193 (1987)], http://jetp.ac.ru/cgi-bin/e/index/e/65/6/ anomalies, Hall viscosity, and bulk-boundary correspon- p1193?a=list. dence in topological states, Phys. Rev. D 88, 025040 [13] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, (2013). Topological semimetal and Fermi-arc surface states in the [32] T. Can, M. Laskin, and P. Wiegmann, Fractional Quantum electronic structure of pyrochlore iridates, Phys. Rev. B 83, Hall Effect in a Curved Space: Gravitational Anomaly and 205101 (2011). Electromagnetic Response, Phys. Rev. Lett. 113, 046803 [14] A. A. Zyuzin and A. A. Burkov, Topological response in (2014). Weyl semimetals and the chiral anomaly, Phys. Rev. B 86, [33] A. G. Abanov and A. Gromov, Electromagnetic and 115133 (2012). gravitational responses of two-dimensional noninteracting [15] D. T. Son and N. Yamamoto, Berry Curvature, Triangle electrons in a background magnetic field, Phys. Rev. B 90, Anomalies, and the Chiral Magnetic Effect in Fermi 014435 (2014). Liquids, Phys. Rev. Lett. 109, 181602 (2012); Kinetic [34] O. Parrikar, T. L. Hughes, and R. G. Leigh, Torsion, parity- theory with Berry curvature from quantum field theories, odd response and anomalies in topological states, Phys. Phys. Rev. D 87, 085016 (2013). Rev. D 90, 105004 (2014). [16] I. Zahed, Anomalous Chiral Fermi Surfaces, Phys. Rev. [35] L. Sun and S. Wan, Chiral viscoelastic response in Weyl Lett. 109, 091603 (2012). semimetals, Europhys. Lett. 108, 37007 (2014). [17] D. T. Son and B. Z. Spivak, Chiral anomaly and classical [36] M. A. Zubkov, Emergent gravity and chiral anomaly in negative magnetoresistance of Weyl metals, Phys. Rev. B Dirac semimetals in the presence of dislocations, Ann. 88, 104412 (2013). Phys. (Amsterdam) 360, 655 (2015). [18] J. E. Avron, R. Seiler, and P. G. Zograf, Viscosity of [37] M. Valle, Torsional response of relativistic fermions in 2 þ Quantum Hall Fluids, Phys. Rev. Lett. 75, 697 (1995); 1 dimensions, J. High Energy Phys. 07 (2015) 006. P. Levay, ´ Berry phases for Landau Hamiltonians [38] B. Bradlyn and N. Read, Low-energy effective theory in on deformed tori, J. Math. Phys. (N.Y.) 36, 2792 the bulk for transport in a topological phase, Phys. Rev. B (1995). 91, 125303 (2015); Topological central charge from Berry

117002-6 PHYSICAL REVIEW LETTERS 124, 117002 (2020)

curvature: Gravitational anomalies in trial wave [56] G. E. Volovik, Exotic Properties of Superfluid 3He (World functions for topological phases, Phys. Rev. B 91, Scientific, Singapore, 1992). 165306 (2015). [57] G. E. Volovik and V.P. Mineev, Orbital angular momentum [39] A. Gromov, G. Y. Cho, Y. You, A. G. Abanov, and E. and orbital dynamics: 3He-A and the Bose liquid, Zh. Eksp. Fradkin, Framing Anomaly in the Effective Theory of the Teor. Fiz. 81, 989 (1981) [Sov. Phys. JETP 54, 524 (1981)], Fractional Quantum Hall Effect, Phys. Rev. Lett. 114, http://jetp.ac.ru/cgi-bin/e/index/e/54/3/p524?a=list. 016805 (2015). [58] R. Combescot and T. Dombre, Twisting in superfluid 3He- [40] A. Gromov and A. G. Abanov, Thermal Hall Effect and A and consequences for hydrodynamics at T ¼ 0, Phys. Geometry with Torsion, Phys. Rev. Lett. 114, 016802 