THERMAL NIEH-YAN ANOMALY in WEYL SUPERFLUIDS PHYSICAL REVIEW RESEARCH 2, 033269 (2020) Not Conﬁrmed in General, See However Refs

PHYSICAL REVIEW RESEARCH 2, 033269 (2020)

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Thermal Nieh-Yan anomaly in Weyl superﬂuids

J. Nissinen 1,* and G. E. Volovik1,2,† 1Low Temperature Laboratory, Department of Applied Physics, Aalto University, FI-00076 AALTO, Finland 2Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

(Received 28 May 2020; revised 17 July 2020; accepted 20 July 2020; published 19 August 2020)

∂ μ = γ 2 T a ∧ T We discuss the possibility of torsional Nieh-Yan anomaly of the type μ(ej5 ) T ( a )inWeyl superfluids, where T is the infrared (IR) temperature scale and T a is the effective or emergent torsion from the superfluid order parameter. As distinct from the dimensionful ultraviolet (UV) parameter 2 in the conventional torsional Nieh-Yan anomaly, the parameter γ is dimensionless in canonical units. This suggests that such dimensionless parameter may be fundamental, being determined by the geometry, topology, and number of chiral quantum fields in the system. By comparing this to a Weyl superfluid with low-temperature corrections,

T 0, we show that such a term does exist in the hydrodynamics of a chiral p-wave superﬂuid, such as 3He-A, or a chiral superconductor. We also discuss and show how other T 2 terms of similar form and of the same order in gradients, coming from, e.g., Fermi-liquid corrections, effective curvature, and the chiral chemical potential, can also be expressed in terms of dimensionless fundamental parameters with emergent low-energy relativistic ﬁelds. Lastly, we discuss our results in comparison to relativistic Weyl fermions and the connection of the torsional gravitational anomalies to thermal transport in Weyl systems.

DOI: 10.1103/PhysRevResearch.2.033269

I. INTRODUCTION These background fields imply the chiral anomaly for the low-energy massless quasiparticles. For the applica- In nonrelativistic topological matter, effectively quasirel- tions of the chiral anomaly in Weyl semimetal and Weyl ativistic description of low-energy quasiparticles with linear superfluids/superconductors, see, e.g., Refs. [1,5–7]. In par- spectrum phenomena may emerge [1,2]. In particular, in three ticular, the nontrivial coordinate dependence (torsion) related spatial dimensions at a generic (twofold) degenerate fermion μ to the tetrads e (x)in(1) can lead to the gravitational Nieh- band crossing at momentum p , the Hamiltonian is of the a W Yan (NY) anomaly [8–18]. Nevertheless, the relativistic NY Weyl form [1,3–5] contribution to the anomaly has remained contentious and = σ a i − +··· , HW ea(p pW )i (1) subtle due to presence of a dimensionful ultraviolet (UV) scale 2 with canonical dimensions of momentum, as required by i = ∂ | a where the ea pi H(p) pW are the linear coefficients of the the canonical dimensions of eμ. In condensed matter, the sim- a Hermitean Pauli matrices σ , a = 1, 2, and 3, in a Taylor plest way to understand this nonuniversal cutoff is the depen- expansion close to the Weyl node(s) at pW . The net chiral- dence of the expansion (1) on microscopics, see Ref. [16]for i ity {p } sgn(det ea ) vanishes in the system [5]. For slowly an example with an explicit provided by the nonrelativistic W σ a μ ∂ ≡ ∂ − varying parameters in the operator ea i μ i t HW ,the Fermi-liquid UV completion, and the discussion below. μ ={ t , i } semiclassical fields ea (x) ea ea are promoted to back- Here we discuss the temperature corrections to the grav- ground space-time tetrad fields with dimensions of unity for itational NY anomaly, such temperature corrections being t i temporal indices ea and velocity for spatial ea.Atthelevelof absent for U(1) gauge fields in the chiral anomaly. We then the Hamiltonian, the shift of the Weyl node pW acts formally compare the anomaly current to finite-temperature terms in as an emergent (axial) gauge field with emergent Lorentz the hydrodynamic momentum transport of chiral Weyl su- symmetry to the linear order. However, the Taylor expansion perfluids at low temperature. The results apply for the chiral (1)atpW is valid at much lower scales pW . If the fermions superconductors as well, when the electromagnetic potential are charged, they can in addition couple to the electromagnetic is added vs → vs − eA/m in some convenient gauge. We vector potential via minimal coupling. show that the momentum current has a term originating from the torsional NY anomaly, among various other temperature terms of the same order but different origin in the free energy. 2 *jaakko.nissinen@aalto.fi All such second order T corrections in the low-temperature †grigori.volovik@aalto.fi expansion of the free energy take an effectively relativistic form. Their dimensionless prefactors do not depend on the Published by the American Physical Society under the terms of the details of the microscopic physics and, instead, are determined Creative Commons Attribution 4.0 International license. Further by geometry, topology, and the number of effective fermionic distribution of this work must maintain attribution to the author(s) species [1,19]. Specifically, we compare the torsional finite- and the published article’s title, journal citation, and DOI. temperature corrections to the lowest order gradient terms

