Non-equilibrium Chiral Magnetic/Vortical Effects in Viscous Fluids

We utilize the chiral kinetic theory in a relaxation-time approximation to investigate the nonlinear anomalous responses of chiral fluids with viscous effects. Unlike the cases in equilibrium, it is found that the chiral magnetic effect and chiral vortical effect are modified by the shear and bulk strengths. Particularly, the shear strength could result in charged Hall currents for chiral magnetic and chiral vortical effects, which propagate perpendicular to applied magnetic fields and vorticity. These quantum corrections stemming from side jumps and anomalies are dissipative and pertinent to interactions. Although the non-equilibrium effects upon charge currents are dissipative, the second law of thermodynamics is still satisfied.

We utilize the chiral kinetic theory in a relaxation-time approximation to investigate the nonlinear anomalous responses of chiral fluids with viscous effects. Unlike the cases in equilibrium, it is found that the chiral magnetic effect and chiral vortical effect are modified by the shear and bulk strengths. Particularly, the shear strength could result in charged Hall currents for chiral magnetic and chiral vortical effects, which propagate perpendicular to applied magnetic fields and vorticity. These quantum corrections stemming from side jumps and anomalies are dissipative and pertinent to interactions. Although the non-equilibrium effects upon charge currents are dissipative, the second law of thermodynamics is still satisfied.
It is generally believed that CME in equilibrium is protected by the chiral anomaly and unaffected by interactions. On the other hand, CVE in equilibrium could be protected by mixed-axial-gravitational anomaly with only background fields [6,33,37], whereas the modification from interactions could emerge in the presence of dynamical gauge fields [38][39][40]. See Refs. [41][42][43] for some following works. In addition, there are studies for the axial currents induced by electric fields and chiral imbalance such as the chiral electric separation effect (CESE) or chiral Hall effect (CHE), which are pertinent to interactions [44][45][46], while these effects are not directly connected to the helicity of Weyl fermions and quantum anomalies. Nonetheless, an interplay between the chiral anomaly, magnetization current stemming from side jumps, and magnetic-moment coupling could result in non-equilibrium corrections involving interactions upon anomalous transport. For example, the alternativecurrent (AC) conductivity for CME with nonzero frequency is modified in the presence of time-dependent magnetic fields [5,14,47]. Also, the nonlinear responses on anomalous transport triggered by fluctuations near local equilibrium in inviscid chiral fluids have been analyzed by some of the authors in this paper [48], where novel anomalous Hall currents led by electric fields and temperature/chemical-potential gradients are found [49]. However, the non-equilibrium corrections on charge currents from magnetic fields or vorticity may appear in the viscous case [50]. Experimentally, in HIC, due to the viscous corrections in the quark gluon plasma, the non-equilibrium effects upon the CME conductivity may impact the charge separation associated with the signal for CME. Moreover, based on recent observations of hydrodynamic transport in graphene [51,52], there exists mounting interest in the possible realization of chiral fluids in Weyl semimetals (see e.g., Refs. [53,54] for relevant studies in theory). The theoretical investigation of relativistic viscous hydrodynamics of chiral fluids is thus imperative.
In this paper, we follow the approach in Ref. [48] to further study the viscous corrections on anomalous transport contributed by the aforementioned quantum effects associated with the helicity of Weyl fermions. It is found that the CME and CVE conductivities of charge currents are modified by shear and bulk strengths. Nevertheless, the non-equilibrium corrections on the energydensity currents should vanish according to the matching condition in the classical relaxation-time approximation (RTA). Although the non-equilibrium quantum corrections are dissipative, the second law of thermodynamics is still satisfied.

II. WIGNER FUNCTIONS AND CHIRAL KINETIC THEORY
We begin with a brief introduction to the Wignerfunction formalism of CKT which will be exploited to study the non-equilibrium transport. As derived in Ref. [36] by solving Dirac equations up to O( ) from the Wigner-function approach, the perturbative solution for the lesser propagators of right-handed Weyl fermions readsS <µ (q, X) = 2πǭ(q · n) q µ δ(q 2 ) + δ(q 2 )S µν (n) D ν whereǭ(q · n) represents the sign of q · n and S µν (n) = ǫ µναβ 2(q · n) q α n β (2) corresponds to the spin tensor depending on a frame vector n µ [35]. The frame vector can be understood as the zeroth component of a vierbein for the local transformation of σ a = (I, σ i ) introduced on the local tangent space to σ µ (X) depending on the global spacetime coordinates, where σ i represent Pauli matrices for i = 1, 2, 3, such that σ µ = e µ a σ a . That is, we define n µ ≡ e µ 0 with µ being the spacetime indices (See Appendix A for details of the choice of the frame vector). In flat spacetime, we may set e µ a (X) = δ µ a such that the spin connection vanishes. Now, the global spacetime coordinate transformation corresponds to the frame transformation n µ → n ′µ .
