Nonlinear Responses of Chiral Fluids from Kinetic Theory

# Nonlinear Responses of Chiral Fluids from Kinetic Theory

Yoshimasa Hidaka, Shi Pu, Di-Lun Yang Theoretical Research Division, Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan.
iTHEMS Program, RIKEN, Wako, Saitama 351-0198, Japan.
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
###### Abstract

The second-order nonlinear responses of inviscid chiral fluids near local equilibrium are investigated by applying the chiral kinetic theory (CKT) incorporating side-jump effects. It is shown that the local equilibrium distribution function can be non-trivially introduced in a co-moving frame with respect to the fluid velocity when the quantum corrections in collisions are involved. For the study of anomalous transport, contributions from both quantum corrections in anomalous hydrodynamic equations of motion and those from the CKT and Wigner functions are considered under the relaxation-time (RT) approximation, which result in anomalous charge Hall currents propagating along the cross product of the background electric field and the temperature (or chemical-potential) gradient and of the temperature and chemical-potential gradients. On the other hand, the nonlinear quantum correction on the charge density vanishes in the classical RT approximation, which in fact satisfies the matching condition given by the anomalous equation obtained from the CKT.

Chiral Kinetic Theory, Chiral Anomalies, Weyl Fermions, Chiral Fluids
preprint: RIKEN-QHP-260, RIKEN-STAMP-36

## I Introduction

In recent years, there have been mounting interests in the transport of relativistic Weyl fermions in both nuclear physics and condensed matter systems. One of the renown examples is the study of chiral magnetic effect (CME) associated with the chiral anomaly, from which a vector-charge current is generated along magnetic fields for Weyl fermions, , where represents the CME conductivity characterized by chiral imbalance and the coefficient of chiral anomaly Vilenkin (1980). Such an effect draws much attention for the research in heavy ion collisions, where a system with approximated chiral symmetry could be realized in quark gluon plasmas (QGP). In addition, a strong magnetic field generated by colliding nuclei and the local chiral imbalance stemming from topological excitations makes heavy ion collisions a suitable testing ground for CME Kharzeev et al. (2008); Fukushima et al. (2008); Kharzeev and Warringa (2009); Kharzeev et al. (2016). On the other hand, in Weyl semimetals, the quasi-particles at Weyl nodes mimic relativistic Weyl fermions. By pumping a nonzero axial chemical potential through applied electric and magnetic fields parallel to each other, longitudinal negative magneto-resistance associated with CME has been recently observed Li et al. (2014). Moreover, not only magnetic fields but also vorticity could trigger an anomalous current, known as the chiral vortical effect (CVE) Vilenkin (1979). As the counter part of vector currents, both magnetic fields and vorticity could also induce axial currents, where the former case is dubbed as chiral separation effect (CSE) Fukushima et al. (2008). The interplay between CME and CSE could further yield propagating charge-density waves Kharzeev and Yee (2011a), which only rely on local fluctuations of vector and axial charges and hence exist even in the absence of net vector/axial chemical potentials. Such charge-density waves known as chiral magnetic waves (CMW) could lead to potentially measurable observables in heavy ion collisions Burnier et al. (2011). There are also some studies for axial currents induced by electric fields via interactions Huang and Liao (2013); Pu et al. (2014, 2015).

