aainstitutetext: Quark Matter Research Center, Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou, Gansu, 073000, China bbinstitutetext:
Department of Physics, Keio University, Yokohama 223-8522, Japan

# Positive magnetoresistance induced by hydrodynamic fluctuations in chiral
media

Noriyuki Sogabe b Naoki Yamamoto a and Yi Yin<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We analyze the combined effects of hydrodynamic fluctuations and chiral
magnetic effect (CME) for a chiral medium in the presence of a background
magnetic field. Based on the recently developed non-equilibrium effective
field theory, we show fluctuations give rise to a CME-related positive
contribution to magnetoresistance, while the early studies without accounting
for the fluctuations find a CME-related negative magnetoresistance. At zero
axial relaxation rate, the fluctuations contribute to the transverse
conductivity in addition to the longitudinal one.

##  1 Introduction

The transport properties of a chiral medium (many-body system involving chiral
fermions) and their deep connection to quantum anomalies have attracted
significant interests recently. Of particular importance is the behavior of
electric conductivity (or its inverse, electric resistance) under the external
magnetic field ${\bm{B}}$. In table-top experiments, the negative
magnetoresistance is proposed as a signature of the chiral magnetic effect
(CME), the anomaly-induced vector current in the presence of magnetic field
and chiral charge imbalance Fukushima:2008xe ; Kharzeev:2007jp ;
Nielsen:1983rb ; Vilenkin:1980fu . Indeed, as shown in refs. Son:2012bg ;
Stephanov:2014dma ; Fukushima:2019ugr , the balance between the axial charge
density $n_{\rm A}$ production due to the chiral anomaly and axial charge
relaxation requires that in a steady state in the presence of electric field
${\bm{E}}$, the axial charge density $n_{\rm A}\propto
C{{\bm{E}}}\cdot{{\bm{B}}}/r$, where $r$ denotes the axial charge relaxation
rate and $C$ is the anomaly coefficient. Then, with the CME, one finds an
additional contribution to the longitudinal conductivity

$\displaystyle\Delta\sigma_{\rm L}\propto\frac{C^{2}{\bm{B}}^{2}}{r}\,.$ (1)

The measurements of magnetoresistance in Weyl and Dirac semimetals have been
reported in refs. Li:2014bha ; Xiong413 ; Huang:2015eia ; Arnold_2016 .

Nevertheless, the fluctuation effects have not yet been taken into account in
eq. (1). It is well-known that in an ordinary fluid, the fluctuations and
interactions among sound and diffusive modes lead to significant effects on
the behavior of transport coefficients POMEAU197563 ; Kovtun:2003vj ;
Kovtun:2012rj ; PhysRevA.16.732 ; Kovtun:2014nsa ; Grossi:2020ezz ;
Grossi:2021gqi . Therefore, one may naturally ask how fluctuations would
modify the magnetoresistance in a chiral medium. Addressing this question is
the primary goal of the present work. For definiteness, we shall consider the
fluctuations of both vector and axial charge densities. As in previous studies
Son:2012bg ; Hattori:2017usa , we assume that $r$ is parametrically small
compared with the microscopic relaxation rate, and hence we include the axial
charge density as a slow mode.

We here use the recently developed non-equilibrium effective field theory
(EFT) for hydrodynamics fluctuations Crossley:2015 ; Crossley:2017  (see ref.
Glorioso:2018wxw  for a review and refs. Haehl:2015foa ; Jensen:2017kzi  for
related developments), including the effects of quantum anomaly
Glorioso:2017lcn , to perform our analysis. Compared with the traditional
methods, the EFTs are derived based on the symmetries and action principle and
provide a basis for the systematic analysis. In some situations, such as the
one considered in ref. Chen-Lin:2018kfl , EFT calculations lead to different
results as compared with traditional analysis. Previous work including the
fluctuations of a single chiral charge and CME can be found in ref.
Delacretaz:2020jis . See refs. Fukushima:2017lvb ; Fukushima:2019ugr  for the
diagrammatic calculation of magnetoresistance for quark-gluon plasma (QGP)
based on perturbative QCD.

In two situations, $r\neq 0$ and $r=0$, we determine specific corrections to
the conductivity due to the combined effects of the CME and fluctuations in
the small ${\bm{B}}$ regime (see eq. (97) and eq. (123), respectively).
Physically, those CME-related contributions have two origins. First, the CME
is proportional to the axial chemical potential which generically depends on
charge density non-linearly and gives rise to non-linear coupling among
density fluctuations. Second, the CME modifies the dispersion relation of
fluctuations modes Kharzeev:2010gd ; Stephanov:2014dma . Given the difference
in physical origin, we should not be surprised to see that the fluctuation
corrections are in marked difference from eq. (1). One important qualitative
feature we observe is that the sign of fluctuation contributions is opposite
to that of eq. (1), meaning they give rise to positive magnetoresistance.
Moreover, we find a non-zero contribution to transverse conductivity when
$r=0$. As already noticed in some references Baumgartner:2017kme ;
Fukushima:2017lvb ; Fukushima:2019ugr , other mechanisms unrelated to the
anomaly could cause magnetoresistance. The present work aims to demonstrate
that even if one only focuses on the effects of the chiral anomaly, the
contribution from fluctuations to magnetoresistance can be qualitatively
different from that at “tree-level.” Our results might apply to physical
systems, such as the QGP created by heavy-ion collisions, Weyl semimetals, and
the electroweak plasma in the primordial Universe.

This paper is organized as follows. After reviewing the construction of the
EFT action in section 2, we determine the relevant Feynman rules and vertices.
In sections 3 and 4, we respectively calculate the conductivity at one-loop at
finite and vanishing axial charge relaxation rate. We conclude in section 5.

In this paper, we use $\hbar=c=k_{\rm B}=e=1$ and the mostly plus metric
$\eta^{\mu\nu}={\rm diag}\,(-1,1,1,1)$. We use the shorthand notation for
space-time and frequency-momentum integrations:$\int_{x}=\int{\rm d}^{4}x$
with $x^{\mu}=(t,{\bm{x}})$; $\int_{Q}=\int_{q^{0}}\int_{\bm{q}}$ with
$Q^{\mu}=(q^{0},{\bm{q}})$ and $q\equiv|{\bm{q}}|$; $\int_{q^{0}}=\int{\rm
d}q^{0}/(2\pi)$; $\int_{\bm{q}}=\int{\rm d}^{3}\bm{q}/(2\pi)^{3}$.

##  2 Non-equilibrium effective field theory

###  2.1 The action

We are interested in the fluctuation dynamics of vector charge density $n_{\rm
V}$ and axial charge density $n_{\rm A}$ in a chiral medium. As already
mentioned in the Introduction, we shall assume the relaxation rate of $n_{\rm
A}$ is small compared with the microscopic relaxation rate. Furthermore, we
shall limit ourselves to situations that temperature is much smaller than
vector chemical potential $\mu_{\rm V}$ and/or axial chemical potential
$\mu_{\rm A}$. In this regime, we could ignore the mixing of $n_{\rm V}$ and
$n_{\rm A}$ with the energy density. We also note in electron systems
including Weyl semimetals, the mean free path of momentum-relaxing scattering
(e.g., impurity scattering) can typically be shorter than the mean free path
of momentum-conserving scattering (electron-electron scattering). In such a
situation, the momentum is not a hydrodynamic variable, and ignoring the
coupling of (charge) density modes to sound/shear modes can be well justified.
Therefore, in long-time and large-distance limits, we can integrate out other
modes and obtain the effective action $I_{{\rm eff}}$ describing the remaining
slow modes $n_{\rm V}$ and $n_{\rm A}$. In general, it is difficult to obtain
$I_{\rm eff}$ directly from microscopic theories. Instead, one should
construct $I_{\rm eff}$ based on the symmetries together with other physical
requirements, as we shall do below following the formalism developed by refs.
Crossley:2015 ; Crossley:2017 ; Glorioso:2017lcn  (see ref. Glorioso:2018wxw
for a pedagogical introduction).

We begin with the path integral representation of the generating functional on
the Schwinger-Keldysh contour,

$\displaystyle{\rm e}^{{\rm i}W[A^{\rm r},A^{\rm a}]}=\int\prod_{\alpha={\rm
V,A}}\prod_{s={\rm a},{\rm r}}\,{\cal D}\psi_{\alpha}^{s}\,{\rm e}^{{\rm
i}I_{{\rm eff}}[\psi^{\rm a},\psi^{\rm r};A^{\rm a},A^{\rm r}]}\,.$ (2)

Here, we have introduced the external gauge fields $A^{\rm r}_{\alpha}$ and
$A^{\rm a}_{\alpha}$ and the dynamical fields $\psi^{\rm r}_{\alpha}$ and
$\psi^{\rm a}_{\alpha}$ associated with charge density $n_{\alpha}$, in the
“r-” and “a-” basis. One can interpret $\psi^{{\rm r}}_{\alpha}$ and
$\psi^{{\rm a}}_{\alpha}$ as the $\rm{U}(1)$ phase rotations of each fluid
element (see refs. Crossley:2015 ; Glorioso:2018wxw  for more details). The
$\rm r$-variables are related to the physical observables, while the $\rm
a$-variables are the associated noise variables.

Next, we list various symmetries and consistency requirements which $I_{{\rm
eff}}$ should satisfy:

  1. 1.

Gauge symmetries: $I_{\rm eff}$ has to be invariant under ${\rm U}(1)_{\rm V}$
gauge symmetry. Furthermore, we require ${\rm U}(1)_{\rm A}$ gauge symmetry
for $I_{\rm eff}$ in the limit that the axial relaxation and the chiral
anomaly are absent. Here, ${\rm U}(1)_{\alpha}$ gauge transformation can be
written explicitly as

$\displaystyle A^{s}_{\mu,\alpha}\to
A^{s}_{\mu,\alpha}-\partial_{\mu}\phi_{\alpha}^{s}\,,\quad\psi_{\alpha}^{s}\to\psi_{\alpha}^{s}+\phi_{\alpha}^{s}\,,$
(3)

where $\phi_{\alpha}^{s}$ is an arbitrary ${\rm U}(1)_{\alpha}$ phase. For a
term invariant under the ${\rm U}(1)_{\alpha}$ symmetry, its dependence on
$A^{\mu,s}_{\alpha}$ and $\psi_{\alpha}^{s}$ should come through the gauge-
invariant combination:

$\displaystyle{\cal
A}^{s}_{\mu,\alpha}=A^{s}_{\mu,\alpha}+\partial_{\mu}\psi_{\alpha}^{s}\,.$ (4)

In particular, the vector and axial chemical potentials are expressed as

$\displaystyle\mu_{\alpha}={\cal A}_{0,\alpha}^{\rm r}.$ (5)

  2. 2.

Shift symmetries: For each fluid element, it should have the freedom of making
independent ${\rm U}(1)_{\rm\alpha}$ phase rotations as far as those phases
$\zeta_{\alpha}(\bm{x})$ are time-independent:

$\displaystyle\psi_{\alpha}^{\rm r}\rightarrow\psi_{\alpha}^{\rm
r}+\zeta_{\alpha}({\bm{x}})\,.$ (6)

Note that shift symmetries will be absent when the global ${\rm
U}(1)_{\rm\alpha}$ symmetry is spontaneously broken (see refs. Dubovsky:2011sj
; Crossley:2015  for further details).

  3. 3.

Dynamical Kubo-Martin-Schwinger (KMS) symmetry: Suppose the microscopic theory
is invariant under a $\mathbb{Z}_{2}$ anti-unitary transformation $\Theta$,
then $I_{{\rm eff}}$ is invariant under the KMS transformation Crossley:2017 ,
which, in the “classical” limit that quantum fluctuations are small compared
with the thermodynamic fluctuations, is defined as

$\displaystyle{\cal A}^{{\rm r}}_{\mu}$ $\displaystyle\rightarrow\Theta{\cal
A}^{{\rm r}}_{\mu}\,,\quad{\cal A}^{{\rm a}}_{\mu}\rightarrow\Theta{\cal
A}^{{\rm a}}_{\mu}+\frac{\rm i}{T}\Theta\partial_{t}{\cal A}^{{\rm
r}}_{\mu}\,,$ (7a) $\displaystyle\mu_{\alpha}$
$\displaystyle\rightarrow\Theta\mu_{\alpha}\,,\quad\psi_{\alpha}^{\rm
a}\rightarrow\Theta\psi_{\alpha}^{\rm a}+\frac{\rm
i}{T}\Theta\partial_{t}\psi_{\alpha}^{\rm r}\,,$ (7b)

where $T$ is the background temperature. The dynamical KMS symmetry is
motivated by the KMS condition satisfied by a thermal system and can be viewed
as a definition of local thermal equilibrium. Generically, one can take
$\Theta$ that includes $\cal T$, i.e., $\Theta$ can be $\cal T$ itself, or any
combination of ${\cal C},{\cal P}$ with ${\cal T}$ Crossley:2017 , depending
on the systems of interest. The presence of background magnetic field and
vector charge will break the symmetries under ${\mathcal{T}}$ and
${\mathcal{C}}$, respectively, so we shall take
$\Theta=\mathcal{C}\mathcal{P}\mathcal{T}$ in this work.

  4. 4.

Unitarity: The unitarity of the underlying system requires that (suppressing
$\alpha$ and $\mu$ indices)

$\displaystyle\,I_{\rm eff}[\psi^{\rm r},A^{\rm r};\psi^{\rm a}=0,A^{\rm
a}=0]=0\,,$ (8) $\displaystyle\,I^{*}_{{\rm eff}}[\psi^{\rm r},A^{\rm
r};\psi^{\rm a},A^{\rm a}]=-I_{\rm eff}[\psi^{\rm r},A^{\rm r};-\psi^{\rm
a},-A^{\rm a}]\,.$ (9)

To consider the low-energy regime of the system, we also perform a derivative
expansion based on the basic philosophy of EFT. For definiteness, we take the
following counting scheme in this paper: ${\cal A}_{0,\alpha}^{\rm
a}\sim\partial_{t}{\cal A}_{0,\alpha}^{\rm r}\sim\epsilon^{2}$, ${\cal
A}_{i,\alpha}^{\rm a}\sim\partial_{t}{\cal A}_{i,\alpha}^{\rm r}\sim\epsilon$
(such that $\partial_{t}\psi_{\alpha}^{\rm a}\sim\epsilon^{2}$,
${\bm{\nabla}}\psi_{\alpha}^{\rm a}\sim\epsilon$), and $\psi_{\alpha}^{\rm
a}\sim\partial_{t}\psi_{\alpha}^{\rm r}\sim\epsilon^{0}$, where $\epsilon$ is
a small expansion parameter.

Now, we are ready to write down the non-equilibrium action $I_{{\rm eff}}$
explicitly. Because of eq. (8), ${\cal L}$ has to contain at least one power
of $\rm a$-field. We shall study ${\cal L}$ up to quadratic order in $\rm
a$-field. More explicitly, we consider the Lagrangian density ${\cal L}$,
which is related to $I_{\rm eff}$ through the standard relation

$\displaystyle I_{\rm eff}[\psi^{\rm r},A^{\rm r};\psi^{\rm a},A^{\rm
a}]=\int_{x}{{\cal L}}[\psi^{\rm r},A^{\rm r};\psi^{\rm a},A^{\rm a}]\,,$ (10)

and divide ${\cal L}$ into three parts:

$\displaystyle{\cal L}={\cal L}_{\rm inv}+{\cal L}_{\rm damp}+{\cal L}_{\rm
anom}\,.$ (11)

Here, ${\cal L}_{\rm inv}$ corresponds to ${\cal L}$ in the limit that both
the effects of the axial charge damping and chiral anomaly are absent. In this
case, $n_{\rm A}$ is also conserved. Hence, $\mathcal{L}_{\rm inv}$ up to
${\cal O}(\epsilon^{2})$ should be of the same form as the hydrodynamic
effective action with two conserved charges as derived in ref. Crossley:2015
(see also ref. Chen-Lin:2018kfl ):

$\displaystyle{\mathcal{L}}_{\rm inv}$
$\displaystyle=\sum_{\alpha}n_{\alpha}{\cal A}_{0,\alpha}^{{\rm
a}}-\sum_{i,j,\alpha,\beta}\sigma^{ij}_{\alpha\beta}\left({\cal A}^{\rm
a}_{i,\alpha}\partial_{t}{\cal A}^{\rm r}_{j,\beta}-{\rm i}T{\cal A}^{\rm
a}_{i,\alpha}{\cal A}^{\rm a}_{j,\beta}\right)\,,$ (12)

where $\sigma^{ij}_{\alpha\beta}$ is the conductivity matrix which is
symmetric with respect to $(i,j)$ and $(\alpha,\beta)$. Because of the shift
symmetries, ${\cal L}_{\rm inv}$ is independent of $\mathcal{A}^{{\rm
r}}_{i,\alpha}$.