2643-1564/2020/2(3)/033269(10) 033269-1 Published by the American Physical Society J. NISSINEN AND G. E. VOLOVIK PHYSICAL REVIEW RESEARCH 2, 033269 (2020) in the free energy of the chiral superfluid, and identify the free-energy in the chiral Weyl superfluid, with the expectation contribution from the chiral NY anomaly in the low-energy that = T , kB = 1, with some dimensionless prefactor γ ,in quasirelativistic Weyl superfluid. This leads us to conjecture units where the tetrads have canonical dimensions of velocity, that the finite temperature IR-scale NY anomaly term can be or the relevant “speed of light” is set unity. Notably, the IR similarly universal in general Weyl systems. Finally, we com- T 2-correction to the anomaly would be absent for U(1) gauge pare these results to relativistic Weyl fermions with positive fields. and negative branches at zero momentum. In terms of effective low-energy actions, the fully relativistic analogs work unambiguously only for terms in the II. TORSIONAL ANOMALY effective action with dimensionless coefficients. Perhaps the most well-known example being the 2+1-dimensional topo- For space-times with torsion (and curvature), Nieh and Yan logical Chern-Simons (CS) terms describing the quantum [8–10] introduced the four-dimensional invariant Hall effect, see, however, Ref. [21]. Gravitational Chern- Simons terms can be similarly quantized in terms of chiral N = T a ∧ T − ea ∧ eb ∧ R , (2) a ab central charge which has relation to thermal transport and the a a μ a where e = eμdx is the local tetrad 1-form field and T = boundary conformal field theory [22,23].TheCSactionwas a + ωa ∧ b a = ωa + ωa ∧ ωc recently generalized to 3+1d (and higher odd space dimen- de b e and R b d b c b, in terms of the ωa = ωa μ sions) crystalline topological insulators, using so-called di- tetrad and spin connection b μbdx . As usual, the space- a b mensionful elasticity tetrads E with dimension [E] = [1/L] = time metric follows as gμν = e e η , where η is the local μ ν ab ab [M]. Topological polarization terms can also be written down orthonormal (Lorentz) metric. This invariant can be written, [24] by dimensional descent. The ensuing higher dimensional using the associated Bianchi identities, as Chern-Simons and polarization terms are expressed as the a ∧ ∧ ∧ ∧ N = dQ , Q = e ∧ Ta . (3) mixed responses E A dA and E E dA with quantized dimensionless coefficients [24–26]. ∧ N is a locally exact 4-from independent√ from tr(R R) and Another such example is the temperature correction to the dual of the scalar curvature gR in the presence of 2 curvature effects, with δSeff = T R in the low-energy nonzero torsion. It can be associated with a difference of action [19]. This represents the analog of the gravitational two topological terms, albeit in terms of an embedding to coupling (inverse Newton constant) in the low-energy action, five dimensions [13–15]. In terms of four-dimensional chiral where R is the effective scalar space-time curvature of the fermions on such a space-time, it has been suggested that this medium felt by the quasiparticles. Since [T ]2[R] = [M]4,the invariant contributes to the axial anomaly, i.e., the anomalous coefficient of this term is dimensionless, and can be given in production of the chiral current: terms of universal constants: it is fully determined by the 2 number of the fermionic and bosonic species in the effective μ ∂μ(ej ) = N(r, t ) , (4) theory on flat background, and thus works both in relativistic 5 π 2 4 and nonrelativistic systems [19]. The same universal behavior = a μ where e det eμ and ej5 is the axial current (pseudotensor) takes place with the terms describing the chiral magnetic density and the nonuniversal parameter has dimension and chiral vortical effects in Weyl superfluid 3He-A, where of relativistic momentum (mass) [] = [1/L] = [M] and is the coefficients are dimensionless [1,27,28]. Similarly, it has determined by some ultraviolet (UV) energy scale. been observed that the coefficient of the tr(R ∧ R) gravita- Given the relativistic anomaly term (4), there has been tional anomaly in chiral Weyl systems affects the thermal several attempts to consider the Nieh-Yan anomaly in con- transport coefficients in flat space [29–35]. These coefficients densed matter systems with Weyl fermions like (1), see e.g. are fundamental, being determined by the underlying degrees [16–18,20]. However, in relativistic systems, the high-energy of freedom in addition to symmetry, topology and geometry. cutoff is not a well defined parameter and, moreover, can be From this perspective especially, since the NY form is second in addition anisotropic in condensed matter systems. There the order in derivatives and can be computed from linear response complete UV theory is, of course, non-Lorentz invariant and on flat space, our findings are very interesting and warrant the linear, quasirelativistic Weyl regime is valid at much lower further research. scales where (1) applies. Moreover, the anomalous hydrody- Our goal in the present paper is to separate different namics of superfluid 3He at zero temperature suggests that the T 2 contributions in the hydrodynamic free energy of Weyl chiral anomaly is either completely exhausted by the emergent superfluids in order to identify the terms responsible for axial gauge field corresponding to the shift of the node or, con- different relativistic phenomena, including a term from the versely, by the gravitational NY anomaly term arising from thermal Nieh-Yan anomaly in the current, as well as simi- the tetrad and spin-connection for local combined orbital- lar gravitational terms corresponding to the scalar curvature gauge invariance along the uniaxial symmetry direction. This and chemical potential, T 2(R + μ2 ), where μ is the chiral was recently shown in Ref. [16], as well as that the low-energy chemical potential μ T . theory satisfies the symmetries and conservations laws related to an emergent quasirelativistic space-time with torsion, III. TEMPERATURE CORRECTION TO being determined from the UV-scale where the linear Weyl THE NIEH-YAN TERM approximation breaks down as dictated by the underlying p-wave BCS Fermi superfluid. Here we will consider the The relativistic zero-temperature anomaly term in the ax- 2 a a b temperature corrections to the anomaly and the hydrodynamic ial current production (T ∧ Ta + e ∧ e ∧ Rab) is still