On the theoretical side, these quantum effects particularly for CME/CVE associated with quantum anomalies have been investigated from various approaches including field theories based on Kubo formula Fukushima et al. (2008); Kharzeev and Warringa (2009); Landsteiner et al. (2011a), kinetic theory Gao et al. (2012); Son and Yamamoto (2012); Stephanov and Yin (2012); Son and Yamamoto (2013); Chen et al. (2013); Manuel and Torres-Rincon (2014a, b); Kharzeev et al. (2017); Huang et al. (2017); Gao et al. (2017), relativistic hydrodynamics Son and Surowka (2009); Neiman and Oz (2011); Sadofyev and Isachenkov (2011); Pu et al. (2011); Kharzeev and Yee (2011b), lattice simulations Abramczyk et al. (2009); Buividovich et al. (2009a, b, 2010); Yamamoto (2011); Müller et al. (2016); Mace et al. (2017), and gauge/gravity duality Erdmenger et al. (2009); Torabian and Yee (2009); Banerjee et al. (2011); Landsteiner et al. (2011b). Peculiarly, recent progress in chiral kinetic theory (CKT) with the manifestation of Lorenz symmetry and the incorporation of collisions has facilitated our understandings on anomalous transport out of equilibrium Chen et al. (2014, 2015); Hidaka et al. (2017). It is found that the presence of side-jump terms in Wigner functions or equivalently the modified Lorentz transformation of distribution functions pertinent to side-jump phenomena of Weyl fermions is crucial for Lorentz covariance and the contribution to CVE. Regarding the Lorentz covariance of CKT, see Refs. Chen et al. (2013); Gao et al. (2017) for the derivation through Wigner functions of relativistic fluids near equilibrium and Refs. Mueller and Venugopalan (2017a, b) for a different approach by using the world-line formalism.

Although it is generally believed that the linear response such as CME conductivity is protected by chiral anomaly and independent of the interactions, the nonlinear responses of Weyl fermions could be affected by interactions. Moreover, it has been recently pointed out that the frequency dependent CME conductivity could be modified by collisions, the energy shift from the magnetic-moment coupling, and magnetization currents associated with side jumps in the non-equilibrium case Kharzeev and Warringa (2009); Satow and Yee (2014); Kharzeev et al. (2017). On the other hand, the coupling independence of the CVE coefficient is in general under debate111For one of the CVE coefficients only depending on temperature and contributing to axial currents, it is proposed that such a coefficient could be protected by mixed-axial-gravitational anomaly Landsteiner et al. (2011b, a), while it is found there exists an exceptional case Hou et al. (2012); Golkar and Son (2015). See Refs.Chowdhury and David (2015); Golkar and Sethi (2016); Chowdhury and David (2016) for some following works.. Similar to the case for background fields, the nonlinear responses involving vorticity may be influenced by interactions as well. In anomalous hydrodynamics, one could classify the possible second-order corrections related to anomalies based on symmetries and thermodynamics Kharzeev and Yee (2011b). However, it is useful to utilize microscopic theories such as CKT to obtain such coefficients and analyze their dependence on thermodynamical properties.The second-order nonlinear transport in Weyl-fermion systems have been recently studied with CKT in Refs. Gorbar et al. (2016); Chen et al. (2016); Gorbar et al. (2017a). Nonetheless, these studies aim at systems in the absence of collective motion for quasi-particles, which are applicable for Weyl semimetals such that the interaction among quasi-particles is suppressed by their collisions with impurities and phonons. In such cases, the energy-momentum conservation could be violated when neglecting the backreaction upon environments. On the contrary, in QGP, the fluid-like properties of Weyl fermions should be taken due to strong coupling and hydrodynamics impose the energy-momentum conservation. Although in reality the QGP coupling could be too strong for the legitimacy of a kinetic-theory description, there exists a temporal window in early times of heavy ion collisions such that the kinetic theory is applicable to delineate the collective motion of quasi-particles, e.g., see Ref. Ebihara et al. (2017) for the boost invariant formation of CKT and the so-called chiral circular displacement, and Refs. Sun et al. (2016); Huang et al. (2017) for the very recent numerical simulations of CKT in heavy ion collisions.

Furthermore, even in Weyl semimetals, the strongly interacting quasi-relativistic plasma could be possibly realized and the hydrodynamics of Weyl fermions should be considered. See e.g., Ref. Lucas et al. (2016) for such a discussion and the references therein for Dirac fluids in graphene. Therefore, in the comparison with the studies Gorbar et al. (2016); Chen et al. (2016), it is also of interest to consider the nonlinear response of CKT under the constraint of hydrodynamics.