Turning to ${\cal L}_{\rm anom}$, which describes the effects of the chiral
anomaly, we explicitly have

$\displaystyle{\mathcal{L}}_{\rm anom}$ $\displaystyle=C\psi_{\rm A}^{\rm
a}{\bm{E}}\cdot{\bm{B}}+C\mu_{{\rm V}}{\bm{B}}\cdot{\bm{\nabla}}\psi_{\rm
A}^{\rm a}+C\mu_{{\rm A}}{\bm{B}}\cdot\bm{{\cal A}}_{{\rm V}}^{\rm a}\,,$ (13)

where $C=1/(2\pi^{2})$ denotes the anomaly coefficient and the electric and
magnetic fields are defined by ${\bm{E}}={\bm{\nabla}}A^{\rm r}_{0,{\rm
V}}-\partial_{t}\bm{A}^{\rm r}_{\rm V}$ and
${\bm{B}}={\bm{\nabla}}\times\bm{A}^{\rm r}_{\rm V}$. To simplify the
expression, we shall consider the cases in the absence of the axial gauge
fields here and from now on. The first term in eq. (13) leads to the anomaly
contribution to the non-conservation of the axial current (see eq. (18)
below). The second and the third terms correspond to the chiral separation
effect (CSE) Son:2004tq ; Metlitski:2005pr  and CME, respectively. In appendix
A, we present the derivation of eq. (13) by generalizing the formulation for a
single chiral charge in ref. Glorioso:2017lcn .

Finally, we postulate to use

$\displaystyle{\mathcal{L}}_{\rm damp}$ $\displaystyle=-\gamma\left[\mu_{{\rm
A}}\psi_{\rm A}^{\rm a}-{\rm i}T(\psi_{\rm A}^{\rm a})^{2}\right]\,,$ (14)

to describe the axial charge relaxation. Here, $\gamma$ denotes the axial
damping coefficient, which is assumed to be ${\cal O}(\epsilon^{2})$ so that
$\mathcal{L}_{\rm damp}$ contributes to the same order as the other terms in
eq. (11). Equation (14) satisfies all requirements as listed above.

We here point out that the equations of motion from $I_{\rm eff}$ is
equivalent to the (non-)conservation equations for the currents. For the
vector charge density, we have

$\displaystyle\frac{\delta I_{\rm eff}}{\delta\psi^{\rm a}_{\rm
V}}=0\leftrightarrow\partial_{\mu}J^{\mu,{\rm r}}=0\,,\quad J^{\mu,{\rm
r}}=\frac{\delta I_{\rm eff}}{\delta A^{{\rm a}}_{\mu,{\rm V}}}\,,$ (15)

since $I_{\rm eff}$ only depends on the combination $A^{\rm a}_{\mu,{\rm
V}}+\partial_{\mu}\psi^{\rm a}_{\rm V}$ but not on $A^{\rm a}_{\mu,{\rm V}}$
and $\psi^{\rm a}_{\rm V}$ individually. For later purpose, we obtain the
expressions for $\bm{J}^{\rm r}$ and $\bm{J}^{\rm a}$ by differentiating (11)
with respect to $\bm{A}^{\rm a}$ and $\bm{A}^{\rm r}$, respectively:

$\displaystyle{\bm{J}}^{\rm r}$ $\displaystyle=-\sigma_{\rm
V}{\bm{\nabla}}\mu_{\rm V}+2{\rm i}T\sigma_{\rm V}{\bm{\nabla}}\psi_{\rm
V}+C\mu_{\rm A}\bm{B}\,,$ (16) $\displaystyle{\bm{J}}^{\rm a}$
$\displaystyle=\partial_{t}\left(\sigma_{\rm V}{\bm{\nabla}}\psi_{\rm
V}+C\psi_{\rm A}\bm{B}\right)+C{\bm{\nabla}}\times\left(\mu_{\rm
A}{\bm{\nabla}}\psi_{{\rm V}}+\mu_{{\rm V}}{\bm{\nabla}}\psi_{\rm
A}+{\bm{E}}\psi_{A}\right)\,.$ (17)

We can also define the axial current $J^{\mu}_{\rm A}$ through the variation
of $I_{\rm eff}$. In that case, we find that the equation of motion for
$\psi^{\rm a}_{\rm A}$ is nothing but the non-conservation equation for the
axial current due to the chiral anomaly and axial charge damping,

$\displaystyle\partial_{\mu}J^{\mu,{\rm r}}_{\rm
A}=C{{\bm{E}}}\cdot{{\bm{B}}}-\gamma\mu_{\rm A}+2{\rm i}\gamma T\psi^{{\rm
a}}_{\rm A}\,.$ (18)

In summary, we shall use the following effective action up to ${\cal
O}(\epsilon^{2})$ for the subsequent analysis for systems with a background
magnetic field ${\bm{B}}$ based on eqs. (12), (13), and (14):

$\displaystyle{\cal L}$
$\displaystyle=\sum_{\alpha}(\partial_{t}\psi_{\alpha})n_{\alpha}-\sum_{\alpha}\sigma_{\alpha}\left[({\bm{\nabla}}\psi_{\alpha})\cdot{\bm{\nabla}}\mu_{\alpha}-{\rm
i}T({\bm{\nabla}}\psi_{\alpha})^{2}\right]$ $\displaystyle\quad+C\mu_{\rm
A}\bm{B}\cdot{\bm{\nabla}}\psi_{\rm V}+C\mu_{\rm
V}\bm{B}\cdot{\bm{\nabla}}\psi_{\rm A}-\gamma\left[\psi_{\rm A}\mu_{\rm
A}-{\rm i}T(\psi_{\rm A})^{2}\right]\,.$ (19)

For notational brevity, here and hereafter, we suppress the ${\rm a}$-index
for $\psi^{\rm a}_{\alpha}$. Note that ${\bm{B}}={\cal O}(\epsilon)$ in our
counting scheme above, and we may ignore the ${\bm{B}}$-dependence of
$\sigma_{\alpha}$ and $\gamma$ at the level of this effective Lagrangian.

In this work, our goal is to showcase the non-trivial interplay among the
axial charge density relaxation, CME, and fluctuations in the simplest
possible settings. In eq. (2.1), we have assumed that at the tree level,
$\sigma^{ij}_{\alpha\beta}=\sigma_{\alpha}\,\delta^{ij}\delta_{\alpha\beta}$,
which is sufficient for the present illustrative purpose. In the same spirit,
we shall use the susceptibility matrix $\chi_{\alpha\beta}\equiv\partial
n_{\alpha}/\partial\mu_{\beta}$ which is also diagonalized,
$\chi_{\alpha\beta}=\chi_{\alpha}\delta_{\alpha\beta}$.

Before closing this section, we point out that the first equation in eq. (15)
and eq. (18) can be matched to the standard stochastic equations for $n_{{\rm
V}}$ and $n_{{\rm A}}$ in the presence of the CME/CSE Iatrakis:2015fma ;
Lin:2018nxj ; Hongo:2018cle . Conversely, one might construct an action of a
similar form to eq. (2.1) from the stochastic equation following the bottom-up
approach of Martin-Siggia-Rose-Janssen-de Dominicis PhysRevA.8.423 ; Janssen ;
PhysRevB.18.4913 , as was done in ref. Hongo:2018cle . However, the formalism
of refs. Crossley:2015 ; Crossley:2017 ; Glorioso:2017lcn  as we employ here
provides a basis for the systematic analysis.

###  2.2 Expansion around thermal equilibrium

Let us consider the fluctuations around the equilibrium state characterized by
a static and homogeneous background vector and axial charge densities
$(n_{{\rm V}})_{0}$ and $(n_{{\rm A}})_{0}$, where the subscript “$0$” denotes
those equilibrium values. In section 3.4, we shall study the situation that
axial charge damping coefficient $\gamma$ is finite so that $(n_{{\rm
A}})_{0}=0$. In section 4, we take the limit $\gamma=0$ and consider the
systems with a finite $(n_{{\rm A}})_{0}$. In both cases, we shall use $\delta
n_{\alpha}=n_{\alpha}-(n_{\alpha})_{0}$ as the dynamical fluctuating fields
for ${\rm r}$-variable and $\psi_{\alpha}$ as the dynamical ${\rm a}$-fields;
the latter vanishes in equilibrium.111 Although we have written down the
action (2.1) explicitly as a functional of $\mu_{\rm V,A}$ to make the
dynamical KMS symmetry (7a) and (7b) manifest, the resulting vertices will
contain time derivatives that would potentially complicate the analysis if we
were using $\delta\mu_{\rm V,A}$ as the dynamical $\rm r$-variables.  In
addition, we rescale $\delta n_{\alpha}$ and $\psi_{\alpha}$ for convenience
by

$\displaystyle\lambda_{\alpha}=g_{\alpha}\delta
n_{\alpha}\,,\quad\psi_{\alpha}\to g_{\alpha}\psi_{\alpha}\,,\quad
g_{\alpha}=\frac{1}{\sqrt{T\chi_{\alpha}}}\,.$ (20)

Note that $g^{-2}_{\alpha}=T\chi_{\alpha}$ is the equilibrium fluctuation of
$\delta n_{\alpha}$ per unit volume. This means that the fluctuations of the
rescaled variables $\lambda_{\alpha}$ are of the order unity, which is real
motivation for the definition (20). Here and throughout this paper, we do not
take the summation over dummy vector/axial indices ($\alpha={\rm V},{\rm A}$)
unless the summation symbol $\Sigma$ is present.

We shall expand the Lagrangian density as

$\displaystyle{\cal L}=({\cal L})_{1}+({\cal L})_{2}+({\cal L})_{3}+\ldots\,,$
(21)

where the subscript of $(\cdots)$ denotes the number of fluctuating fields. By
construction, $({\cal L})_{1}$ is a total derivative. We shall use $({\cal
L})_{2}$ to obtain propagators and read cubic vertices from $({\cal L})_{3}$.
The quartic vertices from $({\cal L})_{4}$ can contribute to one-loop
corrections, but such contributions are simply proportional to the UV cut-off
and will not be of physical importance. In short, ${\cal L}_{2,3}$ are
sufficient for the computing fluctuations corrections at one-loop order. Note
that if we demand $\sigma_{\alpha}$, $\gamma$, and $|{\bm{B}}|$ to be counted
as $g_{\alpha}^{-2}$, then $({\cal L})_{n}\sim g_{\alpha}^{n-2}$ and
$g_{\alpha}$ can be viewed as the effective coupling constant organizing the
fluctuation corrections to the tree-level results.

To determine ${\cal L}_{2,3}$, we need to expand $\mu_{\rm V,A},\sigma$, and
$\gamma$ in terms of $\delta n_{\rm V,A}$:

$\displaystyle\mu_{\alpha}$
$\displaystyle=(\mu_{\alpha})_{1}+(\mu_{\alpha})_{2}+\cdots\,,$ (22a)
$\displaystyle\sigma_{\alpha}$
$\displaystyle=(\sigma_{\alpha})_{0}+(\sigma_{\alpha})_{1}+\cdots\,,$ (22b)
$\displaystyle\gamma$ $\displaystyle=(\gamma)_{0}+(\gamma)_{1}+\cdots\,.$
(22c)

Defining

$\displaystyle\sigma_{\alpha;\beta}\equiv\frac{1}{\sigma_{\alpha}}\sqrt{\frac{\chi_{\alpha}}{\chi_{\beta}}}\frac{\partial\sigma_{\alpha}}{\partial\mu_{\beta}}\,,\quad\gamma_{;\alpha}\equiv\frac{1}{\gamma}\sqrt{\frac{\chi_{\rm
A}}{\chi_{\alpha}}}\frac{\partial\gamma}{\partial\mu_{\alpha}}\,,\quad
n_{\alpha;\beta\gamma}\equiv\frac{1}{\sqrt{\chi_{\beta}\chi_{\gamma}}}\frac{\partial^{2}n_{\alpha}}{\partial\mu_{\beta}\partial\mu_{\gamma}}\,,$
(23)

we have explicitly

$\displaystyle(\mu_{\alpha})_{1}$
$\displaystyle=\sqrt{\frac{T}{\chi_{\alpha}}}\lambda_{\alpha}\,,\quad(\mu_{\alpha})_{2}=-\frac{T}{2\chi_{\alpha}}\sum_{\beta,\gamma}n_{\alpha;\beta\gamma}\lambda_{\beta}\lambda_{\gamma}\,,$
(24a) $\displaystyle(\sigma_{\alpha})_{1}$
$\displaystyle=\sigma_{\alpha}\sqrt{\frac{T}{\chi_{\alpha}}}\sum_{\beta}\sigma_{\alpha;\beta}\lambda_{\beta}\,,\quad(\gamma)_{1}=\gamma\sqrt{\frac{T}{\chi_{\rm
A}}}\sum_{\alpha}\gamma_{;\alpha}\lambda_{\alpha}\,.$ (24b)

Here, the normalizations are chosen to make the counting in terms of
$g_{\alpha}$ manifest in the following expressions. From now on, we omit the
label $(...)_{0}$ for equilibrium quantities when doing so would not lead to
confusion.

Substituting eq. (22) into eq. (2.1) and further using eq. (24), we arrive at
the expressions:

$\displaystyle({{\cal L}})_{2}$
$\displaystyle=\sum_{\alpha}\left[(\partial_{t}\psi_{\alpha})\lambda_{\alpha}-D_{\alpha}({\bm{\nabla}}\psi_{\alpha})\cdot{\bm{\nabla}}\left(\lambda_{\alpha}-{\rm
i}\psi_{\alpha}\right)\right]+{\bm{v}}\cdot\left(\lambda_{{\rm
V}}{\bm{\nabla}}\psi_{{\rm A}}+\lambda_{{\rm A}}{\bm{\nabla}}\psi_{{\rm
V}}\right)-r\psi_{\rm A}\left(\lambda_{{\rm A}}-{\rm i}\psi_{\rm A}\right)\,,$
(25) $\displaystyle({{\cal L}})_{3}$
$\displaystyle=-T\sum_{\alpha,\beta}g_{\alpha}D_{\alpha}\sigma_{\alpha;\beta}\lambda_{\beta}({\bm{\nabla}}\psi_{\alpha})\cdot{\bm{\nabla}}(\lambda_{\alpha}-{\rm
i}\psi_{\alpha})-\frac{T}{2}\sum_{\alpha,\beta,\gamma}g_{\alpha}D_{\alpha}n_{\alpha;\beta\gamma}\lambda_{\beta}\lambda_{\gamma}{\bm{\nabla}}^{2}\psi_{\alpha}$
$\displaystyle\quad-\frac{T}{2}\sum_{\alpha,\beta}\left(g_{\rm V}n_{{\rm
A};\alpha\beta}{\bm{v}}_{\rm A}\cdot{\bm{\nabla}}\psi_{\rm V}+g_{\rm A}n_{{\rm
V};\alpha\beta}{\bm{v}}_{\rm V}\cdot{\bm{\nabla}}\psi_{\rm
A}\right)\lambda_{\alpha}\lambda_{\beta}$ (26) $\displaystyle\quad-Tg_{\rm
A}r\sum_{\alpha}\gamma_{;\alpha}\lambda_{\alpha}\psi_{\rm A}\left(\lambda_{\rm
A}-{\rm i}\psi_{\rm A}\right)+\frac{Tg_{\rm A}r}{2}\sum_{\alpha,\beta}n_{{\rm
A};\alpha\beta}\psi_{\rm A}\lambda_{\alpha}\lambda_{\beta}\,,$

with

$\displaystyle
D_{\alpha}\equiv\frac{\sigma_{\alpha}}{\chi_{\alpha}}\,,\quad{\bm{v}}\equiv\frac{C{\bm{B}}}{\sqrt{\chi_{\rm
V}\chi_{\rm
A}}}\,,\quad{\bm{v}}_{\alpha}\equiv\frac{C{\bm{B}}}{\chi_{\alpha}}\,,\quad
r\equiv\frac{\gamma}{\chi_{\rm A}}\,.$ (27)

Here, $r$ and ${\bm{v}}$ correspond to the bare axial relation rate and the
velocity of the chiral magnetic wave (CMW) Kharzeev:2010gd ; Newman:2005hd ,
respectively. The first and the third lines in eq. (2.2) arise from the non-
linearity due to the charge diffusion and axial charge relaxation,
respectively. Since $\mu_{\rm V,A}$ is generically a non-linear function of
$n_{{\rm V}}$ and $n_{{\rm A}}$, the CME/CSE induce non-linear couplings among
fluctuating fields, as is shown in the second line of eq. (2.2). In the cubic
action (2.2), the terms involving two a-fields correspond to the
multiplicative noise contribution, which is necessary to ensure the KMS
invariance.