033269-2 THERMAL NIEH-YAN ANOMALY IN WEYL SUPERFLUIDS PHYSICAL REVIEW RESEARCH 2, 033269 (2020) not confirmed in general, see however Refs. [16,36]for ∓pF n± at nodes of the two opposite Weyl points sum-up condensed matter Weyl fermions. On one hand, the UV cutoff leading to parameter is not well-defined in relativistic field theory with fundamental chiral fermions. On the other hand, such P˙ anom =−pF ˆl (∂t n5), (10) a cutoff is not in general available in nonrelativistic matter with quasirelativistic low-energy chiral fermions and can be where n5 = n+ − n− is the chiral density. The corresponding anisotropic [16] or even zero. However, a term of the form quasirelativistic momentum density of the Weyl fermions, 2 a a b γ T (T ∧ Ta + e ∧ e ∧ Rab) has the proper dimensionality valid in the vicinity of the node, is [16] [M]4, and its prefactor γ could be a universal constant in =− ˆ 0 . canonical units, being expressed via some invariant related to PNY-node pF lej5 (11) the low-energy degrees of freedom. For concreteness, we focus on the finite temperature Nieh- Thus Eq. (8) gives the temperature correction to this anoma- Yan anomaly in chiral p-wave Weyl superfluid (such as 3He- lous momentum production, leading to a mass current due to A) with the expectation that thermal NY anomaly. Next we discuss this in detail for the μ 2 quasiparticles and the superfluid vacuum in the nonrelativistic ∂μ ej = γ T N(r, t ) , (5) 5 chiral Weyl p-wave superfluid, using a Landau level model and check whether the dimensionless parameter γ can be for the currents [16,39]. For relativistic Weyl fermions, see universal. We now use the result obtained by Khaidukov and [38,40]. For considerations of similar temperature effects in Zubkov [37] and Imaki and Yamamoto [38] for the finite chiral Weyl superfluid, see Ref. [41]. temperature contribution to the chiral current from the chiral vortical effect. For a single (complex) chiral fermion, one has A. Nieh-Yan term from the hydrodynamics of for the chiral current chiral Weyl superfluid T 2 k =− kijT 0 . It is known that the hydrodynamics of chiral gapless 3He-A j5 ij (6) 24 experiences momentum anomalies related to the spectral flow We assume that this current can be covariantly generalized to through the Weyl points. Let us express Eqs. (10) and (11) the 4-current: in terms of the hydrodynamic variables and quasiparticles of the chiral superfluid. The Weyl fermions arise from the BdG T 2 μ =− μναβ T a . ej5 eνa αβ (7) Hamiltonian close to the nodes, 24 Then one obtains the divergence − ∂ 1 {∂ ,i} ( i ) 2i i HBdG(−i∂) = . (12) 2 1 {∂ ,∗i}− (i∂) μ T μναβ a 2i i ∂μ(ej ) =− T μν Tαβ . (8) 5 48 a p2−p2 = F In the presence of curvature R(ω), this becomes the tempera- Here, (p) m is the normal state dispersion minus the μ 2 = + ture correction to the full Nieh-Yan term in Eq. (5), where now Fermi level F ; i c⊥(mˆ inˆ ) is the order parameter of the nonuniversal cutoff is substituted by the well defined the p + ipchiral superfluid; ˆl = mˆ × nˆ is the unit vector in the IR temperature scale T , and the dimensionless parameter γ = direction of the orbital angular momentum of Cooper pairs. = = / 1/12: The anisotropic node velocities are c vF and c⊥ 0 pF , constituting effective speeds of light in Weyl equation along 2 ∂ μ =−T , . and transverse to ˆl, respectively. In the weak coupling BCS μ(ej5 ) N(r t ) (9) 3 −3 12 theory, c⊥ c , and in He-A, their ratio is of order 10 .In Note that it is possible that the local relativistic (Tolman) a chiral superconductor, we in addition perform the minimal = /| 0| coupling (−i∂i ) → (iDi ) + eA0, where Di = ∂i − eAi is the temperature T T0 et enters the local anomaly, while the gauge covariant derivative and Aμ is the electromagnetic constant T0 is the global equilibrium temperature of the con- 0 =− potential. Equivalently this amounts to v → v − eA in the densed matter system [19]. In (1), we have simply et 1. s s m free-energy. In what follows, we ignore the superfluid velocity (giving rise to spin-connection in addition to tetrads) IV. FROM RELATIVISTIC PHYSICS TO in the anomaly considerations. Then the only hydrodynamic CHIRAL WEYL SUPERFLUID variable appearing in the torsion is the unit vector of the In the presence of a finite Weyl node pW = 0, the chiral orbital momentum ˆl [16]. Our results can afterwards then anomaly for the chiral current leads to the anomalous produc- generalized to include the superfluid velocity appearing in tion of the linear momentum [1,16,36]. Even though chiral the free energy, with the expectation that curvature term is current is not well-defined at high-energy, the spectral flow of included in the NY form as in Eqs. (2), (5), and as long chiral quasiparticles is accompanied by the spectral flow of the as we enforce proper normalization due to momentum space 3 linear momentum pW of the Weyl point. In He-A, there are anisotropy of thermal fluctuations, see Appendix A2[16]. See two (spin-degenerate) Weyl points with opposite chirality and also Ref. [42] for two-dimensional finite-temperature effects opposite momenta, pW ± =±pF ˆl, where ˆl is the unit vector of of curvature without torsion. the orbital angular momentum of the superfluid. In particular, The Weyl nodes are at pW ± =±pF ˆl. Expanding the σ a i − i the anomalous production of the linear momentum density Hamiltonian as HBdG ea(ˆp pW )i, the vierbein ea in the