In this paper, we investigate second-order nonlinear responses in a chiral fluid with background fields and vorticity by employing the CKT derived from quantum field theories Hidaka et al. (2017). Due to the involvement of side-jump terms in collisions, it is nontrivial to show the definition of local equilibrium distributions functions in a proper frame. We tackle this issue first and then focus on nonlinear responses with respect to local fluctuations away from equilibrium led by background fields and local temperature/chemical-potential gradients for right-handed Weyl fermions. In the following, we briefly mention our strategy and summarize some important findings. To solve for the corresponding nonlinear responses perturbatively in gradient expansions, we first apply continuity equations to derive anomalous hydrodynamic equations of motion (EOM) given the first-order transport coefficients obtained from equilibrium Wigner functions. Next, we implement CKT to obtain the non-equilibrium corrections of distributions functions in aid of anomalous hydrodynamic EOM. Given the non-equilibrium distribution functions, we directly compute the charge current and charge density defined by Wigner functions. For simplicity, we neglect viscous corrections and utilize the relaxation-time (RT) approximation for the study of nonlinear responses. Through the paper, we refer quantum corrections to the corrections at in the Wigner-function approach, which originate from the spin of Weyl fermions and anomalies, e.g., some nonlinear responses are from combination of the side-jump and anomalous hydrodynamic transports. The higher-order corrections in are beyond the scope of this paper. To give a quick view and simple explanation, here we summarize our findings in short:

1. When quantum corrections from side jumps in collisions are considered, the local equilibrium distribution functions can be defined in the frame in accordance to the fluid velocity.

2. Without applying anomalous hydrodynamic EOM, which corresponds to a system breaking energy-momentum conservation due to the interaction with the environment, the quantum corrections of the second-order responses for charge currents under a “naive” RT approximation give the terms proportional to and , which agree with the findings in Chen et al. (2016); Gorbar et al. (2016). Here and denote an electric field and a chemical potential.

3. By using anomalous hydrodynamic EOM, which corresponds to an isolated system, the transport coefficients for the and terms in charge currents are modified. Furthermore, the and terms emerge from hydrodynamics, where denotes temperature.

4. For an inviscid chiral fluid, vorticity does not affect the nonlinear responses in charge currents up to the second order.

5. Except for the implicit quantum corrections from collisions, the nonlinear quantum correction on charge density vanishes with hydrodynamic EOM in the RT approximation. The result is consistent with the matching condition from the anomalous equation.

We note that a relevant study was previous presented in Ref. Gorbar et al. (2017b), whereas the background fields were not included and the subtlety of local equilibrium stemming from side jumps was not discussed therein.

The paper is organized as following : In Sec. II, we investigate the interacting Weyl fermions in local equilibrium from the Wigner-function approach. In Sec. III, we work out the non-equilibrium distribution functions involving the second-order quantum corrections for an inviscid chiral fluid by using CKT and hydrodynamic EOM. In Sec. IV, we implement the non-equilibrium distribution functions to compute the second-order quantum corrections for the charge current and density. We also analyze the corresponding vector/axial currents in high-temperature and large-chemical-potential limits. In the beginning of each section above, we briefly explain our strategy and highlight the key equations and findings, which could be helpful for readers who are not interested in the details of computations. In Sec. V, we make brief discussions and outlook. For reference, we include the conventions and some widely-used relations in Appendices despite some overlap with the context. We also present the details of some calculations therein.

Throughout this work, we have choose the metric . Therefore, the fluid velocity satisfying , and the projector is given by . We also use the Levi-Civita symbol and choose .

## Ii Local Equilibrium Wigner Functions

Before working on nonlinear responses away from equilibrium, first we would like to review the chiral kinetic theory with side-jump based on the Wigner-function approach in Sec. II.1. Because of side jumps, the distribution function becomes frame dependent stemming from corrections. Therefore, we need to define the local-equilibrium distribution function in a proper way. Consequently, we review the well-defined case in global equilibrium in Sec. II.2, where the explicit form of the global-equilibrium distribution function for an arbitrary frame is shown in Eq. (12). Then we further investigate the case in local equilibrium in Sec. II.3. It turns out that the local-equilibrium distribution function can be introduced in a co-moving frame with the form in Eq. (12) such that the collisional kernel in CKT vanishes for at least 2 to 2 scattering. The explicit expression of the corresponding Wigner function in local equilibrium is presented in Eq. (21).