###  2.3 Propagators and vertices

We now define the two-point correlation functions of the fields:

$\displaystyle{\cal G}_{\alpha\beta}^{\rm rr}(x)$
$\displaystyle=\langle\lambda_{\alpha}(x)\lambda_{\beta}(0)\rangle\,,$ (28a)
$\displaystyle{\cal G}_{\alpha\beta}^{\rm ra}(x)$
$\displaystyle=\langle\lambda_{\alpha}(x)\psi_{\beta}(0)\rangle\,,\quad{\cal
G}_{\alpha\beta}^{\rm
ar}(x)=\langle\psi_{\alpha}(x)\lambda_{\beta}(0)\rangle\,,$ (28b)

while $\langle\psi_{\alpha}(x)\psi_{\beta}(0)\rangle=0$ by causality.

Figure 1:  Diagrammatic representations of (a) $G^{{\rm r}{\rm
a}}_{\alpha\beta}$ , (b) $G^{{\rm r}{\rm r}}_{\alpha\beta}$, and the vertices
(c) $U$ and (d) $V$. The solid and dotted lines correspond to $\lambda$
(r-field) and $\psi$ (a-field), respectively.

To perform the diagrammatic analysis, we shall consider the free propagators
$G^{\rm rr}_{\alpha\beta},G^{\rm ra}_{\alpha\beta}$, and $G^{\rm
ar}_{\alpha\beta}$, which are $\mathcal{G}^{\rm
rr}_{\alpha\beta},\mathcal{G}^{\rm ra}_{\alpha\beta}$, and $\mathcal{G}^{\rm
ar}_{\alpha\beta}$ at the tree level, respectively. Suppressing the indices
$\alpha$ and $\beta$, we can read their expressions from $({\cal L})_{2}$
given by eq. (25) as

$\displaystyle\left(\begin{array}[]{cc}G^{\rm rr}&G^{\rm ra}\\\ G^{\rm
ar}&0\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&-{\rm i}M^{\dagger}\\\ -{\rm
i}M&N\end{array}\right)^{-1}\,,$ (33)

where in the Fourier space with $K^{\mu}=(\omega,{\bm{k}})$,

$\displaystyle M(K)=\left(\begin{array}[]{cc}{\rm i}\omega-D_{\rm
V}{\bm{k}}^{2}&-{\rm i}{\bm{v}}\cdot{\bm{k}}\\\ -{\rm
i}{\bm{v}}\cdot{\bm{k}}&{\rm i}\omega-(r+D_{\rm
A}{\bm{k}}^{2})\end{array}\right)\,,\quad
N({\bm{k}})=2\left(\begin{array}[]{cc}\displaystyle D_{\rm V}{\bm{k}}^{2}&0\\\
0&r+D_{\rm A}{\bm{k}}^{2}\end{array}\right)\,.$ (38)

Note that $M^{\rm T}=M$. From eq. (33), we obtain

$\displaystyle G^{\rm ra}={\rm i}M^{-1}\,,\quad G^{\rm ar}={\rm
i}(M^{-1})^{\dagger}\,,\quad G^{\rm rr}=-G^{\rm ra}NG^{\rm ar}\,.$ (39)

The diagrammatic representations of $G^{{\rm r}{\rm a}}_{\alpha\beta}$ and
$G^{{\rm r}{\rm r}}_{\alpha\beta}$ are given by figures 1 (a) and (b),
respectively.

The retarded propagator $G^{{\rm ra}}$ is the basic building block in the
subsequent diagrammatic computations, whereas $G^{\rm ar}$ and $G^{\rm rr}$
can be expressed in terms of $G^{\rm ra}$:

$\displaystyle G^{{\rm a}{\rm r}}(K)$ $\displaystyle=-(G^{{\rm r}{\rm
a}})^{\dagger}(K)=G^{{\rm r}{\rm a}}(-K)\,,$ (40) $\displaystyle G^{{\rm
r}{\rm r}}(K)$ $\displaystyle={\rm i}\left[G^{{\rm r}{\rm a}}(K)+G^{{\rm
a}{\rm r}}(K)\right]\,,$ (41)

as one can verify explicitly from eqs. (39) and (38). A particular useful form
for $G^{{\rm ra}}$ is that in a Laurent expansion:

$\displaystyle G^{\rm
ra}(K)=\sum_{m=\pm}\frac{R^{m}(\bm{k})}{\omega-\Omega^{m}(\bm{k})}\,,$ (42)

where $m=+,-$ labels two independent collective modes with

$\displaystyle\Omega^{\pm}({\bm{k}})$ $\displaystyle=-\frac{{\rm
i}}{2}(r+D_{\rm V}{\bm{k}}^{2}+D_{\rm
A}{\bm{k}}^{2})\pm\frac{1}{2}\sqrt{4({\bm{v}}\cdot\bm{k})^{2}-(r+D_{\rm
A}{\bm{k}}^{2}-D_{\rm V}{\bm{k}}^{2})^{2}}\,,$ (43) $\displaystyle
R^{\pm}(\bm{k})$ $\displaystyle=\frac{\pm
1}{\Omega_{+}(\bm{k})-\Omega_{-}(\bm{k})}\left(\begin{array}[]{cc}\Omega_{\pm}(\bm{k})+{\rm
i}(r+D_{\rm A}{\bm{k}}^{2})&{\bm{v}}\cdot{\bm{k}}\\\
{\bm{v}}\cdot{\bm{k}}&\Omega_{\pm}(\bm{k})+{\rm i}D_{\rm
V}{\bm{k}}^{2}\end{array}\right)\,.$ (46)

In the limit $r=0$ and for $\bm{k}\cdot\hat{{\bm{B}}}>0$, these two modes
$(m=+,-)$ correspond to the CMW propagating in the same/opposite directions to
the magnetic field, respectively.

Next, we define the interaction vertices from ${\cal L}_{3}$ as

$\displaystyle{\rm i}\int_{x}({\mathcal{L}})_{3}$
$\displaystyle=T\sum_{\alpha,\beta,\gamma}g_{\alpha}\int{\rm
d}t\int_{{\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3}}\left(U_{\alpha\beta\gamma}^{{\bm{k}}_{1};{\bm{k}}_{2},{\bm{k}}_{3}}\psi_{\alpha}^{{\bm{k}}_{1}}\lambda_{\beta}^{{\bm{k}}_{2}}\lambda_{\gamma}^{{\bm{k}}_{3}}+V_{\alpha\beta\gamma}^{{\bm{k}}_{1},{\bm{k}}_{2};{\bm{k}}_{3}}\psi_{\alpha}^{{\bm{k}}_{1}}\psi_{\beta}^{{\bm{k}}_{2}}\lambda_{\gamma}^{{\bm{k}}_{3}}\right)\,,$
(47)

where $\psi^{{\bm{k}}}_{\alpha}\equiv\psi(t,{\bm{k}})$ and
$\lambda^{{\bm{k}}}_{\alpha}\equiv\lambda(t,{\bm{k}})$. There are two types of
vertices: $U$ couples one a-field with two r-fields; $V$ couples two a-fields
with one r-field. They can be read from the cubic action (2.2)

$\displaystyle U_{{\rm
V};\alpha\beta}^{{\bm{k}}_{1};{\bm{k}}_{2},{\bm{k}}_{3}}$
$\displaystyle=\frac{-{\rm i}D_{\rm
V}}{2}\left(\begin{array}[]{cc}\sigma_{{\rm V};{\rm
V}}{\bm{k}}^{2}_{1}&-\sigma_{{\rm V};{\rm A}}{\bm{k}}_{1}\cdot{\bm{k}}_{2}\\\
-\sigma_{{\rm V};{\rm
A}}{\bm{k}}_{1}\cdot{\bm{k}}_{3}&0\end{array}\right)+\frac{{\rm i}}{2}D_{\rm
V}n_{{\rm V};\alpha\beta}{\bm{k}}^{2}_{1}+\frac{1}{2}{\bm{v}}_{\rm
A}\cdot{\bm{k}}_{1}n_{{\rm A};\alpha\beta}\,,$ (48c) $\displaystyle U_{{\rm
A};\alpha\beta}^{{\bm{k}}_{1};{\bm{k}}_{2},{\bm{k}}_{3}}$
$\displaystyle=\frac{-{\rm i}D_{\rm
A}}{2}\left(\begin{array}[]{cc}0&-\sigma_{{\rm A};{\rm
V}}{\bm{k}}_{1}\cdot{\bm{k}}_{3}\\\ -\sigma_{{\rm A};{\rm
V}}{\bm{k}}_{1}\cdot{\bm{k}}_{2}&\sigma_{{\rm A};{\rm
A}}{\bm{k}}_{1}^{2}\end{array}\right)+\frac{{\rm i}}{2}D_{\rm A}n_{{\rm
A};\alpha\beta}{\bm{k}}_{1}^{2}+\frac{1}{2}{\bm{v}}_{\rm
V}\cdot{\bm{k}}_{1}n_{{\rm V};\alpha\beta}$ (48f)
$\displaystyle\quad-\frac{{\rm i}r}{2}\left(\begin{array}[]{cc}0&\gamma_{;{\rm
V}}\\\ \gamma_{;{\rm V}}&2\gamma_{;{\rm A}}\end{array}\right)+\frac{{\rm
i}r}{2}n_{{\rm A};\alpha\beta}\,,$ (48i) $\displaystyle V_{{\rm
V}\alpha;\beta}^{{\bm{k}}_{1},{\bm{k}}_{2};{\bm{k}}_{3}}$
$\displaystyle=D_{\rm
V}{\bm{k}}_{1}\cdot{\bm{k}}_{2}\left(\begin{array}[]{cc}\sigma_{{\rm V};{\rm
V}}&\sigma_{{\rm V};{\rm A}}\\\ 0&0\end{array}\right)\,,$ (48l) $\displaystyle
V_{{\rm A}\alpha;\beta}^{{\bm{k}}_{1},{\bm{k}}_{2};{\bm{k}}_{3}}$
$\displaystyle=D_{\rm
A}{\bm{k}}_{1}\cdot{\bm{k}}_{2}\left(\begin{array}[]{cc}0&0\\\ \sigma_{{\rm
A};{\rm V}}&\sigma_{{\rm A};{\rm
A}}\end{array}\right)-r\left(\begin{array}[]{cc}0&0\\\ \gamma_{;{\rm
V}}&\gamma_{;{\rm A}}\end{array}\right)\,.$ (48q)

Note, the first term in the second line of eq. (48f) and the last term in eq.
(48q) arise from the fact that the axial damping coefficient $\gamma$ depends
on $n_{\rm V}$ and $n_{\rm A}$.

The graphic presentations of
$U^{{\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3}}_{\alpha\beta\gamma}$ and
$V^{{\bm{k}}_{1},{\bm{k}}_{2},{\bm{k}}_{3}}_{\alpha\beta\gamma}$ are shown in
figures 1 (c) and (d), respectively. With the propagators and vertices at
hand, we are ready to compute one-loop corrections to the conductivity.

##  3 Conductivity at finite axial relaxation rate

###  3.1 Conductivity

From the symmetrized correlator of the vector current $J^{\mu}$, ${\cal
C}^{\mu\nu}\sim\langle J^{\mu}J^{\nu}\rangle$, we can determine the (AC)
conductivity tensor through the standard Kubo formula

$\displaystyle\sigma^{ij}(\omega)=\lim_{{\bm{k}}\rightarrow{\bm{0}}}\frac{1}{2T}\mathcal{C}^{ij}(K)\,.$
(49)

Alternatively, we can extract $\sigma^{ij}$ from the retarded correlator (see
eq. (100) in section 4). The conductivity tensor in the presence of the
external magnetic field can be decomposed as

$\displaystyle\sigma^{ij}=\sigma_{\parallel}\hat{B}^{i}\hat{B}^{j}+\sigma_{\perp}(\delta^{ij}-\hat{B}^{i}\hat{B}^{j})\,,$
(50)

where $\sigma_{\parallel}$ and $\sigma_{\perp}$ are the longitudinal and
transverse conductivity, respectively. In this work, we will not consider the
Hall conductivity. From the Ward-Takahashi identity, we also have

$\displaystyle\lim_{{\bm{k}}\rightarrow{\bm{0}}}\frac{1}{2T}\frac{\omega^{2}}{{\bm{k}}^{2}}\mathcal{{\cal
C}}^{00}(K)=\lim_{{\bm{k}}\rightarrow{\bm{0}}}\frac{1}{2T}\hat{k}^{i}\hat{k}^{j}\mathcal{C}^{ij}(K)=\sigma_{\perp}(\omega)+[\sigma_{\parallel}(\omega)-\sigma_{\perp}(\omega)](\hat{\bm{k}}\cdot\hat{\bm{B}})^{2}\,,$
(51)

which allows us to extract $\sigma_{\parallel}$ and $\sigma_{\perp}$ from the
small $k$ behavior of ${\cal C}^{00}$.

In what follows, we determine ${\cal C}^{00}$ and hence $\sigma^{ij}(\omega)$
from the relation,

$\displaystyle{\cal C}^{00}=T\chi_{\rm V}{\cal G}_{\rm VV}^{\rm rr}\,,$ (52)

which follows from the definitions of the rescaled fields, eq. (20). At tree
level, an explicit calculation using eq. (39) yields

$\displaystyle\lim_{\bm{k}\to{\bm{0}}}\frac{\omega^{2}}{2{\bm{k}}^{2}}G^{\rm
rr}_{\rm VV}(K)=D_{\rm V}+\frac{{\bm{v}}^{2}r}{\omega^{2}+r^{2}}\,.$ (53)

It then follows from eqs. (51) and (50) that

$\displaystyle(\sigma_{\parallel})_{{\rm tree}}(\omega)=(\sigma_{\rm
V})_{0}+\frac{C^{2}{\bm{B}}^{2}r}{\chi_{\rm
A}(r^{2}+\omega^{2})}\,,\quad(\sigma_{\perp})_{{\rm
tree}}(\omega)=(\sigma_{\rm V})_{0}\,,$ (54)

which reproduces the well-known CME-induced negative magnetoresistance
Son:2012bg . However, the tree level result (54) does not take into account
the fluctuation effects. We shall study the one-loop corrections to
$\sigma^{ij}$ by computing ${\cal G}^{\rm rr}_{\rm VV}$ dressed by the self-
energies.

###  3.2 Self-energies

Figure 2: Diagrammatic representations of the Dyson equations: (a) for the
$\mathcal{G}^{{\rm r}{\rm a}}$ propagator (62); (b) for the $\mathcal{G}^{{\rm
r}{\rm r}}$ propagator (63). The last expression on (b) is the first-order
expansion with respect to the self-energies. Here, double lines represent the
full propagators. The white bubbles and the bubble with diagonal lines
correspond to $\Sigma^{{\rm a}{\rm a}}$ and $\Sigma^{{\rm a}{\rm r}}$,
respectively. The cross vertex denotes $-N$, which is related to the free
propagators by eq. (39). Note that $(\alpha,\beta)$ labels are suppressed in
these diagrams.

We start from the full propagators which are dressed by self-energies
$\Sigma^{\rm ar},\Sigma^{\rm ra}$, and $\Sigma^{\rm aa}$ through the Dyson
equation:

$\displaystyle\left(\begin{array}[]{cc}{\cal G}^{\rm rr}&{\cal G}^{\rm ra}\\\
{\cal G}^{\rm ar}&0\end{array}\right)^{-1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}G^{\rm rr}&G^{\rm ra}\\\ G^{\rm
ar}&0\end{array}\right)^{-1}-\left(\begin{array}[]{cc}0&\Sigma^{{\rm r}{\rm
a}}\\\ \Sigma^{{\rm a}{\rm r}}&\Sigma^{{\rm r}{\rm r}}\end{array}\right)\,,$
(61)

or equivalently,

$\displaystyle{\cal G}^{{\rm r}{\rm a}}$ $\displaystyle=[(G^{{\rm r}{\rm
a}})^{-1}-\Sigma^{{\rm a}{\rm r}}]^{-1}=(-{\rm i}M-\Sigma^{{\rm a}{\rm
r}})^{-1}\,,$ (62) $\displaystyle{\cal G}^{{\rm r}{\rm r}}$
$\displaystyle=-{\cal G}^{{\rm r}{\rm a}}(N-\Sigma^{{\rm a}{\rm a}}){\cal
G}^{\rm ar}\,.$ (63)

See figures 2 (a) and (b) for graphical representations of eqs. (62) and (63),
respectively.

