aainstitutetext: Theoretisch-Physikalisches Institut, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany.bbinstitutetext:
Institute for Theoretical Physics, Vienna University of Technology (TU Wien),
Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria.ccinstitutetext: Instituto de
Física Téorica UAM/CSIC and Departamento de Física Téorica, Universidad
Autónoma de Madrid, Campus de Cantoblanco, C/Nicolás Cabrera 13–15, ES-28049
Madrid, Spainddinstitutetext: Perimeter Institute for Theoretical Physics, 31
Caroline St N, Waterloo, ON N2L 2Y5, Canadaeeinstitutetext: Department of
Physics and Astronomy, University of Waterloo, 200 University Avenue West,
Waterloo, Ontario, Canada N2L 3G1ffinstitutetext: Department of Physics and
Astronomy, University of Alabama, 514 University Boulevard,Tuscaloosa, AL
35487, USA

#  Chiral hydrodynamics in strong external magnetic fields

Martin Ammon a,b,c Sebastian Grieninger d,e Juan Hernandez f Matthias Kaminski
f Roshan Koirala a Julian Leiber f Jackson Wu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct the general hydrodynamic description of (3+1)-dimensional chiral
charged (quantum) fluids subject to a strong external magnetic field with
effective field theory methods. We determine the constitutive equations for
the energy-momentum tensor and the axial charge current, in part from a
generating functional. Furthermore, we derive the Kubo formulas which relate
two-point functions of the energy-momentum tensor and charge current to 27
transport coefficients: 8 independent thermodynamic, 4 independent non-
dissipative hydrodynamic, and 10 independent dissipative hydrodynamic
transport coefficients. Five Onsager relations render 5 more transport
coefficients dependent. We uncover four novel transport effects, which are
encoded in what we call the shear-induced conductivity, the two expansion-
induced longitudinal conductivities and the shear-induced Hall conductivity.
Remarkably, the shear-induced Hall conductivity constitutes a novel non-
dissipative transport effect. As a demonstration, we compute all transport
coefficients explicitly in a strongly coupled quantum fluid via holography.

††preprint: IFT-UAM/CSIC-20-179

##  1 Introduction

Hydrodynamics is a universal effective field theory description of collective
phenomena in (quantum) systems with many degrees of freedom. The hydrodynamic
description of relativistic fluids has facilitated the physical interpretation
of data from heavy ion collisions. It has also allowed to describe effects in
condensed matter systems from charge conduction in metals to more recent
descriptions of Weyl semimetals and graphene.

In this work, we consider the hydrodynamic description of a (3+1)-dimensional
chiral charged thermal fluid subject to a strong external magnetic field. This
is not to be confused with magnetohydrodynamics, in which the magnetic field
satisfies Maxwell’s equations and hence is dynamical. Our external magnetic
field is not dynamical in that sense. This magnetic field and the associated
gauge potential $A_{\mu}$ can either be related to a vector $U(1)_{V}$
symmetry assumed to be preserved, or to an axial $U(1)_{A}$ symmetry which may
be broken by a chiral anomaly. We point out the differences in the
hydrodynamic description depending on which type of the two $U(1)$ symmetries
is considered. Our focus, however, is the anomalous axial $U(1)_{A}$ case. We
limit our considerations to zeroth and first order in the hydrodynamic
derivative expansion. The magnetic field is referred to as strong if it is
defined to be of zeroth order in derivatives. On a fundamental level, no axial
magnetic fields exist in Nature. However, in low energy electronic
descriptions for condensed matter systems such as Weyl-semimetals effectively
axial magnetic fields and axial potentials can be created Cortijo_2015 ;
Pikulin_2016 ; Grushin_2016 ; Cortijo:2016wnf . Thus we focus here on
exploring the effects of an axial $U(1)_{A}$ symmetry. Based on previous
results Hernandez:2017mch ; Son:2009tf ; Neiman:2010zi , we expect that adding
a conserved vector current associated with a dynamical gauge field will not
qualitatively change the physical effects, merely distribute them over
different currents, introducing copies of analogous effects. Therefore, the
hydrodynamic description derived here can be extended to be applied to the
quark-gluon-plasma generated in heavy-ion-collisions.

Our main result are the Kubo formulas for the transport coefficients of a
charged chiral thermal fluid subject to a strong magnetic field. These are
derived in part from a generating functional and in part from constitutive
relations which we construct in all generality in section 2. A nonzero strong
magnetic field necessarily leads to anisotropic equilibrium states. In
addition, an axial chemical potential breaks parity on the level of the state.
The combination of these two broken symmetries explains the large number of
transport coefficients. Specifically, we find 8 independent thermodynamic
transport coefficients, and initially 19 hydrodynamic transport coefficients.
Among the hydrodynamic transport coefficients we find 5 Onsager relations,
which leaves 14 independent hydrodynamic transport coefficients. Of those
hydrodynamic transport coefficients 3 are non-dissipative and 11 are
dissipative. All independent transport coefficients are listed in tables 3, 4
and 5.

The first subset of the 8 independent thermodynamic transport coefficients are
the well-known chiral conductivities: vortical $\xi=\xi_{TB}$, chiral magnetic
$\xi_{B}$, and chiral thermal $\xi_{T}$. Referring to $\xi_{B}$ as “chiral
magnetic effect” is a slight abuse of language. In most of this paper
$\xi_{B}$ will measure the response of an axial current to an axial $U(1)_{A}$
magnetic field. The term “chiral magnetic effect” was coined for the response
of such a current to a $U(1)_{V}$ vector magnetic field Kharzeev:2004ey , see
also Jensen:2013vta . The remaining 5 thermodynamic transport coefficients are
4 newly non-vanishing susceptibilities: magneto-thermal $M_{1}$, perpendicular
magneto-vortical $M_{2}$, magneto-acceleration $M_{3}$, magneto-electric
$M_{4}$111$M_{1},\,M_{2},\,M_{3},\,M_{4}$ have to vanish in a parity-
preserving microscopic theory when coupled to an external vector $U(1)_{V}$
gauge field. However, they are allowed to be nonzero when the coupling to an
axial $U(1)_{A}$ is considered., and the previously discussed Kovtun:2016lfw ;
Hernandez:2017mch  magneto-vortical susceptibility $M_{5}$.222Note that this
$M_{5}$ was previously referred to as $M_{4}=M_{\Omega}$ Hernandez:2017mch
but considering only a coupling to a external vector $U(1)_{V}$ gauge field,
while we also consider coupling to an axial $U(1)_{A}$ here. Our $M_{5}$ was
also referred to as $M_{20}$ in Kovtun:2016lfw . The magneto-acceleration
susceptibility $M_{3}$ vanishes in conformal field theories. Note that in
addition there are three susceptibilities $\chi_{B}\,,\chi_{33},\,\chi_{13}$
which we do not include in our counting since they are thermodynamic partial
derivatives of the pressure, $p$. In stark contrast to this, the $M_{n}$
($n=1,\,2,\,3,\,4,\,5$) can not be defined as thermodynamic derivatives of the
pressure, thus they are independent coefficients.

There are initially 19 hydrodynamic transport coefficients: the shear
viscosities perpendicular and longitudinal to the magnetic field
$\eta_{\perp},\,\eta_{||}$, the perpendicular and longitudinal Hall
viscosities $\tilde{\eta}_{\perp},\,\tilde{\eta}_{||}$, the bulk viscosities
$\zeta_{1}$, $\zeta_{2}$, $\eta_{1}$, $\eta_{2}$, the perpendicular and
longitudinal charge conductivities $\sigma_{\perp}$, $\sigma_{||}$, the Hall
conductivity $\tilde{\sigma}_{\perp}$ (which can alternately be expressed in
terms of the charge resistivities
$\rho_{\perp},\,\rho_{||},\,\tilde{\rho}_{\perp}$), and the novel $c_{3}$,
$c_{4}$, $c_{5}$, $c_{8}$, $c_{10}$, $c_{14}$, $c_{15}$, $c_{17}$. Due to
their effect on the fluid to be interpreted in section 2.5, we name $c_{8}$
the shear-induced conductivity, $c_{10}$ the shear-induced Hall conductivity,
as well as $c_{4}$ and $c_{5}$ the expansion-induced longitudinal
conductivities. Only 4 of the $c_{n}$’s (with $n=3,4,5,8,10,14,15,17$) are
independent. Only 3 of the bulk viscosities are independent. This follows from
5 Onsager relations derived in this work.

In section 3, we prove the existence of these transport coefficients by direct
computation of their nonzero values within the specific example of a strongly
coupled $\mathcal{N}=4$ Super-Yang-Mills (SYM) theory at a large number of
colors, $N_{c}\to\infty$, coupled to an external axial $U(1)_{A}$ gauge field.
This computation is facilitated by holography Maldacena:1997re . In order to
allow for a charged thermal state subject to a strong magnetic field, the
charged magnetic black brane solutions are considered D'Hoker:2009bc . Within
the classical gravity dual to SYM theory, we compute the frequency or momentum
dependent fluctuations around the branes, which are holographically dual to
field theory correlation functions of the energy momentum tensor and the axial
current. Applying the Kubo formulas derived in section 2, we obtain nonzero
values for most of the transport coefficients. An exception are the transport
coefficients $c_{3}$, $M_{4},\,\tilde{\eta}_{\perp}$, $\zeta_{1}$ and
$\zeta_{2}$ which vanish in the holographic model. The status of $M_{1}$ and
$M_{3}$ is unclear within the holographic model as we only have Kubo relations
for their derivatives. However, $M_{3}$ is expected to vanish due to conformal
invariance.