### ii.1 Wigner Functions and Chiral Kinetic Theory

Wigner functions are defined as the Wigner transformation of lesser/greater propagators,

 S<(>)(q,X)≡∫d4Yeiq⋅YℏS<(>)(x,y), (1)

where and as the expectation values of fermionic correlators with and . Here the gauge link is implicitly embedded to keep gauge invariance and hence denotes the canonical momentum. As shown in Ref. Hidaka et al. (2017), by solving Dirac equations up to , the perturbative solution for the less propagators of right-handed Weyl fermions is given by

 S<(q,X) = ¯σμS<μ(q,X) (2) = ¯σμ2π¯ϵ(q⋅n)(qμδ(q2)f(n)q+ℏδ(q2)Sμν(n)Dνf(n)q+ℏϵμναβqνFαβ∂δ(q2)2∂q2f(n)q),

where represents the sign of , are the spin matrices, and

 Sμν(n)=ϵμναβ2(q⋅n)qαnβ (3)

denotes the spin tensor depending on a frame vector . The choice of a frame corresponds to the choice of an observer and as a timelike vector represents the four velocity of this observer. Here we denote , where , with being less/greater self-energies and and being the distribution functions of incoming and outgoing particles, respectively. In a general case, the distribution functions here are frame dependent, which follows the modified Lorentz transformation between frames222The transformation between different frames here is equivalent to the inverse Lorentz transformation of and . Due to the side-jump term associated with in Wigner functions, the distribution function is no longer a scalar, which thus undergoes the non-trivial frame transformation or equivalently the modified Lorentz transformation upon phase-space coordinates. One may refer to Ref. Hidaka et al. (2017) for more details. As we will discuss later, the explicit expression of distribution functions sometimes may have to be introduced in a particular frame.

 f(n′)q=f(n)q+ℏϵνμαβqαn′βnμ2(q⋅u)(q⋅n′)Dνf(n)q. (4)

Note that both the energy-momentum tensor and currents can be directly obtained from Wigner functions333When performing the explicit computations of currents and energy-momentum tensors from Wigner functions, one actually takes normal ordering and drops infinite constants coming from the anti-commutation relation of fermions.,

 Tμν=∫d4q(2π)4[qμS<ν+qνS<μ],Jμ=2∫d4q(2π)4S<μ. (5)

In Ref. Hidaka et al. (2017), is chosen to be independent of and except for the part inside collisional kernel, which corresponds to the choice of a global observer in the lab frame. However, when choosing a local observer, could depend on spacetime coordinates and the CKT has to be modified. Assuming the frame vector only depends on , we find

 ΔμS<μ = ∂μS<μ+Fνμ∂νqS<μ (6) = 2π¯ϵ(q⋅n){δ(q2)q⋅Δ+ℏδ(q2)[Fρμ(∂ρqSμν(n))Dν+Sμν(n)(∂μFρν)∂ρq+(∂μSμν(n))Dν] +ℏ∂δ(q2)∂q2[2qρFμρSμν(n)Dν+12ϵμναβqνFαβΔμ]}f(n)q

and

 Σ<μS>μ−Σ>μS<μ=2π¯ϵ(q⋅n)(δ(q2)q⋅C+ℏϵμναβCμqνFαβ∂δ(q2)2∂q2). (7)

From , carrying out similar computations as in Ref. Hidaka et al. (2017), the corresponding CKT takes the form,

 δ(q2−ℏB⋅qq⋅n){[q⋅Δ+ℏSμν(n)Eμ(q⋅n)Dν+ℏSμν(n)(∂μFρν)∂ρq+ℏ^Π(n)(q,X)]f(n)q−q⋅C}=0, (8)

where

 ^Π(n)(q,X)=(∂μSμν(n))Dν=ϵνμαβ2q⋅n(qα(∂μnν)−nνqαqρ(∂μnρ)q⋅n)Dβ (9)

comes from the choice of a local observer. Here the electromagnetic fields are defined thorough the frame vector ,

 nνFμν=Eμ,12ϵμναβnνFαβ=Bμ,Fαβ=−ϵμναβBμnν+nβEα−nαEβ. (10)

Note that implicitly incorporates corrections since it contains at least one internal line of Weyl fermions in the self-energy, which is true for most of realistic scattering processes, and the side-jump term will in general be involved. One can alternatively write the CKT as

 (11)

When taking and using the on-shell condition, the CKT reduces to the usual three-momentum form in Refs. Son and Yamamoto (2013); Hidaka et al. (2017).