To determine the small $k$ behavior of ${\cal G}^{\rm rr}_{\rm VV}$ from eq.
(63), we shall first consider the behavior of the self-energies in this limit.
We note $\Sigma^{\rm aa}_{\alpha\beta}$ can be connected to two external legs
associated with $\psi_{\alpha}$ and $\psi_{\beta}$. Since $\psi_{\rm V}$ is
always combined with ${\bm{\nabla}}$ in the cubic Lagrangian density $({\cal
L})_{3}$ in eq. (2.2), we have $\Sigma^{\rm aa}_{\alpha\beta}\sim
k^{\delta_{\alpha{\rm V}}+\delta_{\beta{\rm V}}}$ and $\Sigma^{\rm
ar}_{\alpha\beta}\sim k^{\delta_{\alpha{\rm V}}}$, where
$\delta_{\alpha\beta}$ is the Kronecker delta. Then, the relevant components
of the self-energies can be parametrized as

$\displaystyle(\chi_{\rm V})_{0}\Sigma^{{\rm a}{\rm a}}_{{\rm V}{\rm V}}(K)$
$\displaystyle=-2\Delta\sigma_{\perp}(\omega){\bm{k}}^{2}-2\left[\Delta\sigma_{\parallel}(\omega)-\Delta\sigma_{\perp}(\omega)\right]({\bm{k}}\cdot\hat{\bm{B}})^{2}+\ldots\,,$
(64a) $\displaystyle(\chi_{\rm A})_{0}\Sigma^{{\rm a}{\rm a}}_{\rm AA}(K)$
$\displaystyle=-2\Delta\gamma(\omega)+\ldots\,,$ (64b)
$\displaystyle\Sigma^{{\rm a}{\rm r}}_{\rm AA}(K)$ $\displaystyle=-{\rm
i}\Delta r(\omega)+\ldots\,,$ (64c) $\displaystyle\Sigma_{{\rm V}{\rm
A}}^{{\rm a}{\rm r}}(K)$ $\displaystyle={\bm{k}}\cdot\Delta{\bm{v}}_{\rm
A}(\omega)+\ldots\,,$ (64d)

where $\ldots$ are the terms suppressed by small $k$. Note that
$\Delta\gamma,\Delta r$, and $\Delta{\bm{v}}_{\rm A}$ can be viewed as the
finite frequency corrections to $\gamma,r$, and ${\bm{v}}$, respectively. To
see this, one should keep in mind that $\Sigma^{\rm aa}$ and $\Sigma^{\rm ar}$
enter as the correction to $-N$ and ${\rm i}M$, respectively, in eqs. (63) and
(62) while $(\chi_{\rm A})_{0}N_{\rm AA}=2\gamma+{\cal O}(k^{2})\,,M_{\rm
AA}=-r+{\cal O}(k^{2})$, and $M_{\rm VA}=-{\rm
i}(\bm{k}\cdot{\bm{v}})+\mathcal{O}(k^{2})$.

By substituting eq. (64) into the Dyson equation (63) and evaluate the Kubo
relation (51) with (52), we eventually find

$\displaystyle\sigma_{\perp}(\omega)$ $\displaystyle=(\sigma_{{\rm
V}})_{0}+\Delta\sigma_{\perp}(\omega)\,,$ (65a)
$\displaystyle\sigma_{\parallel}(\omega)$ $\displaystyle=(\sigma_{{\rm
V}})_{0}+\Delta\sigma_{\parallel}(\omega)+\frac{[({\bm{v}})_{0}+\Delta{\bm{v}}_{\rm
A}(\omega)]^{2}[(\chi_{\rm A}r)_{0}+\Delta\gamma(\omega)](\chi_{\rm
V})_{0}}{(\chi_{\rm A})_{0}\left[\omega^{2}+\left((r)_{0}+\Delta
r(\omega)\right)^{2}\right]}\,,$ (65b)

meaning the loop corrections to the conductivity can be expressed in terms of
the following $\omega$-dependent functions:

$\displaystyle\Delta\sigma_{\perp}(\omega)\,,\quad\Delta\sigma_{\parallel}(\omega)\,,\quad\Delta\gamma(\omega)\,,\quad\Delta
r(\omega)\,,\quad\Delta{\bm{v}}_{\rm A}(\omega)\,,$ (66)

which we shall compute in section 3.4. Intuitively, we may understand eq. (65)
by replacing $(\sigma_{\rm V})_{0}$, $\gamma$, $r$, and ${\bm{v}}$ of eq. (54)
into those including fluctuation corrections (66).

###  3.3 One-loop

Figure 3: Self-energies at the one-loop order.

In this subsection, we provide general expressions for computing the self-
energies $\Sigma^{{\rm a}{\rm r}}_{\alpha\beta}$ and $\Sigma^{{\rm a}{\rm
a}}_{\alpha\beta}$ at the one-loop level. We will give the derivation for the
former and present only the results for the latter, which can be derived
similarly (see appendix B for details).

The self-energy $\Sigma^{{\rm a}{\rm r}}_{\alpha\beta}$ consists of three
pieces:

$\displaystyle\Sigma_{\alpha\beta}^{{\rm a}{\rm
r}}=(\Sigma_{\alpha\beta}^{{\rm a}{\rm r}})_{\rm
I}+(\Sigma_{\alpha\beta}^{{\rm a}{\rm r}})_{\rm
II}+(\Sigma_{\alpha\beta}^{{\rm a}{\rm r}})_{\rm III}\,,$ (67)

whose diagrammatic representations are given by figures 3 (a)–(c),
respectively. First of all, the contribution from $(\Sigma_{\alpha\beta}^{{\rm
a}{\rm r}})_{\rm III}$ in figure 3 (c) vanishes. This is because

$\displaystyle\int_{q^{0}}G_{\alpha\beta}^{\rm
ra}(q^{0},\bm{q}_{+})G_{\gamma\delta}^{\rm ar}(\omega-q^{0},-\bm{q}_{-})=0\,,$
(68)

where the integrand has poles only in the upper complex $q^{0}$-plane so that
the contour integral vanishes. Here and hereafter, we use the notation
$\bm{q}_{\pm}=\bm{q}\pm\bm{k}/2$.

For the first term in eq. (67), we have

$\displaystyle(\Sigma_{\alpha\beta}^{{\rm a}{\rm r}})_{\rm I}(K)$
$\displaystyle=4T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}g_{\alpha}U_{\alpha;\gamma\sigma}^{-{\bm{k}};-\bm{q}_{-},\bm{q}_{+}}G^{{\rm
rr}}_{\sigma\tau}(q^{0},\bm{q}_{+})G^{{\rm
ra}}_{\gamma\delta}(\omega-q^{0},-\bm{q}_{-})g_{\delta}U_{\delta;\beta\tau}^{\bm{q}_{-};{\bm{k}},-\bm{q}_{+}}$
$\displaystyle=4{\rm
i}T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}g_{\alpha}U_{\alpha;\gamma\sigma}^{-{\bm{k}};-\bm{q}_{-},\bm{q}_{+}}G^{{\rm
ra}}_{\sigma\tau}(q^{0},\bm{q}_{+})G^{{\rm
ra}}_{\gamma\delta}(\omega-q^{0},-\bm{q}_{-})g_{\delta}U_{\delta;\beta\tau}^{\bm{q}_{-};{\bm{k}},-\bm{q}_{+}}\,.$
(69)

To obtain the second line from the first line, we have replaced $G^{\rm
rr}(q^{0},{\bm{q}}_{+})$ with ${\rm i}G^{\rm ra}(q^{0},{\bm{q}}_{+})$ using
eq. (41) together with eq. (68). Turning to $(\Sigma^{{\rm a}{\rm
r}}_{\alpha\beta})_{\rm II}$, we have

$\displaystyle(\Sigma^{{\rm a}{\rm r}}_{\alpha\beta})_{\rm II}(K)$
$\displaystyle=2T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}g_{\alpha}U_{\alpha;\gamma\sigma}^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}G^{{\rm
ra}}_{\sigma\tau}(q^{0},\bm{q}_{+})G^{{\rm
ra}}_{\gamma\delta}(\omega-q^{0},-\bm{q}_{-})g_{\delta}V_{\delta\tau;\beta}^{\bm{q}_{-},-\bm{q}_{+};{\bm{k}}}\,.$
(70)

Adding $(\Sigma^{{\rm a}{\rm r}}_{\alpha\beta})_{\rm I}$ and $(\Sigma^{{\rm
a}{\rm r}}_{\alpha\beta})_{\rm II}$, the self-energy becomes

$\displaystyle\Sigma^{{\rm a}{\rm r}}_{\alpha\beta}(K)=T^{2}\int_{Q}{\rm
tr}\left[G^{\rm ra}(q^{0},\bm{q}_{+})(\Gamma_{\alpha}^{\rm a})^{\rm T}G^{\rm
ra}(\omega-q^{0},-\bm{q}_{-})\Gamma_{\beta}^{\rm r}\right]\,,$ (71)

where we have defined two-by-two matrices:

$\displaystyle(\Gamma^{\rm
a}_{\alpha})_{\beta\gamma}=g_{\alpha}U_{\alpha;\beta\gamma}^{-\bm{k};-\bm{q}_{-},\bm{q}_{+}}\,,\quad(\Gamma^{\rm
r}_{\beta})_{\gamma\delta}=4{\rm
i}g_{\gamma}U_{\gamma;\beta\delta}^{{\bm{q}}_{-};{\bm{k}},-{\bm{q}}_{+}}+2g_{\gamma}V_{\gamma\delta;\beta}^{\bm{q}_{-},-{\bm{q}}_{+};{\bm{k}}}\,.$
(72)

We now carry out the integration over $q^{0}$ in eq. (71) using the Cauchy
residue theorem, yielding

$\displaystyle\int_{q^{0}}G^{{\rm ra}}_{\sigma\tau}(q^{0},\bm{q}_{+})G^{{\rm
ra}}_{\gamma\delta}(\omega-q^{0},-\bm{q}_{-})=-{\rm
i}\sum_{m,m^{\prime}}R^{m}_{\sigma\tau}(\bm{q}_{+})R^{m^{\prime}}_{\gamma\delta}(-\bm{q}_{-})\Delta^{mm^{\prime}}(K,\bm{q})\,,$
(73)

where we have used eq. (42) and introduced the notation,

$\displaystyle\Delta^{mm^{\prime}}(K,\bm{q})\equiv\frac{1}{\omega-\Omega^{m}(\bm{q}_{+})-\Omega^{m^{\prime}}(-\bm{q}_{-})}\,.$
(74)

Substituting eq. (73) into eq. (71) leads to the final result

$\displaystyle\Sigma^{{\rm a}{\rm r}}_{\alpha\beta}(K)=-{\rm
i}T^{2}\sum_{m,m^{\prime}}\int_{\bm{q}}F^{{\rm a}{\rm
r},mm^{\prime}}_{\alpha\beta}({\bm{k}},{\bm{q}})\Delta^{mm^{\prime}}(K,\bm{q})\,,$
(75)

where we have defined

$\displaystyle F^{{\rm a}{\rm
r},mm^{\prime}}_{\alpha\beta}({\bm{k}},{\bm{q}})={\rm
tr}\left[R^{m}(\bm{q}_{+})(\Gamma_{\alpha}^{\rm a})^{\rm
T}R^{m^{\prime}}(-\bm{q}_{-})\Gamma^{\rm r}_{\beta}\right]\,.$ (76)

One can also analyze $\Sigma^{\rm aa}_{\alpha\beta}$ following the similar
treatment (see appendix B for details). For our purpose, only the diagonal
components of $\Sigma^{\rm aa}_{\alpha\beta}$ are needed; they read

$\displaystyle\Sigma^{{\rm a}{\rm a}}_{\alpha\alpha}(K)$
$\displaystyle=T^{2}\,{\rm Im}\sum_{m,m^{\prime}}\int_{\bm{q}}F^{{\rm a}{\rm
a},mm^{\prime}}_{\alpha\alpha}({\bm{k}},{\bm{q}})\Delta^{mm^{\prime}}(K,\bm{q})\,,$
(77)

where we have defined

$\displaystyle F^{{\rm a}{\rm
a},mm^{\prime}}_{\alpha\beta}({\bm{k}},{\bm{q}})={\rm
tr}\left[R^{m}(\bm{q}_{+})(\Gamma^{\rm a}_{\alpha})^{\rm
T}R^{m^{\prime}}(-\bm{q}_{-})\Gamma^{{}^{\prime}{\rm a}}_{\beta}\right]\,,$
(78)

with

$\displaystyle(\Gamma^{{}^{\prime}{\rm
a}}_{\beta})_{\gamma\delta}=4\left(-g_{\beta}U_{\beta;\gamma\delta}^{\bm{k};\bm{q}_{-},-\bm{q}_{+}}+2{\rm
i}g_{\gamma}V_{\gamma\beta;\delta}^{\bm{q}_{-},\bm{k};-\bm{q}_{+}}\right)\,.$
(79)

Expressions (75) for $\Sigma^{{\rm a}{\rm r}}_{\alpha\beta}$ and (77) for
$\Sigma^{{\rm a}{\rm a}}_{\alpha\alpha}$ are the main results of this
subsection. These are to be used in the following subsection.

###  3.4 Results

We now compute one-loop contributions to the conductivity tensor. We first
note that in the absence of the background axial charge density,

$\displaystyle n_{{\rm A};{\rm A}{\rm A}}=n_{{\rm V};{\rm V}{\rm A}}=n_{{\rm
V};{\rm A}{\rm V}}=n_{{\rm A};{\rm V}{\rm V}}=0,\quad\sigma_{{\rm V};{\rm
A}}=\sigma_{{\rm A};{\rm A}}=0,\quad\gamma_{;{\rm A}}=0\,.$ (80)

Let us further assume, for the present illustrative purpose,

$\displaystyle\chi_{\rm V}=\chi_{\rm A}=\chi,\quad\sigma_{\rm V}=\sigma_{\rm
A}=\sigma\,.$ (81)

Then, the vertices are parameterized by three independent parameters:

$\displaystyle u_{n}\equiv n_{{\rm V};{\rm V}{\rm V}}=n_{{\rm V};{\rm A}{\rm
A}}=n_{{\rm A};{\rm V}{\rm A}}=n_{{\rm A};{\rm A}{\rm V}}\,,\quad
u_{\sigma}\equiv\sigma_{{\rm V};{\rm V}}=\sigma_{{\rm A};{\rm V}}\,,\quad
u_{\gamma}\equiv\gamma_{;{\rm V}}\,.$ (82)

In this subsection, we will drop the subscript in $D,{\bm{v}}$, and $g$
because of eq. (81).

