Nonlinear Responses of Chiral Fluids from Kinetic Theory
Abstract
The secondorder nonlinear responses of inviscid chiral fluids near local equilibrium are investigated by applying the chiral kinetic theory (CKT) incorporating sidejump effects. It is shown that the local equilibrium distribution function can be nontrivially introduced in a comoving 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 relaxationtime (RT) approximation, which result in anomalous charge Hall currents propagating along the cross product of the background electric field and the temperature (or chemicalpotential) gradient and of the temperature and chemicalpotential 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.
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 vectorcharge 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 quasiparticles 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 magnetoresistance 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 chargedensity 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 chargedensity 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 TorresRincon (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 sidejump terms in Wigner functions or equivalently the modified Lorentz transformation of distribution functions pertinent to sidejump 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 worldline 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 magneticmoment coupling, and magnetization currents associated with side jumps in the nonequilibrium 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 debate^{1}^{1}1For 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 mixedaxialgravitational 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 secondorder 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 secondorder nonlinear transport in Weylfermion 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 quasiparticles, which are applicable for Weyl semimetals such that the interaction among quasiparticles is suppressed by their collisions with impurities and phonons. In such cases, the energymomentum conservation could be violated when neglecting the backreaction upon environments. On the contrary, in QGP, the fluidlike properties of Weyl fermions should be taken due to strong coupling and hydrodynamics impose the energymomentum conservation. Although in reality the QGP coupling could be too strong for the legitimacy of a kinetictheory 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 quasiparticles, e.g., see Ref. Ebihara et al. (2017) for the boost invariant formation of CKT and the socalled 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 quasirelativistic 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 secondorder 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 sidejump 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/chemicalpotential gradients for righthanded 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 firstorder transport coefficients obtained from equilibrium Wigner functions. Next, we implement CKT to obtain the nonequilibrium corrections of distributions functions in aid of anomalous hydrodynamic EOM. Given the nonequilibrium distribution functions, we directly compute the charge current and charge density defined by Wigner functions. For simplicity, we neglect viscous corrections and utilize the relaxationtime (RT) approximation for the study of nonlinear responses. Through the paper, we refer quantum corrections to the corrections at in the Wignerfunction approach, which originate from the spin of Weyl fermions and anomalies, e.g., some nonlinear responses are from combination of the sidejump and anomalous hydrodynamic transports. The higherorder corrections in are beyond the scope of this paper. To give a quick view and simple explanation, here we summarize our findings in short:

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.

Without applying anomalous hydrodynamic EOM, which corresponds to a system breaking energymomentum conservation due to the interaction with the environment, the quantum corrections of the secondorder 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.

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.