### ii.2 Global Equilibrium Cases

It is shown from the semi-classical approach that a global equilibrium distribution function of a rotating Weyl fluid could be defined frame-independently Chen et al. (2015). We shall first present an equivalent description in Wigner functions and discuss an obstacle for the generalization to local equilibrium. Following the definition in Ref. Chen et al. (2015), we take the distribution function of right-handed fermions as

 feq(n)q=(eg+1)−1,g=⎛⎝βq⋅u−¯μ+ℏSμν(n)2∂μ(βuν)⎞⎠, (12)

where is the inverse of temperature , with the charge chemical potential, and represents the fluid velocity. In our further calculations, we also often use the ordinary distribution function without side jumps,

 f(0)q=(eg0+1)−1,g0=βq⋅u−¯μ. (13)

We shall find that Eq. (12) gives rise to the distribution functions in global equilibrium with constant and . For general conditions, we may decompose the derivative of into symmetric/anti-symmetric parts , where and . By introducing the fluid vorticity

 ωμ≡12ϵμναβuν(∂αuβ), (14)

we may further rewrite the anti-symmetric part as

 ωαβ=−ϵαβμνωμuν+καβ,καβ=12(uαu⋅∂uβ−uβu⋅∂uα), (15)

and the dual tensor

 ~ωμν=12ϵμναβωαβ=12ϵμναβuαu⋅∂uβ+ωμuν−ωνuμ. (16)

By inserting Eq. (12) into Eq. (2) and using the relation

 Sμν(n)qρωνρ+Sρν(n)2qμωρν=−~ωμαqα2,

we obtain

 S<μ = −ℏqμSρν(n)uν2T2(∂ρT)∂q⋅u−ℏSμν(n)~Eν∂q⋅u)+ℏϵμναβqνFαβ∂δ(q2)2∂q2]f(0)q−ℏδ(q2)Sμν(n)Cν},

where we define

 ~Eν=Eν+(q⋅u)T∂νT−qσ(σνσ+κνσ),Eν=Eν+T∂ν¯μ. (18)

For a rotating fluid with constant and such that , the Wigner function reduces to

 S<μ = (19) −ℏδ(q2)Sμν(n)Cν}.

The original side-jump term combined with the spin-tensor correction in results in a frame-independent contribution associated with vorticity, which suggests that should be also frame independent and the relevant parameters , , and could be defined universally in arbitrary frames. Given that the collisional kernel vanishes in the center of mass (COM) frame as a no-jump frame Chen et al. (2015); Hidaka et al. (2017), it should now vanish in an arbitrary frame for global equilibrium. Therefore, the Wigner functions for a purely rotating Weyl fluid in global equilibrium takes the form444Frame dependence of the sign function does not affect the conclusion.

 S<μgeq = 2π¯ϵ(q⋅n)[δ(q2)(qμ+ℏ2[uμ(q⋅ω)−ωμ(q⋅u)]∂q⋅u)+ℏϵμναβqνFαβ∂δ(q2)2∂q2]f(0)q. (20)

Nonetheless, when , , and are local parameters, which contribute to not only the vorticity, Eq. (II.2) also indicates that these parameters are no longer frame independent under corrections. Although one can introduce local-equilibrium distribution functions in the COM frame such that the collisional kernel vanishes, which is equivalent to introduce multiple observers for different scattering events with different momenta of incoming and outgoing particles, it is impractical since we may only solve for the distribution function with just one observer in CKT. Technically, in CKT, the Wigner function in cannot work in the COM frame when , , and depend on the momenta of other scattered particles as a consequence of their corrections.