To evaluate eqs. (75) and (77), we first consider relevant $\Gamma^{\rm
a}_{\alpha},\Gamma^{{}^{\prime}{\rm a}}_{\alpha},$ and $\Gamma^{\rm
r}_{\alpha}$, which, in the small $k$ limit, are simplified as

$\displaystyle\Gamma_{\rm V}^{\rm a}$
$\displaystyle=-\frac{g}{2}({\bm{v}}\cdot{\bm{k}})\left(\begin{array}[]{cc}0&u_{n}\\\
u_{n}&0\end{array}\right)+\mathcal{O}(k^{2})\,,$ (83c)
$\displaystyle\Gamma_{\rm A}^{\rm a}$ $\displaystyle=\frac{{\rm
i}gr}{2}\left(\begin{array}[]{cc}0&u_{n}-u_{\gamma}\\\
u_{n}-u_{\gamma}&0\end{array}\right)+\mathcal{O}(k)\,,$ (83f)
$\displaystyle\Gamma_{\rm A}^{{}^{\prime}{\rm a}}$ $\displaystyle=-2{\rm
i}gr\left(\begin{array}[]{cc}0&u_{n}-u_{\gamma}\\\
u_{n}+3u_{\gamma}&0\end{array}\right)+\mathcal{O}(k)\,,$ (83i)
$\displaystyle\Gamma_{\rm V}^{{}^{\prime}{\rm a}}$ $\displaystyle=8{\rm
i}gD(\bm{k}\cdot{\bm{q}})\left(\begin{array}[]{cc}u_{\sigma}&0\\\
0&0\end{array}\right)-2g({\bm{v}}\cdot\bm{k})\left(\begin{array}[]{cc}0&u_{n}\\\
u_{n}&0\end{array}\right)+\mathcal{O}(k^{2})\,,$ (83n)
$\displaystyle\Gamma_{\rm A}^{{\rm r}}$
$\displaystyle=-2g{\bm{q}}^{2}D\left(\begin{array}[]{cc}0&u_{n}\\\
u_{n}&0\end{array}\right)+2{\rm
i}g({\bm{v}}\cdot{\bm{q}})\left(\begin{array}[]{cc}u_{n}&0\\\
0&u_{n}\end{array}\right)+2gr\left(\begin{array}[]{cc}0&0\\\
u_{\gamma}-u_{n}&0\end{array}\right)+{\cal O}(k)\,.$ (83u)

Applying eq. (83) to $F^{{\rm a}{\rm a},mm^{\prime}}_{\alpha\beta}$ and
$F^{{\rm a}{\rm r},mm^{\prime}}_{\alpha\beta}$, defined in eqs. (78) and (76),
respectively, one can directly verify the parametric behaviors:

$\displaystyle F_{{\rm V}{\rm V}}^{{\rm a}{\rm a}}$ $\displaystyle\sim
k^{2}\,,\quad F_{{\rm V}{\rm A}}^{{\rm a}{\rm r}}\sim k\,,\quad F_{{\rm A}{\rm
A}}^{{\rm a}{\rm a}}\sim F_{{\rm A}{\rm A}}^{{\rm a}{\rm r}}\sim k^{0}\,\quad\
(m\neq m^{\prime})\,,$ (84a) $\displaystyle F_{{\rm V}{\rm V}}^{{\rm a}{\rm
a}}$ $\displaystyle\sim k^{3}\,,\quad F_{{\rm V}{\rm A}}^{{\rm a}{\rm r}}\sim
k^{2}\,,\quad F_{{\rm A}{\rm A}}^{{\rm a}{\rm a}}\sim F_{{\rm A}{\rm A}}^{{\rm
a}{\rm r}}\sim k\,\quad\ (m=m^{\prime})\,.$ (84b)

Comparing eq. (84) with eq. (64), we observe that only the contribution from
$m\neq m^{\prime}$ are needed to evaluate the functions listed in eq. (66).
Therefore, we can replace $\Delta^{mm^{\prime}}$ in eqs. (75) and (77) with
their expression in the limit ${\bm{k}}\rightarrow{\bm{0}}$:

$\displaystyle\lim_{{\bm{k}}\to{\bm{0}}}\Delta^{+-}(K,\bm{q})=\lim_{{\bm{k}}\to{\bm{0}}}\Delta^{-+}(K,\bm{q})=\frac{1}{\omega+{\rm
i}r+2{\rm i}D{\bm{q}}^{2}}\,.$ (85)

We first evaluate $\Sigma^{{\rm a}{\rm a}}_{\rm VV}$ which in turn determines
$\Delta\sigma_{\parallel}(\omega)$ and $\Delta\sigma_{\perp}(\omega)$. From
eqs. (83) and (78), we have

$\displaystyle F_{\rm VV}^{{\rm a}{\rm a},+-}+F_{\rm VV}^{{\rm a}{\rm
a},-+}=2g^{2}u_{n}^{2}({\bm{v}}\cdot\bm{k})^{2}+{\cal O}(k^{3})\,,$ (86)

so that eq. (77) becomes

$\displaystyle\Sigma_{\rm VV}^{{\rm a}{\rm
a}}(K)=2g^{2}T^{2}u_{n}^{2}({\bm{v}}\cdot\bm{k})^{2}\,{\rm
Im}\int_{\bm{q}}\frac{1}{\omega+{\rm i}r+2{\rm i}D{\bm{q}}^{2}}+{\cal
O}(k^{4})\,.$ (87)

Comparing eq. (87) with eq. (64a), we find

$\displaystyle\Delta\sigma_{\parallel}(\omega)=-\frac{g^{2}T^{2}\chi
u_{n}^{2}{\bm{v}}^{2}}{8\pi D}\,{\rm Re}\sqrt{\frac{r-{\rm
i}\omega}{2D}}\,,\quad\Delta\sigma_{\perp}(\omega)=0\,.$ (88)

Here and below, we use the dimensional regularization to perform the
integration over ${\bm{q}}$,

$\displaystyle\int_{\bm{q}}\frac{1}{\omega+{\rm i}r+2{\rm
i}D{\bm{q}}^{2}}=\frac{{\rm i}}{8\pi D}\sqrt{\frac{r-{\rm
i}\omega}{2D}}\,,\quad\,\int_{\bm{q}}\frac{{\bm{q}}^{2}}{\omega+{\rm i}r+2{\rm
i}D{\bm{q}}^{2}}=\frac{-{\rm i}}{8\pi D}\left(\frac{r-{\rm
i}\omega}{2D}\right)^{\frac{3}{2}}\,.$ (89)

Evaluating loop integrals using dimensional regularization is essentially
picking up the contribution from the IR scale, which only depends on the
physical inputs but does not depend on the unphysical UV cut-off $\Lambda$ of
the EFT (see, e.g., ref. Manohar:2018aog ). For instance, in eqs. (89), the IR
momentum scale is given by

$\displaystyle q^{*}_{r}(\omega)=\left|\sqrt{\frac{r-{\rm
i}\omega}{2D}}\right|\,,$ (90)

which arises from the competition between axial relaxation and diffusion at
finite $r$. Such IR momentum scale can be interpreted as the characteristic
momentum of fluctuation modes and has to be parametrically smaller than the
cut-off scale of the EFT, which, in the present case, is $\Lambda\sim
l^{-1}_{{\rm mfp}}$. Here, $l_{{\rm mfp}}$ denotes the mean free path. Indeed,
from $D\sim l_{{\rm mfp}}$, it is easy to verify that $q^{*}_{r}\ll\Lambda$.
Note that at $r=0$, $q^{*}_{r}\sim\sqrt{\omega/D}$, which leads to
hydrodynamic long-time tail behavior POMEAU197563 ; Kovtun:2003vj ;
Kovtun:2012rj .

Turning to $\Sigma^{\rm aa}_{\rm AA}$, a similar treatment results in

$\displaystyle\Sigma_{\rm AA}^{{\rm a}{\rm
a}}(\omega,{\bm{0}})=2g^{2}T^{2}r^{2}(u_{n}^{2}-u_{\gamma}^{2})\,{\rm
Im}\int_{\bm{q}}\frac{1}{\omega+{\rm i}r+2{\rm i}D{\bm{q}}^{2}}\,.$ (91)

Therefore, $\Delta\gamma(\omega)$, defined by eq. (64b), reads

$\displaystyle\Delta\gamma(\omega)=-\frac{g^{2}T^{2}\chi
r^{2}(u_{n}^{2}-u_{\gamma}^{2})}{8\pi D}\,{\rm Re}\sqrt{\frac{r-{\rm
i}\omega}{2D}}\,.$ (92)

We shall interpret our results for $\Delta\gamma(\omega)$ at the end of this
section.

Finally, we turn to $\Sigma_{\rm VA}^{{\rm a}{\rm r}}$ and $\Sigma_{\rm
AA}^{{\rm a}{\rm r}}$, which can be obtained from the familiar steps. They are

$\displaystyle\Sigma_{\rm AA}^{{\rm a}{\rm r}}(\omega,{\bm{0}})$
$\displaystyle=-g^{2}T^{2}r(u_{n}-u_{\gamma})\int_{\bm{q}}\frac{r(u_{n}-u_{\gamma})+D{\bm{q}}^{2}(2u_{n})}{\omega+{\rm
i}r+2{\rm i}D{\bm{q}}^{2}}\,,$ (93) $\displaystyle\Sigma^{{\rm a}{\rm r}}_{\rm
VA}(\omega,{\bm{k}})$ $\displaystyle=-{\rm
i}g^{2}T^{2}u_{n}({\bm{v}}\cdot{\bm{k}})\int_{\bm{q}}\frac{r(u_{n}-u_{\gamma})+D{\bm{q}}^{2}(2u_{n})}{\omega+{\rm
i}r+2{\rm i}D{\bm{q}}^{2}}+\mathcal{O}(k^{2})\,,$ (94)

which, upon using eqs. (64c) and (64d), gives the expressions,

$\displaystyle\Delta r(\omega)$
$\displaystyle=-\frac{g^{2}T^{2}r(u_{n}-u_{\gamma})(ru_{\gamma}-{\rm i}\omega
u_{n})}{8\pi D}{\rm Re}\sqrt{\frac{r-{\rm i}\omega}{2D}}\,,$ (95)
$\displaystyle{\bm{v}}_{{\rm A}}(\omega)$
$\displaystyle=-\frac{g^{2}T^{2}{\bm{v}}u_{n}(ru_{\gamma}-{\rm i}\omega
u_{n})}{8\pi D}{\rm Re}\sqrt{\frac{r-{\rm i}\omega}{2D}}\,.$ (96)

We now have all the ingredients needed to compute the conductivity from eq.
(65). By the substitution of eqs. (88), (92), (95), and (96) into eq. (65), we
find $\sigma_{\perp}$ does not receive any loop corrections while
$\sigma_{\parallel}$ at the one-loop order is given by

$\displaystyle\sigma_{\parallel}(0)$
$\displaystyle=(\sigma)_{0}+\frac{C^{2}{\bm{B}}^{2}}{\chi
r}\left[1-\frac{g^{2}T^{2}\left(2u_{n}^{2}+u_{\gamma}^{2}\right)}{4\pi}\left(\frac{r}{2D}\right)^{\frac{3}{2}}\right]\,,$
(97)

where the first and second terms in $\left[\ldots\right]$ correspond to the
CME related tree contribution (see eq. (54)) and the one-loop correction,
respectively. Note that the fluctuation correction to $\sigma_{\parallel}$ is
of the opposite sign to the tree one and lead to a positive-magnetoresistance
contribution. Equation (97) is the main result of this section (see section 5
for further discussions).

Before closing this section, we remark that although the loop corrections to
$\sigma$ are resulting from the combination of the set of functions listed in
eq. (66), our expression for each of them might be of interest on its own. Let
us give two examples below.

For the first example, we recall that for a chiral medium microscopically
described by non-Abelian gauge theories, $\gamma$ is referred to as Chern-
Simons (CS) diffusion rate. In the weak-coupling regime, the CS diffusion rate
is computed by accounting for contributions at the microscopic length scale
$g^{2}T$ and time scale $g^{4}T$ Arnold:1996dy . Our result (92) can be
interpreted as the additional contribution to the CS diffusive rate from the
macroscopic length scale $\sim\sqrt{r/D}$.

As for the second example, we consider the pole of $G^{\rm ra}_{\rm AA}$ in
the limit ${\bm{k}}\rightarrow{\bm{0}}$, which, at the tree level, is located
at $\omega=-{\rm i}r$. The one-loop corrections to this relaxation pole can be
determined using the Dyson equation (62),

$\displaystyle(\mathcal{G}^{{\rm r}{\rm a}}_{{\rm A}{\rm
A}})^{-1}(\omega,{\bm{0}})=0\longrightarrow\omega+{\rm i}[r+\Delta
r(\omega)]=0\,.$ (98)

Upon substituting eq. (95), we find

$\displaystyle\omega=-{\rm i}r+\mathcal{O}(g^{4})\,.$ (99)

This should be contrasted with the naive expectation that the correction is of
the order $g^{2}$.

##  4 Conductivity at zero relaxation rate

To complement the analysis in the previous section, we shall work on the limit
that the axial relaxation rate $\gamma$ vanishes in this section. By sending
$r\to 0$ in the tree-level result (54), we notice that the CME-related
contribution to the conductivity vanishes. However, as we shall show below,
there is a CME-related contribution to the conductivity due to the effects of
the fluctuations.

Instead of extracting conductivity tensor $\sigma^{ij}$ from relevant self-
energies as was done in the previous section and in appendix D, in this
section, we will employ a complementary approach which determines
$\sigma^{ij}$ from the retarded correlator of the vector current:

$\displaystyle\sigma^{ij}(\omega)=\lim_{\bm{k}\to{\bm{0}}}\frac{1}{\omega}{\rm
Im}\,{{\cal C}}^{ij}_{\rm R}(K)\,.$ (100)

Here, we have used the fluctuation-dissipation relation

$\displaystyle{\cal C}^{ij}(K)=\frac{2T}{\omega}{\rm Im}\,{\cal C}^{ij}_{\rm
R}(K)\,.$ (101)

Let us begin with the retarded correlator expressed in terms of the vector
currents ${\bm{J}}^{{\rm r}}$ and ${\bm{J}}^{{\rm a}}$:

$\displaystyle{\cal C}^{ij}_{\rm R}(K)={\rm i}\langle J^{i,{\rm
r}}(K)J^{j,{\rm a}}(-K)\rangle\,.$ (102)

Here and hereafter, $\langle\ldots\rangle$ denotes the average weighted by the
path-integral (2) with $\gamma=0$. At one-loop order, ${\cal C}^{ij}_{\rm R}$
is given by the correlation of ${\bm{J}}^{\rm r}$ and ${\bm{J}}^{\rm a}$
expanded to the quadratic order in the fluctuating fields. In details, we use
the expressions given by (16) and (17) with ${\bm{E}}={\bm{0}}$ and have

$\displaystyle\left[C^{ij}_{\rm R}(K)\right]_{{\rm loop}}$ $\displaystyle={\rm
i}\langle(J^{i,{\rm r}})_{1}(K)\,(J^{j,{\rm a}})_{2}(-K)\rangle+{\rm
i}\langle(J^{i,{\rm r}})_{2}(K)\,(J^{j,{\rm a}})_{1}(-K)\rangle+{\rm
i}\langle(J^{i,{\rm r}})_{2}(K)\,(J^{j,{\rm a}})_{2}(-K)\rangle\,,$ (103)

where

$\displaystyle({\bm{J}}^{{\rm r}})_{1}$ $\displaystyle=C\bm{B}(\mu_{\rm
A})_{1}+{\cal O}(\bm{\nabla})\,,$ (104a) $\displaystyle({\bm{J}}^{{\rm
r}})_{2}$ $\displaystyle=C(\mu_{\rm A})_{2}\bm{B}-(\sigma_{\rm
V})_{1}{\bm{\nabla}}(\mu_{\rm V})_{1}+2{\rm i}T(\sigma_{\rm
A})_{1}{\bm{\nabla}}\psi_{\rm V}\,,$ (104b) $\displaystyle({\bm{J}}^{{\rm
a}})_{1}$ $\displaystyle=C\partial_{t}\psi_{\rm A}\bm{B}+{\cal
O}(\bm{\nabla})\,,$ (104c) $\displaystyle({\bm{J}}^{{\rm a}})_{2}$
$\displaystyle=\partial_{t}\left[(\sigma_{\rm V})_{1}{\bm{\nabla}}\psi_{\rm
V}\right]+{\cal O}(\bm{\nabla})\,.$ (104d)

Here, $\mathcal{O}(\bm{\nabla})$ denotes terms which would vanish in the limit
$\bm{k}\rightarrow{\bm{0}}$.