Here we denote D β f being less/greater self-energies and f being the distribution functions of incoming and outgoing particles, respectively. The second term in Eq. (1) associated with S µν (n) as the side-jump term only contributes for the non-equilibrium cases or for a rotating system in global equilibrium, which results in magnetization currents and CVE. On the other hand, the third term in Eq. (1) yields CME in equilibrium.
Given the Wigner functions, one can directly evaluate the charge current and energy-momentum tensor through The distribution function inS <µ has to be solved from the CKT led by the Dirac equation. In light of the study in Ref. [48], we focus on the fluctuations slightly away from the local equilibrium distribution function defined in a comoving frame n µ = u µ with u µ being a fluid velocity.
For an arbitrary frame n µ , the CKT takes the form [48], and Here, we define B µ and E µ by decomposing the field strength into F αβ = −ǫ µναβ B µ n ν + n β E α − n α E β . We will then work in n µ = u µ . The collision term depends on details of the system, so that it is not universal. Here, for simplicity and generality to be applicable for various systems including e.g., HIC or Weyl semimetals and also to make comparisons with previous studies [55,56] on equal footing, we apply the RTA for the collisional kernel in order to capture the qualitative features based on the symmetric properties of collisions. We thus approximate where A µ denotes an operator acting on δf q , which represents possible quantum corrections with O(∂) that depends on the details of collision terms. τ R is the relaxation time charactering the inverse strength of interactions, which will be treated as a constant [57] . Note that here A µ is also frame dependent. The transformation of A µ in different frames should preserve the frame independence (Lorentz invariance) of the CKT with the RTA. We introduce such a frame transformation upon A µ in Appendix B.
We will then follow the computations in Ref. [48] to perturbatively solve for the non-equilibrium distribution function δf q = f (u) q − f eq q incorporating viscous corrections from the CKT in Eq. (4) based on the and derivatives expansions. Here the local equilibrium distribution function takes the form f eq q = (e g + 1) −1 with g = (q ·u − µ+ (ω ·q)(2q ·u) −1 )/T for T and µ being the local temperature and chemical potential, respectively. Also, the vorticity ω µ is defined as ω µ ≡ ǫ µναβ u ν ∂ α u β /2.
Hydrodynamics and matching conditions.-Furthermore, following the charge and energy-momentum conservation with the chiral anomaly, we should also implement the anomalous hydrodynamic EOM led by These two equations provide the physical constrains for CKT, which dictate the dynamics of thermodynamic parameters T , µ, and u µ in f eq . However, by utilizing the equation ∆ µS <µ = Σ < µS >µ − Σ > µS <µ as the origin of the CKT, it is shown in Ref. [48] that the divergence of currents manifests the chiral anomaly, where we denotes q = d 4 q (2π) 3ǭ (q · n). Moreover, performing similar computations as in the case for ∂ µ J µ , the divergence of the energy-momentum gives rise to The detailed derivation is shown in Appendix C. For collisions in practical systems obeying charge and energymomentum conservation, the collisional terms in Eqs. (9) and (10) should automatically vanish. In contrast, in the RTA, the charge and energy-momentum conservations give matching conditions. In particular, when A µ = 0, we find that Eqs. (9) and (10) become In this case, the matching conditions turns to the standard ones, u µ δJ µ = u µ δT µν = 0. The vanishing non-equilibrium modifications upon charge density and energy-density current consequently allow us to consistently define the temperature and chemical potential in equilibrium as alternatively indicated in Ref. [58]. Taking other approximations could result in distinct conserved quantities without consistent physical interpretations.

III. NON-EQUILIBRIUM RESPONSES AND CHARGE CURRENTS
Following Ref. [48], the perturbative solution for the non-equilibrium distribution function is given by where A µ is taken to be a constant here. We may make the decomposition, δf q = δf q , where the superindices (c) and (Q) correspond to the classical and quantum corrections, respectively. We then further separate the part for ideal fluids, and the viscous correction, , where the subindices I and v denote the inviscid and viscous parts. The explicit expression of δ I f q can be found in Ref. [48]. In light of Ref. [48], we decompose the quantum corrections of the non-equilibrium distribution function into three parts as δ v f where δ v f K q is led by the perturbative solution out of equilibrium solved from CKT and δ v f H q is attributed to the corrections of the temporal derivatives (u · ∂) upon T ,μ ≡ µ/T , and u µ from hydrodynamic EOM obtained from Eq. (8). Finally, δ v f C q comes from the corrections in the collisional kernel, while this term depends on the assumption of A µ in the RT approximation, which does not play a significant role in our analysis. In Ref. [48] for an inviscid case, A µ is treated as a constant. We will follow the same convention in the computation of viscous corrections for consistency although A µ can be an operator governed by the frame transformation. On the other hand, as discussed previously, we may set A µ = 0 in the co-moving frame for self-consistency to introduce local equilibrium thermodynamical parameters suggested by the matching conditions.