Since it is formidable to find a general expression of the local equilibrium distribution function for an arbitrary frame such as Eq. (12) in global equilibrium, we may downgrade the problem to seek for the local equilibrium function in a particular frame, from which one can implement the modified Lorentz transformation to write down the corresponding distribution functions in different frames. Fortunately, we find that setting as the co-moving frame with the expression in Eq. (12) fits our purpose, which yields the vanishing collisional kernel in 2 to 2 scattering albeit the proof is somewhat technical as we will show in the following subsection.

### ii.3 Local Equilibrium Cases

By taking with the distribution function in Eq. (12), the Wigner function can be written as

 S<μleq = 2π¯ϵ(q⋅u)δ(q2)(qμ+ℏSμν(u)Δν+ℏϵμναβqνFαβ∂δ(q2)2∂q2)feq(u)q (21) = 2π¯ϵ(q⋅u)[δ(q2)(qμ+ℏ2(uμ(q⋅ω)−ωμ(q⋅u))∂q⋅u−ℏSμν(u)~Eν∂q⋅u) +ℏϵμναβFαβ4∂qνδ(q2)]f(0)q.

Here the collisional corrections in the side-jump term do not contribute to since at should be proportional to either or . Given that , where the bar for here corresponds to the distribution functions for outgoing particles, we can write down a relation between the less and greater propagators,

 S<μleq=e−β(q⋅u−μ)(~S>μleq−ℏ2T(uμω⋅S>leq−ωμS>leq⋅u)+ℏϵμναβuν2T(q⋅u)~EβS>leqα). (22)

For the leading-order 2 to 2 Coulomb scattering as considered in Ref. Hidaka et al. (2017), using Eq. (22), one finds

 S>leq⋅Σ<−S =ℏπδ(q2)∫q′,k,k′|M|24(k⋅k′)¯f(0)q¯f(0)q′f(0)kf(0)k′[ˇχ(q,q′)+ˇχ(q′,q)−ˇχ(k,k′)−ˇχ(k′,k)], (23)

where we introduced a compact notation for the integral,

 ∫q′,k,k′=∫d4q′d4kd4k′(2π)5δ(4)(q+q′−k−k′)δ(q′2)δ(k2)δ(k′2),

and

 ˇχ(q,q′)=−1T((q⋅u)(q′⋅ω)−(q⋅ω)(q′⋅u))+ϵμναβuνT(q′⋅u)qμ~Eβ(q′)q′α. (25)

Here is the squared matrix element for the 2 to 2 scattering process:

 |M|2=4e4[(q⋅q′)(q⋅k)+(q⋅q′)(q⋅k′)]2. (26)

The vorticity cancels out in the sum of and appeared in Eq. (23):

 ˇχ(q,q′)+ˇχ(q′,q)=ϵμναβuνTqμq′α(~Eβ(q′)(q′⋅u)−~Eβ(q)(q⋅u)), (27)

and thus

 S>Leq⋅Σ<−S = ℏπδ(q2)∫q′,k,k′|M|24(k⋅k′)¯f(0)q¯f(0)q′f(0)kf(0)k′ϵμναβuνT (28)

It is clear to see that the vorticity-related part, which is actually independent of frames, vanishes by symmetry. This finding agrees with the case in global equilibrium. Now, we shall deal with the rest part pertinent to .

For convenience, we can work in the local rest frame such that and yet . Then we find

 S>Leq⋅Σ<−S = ℏπδ(q2)∫q′,k,k′|M|24(k⋅k′)T¯f(0)q¯f(0)q′f(0)kf(0)k′ (29) ×[(q×q′)⋅(~E(q′)q′0−~E(q)q0)−(k×k′)⋅(~E(k′)k′0−~E(k)k0)],

where . It turns out that this integral actually vanishes, which can be shown based on the symmetry as discussed below. Apparently, the terms in the integral cancel each other. Now, considering the inversion of spatial momentum and electric fields (, , ,). In Eq. (29), we find that the integral is an odd function under the inversion. On the contrary, from the remaining terms in the integral, a non-vanishing collisional kernel should only be proportional to and and thus should be “even” under the inversion. Accordingly, the integral in Eq. (29) and the full collisional kernel should vanish. We thus conclude that Eq. (21) indeed corresponds to the local equilibrium Wigner function at least when considering only 2 to 2 scattering.

