# Electric and magnetic conductivities in magnetized fermion systems

Hao-Hao Peng  Department of Modern Physics, University of Science and
Technology of China, Hefei 230026, China  Xin-Li Sheng  INFN Sezione di
Firenze, Via Giovanni Sansone, 1, 50019 Sesto Fiorentino  Shi Pu  Department
of Modern Physics, University of Science and Technology of China, Hefei
230026, China  Qun Wang  Department of Modern Physics, University of Science
and Technology of China, Hefei 230026, China  School of Mechanics and Physics,
Anhui University of Science and Technology, Huainan 232001, China

###### Abstract

In Wigner function approach with relaxation time approximation, we calculate
electric and magnetic conductivities of a fermion system in the strong
magnetic field. The linear response has been calculated to the perturbation of
electromagnetic fields on the background constant magnetic field. The Wigner
function is separated into an equilibrium part in the background magnetic
field and an off-equilibrium part induced by perturbative fields. The
analytical expression for the equilibrium part and the corresponding
equilibrium conditions are given. For the off-equilibrium part, we obtain the
kinetic equation at the leading order in $\hbar$ from the master equation of
the Wigner function. When perturbative fields only depend on the proper time,
the off-equilibrium part can be analytically solved from which the vector and
axial vector currents are obtained. We obtain the longitudinal and transverse
Ohm conductivities as well as Hall conductivity as the linear response of the
vector current to the perturbative electric field. The behaviors of these
conductivities as functions of the evolving time, relaxation time, particle
mass, and strength of the background magnetic field are investigated both
analytically and numerically.

##  I Introduction

Relativistic heavy ion collisions provide a unique opportunity to study strong
interaction matter, namely the quark gluon plasma (QGP), under extreme
conditions. Strong electromagnetic (EM) fields are created in high energy
collisions. In peripheral collisions, the dominant component is the magnetic
field along the direction of the reaction plane (Skokov:2009qp, ;
Voronyuk:2011jd, ; Bzdak:2011yy, ; Deng:2012pc, ). In Au+Au collisions at
$\sqrt{s_{\text{NN}}}=200$ GeV, the highest collision energy at the
Relativistic Heavy Ion Collider (RHIC) of Brookhaven National Laboratory, the
magnetic field at the moment after the collision can reach $10^{17}$-$10^{18}$
Gauss (Skokov:2009qp, ; Voronyuk:2011jd, ; Bzdak:2011yy, ; Deng:2012pc, ;
Roy:2015coa, ). In Pb+Pb collisions with $\sqrt{s_{\text{NN}}}=2.76$ TeV at
the Large Hadron Collider (LHC) of the European Organization for Nuclear
Research (CERN), the magnetic field can reach 10 times as high as that at
RHIC. In the early stage of the collision, the magnetic field decays with time
as $\sim t^{-3}$. However, the finite Ohm conductivity of the QGP slows down
the damping of the magnetic field in the later stage, resulting in a $\sim
t^{-1}$ behavior (Tuchin:2013apa, ; Gursoy:2014aka, ; Zakharov:2014dia, ;
Yan:2021zjc, ; Wang:2021oqq, ; Zhang:2022lje, ; Siddique:2019gqh, ;
Wang:2020qpx, ; Pu:2016ayh, ; Pu:2016bxy, ; Roy:2015kma, ). This may
significantly extend the lifetime of the magnetic field, so one can expect
sizeable effects from the interaction between the magnetic field and the QGP.
These effects include the charge-odd directed flow $v_{1}^{\text{odd}}$
induced by the Lorentz force acting on charged particles (Gursoy:2014aka, ;
Inghirami:2019mkc, ; Gursoy:2020jso, ; Dubla:2020bdz, ; Zhang:2022lje, ), the
splitting of $\Lambda$ and $\overline{\Lambda}$ hyperons’ global polarizations
induced by the coupling between the magnetic field and hyperon’s magnetic
moment (Muller:2018ibh, ; Guo:2019joy, ; Buzzegoli:2022qrr, ; Xu:2022hql, ;
Peng:2022cya, ), and anomalous transport phenomena including the chiral
magnetic effect (CME) (Vilenkin:1980fu, ; Kharzeev:2007tn, ; Kharzeev:2007jp,
; Fukushima:2008xe, ; STAR:2021mii, ), the chiral separation effect (CSE)
(Aghababaie:2003iw, ; Son:2004tq, ; Metlitski:2005pr, ), and the chiral
magnetic wave (CMW) (Kharzeev:2010gd, ; Burnier:2011bf, ), see Refs.
(Miransky:2015ava, ; Huang:2015oca, ; Kharzeev:2015znc, ; Hattori:2016emy, ;
Fukushima:2018grm, ; Hattori:2022hyo, ; Gao:2020vbh, ; Hidaka:2022dmn, ) for
reviews. Study of strong interaction matter in strong magnetic fields may also
help us understand compact stars such as magnetars (Duncan:1992hi, ;
Lai:2000at, ; Price:2006fi, ; Enoto:2019vcg, ), in which strong magnetc fields
and QGP coexist in the cores of compact stars.

In strong magnetic fields, quark spectrum will be described by Landau energy
levels (landau2013quantum, ). As a result, the transport properties of the QGP
will be affected when the field strength is much larger than the energy scale
of the system such as the temperature or chemical potential. In past few
years, many studies have been made on magnetic field effects on transport
quantities such as shear and bulk viscosities (Hattori:2017qih, ; Li:2017tgi,
; Kurian:2018dbn, ; Kurian:2018qwb, ; Denicol:2018rbw, ; Denicol:2019iyh, ),
diffusion constants (Mitra:2016zdw, ; Hattori:2017qih, ; Kurian:2018qwb, ;
Denicol:2018rbw, ; Denicol:2019iyh, ; Rath:2019vvi, ), jet quenching
parameters (Li:2016bbh, ; Finazzo:2016mhm, ; Banerjee:2021sjm, ), and
electrical conductivities (Hattori:2016cnt, ; Hattori:2016lqx, ;
Gorbar:2016qfh, ; Fukushima:2017lvb, ; Kurian:2018qwb, ; Rath:2019vvi, ;
Lin:2019fqo, ). There are also some effects of strong magnetic fields on
photon emission in heavy-ion collisions (Wang:2022jxx, ; Wang:2021ebh, ;
Wang:2020dsr, ; Sun:2023pil, ). Among all these quantities, electrical
conductivities are of special importance because they directly control the
lifetime of magnetic fields and are also necessary parameters in the numerical
simulation of magnetohydrodynamics (anile2005relativistic, ;
giulini2015luciano, ).

When the strength of the magnetic field is sufficiently large, the energy gap
between the lowest Landau level and higher Landau levels becomes so large that
nearly all fermions occupy the lowest Landau level. As a consequence, the
system is greatly simplified to a (1+1)-dimension system from a
(3+1)-dimension system, as the transverse motion (perpendicular to the
direction of magnetic field) of fermions is restricted (landau2013quantum, ).
Electrical conductivities in the approximation of the lowest Landau level have
been calculated in Ref. (Hattori:2016cnt, ; Hattori:2016lqx, ) with quantum
field theory at finite temperature, and in Ref. (Lin:2019fqo, ) with chiral
kinetic theory. On the other hand, conductivities in weak magnetic fields have
been studied in the kinetic approach (Rath:2019vvi, ; Gorbar:2016qfh, ),
holograph models (Pu:2014fva, ; Pu:2014cwa, ), and the Lattice QCD
(Astrakhantsev:2019zkr, ; Buividovich:2010tn, ). In the medium magnetic field,
one has to sum over all Landau levels. Electrical conductivities in this
region have been computed by solving the linearized Boltzmann equations with
explicit $1\leftrightarrow 2$ processes in Ref. (Fukushima:2017lvb, ) and by
employing the relaxation time approximation (Das:2019pqd, ) and the
holographic QCD model (Li:2018ufq, ; Fukushima:2021got, ).

The present work is based on covariant Wigner functions (Elze:1986qd, ;
Vasak:1987um, ; Heinz:1983nx, ; Mrowczynski:1992hq, ; Blaizot:2001nr, ;
Wang:2001dm, ; Gao:2012ix, ; Hidaka:2016yjf, ; Weickgenannt:2019dks, ;
Sheng:2021kfc, ), for recent reviews, see e.g. Refs. (Gao:2020pfu, ;
Hidaka:2022dmn, ). In the constant magnetic field, the Wigner function can be
analytically derived using the exact solution to the Dirac equation
(Sheng:2017lfu, ; Gorbar:2017awz, ), which can serve as a tool to calculate
quantities such as the pair production (Sheng:2018jwf, ) and chiral
susceptibility (Das:2019crc, ). For particle collisions or interaction among
particles in the Wigner function approach, we apply the relaxation time
approximation as in Ref. (hakim1992relaxation, ; Wang:2021qnt, ).

This paper is organized as follows. In Sec. II, we give an introduction to
covariant Wigner functions and kinetic equations under the relaxation time
approximation. In Sec. III, we discuss Wigner functions in equilibrium to give
analytical solutions and equilibrium conditions. Section IV is devoted to the
kinetic equation and solution for the off-equilibrium part. One can show that
the off-equilibrium part of the Wigner function is fully determined by its
vector and axial vector components. We focus on these two components in Sec. V
and Sec. VI, respectively. Analytical solutions are given when perturbative
fields only depend on the proper time. The Ohm and Hall conductivities are
studied in Sec. V in respect to their dependence on the evolution and
relaxation time, the particle mass, and the strength of the background
magnetic field. We make conclusions in the last section, Sec. VII.

Throughout this work, we choose the convention for the metric tensor
$g_{\mu\nu}=\mathrm{diag}\\{+,-,-,-\\}$ and the Levi-Civita tensor
$\epsilon^{\mu\nu\rho\sigma}$, $\epsilon^{0123}=-\epsilon_{0123}=+1$. The
local collective velocity $u^{\mu}=\gamma(1,\mathbf{v})$ satisfies
$u^{\mu}u_{\mu}=1$ with $\gamma$ being the Lorentz contraction factor.
Relative to $u^{\mu}$, we define the orthogonal projector
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ . The electric charge $q$ is set
to 1 throughout this paper for simplicity.