The effect of chiral anomalies in hydrodynamics333In a non-hydrodynamic
context, the effect of chiral anomalies on currents was pioneered by
Vilenkin:1978is ; Vilenkin:1979ui ; Vilenkin:1980zv . was first found through
holographic calculations which yielded nonzero anomalous transport in
Erdmenger:2008rm ; Banerjee:2008th ; Torabian:2009qk . More generally, the
existence of anomalous transport as a consequence of chiral anomalies was
elucidated in terms of a local version of the second law of thermodynamics in
Son:2009tf . Subsequent studies of anomalous hydrodynamics include
Neiman:2010zi ; Kharzeev:2009pj ; Kharzeev:2010gd ; Kharzeev:2013ffa ;
Jensen:2012kj ; Jensen:2013kka ; Jensen:2013vta . The equilibrium partition
function formulation of relativistic hydrodynamics was first introduced in
Jensen:2012jh ; Banerjee:2012iz  and subsequently used in a variety of
settings Bhattacharyya:2012xi ; Bhattacharya:2012zx ; Bhattacharyya:2013ida ;
Kovtun:2016lfw ; Hernandez:2017mch ; Kovtun:2018dvd ; Armas:2018zbe ;
Kovtun:2019wjz . In particular, the equilibrium generating functional approach
was used to formulate the hydrodynamic framework of (parity preserving) fluids
subject to strong external as well as the framework of magnetohydrodynamics
(when the electromagnetic field is dynamical) in Hernandez:2017mch . The
relation between the frameworks of hydrodynamics with strong magnetic fields
and magnetohydrodynamics, as well as the dual formulation of
magnetohydrodynamics in terms of two-form fields introduced in
Grozdanov:2016tdf  was elucidated in Hernandez:2017mch . The anomaly inflow
generating functionals have been used for anomalous hydrodynamics in
equilibrium in Jensen:2013kka ; Jensen:2012kj  and for out of equilibrium
hydrodynamics in Haehl:2013hoa . Dispersion relations of hydrodynamic modes
within the system under consideration in this work have been computed
previously at weak magnetic fields of first order in the hydrodynamic
derivative expansion Ammon:2017ded ; Kalaydzhyan:2016dyr ; Abbasi:2016rds .
Anisotropic hydrodynamics has been discussed in the context of heavy ion
collisions, see for example Martinez:2010sc ; Martinez:2010sd ;
Ryblewski:2010bs ; Ryblewski:2011aq ; Ryblewski:2012rr ; Florkowski:2012lba ;
Strickland:2014pga ; Huang:2011dc ; Ammon:2017ded .

Holographic duals of quantum field theories with a chiral anomaly and subject
to weak electromagnetic fields (of first order in the hydrodynamic derivative
expansion) have received much attention due to a host of applications that
ranges from condensed matter physics to heavy ion collisions. Specific
interest was focused on the analytically known Son:2009tf  chiral
conductivities: chiral magnetic effect Newman:2005hd ; Kharzeev:2004ey ;
Kharzeev:2007jp ; Fukushima:2008xe ; Kharzeev:2009pj , the chiral vortical
effect Vilenkin:1978is ; Vilenkin:1979ui ; Erdmenger:2008rm ; Banerjee:2008th
, and later the chiral thermal conductivity, see e.g. Neiman:2010zi ;
Ammon:2017ded . These (DC) conductivities have been shown to be exact in a
multitude of holographic models Gursoy:2014boa ; Grozdanov:2016ala , and based
on field theory arguments Fukushima:2008xe . (Non-)renormalization of these
chiral conductivities was addressed holographically Gursoy:2014ela ; Jimenez-
Alba:2014iia ; Gallegos:2018ozs  and field theoretically Golkar:2012kb . The
frequency dependent (AC) chiral conductivities have been discussed in
Amado:2011zx ; Landsteiner:2013aba ; Li:2018srq , and from the field theory
side in Kharzeev:2009pj . At nonzero value of the anomaly and without a strong
magnetic field, analytic results for helicity-1 correlators in the
hydrodynamic approximation have been obtained in Matsuo:2009xn . Without the
anomaly, in strong magnetic field in an uncharged state Kubo formulas for
seven transport coefficients have been derived and values were calculated
numerically Grozdanov:2017kyl . The shear viscosities have been calculated in
Critelli:2014kra ; PhysRevD.96.019903  under the assumption of the validity of
the membrane paradigm. Dispersion relations of hydrodynamic and non-
hydrodynamic modes within the system under consideration in this work have
been computed from quasinormal modes previously at weak magnetic fields of
first order in the hydrodynamic derivative expansion Ammon:2017ded .
Quasinormal modes of magnetic black branes were calculated in Ammon:2016fru ;
Janiszewski:2015ura ; Ammon:2017ded ; Baggioli:2020edn . In Grozdanov:2017kyl
dynamical gauge fields in the dual field theory are considered within a two-
form field formalism which is distinct from ours. See also Grozdanov:2016tdf ;
Hernandez:2017mch  for the relation between the two formalisms. Anisotropic
effects not related to magnetic fields have also been included in holography
in the hydrodynamic approximation Rebhan:2009vc ; Erdmenger:2010xm ;
Erdmenger:2014jba ; Garbiso:2020puw .

##  2 Hydrodynamics

In this section, the constitutive equations, Kubo formulas, equilibrium
generating functionals, as well as symmetry constraints, Onsager relations,
and the entropy constraints are derived for a charged fluid subjected to a
strong external magnetic field. Chemical potentials and magnetic fields
associated with either an axial $U(1)_{A}$-symmetry or a vector
$U(1)_{V}$-symmetry are considered. Quantities can be classified according to
their charge under a parity transformation of the three spatial coordinates in
a field theory fluid state. It is helpful to notice that there are three
potential sources for parity breaking in the fluids we consider: the chiral
anomaly in the microscopic field theory, the external magnetic field
associated with a vector $U(1)_{V}$-symmetry, or the axial chemical potential
if a global axial $U(1)_{A}$-symmetry is considered.444In most of this work we
consider such an axial $U(1)_{A}$; exceptions are clearly marked. The vector
chemical potential associated with a vector $U(1)_{V}$ does not break parity,
neither does the magnetic field associated with an axial $U(1)_{A}$. In order
to derive constitutive relations and Kubo formulas, we will use generating
functionals among other methods. Note that the generating functionals (2),
(16) and (21) in presence of a global axial $U(1)_{A}$-symmetry transforms
even under charge-inversion, parity, and time-reversal, i.e. it has
$(C,P,T)$-eigenvalues (+/+/+).

###  2.1 Thermodynamics

####  2.1.1 Generating functional and equilibrium constitutive relations

Following the procedure proposed in Banerjee:2012iz ; Jensen:2012jh , we begin
by considering the equilibrium constraints on the hydrodynamic framework
arising from the existence of a static generating functional. These
constraints arise from considering a system with a time-like Killing vector
field $V$ (i.e. ${\cal L}_{V}=0$) coupled to an external metric $g$ and gauge
field $A$. For systems that

  * •

have finite correlation lengths,

  * •

are in equilibrium (${\cal L}_{V}=0)$,

  * •

have sources $g,A$ that vary on scales much longer than the correlation
lengths,

the generating functional $W_{s}[g,A]=-i\,{\rm ln}Z[g,A]$ is a local
functional of the Killing vector field and the sources. The equilibrium
generating functional can be systematically expanded in a derivative
expansion. The temperature $T$, the chemical potential $\mu$ and the fluid
velocity $u^{\mu}$, which are traditionally considered the only zero
derivative terms and are defined in terms of the Killing vector field and the
sources

$T=\frac{T_{0}}{\sqrt{-V^{2}}}\,,\quad
u^{\mu}=\frac{V^{\mu}}{\sqrt{-V^{2}}}\,,\quad\mu=\frac{V^{\mu}A_{\mu}+\Lambda_{V}}{\sqrt{-V^{2}}}\,,$
(1)

where $T_{0}$ is a constant setting the normalization of the temperature, and
$\Lambda_{V}$ is a gauge parameter which ensures that $\mu$ is gauge invariant
Jensen:2013kka . In addition, for a system subject to a strong magnetic field
$B^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_{\nu}F_{\rho\sigma}\sim{\cal
O}(1)$, the scalar $B^{2}=B^{\mu}B^{\nu}g_{\mu\nu}$ is order zero in
derivatives as well. In this paper, we assume the counting
$T,\mu,u^{\mu},B^{\mu}\sim{\cal O}(1)$ and all other terms with a derivative
as ${\cal O}(\partial)$. For example, $E_{\mu}=F_{\mu\nu}u^{\nu}\sim{\cal
O}(\partial)$ and $R^{\mu}_{\ \nu\rho\sigma}\sim\partial\partial g\sim{\cal
O}(\partial^{2})$. In addition, the derivatives of the fluid velocity such as
the vorticity
$\Omega^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}u_{\sigma}$ and
the fluid acceleration $a^{\mu}=u^{\nu}\nabla_{\nu}u^{\mu}$ are
$\mathcal{O}(\partial)$. In table 6, the behavior of hydrodynamic fields,
sources and other quantities under charge conjugation $C$, parity $P$, and
time reversal $T$ are collected. Note that in this subsection on equilibrium
states, the only parity violation stems from the axial gauge field and the
associated axial chemical potential.555 Naively, vector magnetic fields do
also break parity. However, the magnetic field is the only vector valued
quantity characterizing the equilibrium state. Hence, only the scalar $B^{2}$
enters into the pressure $p$ and the transport coefficients (such as $M_{n}$).
Note that $B^{2}$ transforms even under $P$ (as well as under $C$ and $T$).