## Iii Non-Equilibrium Distribution Functions

Our final goal is to evaluate second-order quantum corrections on the charge current and density when the system is slightly away from local equilibrium. To handle this problem, we follow the standard strategy: expanding all the quantities and evolution equations in the power series of and space-time derivative . In the previous section, we have defined the distribution function in local equilibrium. Nevertheless, we have to first derive the non-equilibrium distribution functions led by fluctuations up to the . In addition, since we consider a closed system, we shall impose the energy-momentum conservation through anomalous hydrodynamics dictated by continuity equations in Eq. (36) as constrains for CKT.

In Sec. III.1, we derive the equations of motion (EOM) for anomalous hydrodynamics necessary for the study of corrections on non-equilibrium distribution functions. Those anomalous-hydrodynamic EOM will govern the dynamics of free thermodynamic parameters such as , , and in local-equilibrium distribution functions as shown in Eq. (44). Later, these relations have to be applied when solving the non-equilibrium distribution function perturbtively from CKT.

Subsequently, in Sec. III.2 the non-equilibrium distribution function is perturbatively solved from CKT by using an ansatz in Eq. (56) with the RT approximation. The corresponding solutions are presented in Eqs. (59) and (III.2), where the quantum part is further composed of three pieces coming from the corrections in CKT, hydrodynamic EOM, and postulated terms in collisions, respectively.

### iii.1 Anomalous Hydrodynamic Equations

Now, armed with the local equilibrium Wigner function in Eq. (21), we may first proceed to reproduce first-order anomalous transport coefficients in hydrodynamics, which have been studied from various approaches (e.g., see Refs. Gao et al. (2012); Landsteiner et al. (2013) and the references therein). On the other hand, we will also derive the hydrodynamic EOM with quantum corrections, which further contribute to the second-order transport.

In local equilibrium, the constitutive relations for energy-momentum tensors and charge currents are given by

 Tμν=uμuνϵ−pΘμν+Πμν% non+Πμνdis,Jμ=N0uμ+vμ% non+vμdis, (30)

where and and denote the energy density and pressure, respectively. Here the subindices “non” and “dis” represent the non-dissipative and dissipative corrections, respectively. The non-dissipative corrections come from anomalous transport, which can be written as

 Πμνnon=ℏξω(ωμuν+ωνuμ)+ℏξB(Bμuν+Bνuμ),vμnon=ℏσBBμ+ℏσωωμ, (31)

where and contribute to the heat conductivity and and corresponds to the charge conductivity of CME and CVE, respectively. Note that the electromagnetic fields are defined in Eq. (10). Such decompositions we applied are more convenient to be embedded into CKT, which are distinct from the Landau frame implemented in the previous studies of anomalous hydrodynamics such as in Ref. Son and Surowka (2009); Yamamoto (2015).

We may compute the non-dissipative contributions of energy-momentum tensors and currents from the Wigner functions in local equilibrium,

 Tμνleq = uμuνϵ−pΘμν+Πμνnon=∫d4q(2π)4(qμS<νleq+qνS<μleq), Jμleq = N0uμ+vμnon=2∫d4q(2π)4S<μleq, (32)

whereas the dissipative parts stem from non-equilibrium corrections associated with collisions. In practice, it is more convenient to work in the local rest frame to derive the transport coefficients and plug them back into the constitutive relations. By employing Eq. (21) and carrying out direct computations, we find

 ϵ=3p=T4(7π2120+¯μ24+¯μ48π2),N0=T36(¯μ+¯μ3π2), (33)
 σω=T212(1+3¯μ2π2),σB=μ4π2, (34)

and

 ξω=T36(¯μ+¯μ3π2)=N0,ξB=T224(1+3¯μ2π2)=σω2. (35)

Note that term in does not contribute to and , which can be shown in direct calculations. The anomalous coefficients obtained above agree with what have been found previously e.g., from Kubo formulae Landsteiner et al. (2013) or hydrodynamics with second-law of thermodynamics Neiman and Oz (2011).