To proceed further, we first show that the first two terms in eq. (103) do not
contribute to eq. (102) for $\bm{k}\rightarrow{\bm{0}}$:

$\displaystyle\lim_{\bm{k}\to{\bm{0}}}\langle(J^{i,{\rm r}})_{1}(K)(J^{j,{\rm
a}})_{2}(-K)\rangle=0\,,\quad\lim_{\bm{k}\to{\bm{0}}}\langle(J^{i,{\rm
r}})_{2}(K)(J^{j,{\rm a}})_{1}(-K)\rangle=0\,.$ (105)

To see this, we consider

$\displaystyle\langle\lambda_{\alpha}(x)({\bm{J}}^{{\rm
a}})_{2}(x^{\prime})\rangle$ $\displaystyle=\sum_{\beta}\left(\int_{y}G^{\rm
rr}_{\alpha\beta}(x,y)\left<\frac{\delta I_{{\rm
int}}}{\delta\lambda_{\beta}(y)}({\bm{J}}^{{\rm
a}})_{2}(x^{\prime})\right>_{0}+G^{\rm
ra}_{\alpha\beta}(x,y)\left<\frac{\delta I_{{\rm
int}}}{\delta\psi_{\beta}(y)}({\bm{J}}^{{\rm
a}})_{2}(x^{\prime})\right>_{0}\right)$
$\displaystyle=\sum_{\beta}\int_{y}G^{\rm
ra}_{\alpha\beta}(x,y)\left<\frac{\delta I_{{\rm
int}}}{\delta\psi_{\beta}(y)}({\bm{J}}^{{\rm
a}})_{2}(x^{\prime})\right>_{0}\,.$ (106)

Here, $I_{\rm int}=\int_{x}({\cal L})_{3}$ and $\left<...\right>_{0}$ denotes
the average weighted by the Gaussian part of the effective action, as this
suffices to the present one-loop calculations. In eq. (4), we have also used

$\displaystyle\left<\frac{\delta I_{{\rm
int}}}{\delta\lambda_{\beta}(y)}({\bm{J}}^{{\rm
a}})_{2}(x^{\prime})\right>_{0}=0\,,$ (107)

which can be shown by causality (see ref. Gao:2018bxz  for a general
discussion) and the fact that both $\delta I_{{\rm int}}/\delta\psi_{\beta}$
and $({\bm{J}}^{{\rm a}})_{2}$ contain at least one power of the a-field. On
the other hand, in the absence of $\gamma$, one can confirm from eq. (2.1)
that $\delta I_{{\rm int}}/\delta\psi_{\beta}={\bm{\nabla}}\cdot({\bm{J}}^{\rm
r}_{\beta})_{2}$, meaning the Fourier transform of eq. (4) vanishes in the
small $k$ limit. Therefore, the first term in eq. (103), which can be
expressed as a linear combination of
$\langle\lambda_{\alpha}(x)\,({\bm{J}}^{\rm a})_{2}(x^{\prime})\rangle$,
should also vanish in this limit. A similar analysis applies to the second
term in eq. (103). As a consequence, at one-loop order, eq. (103) is reduced
to

$\displaystyle\lim_{\bm{k}\to{\bm{0}}}{\cal C}^{ij}_{\rm R}(K)={\rm
i}\lim_{\bm{k}\to{\bm{0}}}\left<(J^{i,{\rm r}})_{2}(K)(J^{j,{\rm
a}})_{2}(-K)\right>_{0}\,.$ (108)

To compute eq. (108), we rewrite $({\bm{J}}^{\rm r})_{2}$ and $({\bm{J}}^{\rm
a})_{2}$ in terms of the rescaled fields (20) using eq. (23):

$\displaystyle({\bm{J}}^{{\rm r}})_{2}$
$\displaystyle=-\frac{T{\bm{v}}}{2}\,\tilde{u}_{n}\left(\lambda^{2}_{\rm
V}+\lambda^{2}_{\rm A}\right)-TD_{\rm V}\tilde{u}_{\sigma}\lambda_{\rm
A}\left({\bm{\nabla}}\lambda_{\rm V}-2{\rm i}{\bm{\nabla}}\psi_{\rm
V}\right)\,,$ (109) $\displaystyle({\bm{J}}^{{\rm a}})_{2}$
$\displaystyle=D_{\rm V}\tilde{u}_{\sigma}\partial_{t}\left(\lambda_{\rm
A}{\bm{\nabla}}\psi_{\rm V}\right)\,,$ (110)

where we have further assumed the absence of the background vector chemical
potential,

$\displaystyle(n_{\rm V})_{0}=0\,,\quad(n_{\rm A})_{0}\neq 0\,,$ (111)

so that $n_{\rm A;VA}=n_{\rm A;AV}=0$ and $\sigma_{{\rm V};{\rm V}}=0$. Here,
the parameters which describe the strength of non-linearity are defined by

$\displaystyle\tilde{u}_{n}\equiv n_{\rm A;VV}=n_{\rm
V;AA}\,,\quad\tilde{u}_{\sigma}\equiv\sigma_{{\rm V};{\rm A}}\,.$ (112)

Noting the correlation between the last term in eq. (109) and eq. (110)
vanishes by causality, we now have two remaining contributions to eq. (108):
the correlations between $({\bm{J}}^{a})_{2}$ and the first term (CME part)
and the second term (diffusive part) of eq. (109). We shall first show that
the former contribution vanishes for $\bm{k}\rightarrow{\bm{0}}$. Indeed,

$\displaystyle\lim_{\bm{k}\to{\bm{0}}}\langle(\lambda_{\rm V}\lambda_{\rm
V})(K)(\lambda_{\rm A}{\bm{\nabla}}\lambda_{\rm V})(-K)\rangle$
$\displaystyle=\int_{\bm{q}}\sum_{m,m^{\prime}}{\rm i}{\bm{q}}\left[R^{m}_{\rm
VA}(\bm{q})R^{m^{\prime}}_{\rm VV}(-\bm{q})-R^{m}_{\rm
VV}(\bm{q})R^{m^{\prime}}_{\rm
VA}(-\bm{q})\right]\Delta^{mm^{\prime}}(\omega,\bm{q})\,,$ (113)

and similarly,

$\displaystyle\lim_{\bm{k}\to{\bm{0}}}\langle(\lambda_{\rm A}\lambda_{\rm
A})(K)(\lambda_{\rm A}{\bm{\nabla}}\lambda_{\rm V})(-K)\rangle$
$\displaystyle=\int_{\bm{q}}\sum_{m,m^{\prime}}{\rm i}{\bm{q}}\left[R^{m}_{\rm
AA}(\bm{q})R^{m^{\prime}}_{\rm AV}(-\bm{q})-R^{m}_{\rm
AV}(\bm{q})R^{m^{\prime}}_{\rm
AA}(-\bm{q})\right]\Delta^{mm^{\prime}}(\omega,\bm{q})\,.$ (114)

By evaluating eq. (46) at $r=0$ and further assuming $D_{\rm V}=D_{\rm A}$, we
find

$\displaystyle R^{\pm}(\bm{k})=\frac{1}{2}\left(\begin{array}[]{cc}1&\pm{\rm
sgn}({\bm{v}}\cdot\bm{k})\\\ \pm{\rm
sgn}({\bm{v}}\cdot\bm{k})&1\end{array}\right)\,.$ (117)

Substituting eq. (117) into eqs. (4) and (4) leads to

$\displaystyle\lim_{\bm{k}\to{\bm{0}}}\langle(\lambda^{2}_{\rm
V}+\lambda^{2}_{\rm A})(K)(\lambda_{\rm A}{\bm{\nabla}}\psi_{\rm
V})(-K)\rangle=0\,.$ (118)

The one-loop corrections to the conductivity tensor now becomes

$\displaystyle\left[\sigma^{ij}(0)\right]_{{\rm loop}}$
$\displaystyle=TD^{2}\tilde{u}^{2}_{\sigma}\,{\rm Im}\lim_{\omega\to
0}\lim_{\bm{k}\to{\bm{0}}}\left<(\lambda_{\rm A}\partial^{i}\lambda_{\rm
V})(K)(\lambda_{\rm A}\partial^{j}\psi_{\rm V})(-K)\right>$
$\displaystyle=TD^{2}\tilde{u}^{2}_{\sigma}\,{\rm
Im}\int_{Q}q^{i}q^{j}\left[G^{\rm rr}_{\rm AA}(Q)G^{\rm ra}_{\rm
VV}(-Q)-G^{\rm ra}_{\rm AV}(Q)G^{\rm rr}_{\rm VA}(-Q))\right]$
$\displaystyle=TD^{2}\tilde{u}^{2}_{\sigma}\,{\rm Im}\,{\rm
i}\int_{Q}q^{i}q^{j}\left[G^{\rm ra}_{\rm AA}(Q)G^{\rm ra}_{\rm VV}(-Q)-G^{\rm
ra}_{\rm AV}(Q)G^{\rm ra}_{\rm VA}(-Q)\right]$
$\displaystyle\,=TD^{2}\tilde{u}^{2}_{\sigma}\,{\rm
Im}\,\int_{\bm{q}}\sum_{m,m^{\prime}}q^{i}q^{j}\left[R^{m}_{\rm
AA}(\bm{q})R^{m^{\prime}}_{\rm VV}(-\bm{q})-R^{m}_{\rm
AV}(\bm{q})R^{m^{\prime}}_{\rm
VA}(-\bm{q})\right]\Delta^{mm^{\prime}}(0,\bm{q})\,.$ (119)

It is straightforward to show from eq. (117) that the terms inside $[...]$ of
eq. (4) vanish when $(m,m^{\prime})=(+,-),(-,+)$. On the other hand, the
contributions from $(m,m^{\prime})=(+,+),(-,-)$ give a finite result with

$\displaystyle\Delta^{mm}(0,{\bm{q}})=\frac{1}{2{\rm
i}D{\bm{q}}^{2}-2m|{{\bm{v}}\cdot{\bm{q}}}|}\,.$ (120)

Thus, we finally have

$\displaystyle\left[\sigma^{ij}(0)\right]_{{\rm loop}}$
$\displaystyle=-\frac{T(D\tilde{u}_{\sigma})^{2}}{D}\int_{{\bm{q}}}\frac{D^{2}{\bm{q}}^{2}q^{i}q^{j}}{2[(D{\bm{q}}^{2})^{2}+({\bm{v}}\cdot{\bm{q}})^{2}]}$
$\displaystyle=-\frac{T(D\tilde{u}_{\sigma})^{2}}{192\pi
D}\left(\frac{|{\bm{v}}|}{D}\right)^{3}\left[4\hat{v}^{i}\hat{v}^{j}+(\delta^{ij}-\hat{v}^{i}\hat{v}^{j})\right]\,.$
(121)

From the first line to the second line in eq. (4), we have first used the
dimensional regularization to integrate $q$ and then integrate over the angle
$\hat{{\bm{B}}}\cdot\hat{\bm{q}}$ analytically. The result of doing so gives
rise to an emergent IR scale,

$\displaystyle q_{*B}=\frac{|{\bm{v}}|}{D}\,.$ (122)

In our counting scheme ${\bm{B}}={\cal O}(\epsilon)$, it follows that
$|{\bm{v}}|\ll 1$, and one can again verify that $q_{*B}\ll\Lambda$.

Finally, by comparing eq. (4) with eq. (50), we find the main results of this
section,

$\displaystyle\frac{\Delta\sigma_{{\rm
V}\parallel}(0)}{(\sigma)_{0}}=\frac{4\Delta\sigma_{{\rm
V}\perp}(0)}{(\sigma)_{0}}=-\frac{(gT\tilde{u}_{\sigma})^{2}}{48\pi}\left(\frac{|{\bm{v}}|}{D}\right)^{3}\,.$
(123)

See the subsequent section for the discussion of eq. (123).

In ref. Delacretaz:2020jis , the authors consider the fluctuation effects of a
single chiral charge in the presence of the CME. For spatial dimension $d=3$,
they find finite corrections to the AC conductivity ($\omega\neq 0$) but
vanishing DC conductivity ($\omega=0$). The differences between theirs and the
present results mainly arise from the fact that we have considered the
couplings between the axial and vector charge densities (see also refs.
Kovtun:2014nsa ; Chen-Lin:2018kfl ; Mukerjee_2006  on the studies of
fluctuation dynamics with multiple conserved charges).

##  5 Discussion

###  5.1 Positive magnetoresistance

We presented in this paper the diagrammatic calculation of the modifications
of conductivity (the inverse of resistance) for a chiral medium with the
chiral magnetic effect (CME) based on the non-equilibrium effective field
theory (EFT) approach. We consider a generic vector charge density and axial
charge density as slow variables and study the intertwined effects from their
fluctuations and CME. For the first time, we obtain the CME-related
modifications to the conductivity tensors due to fluctuations for systems with
finite and vanishing axial relaxation rate $r$, as summarized in eq. (97) and
eq. (123), respectively.

Contrary to the common statement that the CME leads to a negative
magnetoresistance, we find that CME-related effects due to fluctuations give
rise to a positive magnetoresistance. Whether the net magnetoresistance is
positive or negative is determined by the competition between the two terms
inside $[...]$ in eq. (97). Note, to apply our one-loop results, we should
require the fluctuation contribution to be much smaller than the tree-level
expression, which includes both the first term and second term on the right-
hand side of eq. (97) (c.f. eq. (54)). Since we are working in the weak
${\bm{B}}$ limit, the tree-level contribution is dominated by $(\sigma)_{0}$,
which is indeed much larger than the one-loop correction, i.e., the third term
on the right-hand side of eq. (97). However, this does not necessarily mean
that the third term has to be smaller than the second term. Therefore, the net
CME-related contribution can in principle be dominated by the fluctuations
effects. We have made a number of simplifications in our analysis, but we hope
this qualitative feature might have some implication to real physical systems.
Interestingly, a positive magnetoresistance might have already been seen in
Weyl semimetals when the magnetic field is small (e.g., ref. Li:2014bha ).

It is truly striking that the CME contributes to the conductivity even in the
limit $r=0$ (see eq. (123)). This is in a marked difference from the result,
which does not account for fluctuations that the CME contribution vanishes in
this limit. Moreover, the fluctuation modifies both longitudinal and
transverse conductivities, while at finite $r$, only longitudinal conductivity
receives the correction from the CME.

The parametric behavior of the ratio of fluctuation effects to the tree-level
contribution is very instructive. From eqs. (97) and (123), we schematically
have

$\displaystyle\frac{\rm(Hydrodynamic\ fluctuation)}{\rm(Bare\
contribution)}\sim(Tu)^{2}g^{2}q_{*}^{3}\,,$ (124)

where $q_{*}$ is characteristic momentum of fluctuating modes. For the case
with $r\neq 0$, $q_{*}=\sqrt{r/D}$, resulting from the competition between the
diffusion of vector charge and the damping of the axial charge, whereas for
$r=0$, $q_{*}=|{\bm{v}}|/D$ originating from the competition between diffusion
and propagation of the chiral magnetic wave (CMW). Equation (124) indicates
that the relative importance of fluctuation corrections is determined by two
factors, a) the strength of non-linearity $(Tu)^{2}$ and b) the ratio between
the phase space volume of the long-wavelength fluctuating modes, $q^{3}_{*}$,
and that of the whole system, $g^{-2}=\chi T$.

Let us end this section by comparing the role the CME played in two cases
under study. For $r\neq 0$, the CME gives rise to non-linear couplings among
charge fluctuations but plays no role in determining the IR scale. In
contrast, for $r=0$ and at vanishing background vector charge, the non-
linearity relevant to the finite corrections to the conductivity solely comes
from the density-dependence of diffusive constant and conductivity but does
not rely on the CME. However, the competition between CMW propagation and
charge diffusion leads to the emergent IR scale. Due to the differences
explained above, $C|{\bm{B}}|$ dependence of $\Delta\sigma$ is different. The
correction to the conductivity scales as $(C{\bm{B}})^{2}$ in the former case
and scales as $(C|{\bm{B}}|)^{3}$ in the latter case.

###  5.2 Remarks on hydrodynamic fluctuations

Another motivation of the present study is to advance our understanding of
general aspects of hydrodynamic fluctuations. Here, we employ the recently
developed non-equilibrium EFT for the present studies. Our exercise here
demonstrates this EFT approach allows us to use powerful (and familiar) field
theory techniques to analyze hydrodynamic fluctuations. By construction, the
EFT automatically takes into account the constraints from symmetries. For
example, in appendix D, we show explicitly how fluctuations-dissipation
theorem and the Ward-Takahashi identity are satisfied at one-loop order. In
the traditional method, special care is needed to ensure those relations (see
the recent work Chao:2020kcf  for the former).

Remarkably, the fluctuation contribution to the conductivity is finite due to
the CME. This should be contrasted with the case of an ordinary fluid that
fluctuation corrections to transport coefficients (at zero frequency limit)
are typically zero. Such difference is related to the emergent IR momentum
scale behavior in the loop integration, $q_{*}$. Generically, the corrections
to transport coefficients should scale with $q_{*}$ to an appropriate (and
positive) power for $d=3$. For a normal fluid, $q_{*}\sim\sqrt{\omega}$,
giving rise to the renowned long-time tail phenomena POMEAU197563 ;
Kovtun:2003vj ; Kovtun:2012rj , and consequently vanishes in the limit
$\omega\rightarrow 0$ (see also refs Akamatsu:2016llw ; Stephanov:2017ghc ;
Jain:2020hcu  for related discussion). Therefore, to obtain finite corrections
to transport coefficients in this limit, there must be additional soft scales.
Such scales are generated by the magnetic field and/or axial relaxation rate
in the present study. Given the generality of the discussion above, we
anticipate that the CME and hydrodynamic fluctuations together might
contribute to other transports coefficients. A natural follow-up would be to
include fluctuations from energy and momentum densities and study the effects
on shear and bulk viscosities. We leave these and other extensions of this
work to future studies.