033269-3 J. NISSINEN AND G. E. VOLOVIK PHYSICAL REVIEW RESEARCH 2, 033269 (2020) vicinity of the Weyl point take the form In addition, the relativistic anomaly results in Sec. III, ⎛ ⎞ 10 with Eq. (7), and the Landau level argument for the chiral ⎜ ⎟ superfluid suggest the following temperature correction to the μ t i ⎜0 c⊥mˆ ⎟ e = e , e = , a = 0, 1, 2, 3. (13) anomalous momentum in 3He-A, in the vicinity of the node, a a a ⎝0 c⊥nˆ ⎠ from the thermal Nieh-Yan anomaly [16]: 0 c ˆl p T 2 i a = δi =− ˆ = F ˆ ˆ · ∇×ˆ . For the inverse vierbein eae j j,wehave PNY-node(T ) pF l n5(T ) 2 l(l ( l)) (19) 12c⊥ a a a 10 0 0 e = e , e = . The quasiparticle density n5(T ) between the two Weyl nodes μ t i 0 1 mˆ 1 nˆ 1 ˆl (14) c⊥ c⊥ c is computed in the LL model of Appendix A2from a similar integral as Eq. (16), In order to compute the thermal contribution from the quasiparticles in the presence of torsion, we assume that the ∞ =− − μ − μ − tetrad gives rise to a constant torsion via mˆ = xˆ, nˆ = yˆ − n5(T ) 2 NLL(pz )dpz[nF ( pz F ) ( pz )] 0 TBxzˆ, where TB 1 is a perturbation [16,36,39]. In this −T ∞ −T 2 case, the three-dimensional spectrum organizes into one- = B dxxn (x) = (ˆl ·∇×ˆl). (20) 2 F 2 dimensional states on two-dimensional Landau levels (LLs) 2π 0 12c⊥ [1,17,36,39,40]. See also e.g. [43,44] for reviews of vector As for the total current, this is the contribution from the and axial U(1) fields in semimetals. In our case, the relevant linear spectrum close to the nodes due to thermal fluctuations, spatial torsion is with T playing the role of the cutoff relevant for the full 1 ˆl dispersion of the chiral superfluid, such as 3He-A. The thermal ijkT 3 i = ˆ · ∇× ≡ , jke3 c l TB (15) 2 c fluctuations for the superfluid are discussed in Appendix A1. What have been calculated above are actually the tem- = ˆ ·∇×ˆ = T z and pzTB pzl l pz xy is playing the role of tor- perature dependence of the anomalous torsional response of sional effective magnetic field with momentum charge. the Weyl superfluid, i.e., the chiral momentum-density and 3 For more on LL spectrum in He-A and the nonrelativis- number densities in response to TB, see, e.g., Ref. [32]. Note tic anisotropic model with torsion, see Appendix A2 and in particular that the leading torsional contribution from the Refs. [16,36,39,45]. NY anomaly in Eqs. (16) and (20) from thermal fluctuations ˆ ≈ −2 The anomalous (chiral) quasiparticle current along l zˆ is suppressed by the factor c⊥ as was found as the leading becomes order term to Eq. (2)inRef.[16] including the curvature con- ∞ qp · ˆ =− − μ tribution to the anomaly. This is in contrast to the subleading P l 2 NLL(pz )dpz pznF ( pz F ) = 1 ijk 3T 3 = TB torsion part S ei jk 2 from (14) that would arise 0 2 c 3 2 in the LL model without the anisotropic thermal fluctuations p pF T =− F + ˆl ·∇×ˆl (16) discussed. Apart from such geometric factors, it is amusing 6π 2 6c2 ⊥ to see that the nonrelativistic thermal integrals in Eqs. (16) = vac + qp , janom janom (T ) and (20) with explicit nonrelativistic vacuum regulatization due to the filled quasiparticle states coincide with similar = βx + −1 where nF (x) (e 1) is the quasiparticle distribution expression for relativistic Weyl fermions with positive and function and a factor of two comes from the spin-degeneracy. negative branches [28,30–32,37,38], see Appendix A3.This In order to compute the integral, we have used T 0 is by the universality of thermal fluctuations close to the node μ F as well as the linear expansion with anisotropic momen- and the underlying reason why the corresponding anomaly tum scaling, where close to the nodes thermal fluctuations contribution can be universal. T T c⊥ T 2 are p⊥ ∼ pF = , p = ( ) for both T, mc⊥. 0 c⊥ c c⊥ 3 This is due to the anisotropy of the linear Weyl regime in He- B. Anomalous vacuum current A with thermal fluctuations ∼ T/2. See Appendixes A1, Now we discuss the anomalous current from the underlying A2, and Refs. [16,36] for more details. We conclude that jvac anom vacuum with torsion fields, i.e., the current in the chiral is the anomalous superfluid current from filled quasiparticle superfluid (or superconductor). states, see Eq. (21) below, The hydrodynamic anomaly in momentum conservation 3 vac pF C0 arises between the quasiparticles and vacuum mass current j =− ˆl(ˆl ·∇×ˆl) =− ˆl(ˆl ·∇×ˆl), (17) anom 6π 2 2 P = j of the superfluid, which at T = 0 has the following whereas the finite temperature contribution to the quasiparti- general form [1,46,47]: cle momentum is = ρ + 1 ∇× ρˆ − C0 ˆ ˆ · ∇×ˆ , 2 j vs ( l) l(l ( l)) (21) qp =−pF T ˆ ˆ ·∇×ˆ . 4m 2 janom (T ) 2 l(l l) (18) 6c⊥ where the last term is anomalous with the parameter C0 from the combined orbital-gauge symmetry of the superfluid [48], and arises due to thermal normal component close to the that fully determines the axial anomaly in the system due to nodes. These contributions to the anomalous vacuum current the Weyl quasiparticles of the superfluid [1] from the perspective of the superfluid are analyzed in the next = = 3 / π 2 = ρ, subsection. C0(T 0) pF 3 (22)