Next, we shall consider the viscous corrections on the hydrodynamic EOM. By solving Eq. (8) with the current and energy-momentum tensor in local equilibrium, it is found that DT = −θ/3 + O( ) and Dμ = O( ) with µ ≡ µ/T , while the viscous correction does not lead to corrections for DT and Dμ. We also include their contributions for the computation of currents. In addition, we find that the viscous correction gives rise to the correction upon D v u µ ⊥ and accordingly the hydrodynamic EOM results in whereŨ B = −T 2 (3μ 2 + π 2 )/(48pπ 2 ) andŨ ω = T 2μ μ 2 + π 2 /(12pπ 2 ) with p being pressure. On the other hand, the corrections in collisions give From Eqs. (1) and (3), the quantum corrections of the non-equilibrium current reads By inserting δf into Eq. (17), we obtain the quantum correction upon the charge current, δJ µ Q⊥ = δJ µ IQ⊥ + δJ µ vQ⊥ , where δJ µ IQ⊥ as the part for ideal fluids is shown in Ref. [48]. The viscous part takes the form When not applying the hydrodynamic EOM, only δ v f K q contributes and one finds where I 1 =μ 2 + π 2 /3. The results suggest that the viscous corrections upon CME and CVE conductivities should exist even for an open system in which the backreaction on environments is neglected and the energymomentum conservation is violated such as the case in Weyl semimetals when the scattering between quasiparticles and impurities dominate the interactions among quasi-particles. Now, for the right-handed chiral fluid as a closed systems with energy-momentum conservation, by implementing the hydrodynamic EOM, the coefficients become where I 2 =μ μ 2 + π 2 . Note that there exist two terms in δf and − τ R (2q 0 ) −2 (q ·∂)(q ·ω), which also contribute to viscous corrections with vorticity when the hydrodynamic EOM are applied in computations. In general, by redefining u µ , one can shift the above corrections on CME/CVE conductivities to the transport coefficients of energy density currents. It is more enlightening to simplify the expressions of δσ µν B and δσ µν ω in distinct limits. In the hightemperature limit (μ ≪ 1), the coefficients in Eq. (18) reduce to On the contrary, in the low-temperature limit (μ ≫ 1), one obtains Despite the complexity of computations, it is worthwhile to note that such viscous corrections actually originate from Bianchi identities ∂ νF µν = ∂ νω µν = 0, wherẽ ω µν = 1 2 ǫ µναβ ∂ α u β , which relate the temporal derivatives of B µ and ω µ (in the fluid co-moving frame) to their couplings with the gradients of u µ as shown in Ref. [48]. Such an origin is a reminiscence of the AC conductivity of CME driven by time-dependent B µ and the viscous corrections upon CME/CVE are expected to be dissipative.
So far, we have only considered the contributions for right-handed fermions. The quantum corrections for lefthanded fermions will yield the same results but with the change of an overall sign for each term at O( ). Since P µν θ and π µν are even under the parity (P) transformation, it is anticipated that the transport coefficients of the non-equilibrium corrections on CME/CVE have the same parity as those in equilibrium, which can be more apparently observed from the simplified expressions shown in two limits above. Therefore, the bulk and shear strengths not only affect the vector currents (J µ V = J µ R + J µ L ) induced by CME/CVE with nonzero axial-charge chemical potentials (µ A = µ R − µ L ) but also the axial currents (J µ A = J µ R − J µ L ) from the chiral separation effect(CSE) [59] and CVE with nonzero vector-charge chemical potentials (µ V = µ R + µ L ). It is worthwhile to note that such second-order quantum corrections on currents have different symmetry properties compared to the second-order classical effects also pertinent to magnetic fields such as the Hall-diffusion currents J µ Ref. [56]. Because there is no sign flipping for right/lefthanded fermions in the classical case, the corresponding axial current can only be generated when µ A = 0(or ∂ ⊥β µ A = 0), which is analogous to CESE [44,45].