In this paper, we consider a weak perturbation field $f^{\mu\nu}$ in
additional to a strong background field $F_{0}^{\mu\nu}$. The background field
is taken as a constant magnetic field in the comoving frame with respect to
the velocity $u^{\mu}$, which is given by

$F_{0}^{\mu\nu}=B_{0}\epsilon^{\mu\nu\alpha\beta}u_{\alpha}b_{\beta},$ (1)

where $B_{0}$ is the background field’s strength with $b^{\mu}$ being its
direction which is orthogonal to $u^{\mu}$, $u_{\mu}b^{\mu}=0$. We define
another projection tensor

$\Xi^{\mu\nu}\equiv g^{\mu\nu}-u^{\mu}u^{\nu}+b^{\mu}b^{\nu}.$ (2)

which can project a four-vector to the direction perpendicular to both
$u^{\mu}$ and $b^{\mu}$.

##  II Covariant Wigner functions

The Wigner function is defined as the Fourier transformation of the two-point
correlation function (Elze:1986qd, ; Vasak:1987um, )

$W_{\alpha\beta}(x,p)=\int\frac{d^{4}y}{(2\pi\hbar)^{4}}\exp\left(-\frac{i}{\hbar}p\cdot
y\right)\left\langle:\overline{\psi}_{\beta}(x_{1})U(x_{1},x_{2})\psi_{\alpha}(x_{2}):\right\rangle,$
(3)

where $\left\langle:O:\right\rangle$ denotes the expectation value of a
normal-ordered operator $O$. Here $x_{1}$ and $x_{2}$ are two space-time
coordinates, with $y^{\mu}=x_{1}^{\mu}-x_{2}^{\mu}$ and
$x^{\mu}=(x_{1}^{\mu}+x_{2}^{\mu})/2$ being the relative and central
coordinate respectively. With the vector potential $\mathbb{A}_{\mu}$ for the
classical electromagnetic field, the gauge link in Eq. (3) is defined as

$\displaystyle U(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{i}{\hbar}y^{\mu}\int_{-1/2}^{1/2}dt\mathbb{A}_{\mu}(x+ty)\right].$
(4)

The free-streaming Dirac field satisfies the Dirac equation and its adjoint
form

$(i\hbar\gamma\cdot D-m)\psi=\overline{\psi}(i\hbar\gamma\cdot
D^{\dagger}+m)=0,$ (5)

where $D_{\mu}=\partial_{x\mu}+(i/\hbar)\mathbb{A}_{\mu}$ is the covariant
derivative. The kinetic equation for the Wigner function has a similar form

$(\gamma\cdot K-m)W(x,p)=0,$ (6)

which can be derived from the Dirac equation. The operator $K^{\mu}$ is given
by

$\displaystyle K^{\mu}$ $\displaystyle=$
$\displaystyle\Pi^{\mu}+\frac{i\hbar}{2}\nabla^{\mu},$ (7)

where the generalized momentum operator $\Pi^{\mu}$ and space-time derivative
$\nabla^{\mu}$ are

$\displaystyle\Pi^{\mu}$ $\displaystyle=$ $\displaystyle
p^{\mu}-\frac{\hbar}{2}j_{1}(\Delta)F^{\mu\nu}(x)\partial_{p\nu},$
$\displaystyle\nabla^{\mu}$ $\displaystyle=$
$\displaystyle\partial_{x}^{\mu}-j_{0}(\Delta)F^{\mu\nu}(x)\partial_{p\nu},$
(8)

with $j_{0}(\Delta)$ and $j_{1}(\Delta)$ are spherical Bessel functions and
$\Delta=(\hbar/2)\partial_{p}^{\nu}\partial_{x\nu}$. We note that the
derivative $\partial_{x\nu}$ in $\Delta$ only acts on $F^{\mu\nu}(x)$ but not
on the Wigner function.

Concerning interaction among particles, we use the relaxation time
approximation in addition to Eq. (6) (hakim1992relaxation, ; Wang:2021qnt, ),

$(\gamma\cdot K-m)W(x,p)=-\frac{i\hbar}{2}\gamma^{\mu}u_{\mu}\frac{\delta
W(x,p)}{\tau},$ (9)

where $\tau$ is the relaxation time, $\delta W$ represents the deviation from
thermodynamical equilibrium

$\delta W(x,p)\equiv W(x,p)-W_{\text{eq}}(x,p),$ (10)

and $u_{\mu}$ is the local collective velocity

$u_{\mu}\propto\int
d^{4}p\,\text{Tr}\left[\gamma^{\mu}W_{\text{eq}}(x,p)\right].$ (11)

In this work, the equilibrium Wigner function is taken as the solution in a
constant $F_{0}^{\mu\nu}$ to the following equation

$\left\\{\gamma_{\mu}\left[p^{\mu}+\frac{i\hbar}{2}\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{\nu}^{p}\right)\right]-m\right\\}W_{\text{eq}}(x,p)=0.$
(12)

Then from Eq. (9) we obtain the equation for $\delta W$

$\left\\{\gamma_{\mu}p^{\mu}+\frac{i\hbar}{2}\gamma_{\mu}\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{\nu}^{p}+\frac{1}{\tau}u^{\mu}\right)-m\right\\}\delta
W(x,p)=\frac{i\hbar}{2}\gamma_{\mu}f^{\mu\nu}\partial_{\nu}^{p}W_{\text{eq}}+\mathcal{O}(\hbar^{2})\>,$
(13)

where we have made a semi-classical expansion with respect to $\hbar$ and
truncate it at second order in $\hbar$.

##  III Wigner functions in constant magnetic field: equilibrium

In this section, we discuss about the equilibrium form of the Wigner function
$W_{\text{eq}}(x,p)$ as the solution to Eq. (12) in the constant magnetic
field (1) without interaction among particles.

The Wigner function satisfies $W^{\dagger}=\gamma^{0}W\gamma^{0}$ and
therefore it can be expanded in terms of the generators of Clifford algebra
$\Gamma_{i}=\\{1_{4},i\gamma^{5},\gamma^{\mu},\gamma^{5}\gamma^{\mu},(1/2)\sigma^{\mu\nu}\\}$
as

$\displaystyle W(x,p)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(1_{4}\mathcal{F}+i\gamma^{5}\mathcal{P}+\gamma^{\mu}\mathcal{V}_{\mu}+\gamma^{5}\gamma^{\mu}\mathcal{A}_{\mu}+\frac{1}{2}\sigma^{\mu\nu}\mathcal{S}_{\mu\nu}\right),$
(14)

where $\sigma^{\mu\nu}=(i/2)[\gamma^{\mu},\gamma^{\nu}]$, and all expansion
coefficients
$\\{\mathcal{F},\mathcal{P},\mathcal{V}^{\mu},\mathcal{A}^{\nu},\mathcal{S}^{\mu\nu}\\}$
are real functions in phase space. Each function contains an equilibrium and
an off-equilibrium part, for example,

$\mathcal{F}=\mathcal{F}_{\text{eq}}+\delta\mathcal{F}.$ (15)

So $W_{\text{eq}}(x,p)$ also gives its equilibrium components
$\\{\mathcal{F}_{\text{eq}},\mathcal{P}_{\text{eq}},\mathcal{V}_{\text{eq}}^{\mu},\mathcal{A}_{\text{eq}}^{\nu},\mathcal{S}_{\text{eq}}^{\mu\nu}\\}$
similarly as in Eq. (14).

Projecting Eq. (12) onto $\Gamma_{i}$ and decoupling real and imaginary parts,
we obtain the following set of partial differential equations

$\displaystyle p_{\mu}\mathcal{V}_{\text{eq}}^{\mu}-m\mathcal{F}_{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (16)
$\displaystyle\frac{\hbar}{2}\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{p\nu}\right)\mathcal{A}_{\mu}^{\text{eq}}+m\mathcal{P}_{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (17) $\displaystyle
p_{\mu}\mathcal{F}_{\text{eq}}-\frac{\hbar}{2}\left(\partial_{x}^{\nu}-F_{0}^{\nu\alpha}\partial_{p\alpha}\right)\mathcal{S}_{\nu\mu}^{\text{eq}}-m\mathcal{V}_{\mu}^{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (18)
$\displaystyle-\frac{\hbar}{2}\left(\partial_{x\mu}-F_{0,\mu\nu}\partial_{p}^{\nu}\right)\mathcal{P}_{\text{eq}}+\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}p^{\nu}\mathcal{S}_{\text{eq}}^{\alpha\beta}+m\mathcal{A}_{\mu}^{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (19)
$\displaystyle\frac{\hbar}{2}\left(\partial_{x\mu}-F_{0,\mu\alpha}\partial_{p}^{\alpha}\right)\mathcal{V}_{\nu}^{\text{eq}}-\frac{\hbar}{2}\left(\partial_{x\nu}-F_{0,\nu\alpha}\partial_{p}^{\alpha}\right)\mathcal{V}_{\mu}^{\text{eq}}$
$\displaystyle-\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}_{\text{eq}}^{\beta}-m\mathcal{S}_{\mu\nu}^{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (20)
$\displaystyle\hbar\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{p\nu}\right)\mathcal{V}_{\mu}^{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (21) $\displaystyle
p^{\mu}\mathcal{A}_{\mu}^{\text{eq}}$ $\displaystyle=$ $\displaystyle 0,$ (22)
$\displaystyle\frac{\hbar}{2}\left(\partial_{x\mu}-F_{0,\mu\nu}\partial_{p}^{\nu}\right)\mathcal{F}_{\text{eq}}+p^{\nu}\mathcal{S}_{\nu\mu}^{\text{eq}}$
$\displaystyle=$ $\displaystyle 0,$ (23) $\displaystyle
p_{\mu}\mathcal{P}_{\text{eq}}+\frac{\hbar}{4}\epsilon_{\mu\nu\alpha\beta}\left(\partial_{x}^{\nu}-F_{0}^{\nu\rho}\partial_{p\rho}\right)\mathcal{S}_{\text{eq}}^{\alpha\beta}$
$\displaystyle=$ $\displaystyle 0,$ (24) $\displaystyle
p_{[\mu}\mathcal{V}_{\nu]}^{\text{eq}}+\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}\left(\partial_{x}^{\alpha}-F_{0}^{\alpha\rho}\partial_{p\rho}\right)\mathcal{A}_{\text{eq}}^{\beta}$
$\displaystyle=$ $\displaystyle 0.$ (25)