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2/ 2 = where we ignore corrections of the order of ( 0 EF ) correspond to hydrodynamics and phenomena with different 2 (c⊥/c ) 1 at zero temperature to the density from pairing origins and scales. Although of similar form in terms of the ρ − ∼ ρ 2/ 2 [ C0 ( 0 EF )]. Notably, this term exists only on the low-energy Goldstone variables of the superfluid, they can be weak coupling side of the topological BEC-BCS Lifshitz expressed in relativistic form with fundamental parameters transition, where the pair of the Weyl points with pW ± = [19] only when carefully keeping track of each individual ±pF ˆl appears in the quasiparticle spectrum [1,48]. contribution to avoid double counting. In particular, in the next Here we are interested in the temperature correction to section we shall see that the nonfundamental parameters m ∗ the anomaly, which corresponds to the Nieh-Yan term. The and m do not enter the free energy or the final results, when extension of the anomalous current to nonzero temperature expressed in terms of the correct relativistic variables valid in gives the quasirelativistic low-energy theory. 1 Panom (T ) =− C0(T )ˆl(ˆl · (∇×ˆl)), (23) 2 V. RELATIVISTIC CORRECTIONS TO FREE-ENERGY where according to Cross [46,47], the anomalous parame- Let us consider the T 2 corrections in the free energy which ter C0(T ) has the following temperature dependence at low are second order in derivatives containing combinations of T Tc: 4 3 (ˆl · vs ) and (ˆl · (∇×ˆl)), neglecting all higher order O(T ,∂ ) p m∗ = − 2 F + . terms. These terms can be distributed into three groups, which C0(T ) C0(0) T 2 1 (24) ∗ 6c⊥ m have different dependence on m and m: ∗ Here, m is the effective mass of quasiparticles in the normal F = F1 + F2 + F3 , (28) Fermi liquid, which differs from the bare mass m of the 3 2 He atom due to the Fermi liquid corrections. In Eqs. (17) ˆ, = pF T ˆ · ˆ · ∇×ˆ , F1[l vs] 2 (l vs )(l ( l)) (29) and (18), corresponding to the simple one-dimensional LL 12 c⊥ model, we have neglected the Fermi-liquid corrections due to p m∗ T 2 interactions in the three-dimensional superfluid: An additional F [ˆl, v ] =− F [4m(ˆl · v ) − (ˆl · (∇×ˆl))]2, (30) ∗ 2 s 2 2 s factor 1 ( m − 1)(ρ(0)/ρ) arises from the reduced quasipar- 96m c⊥ 2 m n 2 ticle momentum flow, due to Galilean invariance. The full ˆ =− pF T . F3[l] ∗ 2 (31) current becomes [47] 288m c⊥ 1 m∗ j = j(0) , (25) The relativistic form of each free-energy contribution above anom + 1 s ρ(0)/ρ m anom arises separately as follows. 1 3 F1 n The term F3 in Eq. (31) describes the universal tempera- 1 s = m∗ − ρ(0) where the Landau parameter 3 F1 m 1 and n ,⊥(T )are ture correction to the Newton gravitational coupling, which the bare thermal quasiparticle densities without Fermi-liquid depends on the number of fermionic species [19]: ˆ corrections, along and perpendicular to l, respectively. They 2 2 c T T √ are given by, see the Appendix A1, R, =− ˆ · ∇×ˆ 2 = − R . F3[ T ] 2 (l ( l)) g (32) 288 c⊥ 144 π 2T 2 7π 4T 4 ρ(0) = ρ, ρ(0) = ρ, (26) n 2 n⊥ 4 Note that being exressed√ in terms of the scalar curvature 0 15 0 2 and metric determinant −g = 1/c c⊥, this term becomes where by Galilean invariance, ρδij = ρsi j + ρni j at all temper- universal: it does not contain the microscopic parameters of ∗ atures. From Eqs. (25) and (26) we gather the system: pF , m, and m . The prefactor is fully determined ∗ by the number of fermionic species. C (T ) p3 p T 2 p T 2 m 0 = F − F − F − 1 + O(T 4), (27) The term F in Eq. (30) is expressed in terms of the 2 2 2 2 2 6π 6c⊥ 12c⊥ m ˆ · = ˆ · − 1 ˆ · ∇×ˆ combination l v l vs 4m l ( l), proportional to the now with the finite temperature and Fermi-liquid corrections ground state current which does not receive corrections due to of Eq. (24) separated, corresponding to the result (18) when Galilean invariance. Here the velocity v = j/ρ is the velocity the Fermi-liquid corrections are ignored m∗/m = 1. of the “total quantum vacuum”, where j ≡ j(T = 0) is the We summarize these findings as follows. While Eq. (23) total vacuum current in Eq. (21)atT = 0. We conclude that with temperature corrections Eq. (24) is not equal to the NY the F2 contribution gives the temperature T and chemical μ anomaly density PNY-node(T )Eq.(19), Panom (T ) represents the potential correction to the free energy of the gas of chiral consistent anomaly [43] vacuum current from all filled states, fermionic particles in the limit |μ|T (see Eqs. (9.12) and which in addition depends on the nonfundamental parameters (10.42) in Ref. [1]): ∗ pF , m, and m via the Fermi-liquid corrections. At the same ∗ 2 pF m T time, we expect that the T 2 contribution to the (covariant) F [μ, T ] =− (4m ˆl · v − ˆl · (∇×ˆl))2 2 m2 2 s NY anomaly arises from contributions close to the linear 96 c⊥ Weyl nodes and should contain fundamental and universal T 2 √ =− −gμ2 , (33) prefactors, as the axial quasiparticle density Eq. (20). 6 Concerning the anomalous vacuum momentum current of where the chiral chemical potential of the superfluid is deter- the superfluid (23), the reason for such parameters is that mined by the Doppler shift there are several different and interrelated T 2 contributions to the current and free energy in the chiral superfluid, and they μR =−μL = pF v · ˆl = pF ˆl · vs − (pF /4m)ˆl · (∇×ˆl). (34)