IV. ENTROPY PRODUCTION
In contrast to the anomalous transport in equilibrium, the non-equilibrium quantum corrections of the charge current δJ µ Q are dissipative. This is foreseen by the time-reversal symmetry (T ). Since the charge current J µ ⊥ , B µ , and ω µ are T -odd and E µ is T -even, from the classical ohmic current led by the RT approximation J µ ⊥ ∝ τ R E µ , one finds that τ R is also T -odd, whereas the CME/CVE conductivities in equilibrium are T -even and thus non-dissipative. Nonetheless, because θP µν and π µν are both T -odd, the corresponding transport coefficients of the viscous corrections upon chiral magnetic/vortical currents are proportional to τ R multiplied by T -even functions of T and µ, which accordingly yield dissipation. The same arguments can be applied to the non-equilibrium transport found in inviscid cases [48]. Notably, in light of symmetry, one finds A µ ∼ B µ or A µ ∼ ω µ for viscous corrections. Consequently, the corrections in practical collisions can only affect the prefactors of transport coefficients without altering their structures. Overall, the non-equilibrium second-order quantum transport is parity-odd and dissipative as opposed to the classical one (e.g., classical Hall effects), which is parity even and non-dissipative.
Albeit the non-equilibrium quantum transport results in dissipation, its entropy production is suppressed by the dissipation from classical effects as shown below. We may introduce the entropy density current in an usual form as for relativistic hydrodynamics, where the non-dissipative corrections proportional to B µ and ω µ originating from the non-dissipative charge and energy-density currents. The coefficients D B and D ω should be determined by the transport coefficients of CME and CVE in equilibrium. See e.g., Ref. [18] for details. The explicit form of D B(ω) is not important in our discussion. The constitutive relation now can be written as T µν = u µ u ν ǫ − pP µν + Π µν dis + Π µν non , where Π µν dis denotes the dissipative corrections characterized by viscous effects and Π µν non corresponds to the non-dissipative corrections led by anomalous effects in equilibrium. When A µ = 0, based on the matching conditions such that u µ δJ µ = u µ δT µν = 0, we find . Although the non-equilibrium quantum corrections here could be either positive or negative, they are always suppressed by the classical contributions and the second law of thermodynamics is satisfied.

V. DISCUSSIONS AND OUTLOOK
Regarding the validity of our findings, due to the gradient expansion, the results should be legitimate for τ R ∂ ≪ 1, which imparts an upper bound for τ R . In addition, given that the CKT itself is subject to weakly coupled systems, τ R cannot be too small. More generally, although the CKT is developed to cope with nonequilibrium conditions, it is applicable for small gradients and weak background fields based on the expansion. To extend the validity beyond such limits, we have to solve for Wigner functions non-perturbatively from Kadanoff-Baym equations (derived from Schwinger-Dyson equations), which could be a formidable analytic problem.
Nonetheless, we can still perturbatively solve for higher-order corrections in the expansion (see Ref. [60] for a relevant study). Alternatively, one may also construct Wigner functions for non-equilibrium situations directly from Landau-level wave functions involving allorder corrections, while it is subject to the case with just constant magnetic fields. Both approaches are systematic derivations of CKT from quantum field theory, which also complement each other. Up to O( 2 ), it is anticipated that novel non-dissipative quantum effects such as the charge modification led by ω · B discovered in Ref. [61] will be found. Furthermore, for more general cases, one may include dynamical gauge fields, which could yield chiral-plasma instabilities [62] and more profound phenomena. Such instabilities can be treated in our hydrodynamics approach if the time and length scales of the instabilities are much larger than the mean free time and length. The typical time and length scales of the instabilities are evaluated as τ inst ∼ τ R m 2 D ℓ 2 inst and ℓ inst ∼ 1/(αµ A ), where α = e 2 /(4π) and m D is the Debye mass, while the mean free time and path are of order τ R [62]. Therefore, τ inst ≫ τ R and ℓ inst ≫ τ R lead to conditions m 2 D ≫ (αµ A ) 2 and τ R ≪ 1/(αµ A ). The former is satisfied at high temperature T ≫ √ αµ A or density The latter depends on the strength of interaction. For example, if we evaluate τ R in a quark gluon plasma, it behaves τ R ∼ α s T ln α −1 s , where α s is the strong coupling constant; thus, the condition reduces to T ≫ αµ A /(α s ln α −1 s ). If the opposite limit is realized, we have to employ the chiral kinetic equation to describe the dynamics of chiral kinetic instabilities.
By parameterizingS < =σ µS< µ , the above equation yield the difference equations, where [A, B] = AB − BA and {A, B} = AB + BA and We should now perturbatively solve forS < µ from Eq. (A3) up to O( ). We thus make the ansatz,S < µ =S It is easy to find that which is independent of a choice for the basis of spins.
When taking n µ = u µ and using the RTA, one finds that Eq. (B4) reduces to which is independent of collisions. Note that the equation above also works for δf which gives the frame transformation upon the collisional terms in a RTA.