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

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 Wignerfunction approach. In Sec. III, we work out the nonequilibrium distribution functions involving the secondorder quantum corrections for an inviscid chiral fluid by using CKT and hydrodynamic EOM. In Sec. IV, we implement the nonequilibrium distribution functions to compute the secondorder quantum corrections for the charge current and density. We also analyze the corresponding vector/axial currents in hightemperature and largechemicalpotential 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 widelyused 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 LeviCivita 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 sidejump based on the Wignerfunction approach in Sec. II.1. Because of side jumps, the distribution function becomes frame dependent stemming from corrections. Therefore, we need to define the localequilibrium distribution function in a proper way. Consequently, we review the welldefined case in global equilibrium in Sec. II.2, where the explicit form of the globalequilibrium 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 localequilibrium distribution function can be introduced in a comoving 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,
(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 righthanded Weyl fermions is given by
(2)  
where represents the sign of , are the spin matrices, and
(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 selfenergies 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 frames^{2}^{2}2The transformation between different frames here is equivalent to the inverse Lorentz transformation of and . Due to the sidejump term associated with in Wigner functions, the distribution function is no longer a scalar, which thus undergoes the nontrivial frame transformation or equivalently the modified Lorentz transformation upon phasespace 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.
(4) 
Note that both the energymomentum tensor and currents can be directly obtained from Wigner functions^{3}^{3}3When performing the explicit computations of currents and energymomentum tensors from Wigner functions, one actually takes normal ordering and drops infinite constants coming from the anticommutation relation of fermions.,
(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
(6)  
and
(7) 
From , carrying out similar computations as in Ref. Hidaka et al. (2017), the corresponding CKT takes the form,
(8) 
where
(9) 
comes from the choice of a local observer. Here the electromagnetic fields are defined thorough the frame vector ,
(10) 
Note that implicitly incorporates corrections since it contains at least one internal line of Weyl fermions in the selfenergy, which is true for most of realistic scattering processes, and the sidejump term will in general be involved. One can alternatively write the CKT as
(11) 
When taking and using the onshell condition, the CKT reduces to the usual threemomentum form in Refs. Son and Yamamoto (2013); Hidaka et al. (2017).
ii.2 Global Equilibrium Cases
It is shown from the semiclassical approach that a global equilibrium distribution function of a rotating Weyl fluid could be defined frameindependently 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 righthanded fermions as
(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,
(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/antisymmetric parts , where and . By introducing the fluid vorticity
(14) 
we may further rewrite the antisymmetric part as
(15) 
and the dual tensor
(16) 
By inserting Eq. (12) into Eq. (2) and using the relation
we obtain
where we define
(18) 
For a rotating fluid with constant and such that , the Wigner function reduces to
(19)  
The original sidejump term combined with the spintensor correction in results in a frameindependent 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 nojump 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 form^{4}^{4}4Frame dependence of the sign function does not affect the conclusion.
(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 localequilibrium 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 comoving 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
(21)  
Here the collisional corrections in the sidejump 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,
(22) 
For the leadingorder 2 to 2 Coulomb scattering as considered in Ref. Hidaka et al. (2017), using Eq. (22), one finds
(23) 
where we introduced a compact notation for the integral,
and
(25) 
Here is the squared matrix element for the 2 to 2 scattering process:
(26) 
The vorticity cancels out in the sum of and appeared in Eq. (23):
(27) 
and thus
(28)  
It is clear to see that the vorticityrelated 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
(29)  
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 nonvanishing 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 NonEquilibrium Distribution Functions
Our final goal is to evaluate secondorder 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 spacetime derivative . In the previous section, we have defined the distribution function in local equilibrium. Nevertheless, we have to first derive the nonequilibrium distribution functions led by fluctuations up to the . In addition, since we consider a closed system, we shall impose the energymomentum 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 nonequilibrium distribution functions. Those anomaloushydrodynamic EOM will govern the dynamics of free thermodynamic parameters such as , , and in localequilibrium distribution functions as shown in Eq. (44). Later, these relations have to be applied when solving the nonequilibrium distribution function perturbtively from CKT.
Subsequently, in Sec. III.2 the nonequilibrium 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 firstorder 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 secondorder transport.
In local equilibrium, the constitutive relations for energymomentum tensors and charge currents are given by
(30) 
where and and denote the energy density and pressure, respectively. Here the subindices “non” and “dis” represent the nondissipative and dissipative corrections, respectively. The nondissipative corrections come from anomalous transport, which can be written as
(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 nondissipative contributions of energymomentum tensors and currents from the Wigner functions in local equilibrium,
(32) 
whereas the dissipative parts stem from nonequilibrium 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
(33) 
(34) 
and
(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 secondlaw of thermodynamics Neiman and Oz (2011).
Subsequently, using the continuity equations,
(36) 
and taking the projections, we obtain
(37)  
and
(38)  
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
(39) 
(40)  
(41) 
In aid of Bianchi identity and , which takes explicit forms as
(42) 
and
(43) 
we perturbatively solve the continuity equations up to and obtain the hydrodynamic EOM,
(44) 
and
(45) 
where the coefficients involved have the following dimensions in energy, , , . The explicit forms of these coefficients read
(46) 
(47) 