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The final form of Eq. (33) does not contain microscopic A similar momentum anomaly in anisotropic system with parameters and also gives rise to the mass of “photon” in nonrelativistic symmetries was also considered in Ref. [54]. 3 δF2 ∗ He-A [49].We note that the current | = is the m /m piece For the zero-temperature case, see Refs. [16,36]aswellas δvs vs 0 of the anomalous current (23) and Eq. (24). In principle, the Refs. [40,55] for a relativistic model related to Weyl semimet- chiral chemical potential may also serve as the parameter als. We, however, stress that the universal gravitational NY in the Nieh-Yan term when superflow and curvature terms are anomaly and thermal physics we discussed arise in flat space included at finite temperature, see however Appendix A3for from the geometric background fields in the low-energy quasi- the comparison of the LL model and relativistic fermions with particle Hamiltonian. These tetrads arise universally in all chemical potential. Weyl systems (1) and couple to the momentum [16,17,36,56] Finally F1 in Eq. (29) is the term in the free energy without as in gravity. In particular, the torsional LL density of states is any factors of m∗/m, corresponding to the contribution from momentum dependent, similar to the (gravitational) response the consistent Nieh-Yan thermal anomaly in Eq. (18), with in the chiral vortical and magnetic effects, and leads to Fermi-liquid corrections Eq. (27), the finite temperature anomaly that does not exist for U(1) gauge fields. The connection of our results and the relation δF p T 2 P (T ) = 1 = F ˆl(ˆl · (∇×ˆl)) of the gravitational NY anomaly, with the coefficient γ ,to anom δ 2 vs 12 c⊥ thermal transport in Weyl system should be further elucidated −p n (T )ˆl, (35) [29–31,33,35,41,57]. F 5 Detailed consideration of the temperature dependent where the last formal equality was derived in Eq. (20). anomaly terms in the hydrodynamics of the nonrelativistic Panom (T ) is part of the total anomalous vacuum current with p-wave chiral superfluid with quasirelativistic Weyl fermions Fermi liquid corrections, whereas the second line is from the demonstrates that in the hydrodynamics of this liquid there are quasiparticle (momentum) density due to thermal fluctuations several T 2 terms which can be assigned to different emergent close to the node and is universal. relativistic phenomena, both anomalous and nonanomalous. In particular, we identified and discussed the term in the VI. CONCLUSIONS vacuum momentum corresponding to the (consistent) thermal Nieh-Yan anomaly [43]. This vacuum current depends We discussed the possibility of thermal Nieh-Yan anomaly on nonuniversal parameters with finite-temperature Fermi- where the role of the nonuniversal dimensionful UV cutoff liquid corrections, but, as expected originates from universal is played by the temperature IR scale, and the dimensionless anomalous thermal fluctuations close to the linear nodes with prefactor γ in the anomaly is universal. Such a contribution emergent quasirelativistic torsion with anisotropy. Note that cannot arise for the chiral anomaly for U(1) gauge fields in terms of the superfluid and Weyl fermions, the T = 0 nor the conventional gravitational anomaly higher order in anomalous vacuum contribution to the current can be assigned gradients, explicitly pertaining to the anomaly from torsion in- only to a nonlocal action [58] which hints the existence of stead. In simple pragmatic terms, the temperature dependence corresponding gravitational Wess-Zumino terms with torsion. results from the universal coupling of momentum to tetrads We further showed how the various T 2 low-temperature cor- and the corresponding momentum dependence LL density of rections to the free energy can be grouped and written as low- states. energy relativistic terms, including also chiral chemical poten- We identified a contribution from this anomaly in the tial and scalar curvature terms. They have dimensionless pref- known low-temperature corrections of a nonrelativistic chiral actors, which do not seem to depend on microscopic physics, p-wave Weyl superfluid (or superconductor). In this system but are fully determined by geometry, topology and the num- [1,16,36], the tetrad gravity and space-time with torsion arises ber of fermionic and bosonic quantum fields with thermal as in, e.g., Refs. [50,51] from the Fermi condensate, while the fluctuations. Detailed comparison of the finite temperature anomaly results from thermal effects of the linear Weyl spec- superfluid hydrodynamics with Fermi-liquid corrections to the trum at finite momentum in the presence of an explicit vacuum anomalous quasiparticle axial current production, correspond- of filled quasiparticles, although the end results are similar to ing to the covariant torsional anomaly, in the presence of arbi- relativistic fermions when interpreted carefully in terms of the trary textures and superfluid velocity remains to be identified anisotropy and cutoff of the quasirelativistic Weyl regime. [39,59]. See however Ref. [16] for the zero-temperature case. What we calculated, via the anisotropic Landau level Note added. After the initial submission of this paper as a model with nonrelativistic symmetries in Sec. IV, are actu- preprint, arXiv:1909.08936v1, with the predicted T 2 contri- ally the temperature dependence of the anomalous torsional bution to the NY anomaly, Eqs. (9) and (35), Refs. [40,55,60] σ conductivities T a of the quasiparticle system, i.e., the chiral discussing related torsional anomaly phenomena at finite momentum-density and number densities (at the nodes at temperatures appeared. General aspects of the temperature ± finite momenta pF ) in response to the spacelike torsion anomaly in Weyl materials were further discussed in the −2 = 1 ijk 3T 3 c TB 2 ei jk. Namely, for example, short paper [61]. In particular, the result Eq. (9) has been confirmed in Ref. [40] by a direct calculation of the spectral σT b (T ) a= a 0i = a 0ijkT b = δa3σ , P ei T ei jk TB TB (36) flow of relativistic Landau levels in the presence of a constant 2 3 torsional magnetic field Tμν [17,39,45] at finite temperature. Pi = T 0i, where T μν is the stress tensor, which are of second Here similar computations for the nonrelativistic Weyl super- order in derivates and can be calculated in linear response, fluid in Eqs. (16) and (20) give corresponding results. While e.g., Kubo formulas, to the background tetrads [29,32,52,53]. the current manuscript was being finalized, also the paper [62]