With this derivative counting, the equilibrium generating functional for a
hydrodynamic system coupled to external gauge field and metric subject to
strong magnetic fields can be expanded as Hernandez:2017mch

$W_{s}=\int
d^{4}x\sqrt{-g}\left(p(T,\mu,B^{2})+\sum_{n=1}^{5}M_{n}(T,\mu,B^{2})s_{n}+O(\partial^{2})\right)\,,$
(2)

where $p$ is the homogeneous equilibrium pressure and $s_{n}$ are the first
order equilibrium scalars, that is, $s_{n}\sim{\cal O}(\partial)$. Their
definitions are listed in table 1, together with their transformation
properties under charge conjugation, parity, time reversal and Weyl
transformations. The magnetovortical susceptibility $M_{5}=M_{\Omega}$ is the
only nonzero first order thermodynamic function for a parity-preserving theory
coupled to vector gauge fields. On the other hand, for a system coupled to an
axial gauge field, the axial chemical potential $\mu$ breaks parity and the
other $M_{n}$ can in principle be nonzero. These will appear along with the
pressure in the thermodynamic/hydrostatic constitutive relations. We stress
that, in absence of the chiral anomaly, the microscopic theory does not break
the parity symmetry, but rather the state in question breaks parity, provided
there is a nonzero axial chemical potential.666A nonzero vector magnetic field
can also break parity. However, the only zeroth order scalar with odd parity
is an axial chemical potential. A parity odd zeroth order scalar is what in
turn allows $M_{n}$ to be nonzero while remaining parity odd. This is
essential for a parity preserving microscopic system to have nonzero $M_{n}$
accompanying the parity odd first order scalars $s_{n}$.  Table 1 highlights
the symmetry properties of the equilibrium scalars when defined in terms of
axial gauge fields. For this paper, we will focus in the case where our system
is coupled to axial gauge fields. This requires us to include the terms that
are usually considered to be parity-violating when considering vector gauge
fields. That is, the $M_{1}$, $M_{2}$, $M_{3}$ and $M_{4}$ can be nonzero for
a parity preserving system coupled to an external axial gauge field. In this
section, we elaborate on the modifications of the hydrodynamic framework when
these terms are included.

We will write the energy-momentum tensor using the decomposition with respect
to the timelike velocity vector $u^{\mu}$,

$\displaystyle T^{\mu\nu}={\cal E}u^{\mu}u^{\nu}+{\cal P}\Delta^{\mu\nu}+{\cal
Q}^{\mu}u^{\nu}+{\cal Q}^{\nu}u^{\mu}+{\cal T}^{\mu\nu}\,,$ (3)

where $\Delta^{\mu\nu}\equiv g^{\mu\nu}+u^{\mu}u^{\nu}$ is the transverse
projector, the energy current ${\cal Q}^{\mu}$ is transverse to $u_{\mu}$, and
${\cal T}^{\mu\nu}$ is transverse to $u_{\mu}$, symmetric, and traceless.
Explicitly, the coefficients are ${\cal E}\equiv u_{\mu}u_{\nu}T^{\mu\nu}$,
${\cal P}\equiv{\textstyle{\frac{1}{3}}}\Delta_{\mu\nu}T^{\mu\nu}$, ${\cal
Q}_{\mu}\equiv-\Delta_{\mu\alpha}u_{\beta}T^{\alpha\beta}$ and ${\cal
T}_{\mu\nu}\equiv{\textstyle{\frac{1}{2}}}(\Delta_{\mu\alpha}\Delta_{\nu\beta}+\Delta_{\nu\alpha}\Delta_{\mu\beta}-{\textstyle{\frac{2}{3}}}\Delta_{\mu\nu}\Delta_{\alpha\beta})T^{\alpha\beta}$.
Similarly, we will write the current as

$\displaystyle J^{\mu}={\cal N}u^{\mu}+{\cal J}^{\mu}\,,$ (4)

where the charge density is ${\cal N}\equiv-u_{\mu}J^{\mu}$, and the spatial
current is ${\cal J}_{\mu}\equiv\Delta_{\mu\lambda}J^{\lambda}$. We also
decompose the field strength tensor with respect to $u^{\mu}$,

$n$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$s_{n}$ | $B^{\mu}\partial_{\mu}(\frac{B^{2}}{T^{4}})$ | $\epsilon^{\mu\nu\rho\sigma}u_{\mu}B_{\nu}\nabla_{\\!\rho}B_{\sigma}$ | $B{\cdot}a$ | $B{\cdot}E$ | $B{\cdot}\Omega$
$(C,P,T)_{\textrm{axial}}$ | $+/+/-$ | $+/-/+$ | $+/+/-$ | $+/-/-$ | $+/-/+$
$(C,P,T)_{\textrm{vector}}$ | $-/-/-$ | $+/-/+$ | $-/-/-$ | $+/-/-$ | $-/+/+$
W | 3 | 5 | n/a | 4 | 3
Table 1: Independent nonzero $O(\partial)$ invariants in equilibrium in 3+1
dimensions for an axial gauge field. We have used the fluid acceleration
$a^{\mu}=u^{\lambda}\nabla_{\lambda}u^{\mu}$ and the vorticity
$\Omega^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}u_{\sigma}$. An
axial chemical potential $\mu$ is $C$-even and $P$-odd. In case of an axial
chemical potential we expect $M_{1}$ and $M_{3}$ to be even functions of $\mu$
while $M_{2}$, $M_{4}$ and $M_{5}$ should be odd functions of $\mu$. Here the
last row labelled “W” indicates the charge under Weyl-transformations and
“n/a” indicates that the tensor structure $s_{3}$ does not have a definite
behavior under Weyl-transformations.

$F_{\mu\nu}=u_{\mu}E_{\nu}-u_{\nu}E_{\mu}-\epsilon_{\mu\nu\rho\sigma}u^{\rho}B^{\sigma}\,,$
(5)

where $E_{\mu}=F_{\mu\nu}u^{\nu}$ is the electric field and
$B^{\mu}={\textstyle{\frac{1}{2}}}\epsilon^{\mu\nu\rho\sigma}u_{\nu}F_{\rho\sigma}$
is the magnetic field. We use the convention
$\epsilon^{\mu\nu\rho\sigma}=\varepsilon^{\mu\nu\rho\sigma}/\sqrt{-g}$, where
$\varepsilon^{0123}=1$. We also use the vorticity
$\Omega^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}u_{\sigma}$.

The equilibrium constitutive relations are found by varying the generating
functional with respect to the metric and the gauge field

$\delta W_{s}[A,g]=\int
d^{4}x\sqrt{-g}\left({\textstyle{\frac{1}{2}}}T^{\mu\nu}_{\rm eq.}\delta
g_{\mu\nu}+J^{\mu}_{\rm eq.}\delta A_{\mu}\right)\,.$ (6)

This was done in Hernandez:2017mch  for a parity-preserving theory coupled to
a vector gauge field. The new terms allowed when considering an axial gauge
field come from the variation of $M_{i}(T,\mu,B^{2})S_{i}$ for $i\neq 5$ and
are given by

${\cal E}_{{\rm eq.\,new}}=\sum_{n=1}^{4}\epsilon_{n}s_{n}\,,\quad{\cal
P}_{\rm eq.\,new}=\sum_{n=1}^{4}\pi_{n}s_{n}\,,\quad{\cal N}_{\rm
eq.\,new}=\sum_{n=1}^{4}s_{n}\,\quad{\cal T}^{\mu\nu}_{{\rm
eq.\,new}}=\sum_{n=1}^{10}\theta_{n}\tau^{\mu\nu}_{n}\,,$ (7)

where

$\displaystyle\epsilon_{3}=-\frac{4B^{2}}{T^{4}}\epsilon_{1}=3\pi_{3}=-4B^{2}\theta_{3}=\frac{4B^{2}}{T^{4}}\left(M_{1}-TM_{1,T}-\mu
M_{1,\mu}-4B^{2}M_{1,B^{2}}-T^{4}M_{3,B^{2}}\right)\,,$ (8)
$\displaystyle\epsilon_{2}=\theta_{6}=-M_{2}+TM_{2,T}+\mu
M_{2,\mu}\,,\quad\epsilon_{4}=TM_{4,T}+\mu
M_{4,\mu}+\frac{4B^{2}}{T^{4}}M_{1,\mu}+M_{3,\mu}\,,$
$\displaystyle\pi_{2}=-{\textstyle{\frac{2}{3}}}M_{2}-{\textstyle{\frac{4}{3}}}B^{2}M_{2,B^{2}}\,,n\quad\pi_{4}=-{\textstyle{\frac{4}{3}}}B^{3}\theta_{4}=\frac{4B^{2}}{3T^{4}}\phi_{1}=\frac{4B^{2}}{3T^{4}}\left(M_{1,\mu}-T^{4}M_{4,B^{2}}\right)\,,$
$\displaystyle\phi_{2}=M_{2,\mu}\,,\quad\phi_{3}=\epsilon_{4}-3\pi_{4}\,,\quad\theta_{2}=M_{2,B^{2}}\,,\quad\theta_{5}=2M_{2}\,,\quad\theta_{7}=M_{2,B^{2}}\,,\quad\theta_{8}=-M_{2,\mu}\,,$
$\displaystyle\pi_{1}=\phi_{4}=\theta_{1}=0\,.$

The comma subscript denotes the derivative with respect to the argument that
follows, and we are using $(T,\mu,B^{2})$ as our three independent variables.
Hence, for example, $M_{1,T}=\left(\frac{\partial M_{1}}{\partial
T}\right)_{\mu,B^{2}}$. The equilibrium vectors and tensors are defined in
table 2. The equilibrium spatial current $\mathcal{J}^{\mu}$ and energy
current $\mathcal{Q}^{\mu}$ do not receive contributions to $O(\partial)$ from
the novel thermodynamic transport coefficients $M_{1}$, $M_{2}$, $M_{3}$ and
$M_{4}$.