Subsequently, using the continuity equations,

 ∂μTμν=FνρJρ,∂μJμ=ℏ4π2(E⋅B), (36)

and taking the projections, we obtain

 uν∂μTμν = (ϵ+p)∂⋅u+u⋅∂ϵ+ℏ(ω⋅∂ξω+ξω∂⋅ω+(ω⋅u)(u⋅∂)ξω+ξω(ω⋅u)(∂⋅u) (37) +ξωuνu⋅∂ων)+(ω↔B)+uν∂μΠμνdis = (ϵ+p)∂⋅u+u⋅∂ϵ+ℏ(ω⋅∂ξω+ξω∂⋅ω+ξωuνu⋅∂ων)+(ω↔B)+uν∂μΠμνdis = uνFνρJρ,

and

 Θα ν∂μTμν = (ϵ+p)u⋅∂uα−∂αp+uα(u⋅∂p)+ℏ[ξω(ω⋅∂)uα+ξωωα∂⋅u+ωα(u⋅∂)ξω (38) +ξω(u⋅∂)ωα−uαξωuνu⋅∂ων+(ω↔B)]+Θα ν∂μΠμνdis = (Fαρ+uαEρ)Jρ.

For convenience, we will work in the local rest frame and our goal is to solve for the time derivatives of parameters , , and . In addition, the incorporation of and will result in corrections on , , and . Consequently, and , which could yield quantum corrections at least for in CKT, will be omitted. The calculations in a covariant form are shown in Appendix B. For simplicity, we further drop the viscous corrections. Working in the local rest frame and implementing the constitutive relations, the continuity equations give rise to

 ∂0N0=−ℏ[ω⋅∇σω+2σωω⋅∂0u+B⋅∇σB+σB∂⋅B]+ℏ4π2(E⋅B), (39)
 ∂0ϵ = −ℏ(ω⋅∇ξω+ξω∂⋅ω+ξωω⋅∂0u)−ℏ(B⋅∇ξB+ξB∂⋅B+ξBB⋅∂0u) (40) +ℏ(σω(E⋅ω)+σB(E⋅B)),
 (ϵ+p)∂0u = ∇p−N0E+ℏ[ω∂0ξω+ξω∂0ω+B∂0ξB+ξB∂0B−(ξB+σω)(ω×B)]. (41)

In aid of Bianchi identity and , which takes explicit forms as

 ∂⋅B−2E⋅ω−B⋅∂0u=0,∂0B+B×ω+(∇×E−E×∂0u)=0, (42)

and

 ∂⋅ω−2ω⋅∂0u=0,∂0ω−12∇×(∂0u)=0, (43)

we perturbatively solve the continuity equations up to and obtain the hydrodynamic EOM,

 ∂0TT=ℏE⋅(~TBB+~TωωT),∂0¯μ=ℏE⋅(~μBB+~μωωT), (44)

and

 ∂0u=∂0u(0)+ℏ∂0δu,∂0u(0)=−∇TT+N0E4p, ℏ∂0δu=ℏ(~UE∇×E+~UTE×∇TT+~U¯μE×∇¯μ), (45)

where the coefficients involved have the following dimensions in energy, , , . The explicit forms of these coefficients read

 ~UE = ~UT=T296pπ2[(3¯μ2+π2)−N0¯μT(¯μ2+π2)2p], ~U¯μ = T3384p2π2[N0(3¯μ2+π2)+¯μT(¯μ2+π2)(2σω−N20p)], (46)
 ~TB = −15¯μ(15¯μ4+50¯μ2π2+19π4)2(15¯μ4+6¯μ2π2+7π4)(15¯μ4+30¯μ2π2+7π4)T3, ~Tω = −5(45¯μ6+75¯μ4π2−9¯μ2π4−7π6)(15¯μ4+6¯μ2π2+7π4)(15¯μ4+30¯μ2π2+7π4)T3, (47)
 ~μB = 3(75¯μ6+375¯μ4π2+285¯μ2π4+49π6)2(15