033269-6 THERMAL NIEH-YAN ANOMALY IN WEYL SUPERFLUIDS PHYSICAL REVIEW RESEARCH 2, 033269 (2020) appeared discussing the anomaly for relativistic fermion at and finite temperature and chemical potential. ∞ ρ(0) πT 1 du −(1 − u2 )2 n⊥ = 3/2 3ρ 0 − 4 2 + − 2 ACKNOWLEDGMENTS n=−∞ 1 xn (1 u ) ∞ G.E.V. thanks M. Zubkov for discussions. J.N. thanks Z.- πT 2|x | 1 = 3|x | − n + 1−3x2 arctan M. Huang for correspondence, T. Ojanen for discussions and n + 2 n | | 4 0 =−∞ 1 xn xn S. Laurila for collaboration in Ref. [36]. This work has been n ∞ supported by the European Research Council (ERC) under πT 16 π = − | |3 + 2 − + 5 . the European Union’s Horizon 2020 research and innovation xn 3xn 1 O xn 40 3 2 programme (Grant Agreement No. 694248). n=−∞ ζ = ∞ + s We use the regularizations, a(s) n=0(n a) , APPENDIX: VACUUM REGULARIZATION, ANISOTROPIC ∞ ∞ LANDAU LEVEL MODEL, AND THERMAL INTEGRALS = 1 + 2 = 1 + 2ζ (0) = 0, In this Appendix, we shortly review the low-temperature n=−∞ n=1 corrections to the chiral superfluid and the quasirelativistic ∞ 4πT πT thermal integrals utilized in the main text. Although the |x | = ζ / (−1) = , n 1 2 6 temperature and Fermi-liquid corrections in Appendix A1 n=−∞ 0 0 chiral superfluid are well-known [1,46,47], it is of interest ∞ π 2 2 2 8 T to compare them to the chiral quasirelativistic Landau level x = ζ / (−2) = 0, n 2 1 2 model with torsion and explicit UV completion [16,36,39], n=−∞ 0 Appendix A2, as well to relativistic Weyl fermions with ∞ π 3 3 π 3 3 positive and negative branches [28,30,32], Appendix A3.The 3 16 T 7 T |x | = ζ / (−3) =− . (A4) last system has particles and antiparticles at finite chemical n 3 1 2 603 n=−∞ 0 0 potential counted from the node and finite temperature with vacuum contributions subtracted, in contrast to nonrelativistic These give the result in Eq. (26). Similarly ρsi j (T ) = ρδij − systems at finite chemical potential. ρni j (T ) by Galilean invariance, valid at all temperatures with or without Fermi liquid corrections. From the superfluid and normal densities, the conclusion is 1. Temperature corrections to normal and superfluid density in ∼ T = that only anisotropic momenta of the order of p⊥ pF chiral superfluid 0 T c⊥ T and p = ( ) , c = v contribute in the vicinity of c⊥ c c⊥ F We start with the corrections to the superfluid and normal ∼ density. The anisotropic chiral p-wave superfluid density is the gap nodes to the thermal fluctuations T . ∞ ∂n (E ) ρ(0) = ρ ˆ ˆ − F k 2. Quasirelativistic chiral fermions with torsion ni j 3 d kik j d (A1) −∞ ∂Ek = k kF a. Anisotropic Landau level model βx −1 where√ nF (x) = (e + 1) with β = 1/T and Ek = The quasirelativistic model for the chiral superfluid with 2 +| |2 ˆ k k is the quasiparticle energy with normal state anisotropic dispersion is, with l zˆ, 2 dispersion = k − μ . We compute the anisotropic k 2m F p2 p2 contributions from, with ω = πT (2n + 1), = − μ → ≡ z − μ , n p F pz F (A5) ∞ 2m 2m ∞ ∂ ∞ ω2 − E 2 nF n kF 3 2 d − = d T which is valid for real He-A for E mc , c⊥ ≡ /p . ∂ 2 ⊥ 0 F −∞ Ek −∞ ω2 + p2−p2 p2−p2 kF n=−∞ n EkF = F z F = We approximate (p) 2m 2m (pz ), strictly valid ∞ 2 c⊥ c⊥ 2 −| | when p⊥ ( )pF , p ≡ pz ( ) pF which coincides = πT kF . (A2) c c 2 2 3/2 =± − (ω +| | ) with the linear Weyl regime pz vF (pz pF ). However, n=−∞ n kF we can fix the anisotropic dispersion to (pz ) for all momenta πT The dimensionless summation variable is xn = (2n + 1), to obtain a convenient UV-complete model. The following 0 giving for T 0 analysis for the total vacuum current is valid, as long as the ∞ correct cutoff for the linear or anisotropic Weyl regime is ρ(0) 1 2 2 3 πT −u (1 − u ) maintained for He-A, see Ref. [36]. n = du ρ 3/2 In particular for the case of the anomalous current, only 3 0 =−∞ −1 x2 + (1 − u2 ) n n the momentum along ˆl zˆ contributes justifying the approx- ∞ πT 1 imation to the dispersion and the Landau level (LL) model. = 3|x | − 3x2 + 1 arctan 2 n n |x | Actually the leading contribution is from curvature [16] per- 0 n=−∞ n pendicular to ˆl, requiring the anisotropic cutoffs below EWeyl ∞ πT π in the LL model, discussed already. The results generalizes | | 2 4 = 4 xn − 3x − 1 + O x (A3) otherwise by the NY form (2). Similarly, for thermal fluctua- n 2 n 0 n=−∞ tions in the LL model, we get the correct contributions, as long