With the help of Eqs. (16), (17), and (20), we are able to express
$\mathcal{F}_{\text{eq}}$, $\mathcal{P}_{\text{eq}}$, and
$\mathcal{S}_{\text{eq}}^{\mu\nu}$ in terms of the remaining components
$\mathcal{V}_{\text{eq}}^{\mu}$ and $\mathcal{A}_{\text{eq}}^{\mu}$ as

$\displaystyle\mathcal{F}_{\text{eq}}$ $\displaystyle=$
$\displaystyle\frac{1}{m}p_{\mu}\mathcal{V}_{\text{eq}}^{\mu},$
$\displaystyle\mathcal{P}_{\text{eq}}$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{2m}\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{p\nu}\right)\mathcal{A}_{\mu}^{\text{eq}},$
$\displaystyle\mathcal{S}_{\mu\nu}^{\text{eq}}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2m}\left(\partial_{x\mu}-F_{0,\mu\alpha}\partial_{p}^{\alpha}\right)\mathcal{V}_{\nu}^{\text{eq}}-\frac{\hbar}{2m}\left(\partial_{x\nu}-F_{0,\nu\alpha}\partial_{p}^{\alpha}\right)\mathcal{V}_{\mu}^{\text{eq}}$
(26)
$\displaystyle-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}_{\text{eq}}^{\beta}.$

Substituting above relations into Eqs. (23) and (24), we obtain the kinetic
equations for $\mathcal{V}_{\text{eq}}^{\mu}$ and
$\mathcal{A}_{\text{eq}}^{\mu}$,

$\displaystyle
p^{\nu}\left(\partial_{x\nu}-F_{0,\nu\alpha}\partial_{p}^{\alpha}\right)\mathcal{V}_{\mu}^{\text{eq}}-F_{0,\mu\nu}\mathcal{V}_{\text{eq}}^{\nu}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
p^{\nu}\left(\partial_{x\nu}-F_{0,\nu\alpha}\partial_{p}^{\alpha}\right)\mathcal{A}_{\mu}^{\text{eq}}-F_{0,\mu\nu}\mathcal{A}_{\text{eq}}^{\nu}$
$\displaystyle=$ $\displaystyle 0.$ (27)

On the other hand, substituting $\mathcal{F}_{\text{eq}}$,
$\mathcal{P}_{\text{eq}}$, and $\mathcal{S}_{\text{eq}}^{\mu\nu}$ into Eqs.
(18) and (19), we obtain the mass-shell constraints

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(p^{2}-m^{2}\right)\mathcal{V}_{\mu}^{\text{eq}}+\frac{\hbar}{2m}\epsilon_{\mu\nu\alpha\beta}F_{0}^{\nu\alpha}\mathcal{A}_{\text{eq}}^{\beta},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(p^{2}-m^{2}\right)\mathcal{A}_{\mu}^{\text{eq}}+\frac{\hbar}{2m}\epsilon_{\mu\nu\alpha\beta}F_{0}^{\nu\alpha}\mathcal{V}_{\text{eq}}^{\beta},$
(28)

which indicate that the particle’s mass-shell is modified by the magnetic
field.

The Wigner function in a constant magnetic field has been given in Ref.
(Sheng:2017lfu, ; Sheng:2019ujr, ) in a non-covariant form. It is
straightforward to generalize the results to the covariant form,

$\displaystyle\mathcal{V}_{\text{eq}}^{\mu}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left[p^{\mu}-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)\Xi^{\mu\nu}p_{\nu}\right]V_{n}(p)\Lambda_{+}^{(n)}(p_{T}),$
$\displaystyle\mathcal{A}_{\text{eq}}^{\mu}$ $\displaystyle=$
$\displaystyle\left[(u\cdot p)b^{\mu}-(b\cdot
p)u^{\mu}\right]\sum_{n=0}^{\infty}V_{n}(p)\Lambda_{-}^{(n)}(p_{T}),$ (29)

where $b^{\mu}$ is the direction of the magnetic field, $\Xi^{\mu\nu}$ is the
projection operator in Eq. (2),
$p_{T}=\sqrt{-p_{\alpha}\Xi^{\alpha\beta}p_{\beta}}$ is the magnitude of the
transverse momentum, and $\Lambda_{\pm}^{(n)}(p_{T})$ are defined as

$\Lambda_{\pm}^{(n)}(p_{T})=(-1)^{n}\left[L_{n}\left(\frac{2p_{T}^{2}}{\hbar
B_{0}}\right)\mp L_{n-1}\left(\frac{2p_{T}^{2}}{\hbar
B_{0}}\right)\right]\exp\left(-\frac{p_{T}^{2}}{\hbar B_{0}}\right),$ (30)

where $L_{n}(x)$ with $n\geq 0$ are Laguerre polynomials and $L_{-1}(x)=0$. In
Eq. (29), $V_{n}(p)$ is a function that only depends on $u\cdot p$ and $b\cdot
p$,

$\displaystyle V_{n}(p)$ $\displaystyle\equiv$
$\displaystyle\frac{2}{(2\pi)^{3}}\left(2-\delta_{n0}\right)\delta\left[(u\cdot
p)^{2}-(b\cdot p)^{2}-m^{2}-2n\hbar B_{0}\right]$ (31)
$\displaystyle\times\left\\{f_{\text{eq}}^{(+)}(u\cdot p)\theta(u\cdot
p)+f_{\text{eq}}^{(-)}(-u\cdot p)\theta(-u\cdot p)\right\\}.$

where $f_{\text{eq}}^{(\pm)}$ are Fermi-Dirac distributions at local
temperature $T=1/\beta$ and chemical potential $\mu=\alpha T$,

$f_{\text{eq}}^{(\pm)}(u\cdot p)=\frac{1}{1+\exp\left[\beta(u\cdot
p)\mp\alpha\right]}.$ (32)

The space-time dependence of $\mathcal{V}_{\text{eq}}^{\mu}$ and
$\mathcal{A}_{\text{eq}}^{\mu}$ is through $\beta(x)$, $u^{\mu}(x)$, and
$\mu(x)$.

It can be easily checked that $\mathcal{V}_{\text{eq}}^{\mu}$ and
$\mathcal{A}_{\text{eq}}^{\mu}$ in Eq. (29) fulfill the mass-shell conditions
in (28), and $\mathcal{V}_{\text{eq}}^{\mu}$ and
$\mathcal{A}_{\text{eq}}^{\mu}$ should also satisfy Eqs. (21), (22), (25), and
(27), leading to following constraints

$\displaystyle\partial_{x\mu}\left\\{\left[p^{\mu}-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)\Xi^{\mu\nu}p_{\nu}\right]V_{n}(p)\Lambda_{+}^{(n)}(p_{T})\right\\}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\alpha}\left\\{\left[(u\cdot
p)b^{\beta}-(b\cdot
p)u^{\beta}\right]V_{n}(p)\Lambda_{-}^{(n)}(p_{T})\right\\}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle
p^{\nu}\partial_{x\nu}\left\\{\left[p^{\mu}-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)\Xi^{\mu\rho}p_{\rho}\right]V_{n}(p)\Lambda_{+}^{(n)}(p_{T})\right\\}$
$\displaystyle=$ $\displaystyle 0.$ (33)

Since $\Lambda_{\pm}^{(n)}(p_{T})$ are independent of space-time coordinates,
the constraints in Eq. (33) can be cast into those for equilibrium
distributions

$\displaystyle\left[p^{\mu}-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)\Xi^{\mu\nu}p_{\nu}\right]\partial_{x\mu}f_{\text{eq}}^{(\pm)}(x;u\cdot
p)$ $\displaystyle-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)\left(\partial_{x\mu}\Xi^{\mu\nu}p_{\nu}\right)f_{\text{eq}}^{(\pm)}(x;u\cdot
p)$ $\displaystyle=$ $\displaystyle 0,$ (34)
$\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\alpha}\left\\{\left[(u\cdot
p)b^{\beta}-(b\cdot p)u^{\beta}\right]f_{\text{eq}}^{(\pm)}(x;u\cdot
p)\right\\}=0,$ (35) $\displaystyle\left[p^{\mu}-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)\Xi^{\mu\rho}p_{\rho}\right]p^{\nu}\partial_{x\nu}f_{\text{eq}}^{(\pm)}(x;u\cdot
p)$ $\displaystyle-\left(1-\frac{2n\hbar
B_{0}}{p_{T}^{2}}\right)p^{\nu}\left(\partial_{x\nu}\Xi^{\mu\rho}p_{\rho}\right)f_{\text{eq}}^{(\pm)}(x;u\cdot
p)$ $\displaystyle=$ $\displaystyle 0.$ (36)

Substituting $f_{\text{eq}}^{(\pm)}$ in Eq. (32) into these equations, we find
that $\alpha$ should be a space-time constant, and $\beta u_{\mu}$ should
satisfy the Killing condition, namely,

$\displaystyle\partial_{x}^{\mu}\alpha$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\partial_{\mu}^{x}(\beta u_{\nu})+\partial_{\nu}^{x}(\beta
u_{\mu})$ $\displaystyle=$ $\displaystyle 0.$ (37)

These are also conditions for global equilibrium without EM fields. In
addition to the conditions in (37), we also obtain

$\displaystyle\partial_{b}u^{\mu}=0,\;\;\;\;\partial_{\perp}^{\nu}u^{\mu}=0,$
$\displaystyle\partial_{b}b^{\mu}=0,\;\;\;\;\partial_{\perp}^{\nu}b^{\mu}=0,$
$\displaystyle\partial_{\perp}^{\mu}\beta=0,\;\;\;\;\Xi_{\nu}^{\mu}\partial_{t}b^{\nu}=0,$
(38)

where $\partial_{t}\equiv u_{\mu}\partial_{x}^{\mu}$, $\partial_{b}\equiv-
b_{\mu}\partial_{x}^{\mu}$, and
$\partial_{\perp}^{\mu}\equiv\Xi_{\nu}^{\mu}\partial_{x}^{\nu}$. These
constraints are satisfied by that $\beta$, $b^{\mu}$, and $u^{\mu}$ are all
space-time constants.