$n$ | 1 | 2 | 3 | 4
---|---|---|---|---
$v_{n}^{\mu}$ | $\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\sigma}B_{\rho}$ | $\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}T/T$ | $\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}B^{2}$ | $\epsilon^{\mu\nu\rho\sigma}u_{\nu}E_{\rho}B_{\sigma}$

$n$ | $1-4$ | 5 | 6 | 7 | 8
---|---|---|---|---|---
$t_{n}^{\mu\nu}$ | $s_{n}B^{\langle\mu}B^{\nu\rangle}$ | $v_{1}^{\langle\mu}B^{\nu\rangle}$ | $v_{2}^{\langle\mu}B^{\nu\rangle}$ | $v_{3}^{\langle\mu}B^{\nu\rangle}$ | $v_{4}^{\langle\mu}B^{\nu\rangle}$

Table 2: Top: nonzero transverse $O(\partial)$ vectors that appear in the
parity-violating equilibrium energy flux ${\cal Q}^{\mu}$ and in the
equilibrium spatial current ${\cal J}^{\mu}$. The vector $v_{4}^{\mu}$ is the
Poynting vector. Bottom: nonzero symmetric transverse traceless $O(\partial)$
tensors that appear in the equilibrium stress ${\cal T}^{\mu\nu}$. For any two
transverse vectors $X^{\mu}$ and $Y^{\mu}$, the angular brackets stand for
$X^{\langle\mu}Y^{\nu\rangle}\equiv
X^{\mu}Y^{\nu}+X^{\nu}Y^{\mu}-{\textstyle{\frac{2}{3}}}\Delta^{\mu\nu}X{\cdot}Y$.

For a diffeomorphism and gauge invariant theory, invariance of the generating
functional gives the following hydrodynamic equations

$\displaystyle\nabla_{\nu}T^{\mu\nu}=F^{\mu\nu}J_{\nu}\,,$ (9a)
$\displaystyle\nabla_{\mu}J^{\mu}=0\,.$ (9b)

The definition of the equilibrium energy-momentum tensor and conserved
currents ensure that the equations of motion are satisfied in equilibrium.

For completeness, let us summarize the equilibrium constitutive relations for
the energy-momentum tensor and the current. The equilibrium energy-momentum
tensor is given by

$\displaystyle{\cal E}_{\rm eq.}$
$\displaystyle=-p+T\,p_{,T}+\mu\,p_{,\mu}+\left(T{{M_{5}}}_{,T}+\mu{{M_{5}}}_{,\mu}-2{M_{5}}\right)B{\cdot}\Omega$
$\displaystyle\ \ \ \,+\left(TM_{1,T}+\mu
M_{1,\mu}+4B^{2}M_{1,B^{2}}+T^{4}M_{3,B^{2}}-M_{1}\right)s_{1}$
$\displaystyle\ \ \ \,+\left(TM_{2,T}+\mu M_{2,\mu}-M_{2}\right)s_{2}$
$\displaystyle\ \ \ \,+\frac{4B^{2}}{T^{4}}\left(M_{1}-TM_{1,T}-\mu
M_{1,\mu}-4B^{2}M_{1,B^{2}}-T^{4}M_{3,B^{2}}\right)s_{3}$ $\displaystyle\ \ \
\,+\left(TM_{4,T}+\mu
M_{4,\mu}+\frac{4B^{2}}{T^{4}}M_{1,\mu}+M_{3,\mu}\right)s_{4}\,,$ (10a)
$\displaystyle{\cal P}_{\rm eq.}$
$\displaystyle=p-{\textstyle{\frac{4}{3}}}\,p_{,B^{2}}B^{2}-{\textstyle{\frac{1}{3}}}({M_{5}}+4{{M_{5}}}_{,B^{2}}B^{2})B{\cdot}\Omega-{\textstyle{\frac{2}{3}}}\left(M_{2}+2B^{2}M_{2,B^{2}}\right)s_{2}$
$\displaystyle\ \ \ \,+\frac{4B^{2}}{3T^{4}}\left(M_{1}-TM_{1,T}-\mu
M_{1,\mu}-4B^{2}M_{1,B^{2}}-T^{4}M_{3,B^{2}}\right)s_{3}$ $\displaystyle\ \ \
\,+\frac{4B^{2}}{3T^{4}}\left(M_{1,\mu}-T^{4}M_{4,B^{2}}\right)s_{4}\,,$ (10b)
$\displaystyle{\cal Q}^{\mu}_{\rm eq.}$
$\displaystyle=-{M_{5}}\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\sigma}B_{\rho}+(2{M_{5}}-T{{M_{5}}}_{,T}-\mu{{M_{5}}}_{,\mu})\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}T/T$
$\displaystyle\ \ \
\,-{{M_{5}}}_{,B^{2}}\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}B^{2}+({{M_{5}}}_{,\mu}-2p_{,B^{2}})\epsilon^{\mu\nu\rho\sigma}u_{\nu}E_{\rho}B_{\sigma}$
(10c) $\displaystyle{\cal T}^{\mu\nu}_{\rm eq.}$
$\displaystyle=2p_{,B^{2}}\left(B^{\mu}B^{\nu}-{\textstyle{\frac{1}{3}}}\Delta^{\mu\nu}B^{2}\right)+B^{\langle\mu}B^{\nu\rangle}\left({{M_{5}}}_{,B^{2}}B{\cdot}\Omega+M_{2,B^{2}}s_{2}+(M_{4,B^{2}}-{\textstyle{\frac{1}{T^{4}}}}M_{1,\mu})s_{4}\right)$
$\displaystyle\ \ \
\,+B^{\langle\mu}B^{\nu\rangle}\frac{1}{T^{4}}\left(TM_{1,T}+\mu
M_{1,\mu}+4B^{2}M_{1,B^{2}}-M_{1}+T^{4}M_{3,B^{2}}\right)s_{3}+{M_{5}}B^{\langle\mu}\Omega^{\nu\rangle}$
$\displaystyle\ \ \
\,+2M_{2}B^{\langle\mu}\epsilon^{\nu\rangle\rho\sigma\alpha}u_{\rho}\partial_{\sigma}B_{\alpha}+\left(TM_{2,T}+\mu
M_{2,\mu}-M_{2}\right)B^{\langle\mu}\epsilon^{\nu\rangle\alpha\rho\sigma}u_{\alpha}B_{\rho}\partial_{\sigma}T/T$
$\displaystyle\ \ \
\,+M_{2,B^{2}}B^{\langle\mu}\epsilon^{\nu\rangle\alpha\rho\sigma}u_{\alpha}B_{\rho}\partial_{\sigma}B^{2}-M_{2,\mu}B^{\langle\mu}\epsilon^{\nu\rangle\rho\sigma\alpha}u_{\rho}E_{\sigma}B_{\alpha}\,,$
(10d)

where we used the vorticity
$\Omega^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}u_{\sigma}$.
The current is given by

$\displaystyle{\cal N}_{\rm eq.}$
$\displaystyle=p_{,\mu}-\nabla{\cdot}\mathfrak{p}+\mathfrak{p}{\cdot}a-\mathfrak{m}{\cdot}\Omega+\left(M_{1,\mu}-T^{4}M_{4,B^{2}}\right)s_{1}+M_{2,\mu}s_{2}$
$\displaystyle\ \ \ \,+\left(M_{3,\mu}+TM_{4,T}+\mu
M_{4,\mu}+4B^{2}M_{4,B^{2}}\right)s_{3}+{{M_{5}}}_{,\mu}s_{5}\,,$ (11a)
$\displaystyle{\cal J}^{\mu}_{\rm eq.}$
$\displaystyle=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\nabla_{\\!\rho}\mathfrak{m}_{\sigma}+\epsilon^{\mu\nu\rho\sigma}u_{\nu}a_{\rho}\mathfrak{m}_{\sigma}\,,$
(11b)

where $a^{\mu}=u^{\lambda}\nabla_{\lambda}u^{\mu}$ defines the acceleration
and the (electric) polarization vector is
$\mathfrak{p}_{\mu}=\frac{1}{\sqrt{-g}}\frac{\delta W_{s}}{\delta
E^{\mu}}=M_{4}B_{\mu}$. The current is written in terms of the magnetic
polarization vector $\mathfrak{m}_{\mu}=\frac{1}{\sqrt{-g}}\frac{\delta
W_{s}}{\delta B^{\mu}}$777Careful comparison with Kovtun:2018dvd  shows an
agreement with their (2.19a) and (2.19b) in the $B={\cal O}(\partial)$ limit.
Note that the $M_{1}$ and $M_{2}$ would be pushed to higher derivative order
and none of the $M_{n}$ would be functions of $B^{2}$. Similarly, $p_{,B^{2}}$
would be a second order term which corresponds to their $f_{6}$.

$\displaystyle\mathfrak{m}^{\mu}$
$\displaystyle=\left(2\,p_{,B^{2}}+2\sum_{n=2}^{5}M_{n,B^{2}}s_{n}+\frac{2}{T^{4}}\left(M_{1}-TM_{1,T}-\mu
M_{1,\mu}-4B^{2}M_{1,B^{2}}\right)B{\cdot}\partial T/T\right)B^{\mu}$ (12)
$\displaystyle\ \ \
\,+{M_{5}}\Omega^{\mu}+M_{3}a^{\mu}+M_{4}E^{\mu}+M_{1}\Delta^{\mu\nu}\partial_{\nu}\frac{B^{2}}{T^{4}}-M_{2,\mu}\epsilon^{\mu\nu\rho\sigma}u_{\nu}E_{\rho}B_{\sigma}+M_{2,B^{2}}\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}B^{2}$
$\displaystyle\ \ \ \,+\left(TM_{2,T}+\mu
M_{2,\mu}-M_{2}\right)\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}T/T+2M_{2}\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}B_{\sigma}\,.$

Note that we are keeping O($\partial^{2}$) thermodynamic terms in the current
(coming from the variation of $\sum_{n=1}^{5}M_{n}s_{n}$ in the generating
functional) that are needed to ensure that the conservation laws (9) are
satisfied to $O(\partial^{2})$ for time-independent background fields.
Including the $O(\partial^{2})$ thermodynamic terms in the energy-momentum
tensor will ensure these are satisfied identically, but we omit them here for
simplicity.