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2 c⊥ as the cutoff T 0 and E mc⊥ = 0 are maintained. Similarly, the thermal density of states is the vicinity of the c In particular, this requires that nodes for T 0 is ∞ p(pF + p) → p (pF + p )(A6) =− − μ − μ − z n5(T ) 2 NLL(pz )dpz[nF ( pz F ) ( pz )] 0 ∞ ∼ c⊥ T ∼ T . T c (A7) =− B dxxn (x) c 2c⊥ 2 2 F 2π 0 ∼ T = T = c⊥ T , 2 That is p⊥ pF , p ( ) for both T T 0 c⊥ c c⊥ =− (ˆl ·∇×ˆl), (A12) 2 2 mc⊥. This follow also by the thermal equipartition principle 12c⊥ corresponding to the superfluid results in Appendix A1.For where the T = 0 vacuum contribution is subtracted, since the anomaly, only the gapless chiral lowest√ LLs are relevant, only contribution from the vicinity of the nodes to the chiral while the gapped levels, with gap ∼c⊥ pF TB are particle- hole symmetric and cancel out [39,45]. The dispersion of the density is well-defined. The relevant thermal integral is similar lowest LL becomes E = =−sgn(p T ) (p ). The density of to Eq. (A2) n 0 z B z state per pz per Landau level is ∞ ∞ ∂ 1 2 nF (x) dxxnF (x) = x − (A13) |p T | 4 −∞ ∂x N (p ) = z B , (A8) 0 LL z π 2 π 2 2 4 2 2 T = 4π T ζ / (−1) = . (A14) 1 2 6 where TB = ˆl ·∇×ˆl. The lowest LL is particlelike (holelike) for pW =∓pF ˆl. For more on this anisotropic chiral fermion with torsion and the corresponding Newton-Cartan 3. Relativistic chiral fermions space-time, see Ref. [16] and the forthcoming paper with the Now we compare these results with relativistic Weyl anisotropic LL model [36]. fermions with both positive and negative energy branches, i.e., particles and antiparticles with the node at pz = 0. At chem- b. Vacuum currents ical potential μ at the node, the corresponding momentum ˆ The quasirelativistic integrals for the chiral currents along l is (16),(20) in the main text are given by the lowest Landau ∞ · ˆ = − μ − − level (LLL) contribution, which is particle- and holelike for PW l NLL(pz )dpz pz[nF ( pz ) ( pz )] < > −∞ pz 0 and pz 0, respectively. In the chiral superfluid, the quasiparticles are spin-degenerate Majorana-Weyl excitations T ∞ =− B dp p2[n ( − μ) − n ( + μ)] giving a compensating factor of two. It follows that the 2 z z F pz F pz 4π 0 quasiparticle momentum along the ˆl ≈ zˆ direction is, with the μ ∞ = βx + −1 TB 2 x quasiparticle distribution function nF (x) (e 1) , =− dpz p + 4μ dxxnF 4π 2 z T ∞ 0 0 P · ˆl =−2 N (p )dp p n ( − μ ) μ3 μT 2 qp LL z z z F pz F =− + ˆl ·∇×ˆl, (A15) 0 12π 2 12 2T ∞ =− B dp p2n ( − μ ) 2 z z F pz F single = p 4π 0 per node, in units where pz z. The thermal chiral ∞ density, corresponding to the left-handed node, T pF 4p =− B dp p2 + F dxxn (x) π 2 z z 2 F ∞ 2 0 c⊥ 0 =− − μ − − nW −(T ) NLL(pz )dpz[nF ( pz ) ( pz )] 3 2 −∞ pF pF T =− + ˆl ·∇×ˆl ∞ π 2 2 T 6 6c⊥ =− B dp p [n ( − μ) + n ( + μ)] π 2 z z F pz F pz = vac + qp , 4 0 janom janom (T ) (A9) μ2 T 2 where we have used T μ and the linearization = − − ˆl ·∇×ˆl. (A16) F 8π 2 24 of the spectrum close to μF with the anisotropic scaling 3 (A7) which applies for (weak coupling) He-A. The vacuum In order to compute the integral, we have used nF (x) = 1 − contribution is nF (−x) as well as the linear expansion close to the nodes ∞ for T 0. Intriguingly, the structure of this momentum- vac TB 2 j =− dpz p (pF − pz ), (A10) density and density contribution are exactly of the same anom 2π 2 z 0 form [28,30,32,37,38]. The difference is, of course, that μ whereas the quasiparticle current due to the thermal fluctua- is counted from the node and has different sign for positive tions is energy solutions corresponding to the antiparticles and that the contribution of filled vacuum states is subtracted. Lastly, p T 2 qp =− F ˆ ˆ ·∇×ˆ . we note the dissimilarity of these relativistic finite chiral janom (T ) 2 l(l l) (A11) 6c⊥ chemical potential and temperature terms to Eqs. (32)–(34).

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