##  IV General equations of Wigner functions: off-equilibrium

The off-equilibrium part of the Wigner function $\delta W$ satisfies Eq. (13).
Similar to Eq. (14), we can decompose $\delta W$ in terms of $\Gamma_{i}$,
then from Eq. (13) we derive the equations for Clifford components,

$\displaystyle p_{\mu}\delta\mathcal{V}^{\mu}-m\delta\mathcal{F}$
$\displaystyle=$ $\displaystyle 0,$ (39)
$\displaystyle\frac{\hbar}{2}\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{p\nu}+\frac{1}{\tau}u^{\mu}\right)\delta\mathcal{A}_{\mu}+m\delta\mathcal{P}$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2}f^{\mu\nu}\partial_{p\nu}\mathcal{A}_{\mu}^{\text{eq}},$
(40) $\displaystyle
p_{\mu}\delta\mathcal{F}-\frac{\hbar}{2}\left(\partial_{x}^{\nu}-F_{0}^{\nu\rho}\partial_{p\rho}+\frac{1}{\tau}u^{\nu}\right)\delta\mathcal{S}_{\nu\mu}-m\delta\mathcal{V}_{\mu}$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2}f^{\nu\rho}\partial_{p\rho}\mathcal{S}_{\mu\nu}^{\text{eq}},$
(41)
$\displaystyle-\frac{\hbar}{2}\left(\partial_{x\mu}-F_{0,\mu\nu}\partial_{p}^{\nu}+\frac{1}{\tau}u_{\mu}\right)\delta\mathcal{P}+\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}p^{\nu}\delta\mathcal{S}^{\alpha\beta}+m\delta\mathcal{A}_{\mu}$
$\displaystyle=$
$\displaystyle-\frac{\hbar}{2}f_{\mu\nu}\partial_{p}^{\nu}\mathcal{P}^{eq},$
(42)
$\displaystyle\frac{\hbar}{2}\left(\partial_{x\mu}-F_{0,\mu\rho}\partial_{p}^{\rho}+\frac{1}{\tau}u_{\mu}\right)\delta\mathcal{V}_{\nu}$
(43)
$\displaystyle-\frac{\hbar}{2}\left(\partial_{x\nu}-F_{0,\nu\rho}\partial_{p}^{\rho}+\frac{1}{\tau}u_{\nu}\right)\delta\mathcal{V}_{\mu}-\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\delta\mathcal{A}^{\beta}-m\delta\mathcal{S}_{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2}f_{\mu\rho}\partial_{p}^{\rho}\mathcal{V}_{\nu}^{\text{eq}}-\frac{\hbar}{2}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{V}_{\mu}^{\text{eq}},$
$\displaystyle\hbar\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{p\nu}+\frac{1}{\tau}u^{\mu}\right)\delta\mathcal{V}_{\mu}$
$\displaystyle=$ $\displaystyle\hbar
f^{\mu\nu}\partial_{p\nu}\mathcal{V}_{\mu}^{\text{eq}},$ (44) $\displaystyle
p^{\mu}\delta\mathcal{A}_{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (45)
$\displaystyle\frac{\hbar}{2}\left(\partial_{x\mu}-F_{0,\mu\nu}\partial_{p}^{\nu}+\frac{1}{\tau}u_{\mu}\right)\delta\mathcal{F}+p^{\nu}\delta\mathcal{S}_{\nu\mu}$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2}f_{\mu\rho}\partial_{p}^{\rho}\mathcal{F}_{\text{eq}},$
(46) $\displaystyle
p_{\mu}\delta\mathcal{P}+\frac{\hbar}{4}\epsilon_{\mu\nu\alpha\beta}\left(\partial_{x}^{\nu}-F_{0}^{\nu\rho}\partial_{p\rho}+\frac{1}{\tau}u^{\nu}\right)\delta\mathcal{S}^{\alpha\beta}$
$\displaystyle=$
$\displaystyle\frac{\hbar}{4}\epsilon_{\mu\nu\alpha\beta}f^{\nu\rho}\partial_{p\text{$\rho$}}\mathcal{S}_{\text{eq}}^{\alpha\beta},$
(47) $\displaystyle
p_{\mu}\delta\mathcal{V}_{\nu}-p_{\nu}\delta\mathcal{V}_{\mu}+\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}\left(\partial_{x}^{\alpha}-F_{0}^{\alpha\rho}\partial_{p\rho}+\frac{1}{\tau}u^{\alpha}\right)\delta\mathcal{A}^{\beta}$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}f^{\alpha\rho}\partial_{p\text{$\rho$}}\mathcal{A}_{\text{eq}}^{\beta}.$
(48)

Taking the vector and axial-vector components $\delta\mathcal{V}_{\mu}$ and
$\delta\mathcal{A}_{\mu}$ as basic ones, the remaining components are then
given by

$\displaystyle\delta\mathcal{F}$ $\displaystyle=$
$\displaystyle\frac{1}{m}p_{\mu}\delta\mathcal{V}^{\mu},$ (49)
$\displaystyle\delta\mathcal{P}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2m}f^{\mu\nu}\partial_{p\nu}\mathcal{A}_{\mu}^{\text{eq}}-\frac{\hbar}{2m}\left(\partial_{x}^{\mu}-F_{0}^{\mu\nu}\partial_{p\nu}+\frac{1}{\tau}u^{\mu}\right)\delta\mathcal{A}_{\mu},$
(50) $\displaystyle\delta\mathcal{S}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2m}\left(\partial_{x\mu}-F_{0,\mu\rho}\partial_{p}^{\rho}+\frac{1}{\tau}u_{\mu}\right)\delta\mathcal{V}_{\nu}-\frac{\hbar}{2m}\left(\partial_{x\nu}-F_{0,\nu\rho}\partial_{p}^{\rho}+\frac{1}{\tau}u_{\nu}\right)\delta\mathcal{V}_{\mu}$
(51)
$\displaystyle-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\delta\mathcal{A}^{\beta}-\frac{\hbar}{2m}f_{\mu\rho}\partial_{p}^{\rho}\mathcal{V}_{\nu}^{\text{eq}}+\frac{\hbar}{2m}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{V}_{\mu}^{\text{eq}}.$

Substituting $\delta\mathcal{F}$, $\delta\mathcal{P}$, and
$\delta\mathcal{S}_{\mu\nu}$ into Eqs. (41) and (42), we obtain the mass-shell
constraints for $\delta\mathcal{V}_{\mu}$ and $\delta\mathcal{A}_{\mu}$,

$\displaystyle(p^{2}-m^{2})\delta\mathcal{V}_{\mu}+\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}F_{0}^{\nu\alpha}\delta\mathcal{A}^{\beta}$
$\displaystyle=$
$\displaystyle-\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}f^{\nu\alpha}\mathcal{A}_{\text{eq}}^{\beta}+\mathcal{O}(\hbar^{2}),$
(52)
$\displaystyle(p^{2}-m^{2})\delta\mathcal{A}_{\mu}-\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}F_{0}^{\alpha\nu}\delta\mathcal{V}^{\beta}$
$\displaystyle=$
$\displaystyle-\frac{\hbar}{2}\epsilon_{\mu\nu\alpha\beta}f^{\nu\alpha}\mathcal{V}_{\text{eq}}^{\beta}+\mathcal{O}(\hbar^{2}).$
(53)

On the other hand, the kinetic or Boltzmann equations for
$\delta\mathcal{V}_{\mu}$ and $\delta\mathcal{A}_{\mu}$ are derived from Eqs.
(46) and (48)

$\displaystyle\hbar
p^{\nu}\left(\partial_{x\nu}-F_{0,\nu\rho}\partial_{p}^{\rho}+\frac{1}{\tau}u_{\nu}\right)\delta\mathcal{V}_{\mu}-\hbar
F_{0,\mu\nu}\delta\mathcal{V}^{\nu}$ $\displaystyle=$ $\displaystyle\hbar
f_{\mu\nu}\mathcal{V}_{\text{eq}}^{\nu}+\hbar
p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{V}_{\mu}^{\text{eq}}+\mathcal{O}(\hbar^{2}),$
$\displaystyle\hbar
p_{\nu}\left(\partial_{x}^{\nu}-F_{0}^{\nu\rho}\partial_{p\rho}+\frac{1}{\tau}u^{\nu}\right)\delta\mathcal{A}_{\mu}-\hbar
F_{0,\mu\nu}\delta\mathcal{A}^{\nu}$ $\displaystyle=$ $\displaystyle\hbar
f_{\mu\nu}\mathcal{A}_{\text{eq}}^{\nu}+\hbar
p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{A}_{\mu}^{\text{eq}}+\mathcal{O}(\hbar^{2}),$
(55)

where we have dropped terms in $\mathcal{O}(\hbar^{2})$ and higher orders.
Since we considered the case of the strong background magnetic field $B_{0}$,
we treat $\hbar B_{0}$ as the zeroth order quantity in power counting of
$\hbar$. As the result, the equilibrium solutions in Eq. (29) are the zeroth
order contributions. Consequently, $\delta\mathcal{V}_{\mu}$ and
$\delta\mathcal{A}_{\mu}$ obtained by solving Eq. (IV) and (55) are also the
zeroth order contributions. In the following discussions, we will set
$\hbar=1$ for simplicity.