####  2.1.2 Incorporating the chiral anomaly

For a theory with a chiral anomaly subject to external axial gauge fields, the
generating functional is no longer gauge invariant,888In curved spacetime, the
gauge non-invariance of the generating functional (13) includes some curvature
terms proportional to the square of the Riemann tensor. However, in this paper
we restrict our attention to the derivative counting $\partial g\sim{\cal
O}(\partial)$ so that these terms are of order four in derivatives. We
therefore neglect these terms for the rest of the paper. Strictly speaking,
the corresponding gravitational Chern-Simons contribution to eq. (22) includes
curvature terms which cannot be taken as ${\cal O}(\partial^{2})$ in the bulk
spacetime $\cal M$. These terms give rise to the effects we will find by
including the term multiplying $c_{1}$ in the consistent generating functional
(16). See, for example, Chen:2012ca ; Jensen:2012kj ; Landsteiner:2011cp ;
Stone:2018zel ; Landsteiner:2011iq  for more careful treatments of the mixed
anomaly term.

$\delta_{\alpha}W_{cons}=\frac{C}{24}\int
d^{4}x\sqrt{-g}\,\alpha\,\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\equiv{\cal
A}\,,$ (13)

leading to the following

$\displaystyle\nabla_{\mu}T^{\mu\nu}_{A}=F^{\mu\nu}J^{cons}_{\nu}-A^{\nu}\nabla_{\mu}J^{\mu}_{cons}\,,$
(14a)
$\displaystyle\nabla_{\mu}J_{cons}^{\mu}=-\frac{C}{24}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}=\frac{C}{3}E{\cdot}B\,,$
(14b)

where $J_{cons}^{\mu}=\frac{1}{\sqrt{-g}}\frac{\delta W_{cons}}{\delta
A_{\mu}}$ is the gauge dependent consistent current. The fact that it is gauge
dependent follows from the commuting of $\frac{\delta}{\delta A_{\mu}}$ with
the BRST operator $s=\int d^{4}x\partial_{\mu}c\frac{\delta}{\delta A_{\mu}}$
generating gauge transformations, from which we get
$\delta_{\alpha}J^{\mu}_{cons}=\frac{1}{\sqrt{-g}}\frac{\delta}{\delta
A_{\mu}}{\cal
A}={\textstyle{\frac{1}{6}}}C\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}\alpha
F_{\rho\sigma}$. Noting that ${\cal A}$ is independent of the metric, a
similar argument shows that the consistent energy-momentum tensor
$T^{\mu\nu}_{A}=\frac{2}{\sqrt{-g}}\frac{\delta W_{cons}}{\delta g_{\mu\nu}}$
is gauge invariant. It is possible to add a Chern-Simons current
$J^{\mu}_{BZ}=-{\textstyle{\frac{1}{6}}}C\epsilon^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}$,
also known as a Bardeen-Zumino polynomial, to the consistent current to get a
gauge invariant current $J^{\mu}_{cov}$, usually named covariant current. The
equations of motion (14) then take the manifestly gauge covariant from

$\displaystyle\nabla_{\nu}T^{\mu\nu}_{A}=F^{\mu\nu}J^{cov}_{\nu}\,,$ (15a)
$\displaystyle\nabla_{\mu}J^{\mu}_{cov}=-\frac{C}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}=CE{\cdot}B\,.$
(15b)

Note that the covariant energy-momentum tensor is the same as the consistent
energy-momentum tensor. See Landsteiner:2016led  for a recent review on
anomalous currents.

To understand how this gauge anomaly affects the hydrodynamic description, we
construct the equilibrium generating functionals for the consistent and for
the covariant currents using the anomaly inflow mechanism Callan:1984sa . The
anomaly inflow generating functionals have been used for anomalous
hydrodynamics in equilibrium in Jensen:2013kka ; Jensen:2012kj  and for out of
equilibrium hydrodynamics in Haehl:2013hoa .

The gauge dependent generating functional for the consistent current of a 3+1
dimensional theory is given by999Note that $c_{1}$ and $c_{2}$ defined here do
not depend on the thermodynamic quantities. They are properties of the
microscopic theory. Hence, they are entirely different from the transport
coefficients which we will name $c_{n}$ later in the text.

$W_{cons}=W_{s}+\int
d^{4}x\sqrt{-g}\left(c_{1}T^{2}\Omega{\cdot}A+c_{2}T\left(B{\cdot}A+\mu\Omega{\cdot}A\right)+\frac{C}{3}\mu\left(B{\cdot}A+{\textstyle{\frac{1}{2}}}\mu\Omega{\cdot}A\right)\right)\,,$
(16)

where $W_{s}$ is the generating functional for a theory without anomalies (2).
We refer to $W_{cons}$ as the consistent generating functional. The vectors
$T^{2}\Omega^{\mu}$ and $TB^{\mu}+T\mu\Omega^{\mu}$ have vanishing divergence
and do not contribute to the gauge anomaly, unless the 3+1-dimensional theory
has a boundary. The gauge dependence of the consistent generating functional
(13) comes from the non-conservation of the vector $\nabla_{\mu}\left(\mu
B^{\mu}+{\textstyle{\frac{1}{2}}}\mu^{2}\Omega^{\mu}\right)=B{\cdot}E$. Note
that the term multiplying $c_{2}$ breaks CPT symmetry and is therefore not
allowed for Lorentz invariant theories Jensen:2013vta . The coefficient
$c_{1}$ is related to the mixed gauge-gravitational anomaly $c_{m}$ by
Chen:2012ca ; Jensen:2012kj ; Landsteiner:2011cp ; Stone:2018zel ;
Landsteiner:2011iq

$c_{1}=-8\pi^{2}c_{m}\,.$ (17)

The variation of the consistent generating functional yields the energy-
momentum tensor and the consistent current. We now focus on the new terms
coming from $W_{cons}-W_{s}$ and write $\Delta
J_{cons}^{\mu}=J^{\mu}_{cons}-J^{\mu}$ where $J^{\mu}$ is the current found in
eq. (6) by varying $W_{s}$. Similarly, we write $\Delta
T^{\mu\nu}=T^{\mu\nu}_{A}-T^{\mu\nu}$ where $T^{\mu\nu}$ comes from varying
$W_{s}$. Taking the source variations we find

$\displaystyle\Delta T^{\mu\nu}$
$\displaystyle=\xi_{T}u^{(\mu}\Omega^{\nu)}+\xi_{TB}\,u^{(\mu}B^{\nu)}\,,$
(18a) $\displaystyle\Delta J^{\mu}_{cons}$
$\displaystyle={\textstyle{\frac{1}{3}}}CB{\cdot}Au^{\mu}+\xi\,\Omega^{\mu}+\left(\xi_{B}-{\textstyle{\frac{1}{3}}}C\mu\right)B^{\mu}+{\textstyle{\frac{1}{3}}}C\epsilon^{\mu\nu\rho\sigma}A_{\nu}u_{\rho}E_{\sigma}\,,$
(18b)

where101010The following conventions for anomalous transport coefficients are
in the thermodynamic frame used, for example, in Jensen:2013vta . This
corresponds in Amado:2011zx  to the “no drag frame” coefficients $\sigma_{B}$
and $\sigma_{V}$.

$\displaystyle\xi$
$\displaystyle={\textstyle{\frac{1}{2}}}C\mu^{2}+c_{1}T^{2}+2c_{2}T\mu\,,\quad$
$\displaystyle\xi_{B}=C\mu+2c_{2}T\,,$ $\displaystyle\xi_{T}$
$\displaystyle={\textstyle{\frac{1}{3}}}C\mu^{3}+2c_{1}T^{2}\mu+2c_{2}T\mu^{2}\,,\quad$
$\displaystyle\xi_{TB}={\textstyle{\frac{1}{2}}}C\mu^{2}+c_{1}T^{2}+2c_{2}T\mu\,.$
(19)

The consistent current $J^{\mu}_{cons}$ and energy-momentum tensor
$T^{\mu\nu}_{A}$ satisfy the consistent equations of motion (14) derived from
the diffeomorphism invariance and gauge-non-invariance of the consistent
generating functional $W_{cons}$. From the consistent current, we can
construct the covariant current $J^{\mu}_{cov}$ by adding to it the Bardeen-
Zumino/Chern-Simons current
$J^{\mu}_{BZ}=-{\textstyle{\frac{1}{6}}}C\epsilon^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}$,

$J^{\mu}_{cov}=J^{\mu}_{cons}+J^{\mu}_{BZ}\,.$ (20)

Alternatively, we can construct a covariant generating functional $W_{cov}$ by
adding a Chern-Simons functional to the consistent generating functional

$W_{cov}=W_{cons}+W_{CS}\,,$ (21)

where111111This Chern-Simons functional contains only gauge terms since we are
omitting the gravitational anomalies which appear at higher order in
hydrodynamic derivatives. See footnote 8.