###### Acknowledgements.

We thank Kenji Fukushima, Yoshimasa Hidaka, Dmitri Kharzeev, Chris Lau and
Derek Teaney for useful discussions and comments. N. S. and Y. Y. would like
to acknowledge financial support by the Strategic Priority Research Program of
Chinese Academy of Sciences, Grant No. XDB34000000. N. Y. was supported by the
Keio Institute of Pure and Applied Sciences (KiPAS) project at Keio University
and JSPS KAKENHI Grant No. 19K03852.

##  Appendix A Derivation of $I_{{\rm anom}}$

In this section, we shall present the derivation of $I_{{\rm anom}}$ in eq.
(13) used in the main text following the method of ref. Glorioso:2017lcn . We
shall see the anomaly relation combined with KMS invariance uniquely fix the
form of CME. This should be contrasted with the derivation of CME in
hydrodynamics from the second law of thermodynamics Son:2009tf ;
Hattori:2017usa .

Let us first consider the low-energy effective action describing the slow mode
associated with a single chiral charge. We consider the action divided into
two parts,

$\displaystyle I=I_{{\rm inv}}+I_{{\rm anom}}\,,$ (125)

where $I_{{\rm inv}}$ is identical to the hydrodynamic action of a conserved
charge (see eq. (12)). We shall focus on the anomaly-related action $I_{{\rm
anom}}$ from now on.

Because of the anomaly, the consistent ${\rm r}$-current,

$\displaystyle J^{\mu}=\frac{\delta I}{\delta A^{\rm a}_{\mu}}\,$ (126)

obeys the anomaly equation

$\displaystyle\partial_{\mu}J^{\mu}=-\frac{\kappa
C}{24}F_{\mu\nu}\tilde{F}^{{\mu\nu}}\,,$ (127)

where $\kappa=\pm 1$ correspond to the right-handed and left-handed charge,
respectively. In this appendix, we shall keep the index “${\rm a}$” but
suppress the index “${\rm r}$.” To obtain eq. (127), we use the consistent
anomaly equation on the two segments of Schwinger-Keldysh contour labeled by
“1,2”:

$\displaystyle\partial_{\mu}J^{\mu}_{1}=-\frac{\kappa
C}{24}F_{\mu\nu,1}\tilde{F}_{1}^{\mu\nu}\,,\quad\partial_{\mu}J^{\mu}_{2}=-\frac{\kappa
C}{24}F_{\mu\nu,2}\tilde{F}_{2}^{\mu\nu}\,,$ (128)

to compute
$\partial_{\mu}J^{\mu}=\partial_{\mu}\left(J^{\mu}_{1}+J^{\mu}_{2}\right)$. We
have not included a contribution quadratic in $\rm a$-field on the right-hand
side of eq. (127) since such contribution should arise from the action
involving three powers of $\rm a$-field. Similar to the discussion presented
in the main text, the equation of motion for $\psi^{\rm a}$,

$\displaystyle\frac{\delta I}{\delta\psi^{\rm a}}=0\,,$ (129)

should be equivalent to the consistent anomaly equation (127). Therefore, the
anomaly part of the action takes the form

$\displaystyle I_{{\rm anom}}$ $\displaystyle=\int_{x}\left(-\frac{\kappa
C}{24}\psi^{\rm a}F_{\mu\nu}\tilde{F}^{\mu\nu}+J_{\rm
cons}^{\mu}\mathcal{A}^{\rm a}_{\mu}\right)\,.$ (130)

Here, ${\cal A}^{\rm a}_{\mu}=A^{\rm a}_{\mu}+\partial_{\mu}\psi^{\rm a}$ as
given in eq. (4) and $J_{\rm cons}$ ($J_{{\rm cov}}$) denotes the contribution
to the consistent (covariant) current only from $I_{{\rm anom}}$,
distinguished with the current $J$ from the total action $I$. The difference
between $J_{\rm cons}$ and $J_{{\rm cov}}$ defines the Chern-Simons (CS)
current as

$\displaystyle J_{{\rm cov}}=J_{{\rm cons}}+J_{{\rm CS}}\,,$ (131)

where

$\displaystyle J^{\mu}_{{\rm CS}}=-\frac{\kappa
C}{6}\tilde{F}^{\mu\nu}\,A_{\nu}\,.$ (132)

Similar to our previous treatment of the consistent anomaly equation (127), we
have dropped terms quadratic in a-fields in eq. (132). Substituting eqs. (132)
and (131) into eq. (130), we now have

$\displaystyle I_{{\rm anom}}=\int_{x}\,\left(-\frac{C\kappa}{24}\psi^{\rm
a}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{\kappa
C}{6}\tilde{F}^{\mu\nu}A_{\nu}{\cal A}^{\rm a}_{\mu}+J_{{\rm cov}}^{\mu}{\cal
A}^{\rm a}_{\mu}\right)\,.$ (133)

We shall consider the following form for $J^{\mu}_{\rm cov}$:

$\displaystyle J^{\mu}_{\rm cov}=\xi(\mu)B^{\mu}\,,$ (134)

where $\xi(\mu)$ is a function of $\mu$ and we have defined
$E^{\mu}=F^{\mu\nu}\ell_{\nu}$ and $B^{\mu}=\tilde{F}^{\mu\nu}\ell_{\nu}$ with
$\ell^{\mu}=(1,{\bm{0}})$ denoting the frame of the medium. We shall also use
below that $\mu=\ell\cdot{\cal A}$ (see eq. (5)). Since $I_{{\rm anom}}$ is
invariant under the KMS transformations (7a) and (7b),

$\displaystyle{\cal A}^{\rm r}_{\mu}$ $\displaystyle\to\Theta{\cal A}^{\rm
r}_{\mu}\,,\quad\psi^{\rm r}\to\Theta\psi^{\rm r}\,,$ $\displaystyle{\cal
A}^{\rm a}_{\mu}$ $\displaystyle\to\Theta{\cal A}^{\rm a}_{\mu}+\frac{\rm
i}{T}\Theta\left(\ell^{\alpha}\partial_{\alpha}{\cal A}_{\mu}\right)$
$\displaystyle=\Theta{\cal A}^{\rm a}_{\mu}+\frac{\rm
i}{T}\Theta\ell^{\alpha}\left(F_{\alpha\mu}+\partial_{\mu}{\cal
A}_{\alpha}\right)=\Theta{\cal A}^{\rm a}_{\mu}+\frac{\rm
i}{T}\Theta\left(-E_{\mu}+\partial_{\mu}\mu\right)\,,$ $\displaystyle\psi^{\rm
a}$ $\displaystyle\to\Theta\psi^{\rm a}+\frac{\rm
i}{T}\Theta\left(\ell^{\alpha}\partial_{\alpha}\psi\right)\,.$ (135)

The variance of the first two terms in eq. (133) under the KMS transformation
should precisely cancel that from the last term. This requirement uniquely
fixes $\xi(\mu)$, as we shall explain below.

Indeed, for $\Theta={\cal CPT}$, we have

$\displaystyle\delta_{{\rm KMS}}\,\int_{x}\,\psi^{\rm
a}F_{\mu\nu}\tilde{F}^{\mu\nu}=-\frac{\rm
i}{T}\,\int_{x}\,(\ell^{\alpha}\partial_{\alpha}\psi)F_{\mu\nu}\tilde{F}^{\mu\nu}\,,$
(136)

where we have used $\Theta\mathcal{A}^{\mu}(x)=-\mathcal{A}^{\mu}(-x)$,
$\Theta\tilde{F}^{\mu\nu}(x)=\tilde{F}^{\mu\nu}(-x)$, and
$\Theta\ell^{\mu}(x)=\ell^{\mu}(-x)$. Similarly,

$\displaystyle\,\delta_{{\rm KMS}}\int_{x}\,\tilde{F}^{\mu\nu}A_{\nu}{\cal
A}^{\rm a}_{\mu}=-\frac{\rm
i}{T}\int_{x}\tilde{F}^{\mu\nu}A_{\nu}(-E_{\mu}+\partial_{\mu}\mu)=\frac{\rm
i}{T}\int_{x}\left[\frac{1}{4}(\ell\cdot
A)+\frac{\mu}{2}\right]F_{\mu\nu}\tilde{F}^{\mu\nu}\,,$ (137)

where we have used the identity

$\displaystyle\tilde{F}^{\mu\nu}A_{\nu}E_{\mu}=\left(\ell^{\mu}B^{\nu}-\ell^{\nu}B^{\mu}+\epsilon^{\mu\nu\alpha\beta}\ell_{\alpha}E_{\beta}\right)\,A_{\nu}E_{\mu}=-B\cdot
E(\ell\cdot A)=\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}(\ell\cdot A)\,.$ (138)

Putting all pieces together, we find

$\displaystyle\,\delta_{{\rm KMS}}\int_{x}\,\left(-\frac{C\kappa}{24}\psi^{\rm
a}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{\kappa
C}{6}\tilde{F}^{\mu\nu}A_{\nu}{\cal A}^{\rm a}_{\mu}\right)=\frac{{\rm
i}\kappa C}{6T}\int_{x}\left[\frac{1}{4}(\ell\cdot\partial)\psi^{\rm
a}+\frac{1}{4}(\ell\cdot A)+\frac{\mu}{2}\right]F_{\mu\nu}\tilde{F}^{\mu\nu}$
$\displaystyle=\frac{{\rm i}\kappa C}{8T}\int_{x}\mu
F_{\mu\nu}\tilde{F}^{\mu\nu}\,.$ (139)

On the other hand,

$\displaystyle\delta_{{\rm KMS}}\,\int_{x}J^{\mu}_{\rm cov}{\cal A}^{\rm
a}_{\mu}=\frac{\rm
i}{T}\int_{x}\xi(-\mu)B^{\mu}\left(-E_{\mu}+\partial_{\mu}\mu\right)=\frac{\rm
i}{4T}\int_{x}\xi(-\mu)F_{\mu\nu}\tilde{F}^{\mu\nu}\,,$ (140)

where we used the fact that $\xi(-\mu)B^{\mu}\partial_{\mu}\mu$ is a total
derivative. For eq. (A) to cancel eq. (140), we have the appropriate form for
the CME:

$\displaystyle\xi(\mu)=\frac{\kappa C}{2}\mu\,,\quad J^{\mu}_{\rm
cov}=\frac{\kappa C}{2}\mu B^{\mu}\,.$ (141)

Now, we generalize the discussion above to the system with both axial and
vector charges. From the consistent anomaly relation,

$\displaystyle\partial_{\mu}J^{\mu}_{{\rm V}}$ $\displaystyle=0\,,$ (142)
$\displaystyle\partial_{\mu}J^{\mu}_{{\rm A}}$
$\displaystyle=-\frac{C}{12}\left(3F_{\mu\nu,{\rm V}}\tilde{F}^{\mu\nu}_{\rm
V}+F_{\mu\nu,{\rm A}}\tilde{F}^{\mu\nu}_{\rm A}\right)\,.$ (143)

Thus, we have, in analogous to eq. (130), the expression for $I_{{\rm anom}}$:

$\displaystyle I_{\rm anom}$
$\displaystyle=\int_{x}\left[-\frac{C}{12}\psi_{\rm A}^{\rm
a}\left(3F_{\mu\nu,{\rm V}}\tilde{F}^{\mu\nu}_{\rm V}+F_{\mu\nu,{\rm
A}}\tilde{F}^{\mu\nu}_{\rm A}\right)\right.$
$\displaystyle\qquad\quad\left.+\left(J^{\mu}_{\rm cov,{\rm
V}}+C\tilde{F}^{\mu\nu}_{\rm V}A_{\nu,{\rm A}}\right)\mathcal{A}^{\rm
a}_{\mu,{\rm V}}+\left(J^{\mu}_{\rm cov,{\rm
A}}+\frac{C}{3}\tilde{F}^{\mu\nu}_{\rm A}A_{\nu,{\rm
A}}\right)\mathcal{A}^{\rm a}_{\mu,{\rm A}}\right]\,.$ (144)

By imposing the KMS symmetry to eq. (A), we find

$\displaystyle J^{\mu}_{\rm cov,V}=C\left(\mu_{{\rm A}}B^{\mu}_{\rm
V}+\mu_{{\rm V}}B^{\mu}_{\rm A}\right)\,,\quad J^{\mu}_{\rm
cov,A}=C\left(\mu_{{\rm V}}B^{\mu}_{\rm V}+\mu_{{\rm A}}B^{\mu}_{\rm
A}\right)\,.$ (145)

Substituting eq. (145) into eq. (A) leads to the desired expression for
$I_{{\rm anom}}$, which reduces to eq. (13) when the external axial gauge
field is absent.

##  Appendix B One-loop expression for self-energy $\Sigma^{\rm
aa}_{\alpha\beta}$

We here show the expression for $\Sigma^{\rm aa}_{\alpha\beta}$ defined in eq.
(63) at the one-loop level. There are three contributions

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})=(\Sigma^{\rm
aa}_{\alpha\beta})_{\rm I}+(\Sigma^{\rm aa}_{\alpha\beta})_{\rm
II}+(\Sigma^{\rm aa}_{\alpha\beta})_{\rm III}\,,$ (146)

which correspond to the diagrams given by figures 3 (d), (e), and (f),
respectively. Let us start with the first two contributions in eq. (146):

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm I}(K)$
$\displaystyle=4T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha;\gamma\sigma}G^{\rm
rr}_{\sigma\tau}(q^{0},{\bm{q}}_{+})G^{\rm
ra}_{\gamma\delta}(\omega-q^{0},-{\bm{q}}_{-})g_{\delta}V^{{\bm{q}}_{-},{\bm{k}};-{\bm{q}}_{+}}_{\delta\beta;\tau}\,,$
(147) $\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm II}(K)$
$\displaystyle=4T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}g_{\gamma}V^{-{\bm{k}},-{\bm{q}}_{-};{\bm{q}}_{+}}_{\alpha\gamma;\sigma}G^{\rm
rr}_{\sigma\tau}(q^{0},{\bm{q}}_{+})G^{\rm
ar}_{\gamma\delta}(\omega-q^{0},-{\bm{q}}_{-})g_{\beta}U^{{\bm{k}};{\bm{q}}_{-},-{\bm{q}}_{+}}_{\beta;\delta\tau}\,.$
(148)

One can show that

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm II}=(\Sigma^{\rm
aa}_{\beta\alpha})_{\rm I}^{*}\,,$ (149)

using eq. (40),

$\displaystyle G^{\rm rr}(K)=(G^{\rm rr})^{\dagger}(K)=G^{\rm rr}(-K)\,,$
(150)

and

$\displaystyle\left(U^{{\bm{k}}_{1};{\bm{k}}_{2},{\bm{k}}_{3}}_{\alpha;\beta\gamma}\right)^{*}=-U^{-{\bm{k}}_{1};-{\bm{k}}_{2},-{\bm{k}}_{3}}_{\alpha;\beta\gamma}\,,\quad\left(V_{{\rm
A}\alpha;\beta}^{{\bm{k}}_{1},{\bm{k}}_{2};{\bm{k}}_{3}}\right)^{*}=V_{{\rm
A}\alpha;\beta}^{{\bm{k}}_{1},{\bm{k}}_{2};{\bm{k}}_{3}}\,,$ (151)

which can be verified from eq. (48).