In the strong magnetic field, particle’s energy spectra are described by the
Landau levels. However, the perturbation $f^{\mu\nu}$ will modify the energy
spectra, as shown in Eqs. (52) and (53). Since the perturbation $f^{\mu\nu}$
is assumed to be much weaker than $F_{0}^{\mu\nu}$, one can safely expect that
the deviation from Landau levels is negligible. Therefore in the following
discussion, we neglect the mass-shell constraints for
$\delta\mathcal{V}_{\mu}$ and $\delta\mathcal{A}_{\mu}$ in Eqs. (52) and (53),
and focusing on the Boltzmann equations (IV) and (55). For simplicity, we set
$B_{0}$, $\beta$, $b^{\mu}$, and $u^{\mu}$ as spacetime-independent
quantities, and assume that $f^{\mu\nu}$ only depend on the proper time
$t\equiv u\cdot x$. We decompose the perturbation $f^{\mu\nu}$ into the
electric and magnetic part as

$f^{\mu\nu}=\delta E^{\mu}u^{\nu}-\delta
E^{\nu}u^{\mu}+\epsilon^{\mu\nu\alpha\beta}u_{\alpha}\delta B_{\beta}.$ (56)

In Eqs. (IV) and (55), $f^{\mu\nu}$ plays the role of a source term for
$\delta\mathcal{V}_{\mu}$ and $\delta\mathcal{A}_{\mu}$. In general, the
obtained $\delta\mathcal{V}_{\mu}$ and $\delta\mathcal{A}_{\mu}$ can be
expressed as linear responses to $\delta E^{\mu}$ and $\delta B^{\mu}$. The
induced vector and axial vector currents, defined as momentum integrals of
$\delta\mathcal{V}^{\mu}$ and $\delta\mathcal{A}^{\mu}$, are given by

$\displaystyle\delta J^{\mu}$ $\displaystyle\equiv$ $\displaystyle\int
d^{4}p\,\delta\mathcal{V}^{\mu}=\sigma_{E}^{\mu\nu}\delta
E_{\nu}+\sigma_{B}^{\mu\nu}\delta B_{\nu},$ $\displaystyle\delta J_{A}^{\mu}$
$\displaystyle\equiv$ $\displaystyle\int
d^{4}p\,\delta\mathcal{A}^{\mu}=\chi_{A,E}^{\mu\nu}\delta
E_{\nu}+\chi_{A,B}^{\mu\nu}\delta B_{\nu},$ (57)

where the linear responses are described by conductivities
$\sigma_{E}^{\mu\nu}$, $\sigma_{B}^{\mu\nu}$, $\chi_{A,E}^{\mu\nu}$, and
$\chi_{A,B}^{\mu\nu}$. Since we assumed that $\delta E^{\mu}$ and $\delta
B^{\mu}$ are functions of $t$, these conductivitites are also proper time-
dependent quantities. In the following two sections, we will show

$\displaystyle\sigma_{E}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\Xi^{\mu\nu}\sigma_{\perp}-b^{\mu}b^{\nu}\sigma_{\parallel}+\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\sigma_{H},$
$\displaystyle\sigma_{B}^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\chi_{A,E}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-u^{\mu}b^{\nu}\chi_{A},$ $\displaystyle\chi_{A,B}^{\mu\nu}$
$\displaystyle=$
$\displaystyle\Xi^{\mu\nu}\chi_{\parallel}+\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\chi_{\perp},$
(58)

where $\sigma_{\perp}$, $\sigma_{\parallel}$ are Ohm conductivities in the
transverse and longitudinal directions with respect to the direction of the
background magnetic field, $\sigma_{H}$ is the Hall conductivity, $\chi_{A}$
is the coefficient for the axial-charge induced by parallel electric and
background magnetic field, and $\chi_{\parallel}$ and $\chi_{\perp}$ describe
the axial current induced by the perturbative magnetic field $\delta B^{\mu}$.
The explicit expressions for these conductivities will be given in following
sections.

##  V Vector current induced by perturbation fields

###  V.1 Analytical solutions

In this section, we solve the Boltzmann equation (IV) to obtain the vector
current. With $F_{0}^{\mu\nu}$ in Eq. (1), the Boltzmann equation (IV) can be
written as

$\left[(u\cdot
p)\left(\frac{d}{dt}+\frac{1}{\tau}\right)-B_{0}\epsilon^{\nu\lambda\rho\sigma}p_{\nu}^{\perp}\partial_{\lambda}^{p,\perp}u_{\rho}b_{\sigma}\right]\delta\mathcal{V}_{\mu}-B_{0}\epsilon_{\mu\nu\rho\sigma}u^{\rho}b^{\sigma}\delta\mathcal{V}^{\nu}=f_{\mu\nu}\mathcal{V}_{\text{eq}}^{\nu}+p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{V}_{\mu}^{\text{eq}}.$
(59)

where we made approximation that $\delta\mathcal{V}_{\mu}$ only depends on the
proper time $t\equiv u\cdot x$ because $f^{\mu\nu}$ is assumed to be only
$t$-dependent. The transverse momentum and the transverse momentum derivative
are defined with the help of the projection operator $\Xi^{\mu\nu}$ in Eq.
(2),

$p_{\perp}^{\mu}\equiv\Xi^{\mu\nu}p_{\nu},\ \
\partial_{p,\perp}^{\mu}\equiv\Xi^{\mu\nu}\partial_{\nu}^{p},$ (60)

while the longitudinal terms are

$p_{\parallel}^{\mu}\equiv p^{\mu}-p_{\perp}^{\mu},\ \
\partial_{\parallel,p}^{\mu}\equiv\partial_{p}^{\mu}-\partial_{\perp,p}^{\mu}.$
(61)

The vectors $u^{\mu}$, $b^{\mu}$, $p_{\perp}^{\mu}$, and
$\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}$ are orthogonal
to each other when $p_{\perp}^{\mu}\neq 0$ and therefore they form a complete
set of bases
$\\{e_{i}^{\mu},i=1,2,3,4\\}\equiv\\{u^{\mu},b^{\mu},p_{\perp}^{\mu},\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}\\}$
for any four-vector. The vector $\delta\mathcal{V}^{\mu}$ can then be
decomposed as

$\delta\mathcal{V}^{\mu}=u^{\mu}\delta\mathcal{V}_{1}-b^{\mu}\delta\mathcal{V}_{2}-\frac{1}{p_{T}^{2}}\left[p_{\perp}^{\mu}\delta\mathcal{V}_{3}^{\perp}+\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}\delta\mathcal{V}_{4}^{\perp}\right],$
(62)

where we have used $p_{T}=\sqrt{-p_{\perp}^{\mu}p_{\mu}^{\perp}}$. One can
extract all components as
$\delta\mathcal{V}_{i}=e_{i}^{\mu}\delta\mathcal{V}_{\mu}$.

Substituting $\delta\mathcal{V}_{\mu}$ in Eq. (62) into the Boltzmann equation
(59), we obtain an equation for each component

$\left[(u\cdot
p)\left(\frac{d}{dt}+\frac{1}{\tau}\right)-B_{0}\epsilon^{\nu\lambda\rho\sigma}p_{\nu}^{\perp}\partial_{\lambda}^{p,\perp}u_{\rho}b_{\sigma}\right]\delta\mathcal{V}_{i}=e_{i}^{\mu}\left(f_{\mu\nu}\mathcal{V}_{\text{eq}}^{\nu}+p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{V}_{\mu}^{\text{eq}}\right).$
(63)

The source term on the right-hand-side can be put into the form

$e_{i}^{\mu}\left(f_{\mu\nu}\mathcal{V}_{\text{eq}}^{\nu}+p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{V}_{\mu}^{\text{eq}}\right)=C_{i}^{(1)}(t,p_{T},u\cdot
p,b\cdot p)+p_{\perp}^{\mu}C_{i,\mu}^{(2)}(t,p_{T},u\cdot p,b\cdot p),$ (64)

where $C_{i}^{(1)}$ and $C_{i,\mu}^{(2)}$ are even functions of
$p_{\perp}^{\mu}$. With Eqs. (29) and (64), we obtain the source terms that
correspond to $f_{i=1,2}^{(1)}$ and $f_{i=3,4}^{(2)\mu}$ as

$\displaystyle C_{1}^{(1)}$ $\displaystyle=$ $\displaystyle(u\cdot
p)p_{\parallel}^{\nu}f_{\nu\rho}\sum_{n=0}^{\infty}\left[\partial_{\parallel,p}^{\rho}V_{n}(p)\right]\Lambda_{+}^{(n)}(p_{T}),$
$\displaystyle C_{2}^{(1)}$ $\displaystyle=$ $\displaystyle(b\cdot
p)p_{\parallel}^{\nu}f_{\nu\rho}\sum_{n=0}^{\infty}\left[\partial_{\parallel,p}^{\rho}V_{n}(p)\right]\Lambda_{+}^{(n)}(p_{T}),$
$\displaystyle C_{3}^{(2)\mu}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\Xi^{\mu\nu}f_{\nu\rho}p_{\parallel}^{\rho}\left(1+\frac{2nB_{0}}{p_{T}^{2}}\right)V_{n}(p)\Lambda_{+}^{(n)}(p_{T})$
$\displaystyle-\Xi^{\mu\nu}f_{\nu\rho}\sum_{n=0}^{\infty}2nB_{0}\left[\partial_{\parallel,p}^{\rho}V_{n}(p)\right]\Lambda_{+}^{(n)}(p_{T})$
$\displaystyle-2\Xi^{\mu\nu}f_{\nu\rho}p_{\parallel}^{\rho}\sum_{n=0}^{\infty}2nB_{0}V_{n}(p)\left[\frac{\partial}{\partial
p_{T}^{2}}\Lambda_{+}^{(n)}(p_{T})\right],$ $\displaystyle C_{4}^{(2)\mu}$
$\displaystyle=$
$\displaystyle-\sum_{n=0}^{\infty}\left(1-\frac{2nB_{0}}{p_{T}^{2}}\right)\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}f_{\nu\lambda}p_{\parallel}^{\lambda}V_{n}(p)\Lambda_{+}^{(n)}(p_{T}).$
(65)

Assuming $\delta\mathcal{V}_{i}(t_{0})=0$ at a given initial moment $t_{0}$,
the solution to Eq. (63) reads

$\delta\mathcal{V}_{i}(t,p)=f_{i}^{(1)}(t,p_{T},u\cdot p,b\cdot
p)+p_{\perp}^{\mu}f_{i,\mu}^{(2)}(t,p_{T},u\cdot p,b\cdot p),$ (66)

where $f_{i}^{(1)}$ and $f_{i,\mu}^{(2)}$ are obtained from $C_{i}^{(1)}$ and
$C_{i}^{(2)\mu}$ as

$\displaystyle f_{i}^{(1)}(t)$ $\displaystyle=$ $\displaystyle\frac{1}{u\cdot
p}\int_{t_{0}}^{t}dt^{\prime}e^{(t^{\prime}-t)/\tau}C_{i}^{(1)}(t^{\prime}),$
$\displaystyle f_{i,\mu}^{(2)}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{B_{0}(u\cdot
p)}\int_{t_{0}}^{t}dt^{\prime}\sin\left[\frac{B_{0}}{u\cdot
p}(t^{\prime}-t)\right]e^{(t^{\prime}-t)/\tau}$ (67)
$\displaystyle\times\left[(u\cdot
p)\left(\frac{d}{dt^{\prime}}+\frac{1}{\tau}\right)C_{i,\mu}^{(2)}(t^{\prime})+B_{0}\epsilon_{\mu\nu\alpha\beta}u^{\alpha}b^{\beta}C_{i}^{(2)\nu}(t^{\prime})\right].$