Rev. B 33, 79 (1986). (2015). [59] N. D. Mermin and Paul Muzikar, Cooper pairs versus Bose [41] A. Lucas, R. A. Davison, and S. Sachdev, Hydrodynamic condensed molecules: The ground-state current in super- theory of thermoelectric transport and negative magneto- fluid 3He − A, Phys. Rev. B 21, 980 (1980). resistance in Weyl semimetals, Proc. Natl. Acad. Sci. [60] R. Combescot and T. Dombre, Superfluid current in 3He-A U.S.A. 113, 9463 (2016). at T ¼ 0, Phys. Rev. B 28, 5140 (1983). [42] G. Palumbo and J. K. Pachos, Holographic correspondence [61] R. Combescot and T. Dombre, Excitation spectrum and in topological superconductors, Ann. Phys. (Amsterdam) superfluid density of 3He-A at T ¼ 0, Phys. Rev. B 30, 372, 175 (2016). 3765 (1984). [43] J. Gooth et al., Experimental signatures of the mixed axial- [62] G. E. Volovik, Superfluid properties of 3He-A, Sov. Phys. gravitational anomaly in the Weyl semimetal NbP, Nature Usp. 27, 363 (1984). (London) 547, 324 (2017). [63] G. E. Volovik, Orbital angular momentum of vortices and [44] Z. V. Khaidukov and M. A. Zubkov, Chiral torsional effect, textures due to spectral flow through the gap nodes: JETP Lett. 108, 670 (2018). Example of the 3He-A continuous vortex, Pis’ma Zh. [45] P. Maraner, J. K. Pachos, and G. Palumbo, Specific heat of Eksp. Teor. Fiz. 61, 935 (1995) [, JETP Lett. 61, 2D interacting Majorana fermions from holography, Sci. 958 (1995)], http://www.jetpletters.ac.ru/ps/1210/article_ Rep. 9, 17308 (2019). 18297.shtml. [46] A. Gromov and D. Thanh Son, Bimetric Theory of [64] T. D. C. Bevan, A. J. Manninen, J. B. Cook, J. R. Hook, Fractional Quantum Hall States, Phys. Rev. X 7, H. E. Hall, T. Vachaspati, and G. E. Volovik, Momentum 041032 (2017); A. Gromov, S. D. Geraedts, and B. creation by vortices in 3He as a model of primordial Bradlyn, Investigating Anisotropic Quantum Hall States baryogenesis, Nature (London) 386, 689 (1997). with Bimetric Geometry, Phys. Rev. Lett. 119, 146602 [65] See Supplemental Material at http://link.aps.org/ (2017). supplemental/10.1103/PhysRevLett.124.117002 for fur- [47] P. Wiegmann, Inner Nonlinear Waves and Inelastic Light ther details on the model, symmetries and conservation Scattering of Fractional Quantum Hall States as Evidence laws, the momentum anomaly of Eq. (1), as well as for a of the Gravitational Anomaly, Phys. Rev. Lett. 120, glossary on Riemann-Cartan spacetimes and our conven- 086601 (2018). tions regarding relativistic fermions. [48] K. Fujii and Y. Nishida, Low-energy effective field theory [66] M. G. McClure and S. Takagi, Angular Momentum of superfluid 3He-B and its gyromagnetic and Hall re- of Anisotropic Superfluids, Phys. Rev. Lett. 43, 596 sponses, Ann. Phys. 395, 170 (2018). (1979). [49] O. Golan, C. Hoyos, and S. Moroz, Boundary central [67] Y. Ishihara, T. Mizushima, A. Tsuruta, and S. Fujimoto, charge from bulk odd viscosity—chiral superfluids, Phys. Torsional chiral magnetic effect due to skyrmion textures Rev. B 100, 104512 (2019). in a Weyl superfluid 3He-A, Phys. Rev. B 99, 024513 [50] Z.-M. Huang, L. Li, J. Zhou, and H.