To calculate $(\Sigma^{\rm aa}_{\alpha\beta})_{\rm I}$, we use eq. (41) and
write $G^{\rm rr}$ as a sum of $G^{\rm ra}$ and $G^{\rm ar}$. Then, we drop
the latter contribution using eq. (68) and carry out the $q^{0}$ integral from
the former contribution using eq. (73). We obtain

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm I}(K)$ $\displaystyle=-{\rm
i}T^{2}\sum_{\gamma\delta\sigma\tau}\sum_{mm^{\prime}}\int_{Q}\left(F^{{\rm
aa},mm^{\prime}}_{\alpha\beta}\right)_{\rm
I}({\bm{k}},{\bm{q}})\Delta^{mm^{\prime}}(k,{\bm{q}})\,,$ (152)

with

$\displaystyle\left(F^{{\rm aa},mm^{\prime}}_{\alpha\beta}\right)_{\rm
I}({\bm{k}},{\bm{q}})={\rm
tr}\left[R^{m}({\bm{q}}_{+})\left(g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha}\right)^{\rm
T}R^{m^{\prime}}(-{\bm{q}}_{-})\left(4{\rm
i}gV^{{\bm{q}}_{-},{\bm{k}};-{\bm{q}}_{+}}_{\beta}\right)\right]\,,$ (153)

where we have used the same matrix form as eq. (72):

$\displaystyle\left(g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha}\right)_{\beta\gamma}=g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha;\beta\gamma}\,,\quad\left(4{\rm
i}gV^{{\bm{q}}_{-},{\bm{k}};-{\bm{q}}_{+}}_{\beta}\right)_{\gamma\delta}=4{\rm
i}g_{\gamma}V^{{\bm{q}}_{-},{\bm{k}};-{\bm{q}}_{+}}_{\gamma\beta;\delta}\,.$
(154)

By using eq. (149) we have

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm I}(K)+(\Sigma^{\rm
aa}_{\alpha\beta})_{\rm II}(K)$ $\displaystyle=-{\rm
i}T^{2}\sum_{mm^{\prime}}\int_{{\bm{q}}}\left[\left(F^{{\rm
aa},mm^{\prime}}_{\alpha\beta}\right)_{\rm
I}({\bm{k}},{\bm{q}})\Delta^{mm^{\prime}}(k,{\bm{q}})-(\alpha\leftrightarrow\beta+{\rm
c.c.})\right]\,.$ (155)

Here, $(\alpha\leftrightarrow\beta+{\rm c.c.})$ denotes the complex conjugate
of the first term with the interchanged ($\alpha,\beta$) labels.

The third contribution, figure 3 (f), can be written as

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm III}(K)$
$\displaystyle=2T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha;\gamma\sigma}G^{\rm
rr}_{\sigma\tau}(q^{0},{\bm{q}}_{+})G^{\rm
rr}_{\gamma\delta}(\omega-q^{0},-{\bm{q}}_{-})g_{\beta}U^{{\bm{k}};{\bm{q}}_{-},-{\bm{q}}_{+}}_{\beta;\delta\tau}\,.$
(156)

Using eqs. (40) and (41), the $q^{0}$ integral relevant to (156) can be
written as

$\displaystyle\int_{q^{0}}G^{\rm rr}_{\sigma\tau}(q^{0},{\bm{q}}_{+})G^{\rm
rr}_{\gamma\delta}(\omega-q^{0},-{\bm{q}}_{-})$ $\displaystyle={\rm
i}\int_{q^{0}}\left[{\rm i}G^{\rm ra}_{\sigma\tau}(q^{0},{\bm{q}}_{+})G^{\rm
ra}_{\gamma\delta}(\omega-q^{0},-{\bm{q}}_{-})-({\rm c.c.})\right]\,.$ (157)

Further using

$\displaystyle
g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha;\gamma\sigma}g_{\beta}U^{{\bm{k}};{\bm{q}}_{-},-{\bm{q}}_{+}}_{\beta;\delta\tau}$
$\displaystyle=\left(g_{\beta}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\beta;\delta\tau}g_{\alpha}U^{{\bm{k}};{\bm{q}}_{-},-{\bm{q}}_{+}}_{\alpha;\gamma\sigma}\right)^{*}\,,$
(158)

which can be readily checked from eq. (151), we express eq. (156) as

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})_{\rm III}(K)$
$\displaystyle=2{\rm
i}T^{2}\sum_{\gamma\delta\sigma\tau}\int_{Q}\left[g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha;\gamma\sigma}G^{\rm
ra}_{\sigma\tau}(q^{0},{\bm{q}}_{+}){\rm i}G^{\rm
ra}_{\gamma\delta}(\omega-q^{0},-{\bm{q}}_{-})g_{\beta}U^{{\bm{k}};{\bm{q}}_{-},-{\bm{q}}_{+}}_{\beta;\delta\tau}\right.$
$\displaystyle\hskip 250.0pt\left.-(\alpha\leftrightarrow\beta+{\rm
c.c.})\right]\,.$ (159)

Finally, we can combine eqs. (155) and (B) into the form:

$\displaystyle(\Sigma^{\rm aa}_{\alpha\beta})(K)$ $\displaystyle=-\frac{{\rm
i}T^{2}}{2}\sum_{mm^{\prime}}\int_{{\bm{q}}}\left[F^{{\rm
aa},mm^{\prime}}_{\alpha\beta}({\bm{k}},{\bm{q}})\Delta^{mm^{\prime}}(k,{\bm{q}})-(\,\alpha\leftrightarrow\beta\,+\,{\rm
c.c.})\right]\,,$ (160)

where

$\displaystyle F^{{\rm aa},mm^{\prime}}_{\alpha\beta}({\bm{k}},{\bm{q}})={\rm
tr}\left[R^{m}({\bm{q}}_{+})\left(\Gamma^{{\rm a}}_{\alpha}\right)^{\rm
T}R^{m^{\prime}}(-{\bm{q}}_{-})\Gamma^{\prime{\rm a}}_{\beta}\right]\,,$ (161)

with

$\displaystyle(\Gamma^{\rm a}_{\alpha})_{\beta\gamma}$
$\displaystyle=g_{\alpha}U^{-{\bm{k}};-{\bm{q}}_{-},{\bm{q}}_{+}}_{\alpha;\beta\gamma}\,,$
(162) $\displaystyle(\Gamma^{\prime{\rm a}}_{\beta})_{\gamma\delta}$
$\displaystyle=4\left(-g_{\beta}U^{{\bm{k}};{\bm{q}}_{-},-{\bm{q}}_{+}}_{\gamma;\beta\delta}+2{\rm
i}g_{\gamma}V^{{\bm{q}}_{-},{\bm{k}};-{\bm{q}}_{+}}_{\gamma\beta;\delta}\right)\,.$
(163)

If $\alpha=\beta$, eq. (160) reduces to the form (77) in the main text.

##  Appendix C Conductivity tensor from density-density correlator at zero
axial relaxation rate

We here show another derivation of the one-loop conductivity, using the
symmetrized density-density correlator $\mathcal{C}^{00}_{\rm S}$, at the
vanishing axial relaxation rate with zero background vector charge density but
finite axial charge density. The derivation here is more parallel to the
analysis in section 3.4, whereas it will give consistent results with those in
section 4, namely eq. (123).

First of all, by taking the limit $r\to 0$ followed by $\omega\to 0$, eqs.
(65a) and (65b) are reduced to

$\displaystyle\sigma_{\perp}(0)=(\sigma_{\rm
V})_{0}+\Delta\sigma_{\perp}(0)\,,\quad\sigma_{\parallel}(0)=(\sigma_{\rm
V})_{0}+\Delta\sigma_{\parallel}(0)\,,$ (164)

where $\Delta\sigma_{\perp}(0)$ and $\Delta\sigma_{\parallel}(0)$ can be
obtained from eq. (64a) by calculating $\Sigma^{\rm aa}_{\rm VV}$ using eq.
(77). In parallel to the analysis on eq. (83), we consider $\Gamma^{\rm
a}_{\rm V}$ and $\Gamma^{{\rm a}^{\prime}}_{\rm V}$ defined in eqs. (72) and
(79) at the small $k$ limit:

$\displaystyle\Gamma^{\rm a}_{\rm V}$ $\displaystyle=-\frac{{\rm
i}gD}{2}({\bm{k}}\cdot{\bm{q}})\left(\begin{array}[]{cc}0&-\tilde{u}_{\sigma}\\\
\tilde{u}_{\sigma}&0\end{array}\right)-\frac{g}{2}({\bm{v}}_{\rm
A}\cdot{\bm{k}})\left(\begin{array}[]{cc}\tilde{u}_{n}&0\\\
0&\tilde{u}_{n}\end{array}\right)\,,$ (169) $\displaystyle\Gamma^{{\rm
a}^{\prime}}_{\rm V}$ $\displaystyle=2{\rm
i}gD({\bm{k}}\cdot{\bm{q}})\left(\begin{array}[]{cc}0&3\tilde{u}_{\sigma}\\\
\tilde{u}_{\sigma}&0\end{array}\right)-2g({\bm{v}}_{\rm
A}\cdot{\bm{k}})\left(\begin{array}[]{cc}\tilde{u}_{n}&0\\\
0&\tilde{u}_{n}\end{array}\right)\,,$ (174)

where we have assumed eq. (111) and used eq. (112). It is easy to check that
finite contribution to the DC conductivity comes from
$(m,m^{\prime})=(+,+),(-,-)$ contribution with

$\displaystyle F^{\rm aa,++}_{{\rm V}{\rm V}}=F^{\rm aa,--}_{{\rm V}{\rm
V}}=-(Dg\tilde{u}_{\sigma}{\bm{k}}\cdot{\bm{q}})^{2}\,.$ (175)

We thus find

$\displaystyle\Sigma^{\rm aa}_{\rm
VV}(0,{\bm{k}})=(DgT\tilde{u}_{\sigma})^{2}\int_{\bm{q}}\frac{({\bm{k}}\cdot{\bm{q}})^{2}D{\bm{q}}^{2}}{(D{\bm{q}}^{2})^{2}+({\bm{v}}\cdot{\bm{q}})^{2}}+{\cal
O}(k^{3})\,.$ (176)

Using eq. (64a), we obtain eq. (4) and resulting eq. (123), as it should be.

##  Appendix D Explicit verification of the fluctuation-dissipation relation
and Ward-Takahashi identity at one-loop

In this appendix, we show explicitly that at one-loop order, the symmetrized
current-current correlator ${\mathcal{C}}^{ij}$ is related to the retarded
correlator ${\mathcal{C}}^{ij}_{\rm R}$ through the fluctuation-dissipation
relation (101). Since we have already demonstrated that conductivity tensor
obtained from ${\mathcal{C}}^{00}$ coincides with that from
${\mathcal{C}}^{ij}_{\rm R}$ in appendix C, we therefore verify the constraint
imposed by Ward-Takahashi identity at one-loop order between ${\cal C}^{00}$
and ${\cal C}^{ij}$.

For illustrative purpose, we shall focus on the diffusive part of the current
at quadratic order in fluctuations,

$\displaystyle({\bm{J}}^{{\rm r}})^{\rm dif}_{2}$ $\displaystyle=-TD_{\rm
V}\tilde{u}_{\sigma}\lambda_{\rm A}\left({\bm{\nabla}}\lambda_{\rm V}-2{\rm
i}{\bm{\nabla}}\psi_{\rm V}\right)\,,$ (177) $\displaystyle({\bm{J}}^{{\rm
a}})^{\rm dif}_{2}$ $\displaystyle=D_{\rm
V}\tilde{u}_{\sigma}\partial_{t}\left(\lambda_{\rm A}{\bm{\nabla}}\psi_{\rm
V}\right)\,.$ (178)

The second term in eq. (177), which is proportional to the a-field $\psi$,
arises from the multiplicative noise. We shall see this multiplicative noise
contribution is crucial to ensure the fluctuation-dissipation relation and the
Ward-Takahashi identity at one-loop order.

We begin by computing the one-loop corrections to the symmetrized correlator

$\displaystyle\left[{\cal C}^{ij,{\rm dif}}(K)\right]_{\rm loop}$
$\displaystyle=\left<(J^{i,{\rm r}})^{\rm dif}_{2}(K)(J^{j,{\rm r}})^{\rm
dif}_{2}(-K)\right>$ $\displaystyle=(TD_{\rm
V}\tilde{u}_{\sigma})^{2}\left<\left[\lambda_{\rm
A}\partial^{i}\left(\lambda_{\rm V}-2{\rm i}\psi_{\rm
V}\right)\right](K)\left[\lambda_{\rm A}\partial^{j}\left(\lambda_{\rm
V}-2{\rm i}\psi_{\rm V}\right)\right](-K)\right>$ $\displaystyle=(TD_{\rm
V}\tilde{u}_{\sigma})^{2}\left(\mathcal{F}^{ij,\rm rr}-2{\rm
i}\mathcal{F}^{ij,\rm ra}-2{\rm i}\mathcal{F}^{ij,\rm ar}\right)\,,$ (179)

where

$\displaystyle{\cal F}^{ij,\rm rr}$ $\displaystyle=\left<\lambda_{\rm
A}\partial^{i}\lambda_{\rm V}(K)\left(\lambda_{\rm A}\partial^{j}\lambda_{\rm
V}\right)(-K)\right>\,,$ (180) $\displaystyle{\cal F}^{ij,\rm ra}$
$\displaystyle=\left<\lambda_{\rm A}\partial^{i}\lambda_{\rm
V}(K)\left(\lambda_{\rm A}\partial^{j}\psi_{\rm
V}\right)(-K)\right>\,,\quad{\cal F}^{ij,\rm ar}=\left<\lambda_{\rm
A}\partial^{i}\psi_{\rm V}(K)\left(\lambda_{\rm A}\partial^{j}\lambda_{\rm
V}\right)(-K)\right>\,.$ (181)

From eqs. (41) and (40), it is straightforward to show that

$\displaystyle{\cal F}^{ij,\rm rr}(K)={\rm i}{\cal F}^{ij,{\rm ra}}(K)+{\rm
i}{\cal F}^{ij,{\rm ar}}(K)\,,\quad{\cal F}^{ij,{\rm ra}}(K)=-({\cal
F}^{ij,{\rm ar}})^{*}(K)\,.$ (182)

We therefore have

$\displaystyle\left[{{\cal C}}^{ij,{\rm dif}}(K)\right]_{\rm
loop}=(TD{\tilde{u}}_{\sigma})^{2}\left[-{\rm i}{\cal F}^{ij,{\rm ra}}(K)-{\rm
i}{\cal F}^{ij,{\rm ar}}(K)\right]=2(TD{\tilde{u}}_{\sigma})^{2}\,{\rm
Im}\,{\cal F}^{ij,{\rm ra}}(K)\,.$ (183)

On the other hand, the one-loop corrections to the retarded correlator is
given by

$\displaystyle\left[{\cal C}^{ij,{\rm dif}}_{\rm R}(K)\right]_{\rm loop}$
$\displaystyle={\rm i}\langle(J^{i,{\rm r}})^{\rm dif}_{2}(K)\,(J^{j,{\rm
a}})^{\rm dif}_{2}(-K)\rangle$ $\displaystyle=\omega T(D_{\rm
V}\tilde{u}_{\sigma})^{2}\langle\left(\lambda_{\rm A}\partial^{i}(\lambda_{\rm
V}-2{\rm i}\psi_{\rm A})\right)(K)\,(\lambda_{\rm A}\partial^{j}\psi_{\rm
V})(-K)\rangle$ $\displaystyle=\omega T(D_{\rm
V}\tilde{u}_{\sigma})^{2}\mathcal{F}^{ij,{\rm ra}}(K)\,.$ (184)

Comparing eq. (183) with eq. (D), we immediately verify

$\displaystyle\left({{\cal C}}^{ij,{\rm dif}}\right)_{\rm
loop}=\frac{2T}{\omega}{\rm Im}\left({\cal C}^{ij,{\rm dif}}_{\rm
R}\right)_{\rm loop}\,.$ (185)

Note that if we had ignored the multiplicative noise contributions, i.e., the
last two terms in eq. (D), we would obtain the wrong relation

$\displaystyle\left[{\cal C}^{ij,{\rm dif}}(K)\right]_{{\rm
loop}}=(TD{\tilde{u}}_{\sigma})^{2}{\cal F}^{\rm rr}(K)=-\frac{2T}{\omega}{\rm
Im}\left[C^{ij,{\rm dif}}_{\rm R}(K)\right]_{{\rm loop}}\,.$ (186)

Furthermore, one would also get a wrong relation between $\mathcal{C}^{00}$
obtained in appendix C and ${\cal C}^{ij}$:

$\displaystyle\lim_{{\bm{k}}\rightarrow{\bm{0}}}\frac{1}{2T}\hat{k}^{i}\hat{k}^{j}\left[{\cal
C}^{ij,{\rm dif}}(K)\right]_{\rm
loop}=-\lim_{{\bm{k}}\rightarrow{\bm{0}}}\frac{1}{2T}\frac{\omega^{2}}{{\bm{k}}^{2}}\mathcal{C}^{00}(K)\,,$
(187)

which contradicts with eq. (51) based on the Ward-Takahashi identity.

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