Here we have suppressed the dependence of all functions on $p_{T}$, $u\cdot
p$, and $b\cdot p$ for notational simplicity. Substituting Eq. (66) into Eq.
(62), we obtain

$\displaystyle\delta\mathcal{V}^{\mu}$ $\displaystyle=$ $\displaystyle
u^{\mu}f_{1}^{(1)}-b^{\mu}f_{2}^{(1)}$ (68)
$\displaystyle+\left[u^{\mu}f_{1,\nu}^{(2)}-b^{\mu}f_{2,\nu}^{(2)}\right]p_{\perp}^{\nu}+\frac{1}{p_{T}^{2}}\left[p_{\perp}^{\mu}f_{3}^{(1)}+\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}f_{4}^{(1)}\right]$
$\displaystyle-\frac{1}{p_{T}^{2}}\left[p_{\perp}^{\mu}p_{\nu}^{\perp}f_{3}^{(2)\nu}+\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}p_{\lambda}^{\perp}f_{4}^{(2)\lambda}\right].$

Note that $f_{i}^{(1)}$ and $f_{i}^{(2)\mu}$ are even functions of
$p_{\mu}^{\perp}$, thus when calculating the vector current $J^{\mu}(x)$
through integration $\delta\mathcal{V}_{\mu}$ over four-momentum, only terms
with $f_{1}^{(1)}$, $f_{2}^{(1)}$, $f_{3}^{(2)\mu}$ and $f_{4}^{(2)\mu}$
contribute but other terms do not. So we have

$\displaystyle J^{\mu}(x)$ $\displaystyle=$ $\displaystyle u^{\mu}\int
d^{4}p\,f_{1}^{(1)}-b^{\mu}\int d^{4}p\,f_{2}^{(1)}$ (69)
$\displaystyle+\frac{1}{2}\int
d^{4}p\,\left[f_{3}^{(2)\mu}+\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}f_{4,\nu}^{(2)}\right],$

where we have used the relation

$\int
d^{4}p\frac{p_{\perp}^{\mu}p_{\nu}^{\perp}}{p_{T}^{2}}f_{i}^{(2)\nu}=-\frac{1}{2}\int
d^{4}p\,f_{i}^{(2)\mu}.$ (70)

The vector current density $J_{\mu}(x)$ can be calculated using Eqs. (67),
(69), and (65) and the decomposition of $f^{\mu\nu}$ in Eq. (56),

$\displaystyle J^{\mu}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{(2\pi)^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot
p)\int_{t_{0}}^{t}dt^{\prime}e^{(t^{\prime}-t)/\tau}\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right]$
(71) $\displaystyle\times\left\\{b^{\mu}b^{\nu}\delta
E_{\nu}(t^{\prime})\frac{B_{0}(m^{2}+2nB_{0})}{\varepsilon^{3}}\right.$
$\displaystyle\left.+\left[\Xi^{\mu\nu}\left(\frac{d}{dt^{\prime}}+\frac{1}{\tau}\right)\delta
E_{\nu}(t^{\prime})\right]\left(1+\frac{nB_{0}}{\varepsilon}\frac{\partial}{\partial\varepsilon}\right)\sin\left[\frac{B_{0}}{\varepsilon}(t^{\prime}-t)\right]\right\\}$
$\displaystyle-\frac{1}{(2\pi)^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot
p)\int_{t_{0}}^{t}dt^{\prime}e^{(t^{\prime}-t)/\tau}\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\left[f_{\text{eq}}^{(+)}(\varepsilon)-f_{\text{eq}}^{(-)}(\varepsilon)\right]$
$\displaystyle\times\frac{B_{0}}{\varepsilon}\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\delta
E_{\nu}(t^{\prime})\left(1+nB_{0}\frac{\partial}{\partial\varepsilon}\frac{1}{\varepsilon}\right)\sin\left[\frac{B_{0}}{\varepsilon}(t^{\prime}-t)\right],$

where $f_{\text{eq}}^{(\pm)}(\varepsilon)\equiv f_{\text{eq}}^{(\pm)}(u\cdot
p=\varepsilon)$ are given in Eq. (32), $\varepsilon_{b\cdot
p}^{(n)}\equiv\sqrt{m^{2}+(b\cdot p)^{2}+2nB_{0}}$ is the on-shell energy at
$n$-th Landau level, and we have completed the integrals over $u\cdot p$ and
$p_{\perp}^{\mu}$ using explicit expressions for
$\mathcal{V}_{\text{eq}}^{\mu}$, $\Lambda_{\pm}^{(n)}(p_{T})$ and $V_{n}(p)$
in Eqs. (67), (30), and (31).

Assuming that the electric fields are suddenly switched on at $t_{0}=0$,
$\delta E^{\mu}\propto\theta(t)$, and $\delta E^{\mu}$ is a constant for
$t>0$. Then we can write the vector current as

$J^{\mu}(x)=\sigma_{\perp}\Xi^{\mu\nu}\delta
E_{\nu}-\sigma_{\parallel}b^{\mu}b^{\nu}\delta
E_{\nu}+\sigma_{H}\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\delta
E_{\nu},$ (72)

where we still used $\delta E_{\nu}$ but to represent the constant magnitude
of the electric field perturbation, $\sigma_{\parallel}$ and $\sigma_{\perp}$
are the electric conductivities in the longitudinal and transverse direction
respectively, and $\sigma_{H}$ is the Hall conductivity. These conductivities
are given by

$\displaystyle\sigma_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{B_{0}}{(2\pi)^{2}}\mathcal{R}_{1}(t)\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot p)$ (73)
$\displaystyle\times\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\frac{m^{2}+2nB_{0}}{\varepsilon^{3}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
$\displaystyle\sigma_{\perp}$ $\displaystyle=$
$\displaystyle-\frac{1}{(2\pi)^{2}\tau}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot p)$ (74)
$\displaystyle\times\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\left[\left(1+\frac{nB_{0}}{\varepsilon}\frac{\partial}{\partial\varepsilon}\right)\mathcal{R}_{2}(\varepsilon,B_{0},t)\right]\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
$\displaystyle\sigma_{H}$ $\displaystyle=$
$\displaystyle-\frac{B_{0}}{(2\pi)^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot p)$ (75)
$\displaystyle\times\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\frac{1}{\varepsilon}\left[\left(1+nB_{0}\frac{\partial}{\partial\varepsilon}\frac{1}{\varepsilon}\right)\mathcal{R}_{2}(\varepsilon,B_{0},t)\right]\left[f_{\text{eq}}^{(+)}(\varepsilon)-f_{\text{eq}}^{(-)}(\varepsilon)\right].$

The auxiliary functions $\mathcal{R}_{1}(t)$ and
$\mathcal{R}_{2}(\varepsilon,B_{0},t)$ are defined as

$\displaystyle\mathcal{R}_{1}(t)$ $\displaystyle\equiv$
$\displaystyle\tau(1-e^{-t/\tau}),$
$\displaystyle\mathcal{R}_{2}(\varepsilon,B_{0},t)$ $\displaystyle\equiv$
$\displaystyle-\frac{\varepsilon\tau}{\varepsilon^{2}+B_{0}^{2}\tau^{2}}\left\\{B_{0}\tau-e^{-t/\tau}\left[B_{0}\tau\cos\left(\frac{B_{0}}{\varepsilon}t\right)+\varepsilon\sin\left(\frac{B_{0}}{\varepsilon}t\right)\right]\right\\},$
(76)

which controls the time evolution. Since $f_{\text{eq}}^{(\pm)}\rightarrow 0$
at $\varepsilon\rightarrow\infty$, we find that the sum over Landau levels
converges. This ensures that all these conductivities have finite values.

At the long time limit $t\rightarrow\infty$, the auxiliary functions
$\mathcal{R}_{1}(t)$ and $\mathcal{R}_{2}(\varepsilon,B_{0},t)$ in Eq. (76)
are reduced to $\tau$ and $-\varepsilon
B_{0}\tau^{2}/(\varepsilon^{2}+B_{0}^{2}\tau^{2})$, respectively. The
asymptotic values of conductivitites are

$\displaystyle\lim_{t\rightarrow\infty}\sigma_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{B_{0}\tau}{(2\pi)^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\frac{m^{2}+2nB_{0}}{\varepsilon^{3}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
(77) $\displaystyle\lim_{t\rightarrow\infty}\sigma_{\perp}$ $\displaystyle=$
$\displaystyle\frac{B_{0}\tau}{(2\pi)^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\left[\left(1+\frac{nB_{0}}{\varepsilon}\frac{\partial}{\partial\varepsilon}\right)\frac{\varepsilon}{\varepsilon^{2}+B_{0}^{2}\tau^{2}}\right]$
(78)
$\displaystyle\times\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
$\displaystyle\lim_{t\rightarrow\infty}\sigma_{H}$ $\displaystyle=$
$\displaystyle\frac{B_{0}^{2}\tau^{2}}{(2\pi)^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n0}\right)\int
d(b\cdot p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(n)}}\frac{1}{\varepsilon^{2}+B_{0}^{2}\tau^{2}}\left(1-\frac{2nB_{0}}{\varepsilon^{2}+B_{0}^{2}\tau^{2}}\right)$
(79)
$\displaystyle\times\left[f_{\text{eq}}^{(+)}(\varepsilon)-f_{\text{eq}}^{(-)}(\varepsilon)\right].$

We observe that $\sigma_{\parallel}(t\rightarrow\infty)$ is always
proportional to the relaxation time $\tau$, which is consistent with the
result in materials (sommerfield1928elektronentheorie, ). For weak magnetic
fields, the transverse conductivity $\sigma_{\perp}(t\rightarrow\infty)$ is
also proportional to $\tau$, while the Hall conductivity $\sigma_{H}$ is
proportional to $\tau^{2}$. Strong magnetic fields break the linear
proportionality between $\sigma_{\perp}(t\rightarrow\infty)$ and $\tau$, and
that between $\sigma_{H}$ and $\tau^{2}$ due to the term $B_{0}^{2}\tau^{2}$
in the denominators in Eqs. (78) and (79).