-H. Zhang, Torsional (2019). response and Liouville anomaly in Weyl semimetals with [68] H. T. Nieh and M. L. Yan, An identity in Riemann-Cartan dislocations, Phys. Rev. B 99, 155152 (2019). geometry, J. Math. Phys. (N.Y.) 23, 373 (1982); H. T. Nieh, [51] J. Nissinen and G. E. Volovik, Elasticity tetrads, mixed A torsional topological invariant, Int. J. Mod. Phys. A 22, axial-gravitational anomalies, and (3 þ 1)-d quantum Hall 5237 (2007). effect, Phys. Rev. Research 1, 023007 (2019). [69] H. T. Nieh and M. L. Yan, Quantized Dirac field in curved [52] T. Kvorning, T. H. Hansson, A. Quelle, and C. M. Smith, Riemann-Cartan background. I. Symmetry properties, Proposed Spontaneous Generation of Magnetic Fields by Green’s function, Ann. Phys. (Amsterdam) 138, 237 Curved Layers of a Chiral Superconductor, Phys. Rev. (1982). Lett. 120, 217002 (2018). [70] S. Yajima, Evaluation of the heat kernel in Riemann— [53] O. Golan and A. Stern, Probing topological superconduc- Cartan space, Classical Quantum Gravity 13, 2423 (1996). tors with emergent gravity, Phys. Rev. B 98, 064503 [71] Y. N. Obukhov, E. W. Mielke, J. Budczies, and F. W. Hehl, (2018). On the chiral anomaly in non-Riemannian spacetimes, [54] Y. Ferreiros, Y. Kedem, E. J. Bergholtz, and J. H. Bardar- Found. Phys. 27, 1221 (1997). son, Mixed Axial-Torsional Anomaly in Weyl Semimetals, [72] O. Chandia and J. Zanelli, Topological invariants, instan- Phys. Rev. Lett. 122, 056601 (2019). tons, and the chiral anomaly on spaces with torsion, Phys. [55] C. Copetti and K. Landsteiner, Anomalous Hall viscosity Rev. D 55, 7580 (1997); Torsional topological invariants at the Weyl-semimetal-insulator transition, Phys. Rev. B (and their relevance for real life), Phys. Lett. B 416,85 99, 195146 (2019). (1998); Supersymmetric particle in a spacetime with

117002-7 PHYSICAL REVIEW LETTERS 124, 117002 (2020)

torsion and the index theorem, Phys. Rev. D 58, 045014 [87] K. Fujikawa, Path-Integral Measure for Gauge-Invariant (1998). Fermion Theories, Phys. Rev. Lett. 42, 1195 (1979); Chiral [73] C. Soo, Adler-Bell-Jackiw anomaly, the Nieh-Yan form, anomaly and the Wess-Zumino condition, Phys. Rev. D 31, and vacuum polarization, Phys. Rev. D 59, 045006 (1999). 341 (1985). [74] K. Peeters and A. Waldron, Spinors on manifolds with [88] I. L. Shapiro, Physical aspects of the space-time torsion, boundary: APS index theorems with torsion, J. High Phys. Rep. 357, 113 (2002). Energy Phys. 02 (1999) 024. [89] R. T. Hammond, Torsion gravity, Rep. Prog. Phys. 65, 599 [75] D. Kreimer and E. W. Mielke, Comment on “Topological (2002). invariants, instantons, and the chiral anomaly on spaces [90] This is part of the ambiguity of assigning emergent fields with torsion”, Phys. Rev. D 63, 048501 (2001); O. Chandia to the Weyl quasiparticles without the UV completion. In and J. Zanelli, Reply to “Comment on ‘Topological particular, the UV scale Λ is undetermined in the quasir- invariants, instantons, and the chiral anomaly on spaces elativistic low-energy theory. I thank G. E. Volovik for with torsion’”, Phys. Rev. D 63, 048502 (2001). repeatedly emphasizing this point to me. The Landau level [76] T. Kobayashi, T. Matsushita, T. Mizushima, A. Tsuruta, spectral flow calculations in [9,11] arise from the torsional and S. Fujimoto, Negative Thermal Magnetoresistivity as a magnetic field due to a nontrivial tetrad [34]. Signature of a Chiral Anomaly in Weyl Superconductors, [91] E. V. Castro, A. Flachi, P. Ribeiro, and V. Vitagliano, Phys. Rev. Lett. 121, 207002 (2018). Symmetry Breaking and Lattice Kirigami, Phys. Rev. Lett. [77] A. G. Grushin, J. W. F. Venderbos, A. Vishwanath, and R. 121, 221601 (2018). Ilan, Inhomogeneous Weyl and Dirac Semimetals: Trans- [92] S. Imaki and A. Yamamoto, Lattice field theory with port in Axial Magnetic Fields and Fermi Arc Surface States torsion, Phys. Rev. D 100, 054509 (2019). from Pseudo-Landau Levels, Phys. Rev. X 6, 041046 [93] C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, (2016). Bargmann structures and Newton-Cartan theory, Phys. [78] D. I. Pikulin, A. Chen, and M. Franz, Chiral Anomaly from Rev. D 31, 1841 (1985). Strain-Induced Gauge Fields in Dirac and Weyl Semimet- [94] D. T. Son, Newton-Cartan geometry and the Quantum Hall als, Phys. Rev. X 6, 041021 (2016). Effect, arXiv:1306.0638. [79] A. Westström and T. Ojanen, Designer Curved-Space [95] M. H. Christensen, J. Hartong, N. A. Obers, and B. Rollier, Geometry for Relativistic Fermions in Weyl Metamaterial, Torsional Newton-Cartan geometry and Lifshitz hologra- Phys. Rev. X 7, 041026 (2017). phy, Phys. Rev. D 89, 061901(R) (2014). [80] L. Liang and T. Ojanen, Curved spacetime theory of [96] C. Copetti, Torsion and anomalies in the warped limit of inhomogeneous Weyl materials, Phys. Rev. Research 1, Lifschitz theories, J. High Energy Phys. 01 (2020) 190 032006 (2019). (2020). [81] R. Ilan, A. G. Grushin, and D. I. Pikulin, Pseudo- [97] J. Nissinen and G. E. Volovik, Dimensional crossover of electromagnetic fields in topological semimetals, Nat. effective orbital dynamics in polar distorted 3He-A: Tran- Rev. Phys. 2, 29 (2020). sitions to antispacetime, Phys. Rev. D 97, 025018 (2018). [82] D. Vollhardt and P. Wölfle, The Superfluid Phases of [98] G. E. Volovik, Wess-Zumino action for the orbital dynam- Helium 3 (Taylor & Francis, London, 1990). ics of 3He-A, Pis’ma Zh. Eksp. Teor. Fiz. 44, 144 (1986); [83] M. Liu and M. C. Cross, Gauge Wheel of Superfluid 3He, [JETP Lett. 44, 185 (1986)], http://www.jetpletters.ac.ru/ Phys. Rev. Lett. 43, 296 (1979). ps/1376/article_20873.shtml. [84] D. Son and M. Wingate, General coordinate invariance and [99] A. V. Balatsky, Microscopically derived Wess-Zumino conformal invariance in nonrelativistic physics: Unitary action for 3He-A, Phys. Lett. A 123, 27 (1987). Fermi gas, Ann. Phys. (Amsterdam) 321, 197 (2006). [100] J. Dziarmaga, Low-temperature effective electromagnet- [85] N. Mermin and T.-L. Ho, Circulation and Angular ism in superfluid 3He-A, Pis’ma Zh. Eksp. Teor. Fiz. 75, Momentum in the A Phase of Superfluid Helium-3, Phys. 323 (2002) [JETP Lett. 75, 273 (2002)]. Rev. Lett. 36, 594 (1976). [101] J. Nissinen and G. E. Volovik, On thermal Nieh-Yan [86] N. Kopnin, Theory of Nonequilibrium Superconductivity anomaly in Weyl superfluids, arXiv:1909.08936. (Oxford University Press, Oxford, 2001).

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