$W_{CS}=-\frac{C}{6}\int A\wedge F\wedge F=-\frac{C}{24}\int
d^{5}x\sqrt{-G}\epsilon^{mnopq}A_{m}F_{no}F_{pq}\,.$ (22)

We take our Chern-Simons theory to live in a 4+1 dimensional space-time ${\cal
M}$ with a boundary $\partial{\cal M}$ which corresponds to the space-time
where $W_{cons}$ is defined. The five dimensional field strength
$F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$ is defined in terms of the five
dimensional gauge field $A_{m}$. We take the gauge field $A_{\mu}$ appearing
in $W_{cons}$ as the induced gauge field on $\partial{\cal M}$ from $A_{m}$.
The Chern-Simons functional is independent of the five dimensional metric
$G_{mn}$, and we use the convention
$\epsilon^{mnopq}=\varepsilon^{mnopq}/\sqrt{-G}$, where
$\varepsilon^{0123z}=1$ and z is the coordinate normal to $\partial{\cal M}$.
We also take $G_{mn}$ so that the induced metric in $\partial{\cal M}$ is
$g_{\mu\nu}$, the metric used in the consistent generating functional. The
Chern-Simons theory is gauge invariant up to a boundary term

$\delta_{\alpha}W_{CS}=-\frac{C}{24}\int_{\partial{\cal
M}}d^{4}x\sqrt{-g}\alpha\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\,,$
(23)

which cancels the gauge dependence of $W_{cons}$. Taking source variations of
the covariant generating functional gives the covariant energy-momentum tensor
and current as well as the bulk current $J^{m}_{H}$,

$\delta W_{cov}=\int_{\partial{\cal
M}}d^{d}x\sqrt{-g}\left({\textstyle{\frac{1}{2}}}T^{\mu\nu}_{A}\delta
g_{\mu\nu}+J^{\mu}_{cov}\delta A_{\mu}\right)+\int_{\cal
M}d^{5}x\sqrt{-G}J^{m}_{H}\delta A_{m}\,.$ (24)

Note that the bulk energy-momentum tensor
$T^{mn}_{H}=\frac{2}{\sqrt{-G}}\frac{\delta W_{cov}}{\delta G_{mn}}$ vanishes
since $W_{CS}$ is independent of $G_{mn}$. Variations of the Chern-Simons
functional give the Bardeen-Zumino current and the bulk current

$\delta W_{CS}=\int_{\partial{\cal M}}d^{4}x\sqrt{-g}J^{\mu}_{BZ}\delta
A_{\mu}+\int_{\cal M}d^{5}x\sqrt{-G}J^{m}_{H}\delta A_{m}\,.$ (25)

Explicitly, these currents are

$\displaystyle J^{\mu}_{BZ}$
$\displaystyle=-\frac{C}{6}\epsilon^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}\,,$
(26a) $\displaystyle J^{m}_{H}$
$\displaystyle=-\frac{C}{8}\epsilon^{mnopq}F_{no}F_{pq}\,.$ (26b)

Notice that
$J_{H}^{z}=-\frac{C}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}=CB{\cdot}E$.
Diffeomorphism and gauge invariance of $W_{cov}$ then lead to the covariant
equations of motion (15) together with

$\nabla_{m}J^{m}_{H}=0\,,$ (27)

which follows directly from the Bianchi identity. The covariant current can be
found from (20). Using
${\textstyle{\frac{1}{2}}}\epsilon^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}=B{\cdot}Au^{\mu}+\epsilon^{\mu\nu\rho\sigma}A_{\nu}u_{\rho}E_{\sigma}-\mu
B^{\mu}$, we get

$J^{\mu}_{cov}=J^{\mu}+\xi\,\Omega^{\mu}+\xi_{B}\,B^{\mu}$ (28)

Equations (18a) and (28) show how the covariant current and the energy-
momentum tensor have to be modified in the presence of a chiral anomaly. The
transport coefficients determined by the anomaly coefficient $C$ first
appeared in holographic calculations Erdmenger:2008rm ; Banerjee:2008th .
Their first derivation in the hydrodynamic framework was done in Son:2009tf
using entropy current arguments. In Neiman:2010zi , the result was generalized
for theories with general triangle anomalies and the coefficients $c_{1}$ and
$c_{2}$ appear as integration constants from solving the entropy constraints.
These results were then derived using equilibrium generating functionals in
Banerjee:2012iz ; Jensen:2013kka . The anomaly induced transport terms found
in the thermodynamic frame are exact Loganayagam:2011mu  and can be brought to
the Landau-Lifshitz frame by a redefinition of the hydrodynamic variables,
using $u^{\mu}\to u^{\mu}+\delta u^{\mu}$ so that ${\cal Q}_{A}^{\mu}=0$.

####  2.1.3 Thermodynamic correlation functions and Kubo formulas

The Kubo formulas relate the transport coefficients to two-point functions of
conserved currents and stress tensors of the underlying microscopic theory.
For a system in equilibrium $({\cal L}_{V}=0)$, the static correlation
functions can be found by taking second order variations of the generating
functional $W_{s}[g,A]$ with respect to the external sources $g_{\mu\nu}$ and
$A_{\mu}$. Concretely, for $A_{\mu}\to A_{\mu}+\delta A_{\mu}$ and
$g_{\mu\nu}\to g_{\mu\nu}+\delta g_{\mu\nu}$ such that ${\cal L}_{V}\delta
A_{\mu}={\cal L}_{V}\delta g_{\mu\nu}=0$ , we have

$\delta
W_{s}^{(2)}=\int\sqrt{-g}\left({\textstyle{\frac{1}{4}}}G_{T^{\mu\nu}T^{\rho\sigma}}\delta
g_{\mu\nu}\delta
g_{\rho\sigma}+{\textstyle{\frac{1}{2}}}G_{T^{\mu\nu}J^{\rho}}\delta
g_{\mu\nu}\delta A_{\rho}+G_{J^{\mu}J^{\nu}}\delta A_{\mu}\delta
A_{\nu}\right)\,,$ (29)

where $\delta W_{s}^{(2)}$ is the second order variation121212The first order
variation is simply eq. (6). of $W_{s}$ in eq. (2) and
$G_{J^{\rho}T^{\mu\nu}}=G_{T^{\mu\nu}J^{\rho}}$. Note that this is equivalent
to taking the first order variations of the equilibrium current $J^{\mu}[g,A]$
and stress tensor $T^{\mu\nu}[g,A]$ in eq. (6) with respect to the sources.
From here on, we work within an equilibrium state defined in flat space with
metric $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$, with background magnetic field
$B=(0,0,0,B_{0})$, and in the fluid rest frame with velocity
$u^{\mu}=(1,0,0,0)$. We then consider plane wave fluctuations ($\delta
A,\delta g\sim{\rm exp}\left(ik{\cdot}x\right)$) parallel ($k=(0,0,0,k_{z})$)
and perpendicular ($k=(0,0,k_{y},0)$) about such a background.131313The fact
that these correlation functions are evaluated at zero frequency ensures that
the fluctuations satisfy the equilibrium constraint (${\cal L}_{V}=0$).

Let us begin with the Kubo formulas for a thermodynamic transport coefficient
which was previously considered, $M_{5}$, and a novel one $M_{2}$. Both are
expressed in terms of static correlation functions as follows

$\displaystyle\frac{1}{k_{z}}{\rm
Im}\,G_{T^{xz}T^{yz}}(\omega=0,k_{z}\mathbf{\hat{z}})=-2\,B_{0}^{2}\,M_{2}\,,$
(30) $\displaystyle\frac{1}{k_{z}}{\rm
Im}\,G_{T^{tx}T^{yz}}(\omega=0,k_{z}\mathbf{\hat{z}})=-B_{0}\,M_{5}\,,$

in the limit of first setting $\omega=0$, and then taking $k_{z}\to 0$, and
$\mathbf{\hat{z}}$ is the unit vector in $z$-direction. In what follows, we
take this limit in all the Kubo relations for thermodynamic transport
coefficients.

For zero background magnetic field, it is still possible to find Kubo formulas
for the magneto-vortical susceptibility141414This Kubo formula agrees with
(2.26) of Kovtun:2018dvd

$\frac{1}{k_{z}^{2}}G_{J^{x}T^{tx}}(\omega=0,k_{z}\hat{k})=M_{5}\,.$ (31)

While in principle the second order expression (31) could require corrections
from ${\cal O}(\partial^{2})$ thermodynamic transport coefficients which we
have omitted here (such as a coefficient multiplying $E{\cdot}a$ in the
generating functional), this was shown not to be the case in Kovtun:2018dvd
.151515One might worry that the anomaly could cause (31) to receive other
${\cal O}(\partial^{2})$ corrections. However, thermodynamic Kubo formulas are
“protected” from the anomaly in the sense that one can write a static
generating functional $W_{s}$ which includes the pressure and the $M_{n}$ and
simply add gauge dependent term in (16) to account for the anomaly. The
resulting equilibrium correlation functions, which are simply variations of
$W_{cons}$ with respect to the sources, keep the anomalous sector separate
from the other thermodynamic transport coefficients.