###  V.2 Numerical results

Now we present numerical results for conductivitites in Eqs. (73)-(75). In
displaying numerical results, we regard the temperature $T$ as the energy
scale so that all quantities and parameters can be made dimensionless by
multiplying powers of $T$. We choose values of dimensionless parameters in
numerical calculation.

In Fig. 1 (a), we show dimensionless conductivities
$\sigma_{\parallel}/(T^{2}\tau)$ and $\sigma_{\perp}/(T^{2}\tau)$ as functions
of the time $t/\tau$ for $T\tau=$0.1, 1.0 and 2.0. As an example, we set
$\mu/T=0$, $B_{0}/T^{2}=1$, and $m/T=1$. We observe that
$\sigma_{\parallel}/(T^{2}\tau)$ does not depend on $T\tau$, as expected,
while $\sigma_{\perp}/(T^{2}\tau)$ is larger for smaller $T\tau$. As $t$
increases, both $\sigma_{\parallel}$ and $\sigma_{\perp}$ increase and
eventually reach saturated or asymptotic values when $t$ is sufficiently
large. We have verified that these behaviors hold for any set of parameters
$\\{\mu/T,\ B_{0}/T^{2},\ m/T\\}$. In Fig. 1 (b), we plot
$\sigma_{H}/(T^{3}\tau^{2})$ as functions of $t/\tau$ for $T\tau=$0.1, 1.0 and
2.0. We set $\mu/T=0.1$ because the Hall conductivity vanishes at $\mu=0$. We
observe that $\sigma_{H}/(T^{3}\tau^{2})$ increases with $t/\tau$ and reaches
saturated or asymptotic values when $t$ is large enough. For any set of
$\\{\mu/T,\ B_{0}/T^{2},\ m/T\\}$, the Hall conductivity is larger for smaller
$T\tau$, which is due to the presence of $B_{0}^{2}\tau^{2}$ in the
denominator of Eq. (76).

Figure 1:  (a) The Ohm conductivities $\sigma_{\parallel}/(T^{2}\tau)$ and
$\sigma_{\perp}/(T^{2}\tau)$ as functions of $t/\tau$ for different values of
$T\tau$. The parameters are set to $\mu/T=0$, $B_{0}/T^{2}=1.0$ and $m/T=1.0$.
Note that $\sigma_{\parallel}/(T^{2}\tau)$ does not depend on $T\tau$ and is
shown in black solid line. The red, green and blue dotted lines are
$\sigma_{\perp}/(T^{2}\tau)$ for $T\tau=$0.1, 1.0 and 2.0, respectively. (b)
The Hall conductivity $\sigma_{H}/(T^{3}\tau^{2})$ as functions of $t/\tau$
for different values of $T\tau$. The parameters are set to $\mu/T=0$.1,
$B_{0}/T^{2}=1.0$ and $m/T=1.0$.

The long-time asymptotic values of $\sigma_{\parallel}$, $\sigma_{\perp}$, and
$\sigma_{H}$ are presented as functions of $B_{0}/T^{2}$ in Figs. 2 and 3. The
results for $\sigma_{\parallel}/(T^{2}\tau)$ and $\sigma_{\perp}/(T^{2}\tau)$
are shown in solid and dotted lines respectively. In Fig. 2(a), we set
$\mu/T=0$, $m/T=1$, the black solid line represents
$\sigma_{\parallel}/(T^{2}\tau)$ which does not depend on $T\tau$, as can be
seen in Eq. (77), and the red, green and blue dotted lines represent
$\sigma_{\perp}/(T^{2}\tau)$ for $T\tau=$0.1, 1.0 and 2.0 respectively. As
$B_{0}/T^{2}\approx 0$, we observe that $\sigma_{\parallel}$ and
$\sigma_{\perp}$ are equal. At finite values of $B_{0}/T^{2}$, there is a
difference between $\sigma_{\parallel}$ and $\sigma_{\perp}$. We find that
$\sigma_{\perp}/(T^{2}\tau)$ increases with $B_{0}/T^{2}$ for small $T\tau$,
while it decreases with $B_{0}/T^{2}$ and finally approaches to zero at
$T\tau\rightarrow\infty$. Figure 2(b) shows the particle mass effects. We set
$m/T=$0.1, 1.0 and 2.0 represented by red, green and blue lines respectively.
For all three cases, we find that $\sigma_{\parallel}/(T^{2}\tau)$ increases
while $\sigma_{\perp}/(T^{2}\tau)$ decreases with $B_{0}/T^{2}$. Such results
are qualitatively consistent with those by the lattice QCD calculation
(Astrakhantsev:2019zkr, ) and the holographic model (Fukushima:2021got, ).

Figure 2:  Asymptotic values at long time limit for
$\sigma_{\parallel}/(T^{2}\tau)$ (solid lines) and
$\sigma_{\perp}/(T^{2}\tau)$ (dotted lines) as functions of $B_{0}/T^{2}$. The
chemical potential is set zero. (a) $m/T=1$, black solid line is for
$\sigma_{\parallel}/(T^{2}\tau)$ which does not depend on $T\tau$, and the
red, green and blue dotted lines are $\sigma_{\perp}/(T^{2}\tau)$ for
$T\tau=$0.1, 1.0 and 2.0, respectively. (b) $T\tau=1$ and $m/T=$0.1 (red),
$m/T=$1.0 (green), and $m/T=$2.0 (blue).

In Fig. 3, we show the long-time asymptotic values of
$\sigma_{H}/(T^{3}\tau^{2})$ in Eq. (79) as functions of $B_{0}/T^{2}$. In
Fig. 3(a), we set $\mu/T=$0.1, $m/T=1$, and $T\tau=$0.1, 1.0, 2.0. In the
weak-field limit we have $\sigma_{H}\propto\tau^{2}$, and then
$\sigma_{H}/(T^{3}\tau^{2})$ is independent of $T\tau$. The behaviors of
$\sigma_{H}/(T^{3}\tau^{2})$ versus $B_{0}/T^{2}$ are sensitive to the values
of $T\tau$. When $T\tau=2.0$, $\sigma_{H}/(T^{3}\tau^{2})$ increases and then
decreases with $B_{0}/T^{2}$. When $T\tau=1.0$, $\sigma_{H}/(T^{3}\tau^{2})$
increases and then approaches a constant value with $B_{0}/T^{2}$. When
$T\tau=0.1$, $\sigma_{H}/(T^{3}\tau^{2})$ always increases with $B_{0}/T^{2}$.
Figure 3(b) shows $\sigma_{H}/(T^{3}\tau^{2})$ as functions of $B_{0}/T^{2}$
at $\mu/T=0.1$, $T\tau=1.0$, and $m/T=$0.1, 1.0, 2.0. For all cases of Fig.
3(a,b), $\sigma_{H}/(T^{3}\tau^{2})$ is proportional to $B_{0}/T^{2}$ in the
weak field limit. We also find that a large particle mass gives a small
$\sigma_{H}/(T^{3}\tau^{2})$.

Figure 3: The long-time asymptotic values of $\sigma_{H}/(T^{3}\tau^{2})$ (79)
as functions of $B_{0}/T^{2}$ with different set of parameter values. (a)
$\mu/T=$0.1, $m/T=1$, $T\tau=$0.1 (red), 1.0 (green), 2.0 (blue); (b)
$\mu/T=0.1$, $T\tau=1.0$, $m/T=$0.1 (red), 1.0 (green), 2.0 (blue).

##  VI Axial-vector current induced by perturbation fields

In this section, we solve the Boltzmann equation (55) for
$\delta\mathcal{A}^{\mu}$ and calculate the axial-current defined as

$J_{A}^{\mu}(x)\equiv\int d^{4}p\,\delta\mathcal{A}^{\mu}(x,p).$ (80)

Similar to the decomposition for $\delta\mathcal{V}^{\mu}$ in Eq. (62) on the
bases
$e_{i}^{\mu}\equiv\\{u^{\mu},b^{\mu},p_{\perp}^{\mu},\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}\\}$,
we can also decompose $\delta\mathcal{A}^{\mu}$ as

$\delta\mathcal{A}^{\mu}=u_{\mu}\delta\mathcal{A}_{1}-b_{\mu}\delta\mathcal{A}_{2}-\frac{1}{p_{T}^{2}}\left[p_{\perp}^{\mu}\delta\mathcal{A}_{3}^{\perp}+\epsilon^{\mu\nu\rho\sigma}p_{\nu}^{\perp}u_{\rho}b_{\sigma}\delta\mathcal{A}_{4}^{\perp}\right].$
(81)

The dynamical equation for each component reads

$\left[(u\cdot
p)\left(\frac{d}{dt}+\frac{1}{\tau}\right)-B_{0}\epsilon^{\nu\rho\alpha\beta}p_{\nu}\partial_{\rho}^{p}u_{\alpha}b_{\beta}\right]\delta\mathcal{A}_{i}=e_{i}^{\mu}\left(f_{\mu\rho}\mathcal{A}_{\text{eq}}^{\rho}+p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{A}_{\mu}^{\text{eq}}\right),$
(82)

where the source term on the right-hand-side can be expressed as

$e_{i}^{\mu}\left(f_{\mu\rho}\mathcal{A}_{\text{eq}}^{\rho}+p^{\nu}f_{\nu\rho}\partial_{p}^{\rho}\mathcal{A}_{\mu}^{\text{eq}}\right)=D_{i}^{(1)}(t,p_{T},u\cdot
p,b\cdot p)+p_{\perp}^{\mu}D_{i,\mu}^{(2)}(t,p_{T},u\cdot p,b\cdot p).$ (83)