The remaining thermodynamic transport coefficients $M_{1},\,M_{3},$ and
$M_{4}$ are also expressed in terms of static correlation functions. However,
in terms of two-point functions, we only find Kubo relations involving
thermodynamic derivatives of the transport coefficients

$\displaystyle\frac{1}{k_{z}}{\rm
Im}\,G_{J^{t}T^{xx}}(\omega=0,k_{z}\mathbf{\hat{z}})=-\frac{2\,B_{0}^{3}}{T_{0}^{4}}\,\frac{\partial
M_{1}}{\partial\mu}\,,$ (32) $\displaystyle\frac{1}{k_{z}}{\rm
Im}\,G_{J^{t}J^{t}}(\omega=0,k_{z}\mathbf{\hat{z}})=B_{0}\frac{\partial
M_{4}}{\partial\mu}\,,$ $\displaystyle\frac{1}{k_{z}}{\rm
Im}\,G_{J^{t}T^{tt}}(\omega=0,k_{z}\mathbf{\hat{z}})=-B_{0}\left(\frac{\partial
M_{3}}{\partial\mu}+\frac{4B_{0}^{2}}{T_{0}^{4}}\frac{\partial
M_{1}}{\partial\mu}\right)\,.$

The transport coefficient $M_{1}$ can be found without derivatives in the
following combination

$\frac{1}{k_{z}}{\rm
Im}\,G_{T^{tt}T^{xx}}(\omega=0,k_{z}\hat{k})=2\frac{B_{0}^{3}}{T_{0}^{4}}\left(M_{1}-T_{0}\frac{\partial
M_{1}}{\partial T}-\mu\frac{\partial
M_{1}}{\partial\mu}-4B_{0}^{2}\frac{\partial M_{1}}{\partial
B^{2}}-T_{0}^{4}\frac{\partial M_{3}}{\partial B^{2}}\right)\,.$ (33)

The susceptibility matrix may be defined as

$\chi_{ab}=\frac{\delta\langle\varphi_{a}\rangle}{\delta\lambda^{b}}\,,$ (34)

where $\varphi_{a}=(T^{tt},T^{ti},J^{t})$, and $\lambda^{a}=(\delta
T/T,u^{i},T\delta\frac{\mu}{T})$.161616Note that since the magnetic field
breaks rotation invariance, we should have separated $T^{ti}$ into
$T^{t\perp}$ and $T^{tz}$ where $\perp$ labels the orthogonal part of the
momentum to the magnetic field. However, we find that at ${\cal O}(1)$ they
coincide, and therefore simply use $T^{ti}$ here. Explicitly, we have

$\chi_{ab}=\begin{pmatrix}T\left(\frac{\partial\epsilon}{\partial
T}\right)_{\mu/T}&0&\left(\frac{\partial\epsilon}{\partial\mu}\right)_{T}\\\
0&w_{0}&0\\\ T\left(\frac{\partial n}{\partial
T}\right)_{\mu/T}&0&\left(\frac{\partial
n}{\partial\mu}\right)_{T}\end{pmatrix}\,.$ (35)

The susceptibility matrix is symmetric since $T\left(\frac{\partial
n}{\partial
T}\right)_{\mu/T}=\left(\frac{\partial\epsilon}{\partial\mu}\right)_{T}$. The
Kubo formulas for these terms are

$\displaystyle G_{J^{t}J^{t}}(\omega=0,\mathbf{k}\to 0)=\chi_{33}\,,$ (36a)
$\displaystyle G_{T^{tt}J^{t}}(\omega=0,\mathbf{k}\to 0)=\chi_{13}\,,$ (36b)
$\displaystyle G_{T^{tt}T^{tt}}(\omega=0,\mathbf{k}\to 0)=\chi_{11}\,.$ (36c)

These susceptibilities simplify some of the expressions for the transport
coefficients. The enthalpy $w_{0}$ can be read off from the one point
functions $\langle T^{tt}\rangle+\langle T^{zz}\rangle=w_{0}$. In addition,
the magnetic susceptibility ${\chi_{B}}=2p_{,B^{2}}$ can be found by

$\frac{1}{k_{z}}{\rm
Im}\,G_{J^{x}T^{yz}}(\omega=0,k_{z}\hat{k})=-B_{0}\,{\chi_{B}}\,.$ (37)

As discussed in appendix A.3, we can interpret ${\chi_{B}}$ as a
susceptibility.

The anomalous transport coefficients can be found from static correlation
functions Amado:2011zx ; Jensen:2013vta . For example, in flat space with
constant temperature, constant chemical potential and constant magnetic field
in the $z$-direction, we find the following static correlation functions at
small momentum171717We write our Kubo formulas in terms of the covariant-
consistent correlation functions. These can also be written in terms of the
consistent-consistent correlation functions, which we summarize in appendix
A.1.

$\displaystyle\langle
J_{cov}^{x}(\mathbf{k})T^{tz}(-\mathbf{k})\rangle=-i\xi\,k_{y}\,,\quad\langle
J_{cov}^{x}(\mathbf{k})J_{cons}^{z}(-\mathbf{k}))\rangle=-i\xi_{B}\,k_{y}\,,$
(38) $\displaystyle\langle
T^{tx}(\mathbf{k})T^{tz}(-\mathbf{k})\rangle=-i\xi_{T}\,k_{y}\,,\quad\langle
T^{tx}(\mathbf{k})J_{cons}^{z}(-\mathbf{k})\rangle=-i\xi_{TB}\,k_{y}\,,$

where we take the momentum in $y$-direction which is perpendicular to the
magnetic field. We can instead take the momentum to point in the direction of
the magnetic field, in which case we find

$\displaystyle\langle
J_{cov}^{x}(\mathbf{k})T^{ty}(-\mathbf{k})\rangle=-i\xi\,k_{z}\,,\quad\langle
J_{cov}^{x}(\mathbf{k})J_{cons}^{y}(-\mathbf{k}))\rangle=-i\xi_{B}\,k_{z}\,,$
(39) $\displaystyle\langle
T^{tx}(\mathbf{k})T^{ty}(-\mathbf{k})\rangle=-i\xi_{T}\,k_{z}\,,\quad\langle
T^{tx}(\mathbf{k})J_{cons}^{y}(-\mathbf{k})\rangle=-i\xi_{TB}\,k_{z}\,.$

####  2.1.4 A comment about thermodynamic Kubo formulas

In our equilibrium setup with homogeneous magnetic fields, the thermodynamic
functions, $M_{1,3,4}$, unfortunately cannot be isolated using only first
order, static two-point functions. Nevertheless, it is still possible to
isolate their derivatives with respect to the chemical potential, $M_{n,\mu}$.

Now if a given $M_{n}$ is parity odd such that $M_{n,\mu}=0$, then $M_{n}=0$.
The reason for this is quite simple. In a microscopic system where parity is
broken only by the presence of some axial chemical potential, $\mu$, the
generating functional $W_{s}$, see eq. (2), is parity invariant. The
coefficient in front of a parity odd $M_{n}$ must then also be parity odd, and
with $\mu$ the only parity breaking term in the hydrodynamic system, we must
have $M_{n}(\mu=0)=0$. Thus for finite $\mu$, we can write any parity odd
$M_{n}$ as

$M_{n}(\mu)=\int_{0}^{\mu}d\mu^{\prime}M_{n,\mu^{\prime}}(\mu^{\prime})\,,$
(40)

and the statement follows. As shall be seen,
$M_{1,\tilde{\mu}}=M_{3,\tilde{\mu}}=M_{4,\tilde{\mu}}=0$ in our holographic
model, where $\tilde{\mu}=\mu/T$ for fixed $T$, see section 3.2. Since we are
dealing with external axial gauge fields, only $M_{4}$ is odd in $\mu$, and we
conclude that $M_{4}=0$ by the argument above. However, if the system were
coupled to vector gauge fields, the same argument would hold for
$M_{1,3,5}=0$.

###  2.2 Hydrodynamics

####  2.2.1 Non-equilibrium constitutive relations

With the equilibrium terms out of the way, the next step is to add the non-
equilibrium terms to our constitutive relations. The non-equilibrium terms are
the scalar, vector and tensor structures which are required to vanish in
equilibrium by the constraint ${\cal L}_{V}=0$181818See table 3 in
Hernandez:2017mch  for an exhaustive list.. These terms can be derived from a
non-local effective Schwinger-Keldysh action. Recent reviews on the non-
equilibrium formalism for hydrodynamics can be found in Haehl:2018lcu ;
Glorioso:2018wxw ; Jensen:2018hse . For the purposes of our analysis, we use
the effective field theory approach of adding all the non-equilibrium terms
allowed by our symmetries to the constitutive relations, and constraining
their transport coefficients via the Onsager relations and the entropy
constraints.

The definition of the thermodynamic quantities (1) is ambiguous when out of
equilibrium. The redefinition of $T$, $\mu$ and $u^{\mu}$ are referred to as
hydrodynamic frame transformations. An introductory review of this ambiguity
in the hydrodynamic framework can be found in Kovtun:2011np , implications of
frame-choice on the stability of hydrodynamics were discussed recently
Kovtun:2019hdm ; Hoult:2020eho ; Bemfica:2017wps ; Bemfica:2019knx ;
Bemfica:2020zjp , and the modifications required for fluids in strong magnetic
fields are explained in Hernandez:2017mch . For our purposes, we use the
approach in Hernandez:2017mch  to add the non-equilibrium terms in a
systematic way to the hydrodynamic frame invariants.

We begin by isolating $O(1)$ and $O(\partial)$ contributions to the energy-
momentum tensor (3) and the current (4). The spatial part of the current
${\cal J}^{\mu}$ has no ${\cal O}(1)$ term, neither does the energy current
${\cal Q}^{\mu}$. So we are left with the ${\cal O}(1)$ quantities

$\displaystyle{\cal E}=\epsilon(T,\mu,B^{2})+f_{\cal E}\,,$
$\displaystyle{\cal P}=\Pi(T,\mu,B^{2})+f_{\cal P}\,,$ $\displaystyle{\cal
N}=n(T,\mu,B^{2})+f_{\cal N}\,,$ $\displaystyle{\cal
T}^{\mu\nu}={\chi_{B}}(T,\mu,B^{2})\left(B^{\mu}B^{\nu}-{\textstyle{\frac{1}{3}}}\Delta^{\mu\nu}B^{2}\right)+f^{\mu\nu}_{\cal
T}\,,$

where $\epsilon=-p+T(\partial p/\partial T)+\mu(\partial p/\partial\mu)$,
$\Pi=p-{\textstyle{\frac{2}{3}}}{\chi_{B}}B^{2}$, $n=\partial p/\partial\mu$,
and the magnetic susceptibility is ${\chi_{B}}=2\partial p/\partial B^{2}$.
The terms $f_{\cal E}$, $f_{\cal P}$, $f_{\cal N}$, $f^{\mu\nu}_{\cal T}$,
${\cal Q}^{\mu}$, and ${\cal J}^{\mu}$ are all $O(\partial)$, and contain both
equilibrium and non-equilibrium contributions, $f_{\cal E}=\bar{f}_{\cal
E}+f_{\cal E}^{\textrm{non-eq.}}$ etc, where the bar denotes $O(\partial)$
contributions coming from the variation of $W_{s}$.