Similar to $J^{\mu}(x)$ in Eq. (69), the axial-vector current is given by

$\displaystyle J_{A}^{\mu}(x)$ $\displaystyle=$ $\displaystyle u^{\mu}\int
d^{4}p\,g_{1}^{(1)}-b^{\mu}\int d^{4}p\,g_{2}^{(1)}$ (84)
$\displaystyle+\frac{1}{2}\int
d^{4}p\,\left[g_{3}^{(2)\mu}+\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}g_{4,\nu}^{(2)}\right],$

where the functions $g_{i}^{(1)}$ and $g_{i,\mu}^{(2)}$ are derived from
$D_{i}^{(1)}$ and $D_{i,\mu}^{(2)}$ in a similar way to Eq. (67). Using
$\mathcal{A}_{\text{eq}}^{\mu}$ in Eq. (29), we obtain explicit expressions
for $D_{1}^{(1)}$, $D_{2}^{(1)}$, $D_{3}^{(2)\mu}$, and $D_{4}^{(2)\mu}$ as

$\displaystyle D_{1}^{(1)}$ $\displaystyle=$ $\displaystyle-(b\cdot
p)p_{\parallel}^{\nu}f_{\nu\rho}\sum_{n=0}^{\infty}\left[\partial_{p,\parallel}^{\rho}V_{n}(p)\right]\Lambda_{-}^{(n)}(p_{T}),$
$\displaystyle D_{2}^{(1)}$ $\displaystyle=$ $\displaystyle-(u\cdot
p)p_{\parallel}^{\nu}f_{\nu\rho}\sum_{n=0}^{\infty}\left[\partial_{p,\parallel}^{\rho}V_{n}(p)\right]\Lambda_{-}^{(n)}(p_{T}),$
$\displaystyle D_{3}^{(2)\mu}$ $\displaystyle=$
$\displaystyle\Xi^{\mu\nu}f_{\nu\rho}\left[(u\cdot p)b^{\rho}-(b\cdot
p)u^{\rho}\right]\sum_{n=0}^{\infty}V_{n}(p)\Lambda_{-}^{(n)}(p_{T}),$
$\displaystyle D_{4}^{(2)\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Xi_{\lambda}^{\mu}\epsilon^{\lambda\nu\rho\sigma}p_{\nu}^{\parallel}f_{\rho\sigma}\sum_{n=0}^{\infty}V_{n}(p)\Lambda_{-}^{(n)}(p_{T}).$
(85)

Then by making replacements $C_{i}^{(1)}\rightarrow D_{i}^{(1)}$,
$C_{i,\mu}^{(2)}\rightarrow D_{i,\mu}^{(2)}$, $f_{i}^{(1)}\rightarrow
g_{i}^{(1)}$ and $f_{i,\mu}^{(2)}\rightarrow g_{i,\mu}^{(2)}$ in Eq. (67), we
are able to derive $g_{1}^{(1)}$, $g_{2}^{(1)}$, $g_{3,\mu}^{(2)}$ and
$g_{4,\mu}^{(2)}$. Substituting them into Eq. (84), we obtain

$\displaystyle J_{A}^{\mu}(x)$ $\displaystyle=$
$\displaystyle-u^{\mu}b^{\nu}\frac{B_{0}}{(2\pi)^{2}}\int_{t_{0}}^{t}dt^{\prime}e^{(t^{\prime}-t)/\tau}\delta
E_{\nu}\int d(b\cdot p)$ (86)
$\displaystyle\times\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\,\frac{m^{2}}{\varepsilon^{3}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right]$
$\displaystyle-\frac{B_{0}}{(2\pi)^{2}}\Xi^{\mu\nu}\int_{t_{0}}^{t}dt^{\prime}e^{(t^{\prime}-t)/\tau}\delta
B_{\nu}\int d(b\cdot p)$
$\displaystyle\times\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\,\frac{1}{\varepsilon}\sin\left[\frac{B_{0}}{\varepsilon}(t^{\prime}-t)\right]\left[f_{\text{eq}}^{(+)}(\varepsilon)-f_{\text{eq}}^{(-)}(\varepsilon)\right]$
$\displaystyle+\frac{1}{(2\pi)^{2}\tau}\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\int_{t_{0}}^{t}dt^{\prime}e^{(t^{\prime}-t)/\tau}\left[\left(\frac{d}{dt^{\prime}}+\frac{1}{\tau}\right)\delta
B_{\nu}\right]\int d(b\cdot p)$
$\displaystyle\times\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\,\sin\left[\frac{B_{0}}{\varepsilon}(t^{\prime}-t)\right]\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right].$

Assuming that $\delta E^{\mu}$ and $\delta B^{\mu}$ are proportional to
$\theta(t)$ and they maintain constants at $t>0$, we can express $J_{A}^{\mu}$
as

$J_{A}^{\mu}=-u^{\mu}b^{\nu}\delta E_{\nu}\chi_{A}+\Xi^{\mu\nu}\delta
B_{\nu}\chi_{\parallel}+\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\delta
B_{\nu}\chi_{\perp},$ (87)

where we still used $\delta E_{\nu}$ and $\delta B_{\nu}$ but to represent the
constant magnitudes of the electric and magentic field pertubation
respectively. The conductivity $\chi_{A}$ denotes the axial charge induced by
the longitudinal electric field $b^{\nu}\delta E_{\nu}$, and conductivities
$\chi_{\parallel}$ and $\chi_{\perp}$ correspond to the axial current induced
by $\delta B_{\nu}$ in the direction $\Xi^{\mu\nu}\delta B_{\nu}$ and
$\epsilon^{\mu\nu\rho\sigma}u_{\rho}b_{\sigma}\delta B_{\nu}$, respectively,
which are given by

$\displaystyle\chi_{A}$ $\displaystyle=$
$\displaystyle\frac{B_{0}\mathcal{R}_{1}(t)}{(2\pi)^{2}}\int d(b\cdot
p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\frac{m^{2}}{\varepsilon^{3}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
(88) $\displaystyle\chi_{\parallel}$ $\displaystyle=$
$\displaystyle-\frac{B_{0}}{(2\pi)^{2}}\int d(b\cdot
p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\frac{1}{\varepsilon}\mathcal{R}_{2}(\varepsilon,B_{0},t)\left[f_{\text{eq}}^{(+)}(\varepsilon)-f_{\text{eq}}^{(-)}(\varepsilon)\right],$
(89) $\displaystyle\chi_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}\tau}\int d(b\cdot
p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\mathcal{R}_{2}(\varepsilon,B_{0},t)\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
(90)

where $\varepsilon_{b\cdot p}^{(0)}\equiv\sqrt{m^{2}+(b\cdot p)^{2}}$ is the
on-shell energy at the lowest Landau level and $\mathcal{R}_{1}(t)$,
$\mathcal{R}_{2}(\varepsilon,B_{0},t)$ are defined in Eq. (76). We note that
only the lowest Landau level contributes to these quantities, because higher
Landau levels are degenerated with respect to spin and thus do not contribute
to the axial current.

Similar to $\sigma_{\parallel}$, $\sigma_{\perp}$, and $\sigma_{H}$ in Eqs.
(77)-(79), the conductivities $\chi_{A}$, $\chi_{\parallel}$, and
$\chi_{\perp}$ also approach to their long-time asymptotic values that are
given by

$\displaystyle\lim_{t\rightarrow\infty}\chi_{A}$ $\displaystyle=$
$\displaystyle\frac{B_{0}\tau}{(2\pi)^{2}}\int d(b\cdot
p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\frac{m^{2}}{\varepsilon^{3}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right],$
(91) $\displaystyle\lim_{t\rightarrow\infty}\chi_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{B_{0}^{2}\tau^{2}}{(2\pi)^{2}}\int d(b\cdot
p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\frac{1}{\varepsilon^{2}+B_{0}^{2}\tau^{2}}\left[f_{\text{eq}}^{(+)}(\varepsilon)-f_{\text{eq}}^{(-)}(\varepsilon)\right],$
(92) $\displaystyle\lim_{t\rightarrow\infty}\chi_{\perp}$ $\displaystyle=$
$\displaystyle-\frac{B_{0}\tau}{(2\pi)^{2}}\int d(b\cdot
p)\lim_{\varepsilon\rightarrow\varepsilon_{b\cdot
p}^{(0)}}\frac{\varepsilon}{\varepsilon^{2}+B_{0}^{2}\tau^{2}}\left[f_{\text{eq}}^{(+)}(\varepsilon)+f_{\text{eq}}^{(-)}(\varepsilon)\right].$
(93)

Note that $\chi_{A}$ is exactly the same as the lowest Landau level
contribution to $\sigma_{\parallel}$. This is because particles at the lowest
Landau level always have spin along the direction of $b^{\mu}$, thus they are
right-handed when their momenta are parallel to $b^{\mu}$, and left-handed
when their momenta are anti-parallel to $b^{\mu}$. Therefore these particles
have the same contribution to the longitudinal vector current and the axial
charge.

##  VII Conclusion

We study the linear response of a fermion system to the perturbative
electromagnetic field in a constant magnetic background field based on the
Wigner function in relaxation time approximation. The Wigner function is
separated into an equilibrium and an off-equilibrium part. The equilibrium
part is the solution to the Wigner function in the constant magnetic
background field, which depends on the thermodynamical variables.

The off-equilibrium part of the vector and axial vector components induced by
the field perturbation can be determined by solving kinetic or Boltzmann
equations. They also satisfy the mass-shell constraints. In the case that the
field perturbation only depends on the proper time, the kinetic equations can
be solved analytically. The vector current only depends on the perturbative
electric field, with the linear response are described by longitudinal and
transverse Ohm conductivities as well as the Hall conductivity. As an example,
we assume that the perturbative electric field is turned on at the initial
proper time and maintains constant at later time, we numerically calculate
these conductivities. In the long time limit, these conductivities approach to
asymptotic values which depend on the relaxation time, the particle mass and
the background magnetic field. The contributions from all Landau levels are
included and therefore the results in this paper can be applied to any
strength of the magnetic field. The results for Ohm conductivities
qualitatively agree with those from the lattice QCD (Astrakhantsev:2019zkr, )
and the holographic model (Fukushima:2021got, ). We also obtain the analytical
solution to the axial vector current as the linear response to the
perturbative electric and magnetic fields.

###### Acknowledgements.

The authors thank Lixin Yang for enlightening discussions. H.H.P, S.P. and
Q.W. are supported by National Natural Science Foundation of China (NSFC)
under Grants No. 12135011, 12075235 and 11890713 (a subgrant of 11890710).
X.L.S. is supported by the Project funded by China Postdoctoral Science
Foundation under grant No. 2021M701369.

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