We can then write down the following quantities which are invariant under
hydrodynamic frame transformations

$\displaystyle f\equiv f_{\cal
P}-\left(\frac{\partial\Pi}{\partial\epsilon}\right)_{\\!n}f_{\cal
E}-\left(\frac{\partial\Pi}{\partial n}\right)_{\\!\epsilon}f_{\cal N}\,,$
(41a) $\displaystyle\ell\equiv\frac{B^{\alpha}}{B}\left({\cal
J}_{\alpha}-\frac{n}{\epsilon+p}{\cal Q}_{\alpha}\right)\,,$ (41b)
$\displaystyle\ell^{\mu}_{\perp}\equiv\mathbb{B}^{\mu\alpha}\left({\cal
J}_{\alpha}-\frac{n}{\epsilon+p-{\chi_{B}}B^{2}}{\cal Q}_{\alpha}\right)\,,$
(41c) $\displaystyle t^{\mu\nu}\equiv f^{\mu\nu}_{\cal
T}-\left(B^{\mu}B^{\nu}-{\textstyle{\frac{1}{3}}}\Delta^{\mu\nu}B^{2}\right)\left[\left(\frac{\partial{\chi_{B}}}{\partial\epsilon}\right)_{\\!n}f_{\cal
E}+\left(\frac{\partial{\chi_{B}}}{\partial n}\right)_{\\!\epsilon}f_{\cal
N}\right]\,.$ (41d)

Here $\mathbb{B}^{\mu\nu}\equiv\Delta^{\mu\nu}-B^{\mu}B^{\nu}/B^{2}$ is the
projector onto a plane orthogonal to both $u^{\mu}$ and $B^{\mu}$, all
thermodynamic derivatives are evaluated at fixed $B^{2}$, and
$B\equiv\sqrt{B^{2}}$. When the magnetic susceptibility ${\chi_{B}}$ is $T$\-
and $\mu$-independent, the stress $f^{\mu\nu}_{\cal T}$ is frame-invariant.

Following the notation of Hernandez:2017mch  with the slight modification
$c_{14}\to c_{14}-c_{15}$, the terms in the non-equilibrium frame invariants
are

$\displaystyle f_{\text{non-eq.}}$
$\displaystyle=-\zeta_{1}\,\nabla{\cdot}u-\zeta_{2}\,{b^{\mu}b^{\nu}}\nabla_{\\!\mu}u_{\nu}+c_{3}\,b{\cdot}V\,,$
(42a) $\displaystyle\ell_{\text{non-eq.}}$
$\displaystyle=\sigma_{\parallel}\,b{\cdot}V+c_{4}\,\nabla{\cdot}u+c_{5}\,b^{\mu}b^{\nu}\nabla_{\mu}u_{\nu}\,,$
(42b) $\displaystyle\ell_{\perp,\text{non-eq.}}^{\mu}$
$\displaystyle=\sigma_{\perp}V^{\mu}_{\perp}+\tilde{\sigma}_{\perp}\tilde{V}^{\mu}+c_{8}\Sigma^{\mu}+c_{10}\tilde{\Sigma}^{\mu}\,,$
(42c) $\displaystyle\tau^{\mu\nu}_{\text{non-eq.}}$
$\displaystyle=-\eta_{\perp}\sigma^{\mu\nu}_{\perp}-\eta_{\parallel}(b^{\mu}\Sigma^{\nu}+b^{\nu}\Sigma^{\mu})-b^{\langle\mu}b^{\nu\rangle}\left(\eta_{1}\nabla{\cdot}u+\eta_{2}b^{\alpha}b^{\beta}\nabla_{\alpha}u_{\beta}-c_{14}b{\cdot
V}\right)$
$\displaystyle-\tilde{\eta}_{\perp}\tilde{\sigma}^{\mu\nu}_{\perp}-\tilde{\eta}_{\parallel}(b^{\mu}\tilde{\Sigma}^{\nu}+b^{\nu}\tilde{\Sigma}^{\mu})+c_{15}(b^{\mu}V_{\perp}^{\nu}+b^{\nu}V^{\mu}_{\perp})+c_{17}(b^{\mu}\tilde{V}^{\nu}+b^{\nu}\tilde{V}^{\mu})\,,$
(42d)

where $b^{\mu}=B^{\mu}/\sqrt{B^{2}}$,
$V^{\mu}=E^{\mu}-T\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}$,
$V_{\perp}^{\mu}=(\Delta^{\mu\nu}-b^{\mu}b^{\nu})V_{\nu}$,
$\Sigma^{\mu}=(\Delta^{\mu\nu}-b^{\mu}b^{\nu})\sigma_{\nu\rho}b^{\rho}$ and
for any vectors
$\tilde{v}^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}b_{\rho}v_{\sigma}$,
$v^{(\mu}_{1}v^{\nu)}_{2}=v^{\mu}_{1}v^{\nu}_{2}+v^{\nu}_{1}v^{\mu}_{2}$ and
$v_{1}^{\langle\mu}v_{2}^{\nu\rangle}=v_{1}^{\mu}v_{2}^{\nu}+v_{2}^{\mu}v_{1}^{\nu}-{\textstyle{\frac{2}{3}}}\Delta^{\mu\nu}v_{1}{\cdot}v_{2}$.
The shear tensor and the projector orthogonal to $u^{\mu}$ have the usual
definitions
$\sigma^{\mu\nu}=\Delta^{\mu\alpha}\Delta^{\nu\beta}(\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{\alpha}-{\textstyle{\frac{2}{3}}}g_{\alpha\beta}\nabla{\cdot}u)$
and $\Delta^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$. The transverse component of
the shear tensor is
$\sigma_{\perp}^{\mu\nu}=\frac{1}{2}\left(\mathbb{B}^{\mu\alpha}\mathbb{B}^{\nu\beta}+\mathbb{B}^{\mu\beta}\mathbb{B}^{\nu\alpha}-\mathbb{B}^{\mu\nu}\mathbb{B}^{\alpha\beta}\right)\sigma_{\alpha_{\beta}}$
and the tilded version is
$\tilde{\sigma}_{\perp}^{\mu\nu}=\frac{1}{2}\left(\epsilon^{\mu\alpha\beta\gamma}u_{\alpha}b_{\beta}\sigma_{\perp\gamma}^{\nu}+\epsilon^{\nu\alpha\beta\gamma}u_{\alpha}b_{\beta}\sigma_{\perp\gamma}^{\mu}\right)$.
The coefficients in front of the first order hydrodynamic terms are
hydrodynamic transport coefficients. These are functions of the ${\cal O}(1)$
hydrodynamic quantities (For example, $\zeta_{1}=\zeta_{1}(T,\mu,B^{2})$).
These transport coefficients will be subject to four equality constraints
coming from the Onsager relations, as well as some inequality constraints
coming from the entropy/correlation function argument.

Furthermore, considering the parity eigenvalues of the quantities in front of
the transport coefficients, we can predict the parity eigenvalue of the
transport coefficient themselves, since the combination of the two must match
the parity of the stress tensor or axial current. Since $\mu$ is the only
parity pseudo-scalar, this allows us to constraint these transport
coefficients as even or odd functions of $\mu$. From the previously explored
transport coefficients, the tilded ones ($\tilde{\eta}_{\perp}$,
$\tilde{\eta}_{\parallel}$, and $\tilde{\sigma}_{\perp}$) are odd functions of
the chemical potential, while the rest ($\sigma_{\parallel}$, $\rho_{\perp}$,
$\eta_{\perp}$, $\eta_{\parallel}$, $\zeta_{1}$, $\zeta_{2}$, $\eta_{1}$ and
$\eta_{2}$) are even functions of the chemical potential. From the previously
unexplored transport coefficients, $c_{10}$ and $c_{17}$ are even functions of
the chemical potential, while $c_{3}$, $c_{4}$, $c_{5}$, $c_{8}$, $c_{14}$ and
$c_{15}$ are odd functions of the chemical potential.

For completeness, let us summarize the constitutive relations for a parity-
violating theory in the thermodynamic frame. The energy-momentum tensor is
given by

$\displaystyle{\cal E}$ $\displaystyle={\cal E}_{\rm eq.}\,,\quad{\cal
Q}^{\mu}={\cal Q}^{\mu}_{\rm eq.}\,\quad{\cal P}={\cal P}_{\rm
eq.}-\zeta_{1}\nabla{\cdot}u-\zeta_{2}{b^{\mu}b^{\nu}}\nabla_{\\!\mu}u_{\nu}+c_{3}b{\cdot}V\,,$
(43a) $\displaystyle{\cal T}^{\mu\nu}$ $\displaystyle={\cal T}^{\mu\nu}_{\rm
eq.}-\eta_{\perp}\sigma^{\mu\nu}_{\perp}-\eta_{\parallel}(b^{\mu}\Sigma^{\nu}+b^{\nu}\Sigma^{\mu})-b^{\langle\mu}b^{\nu\rangle}\left(\eta_{1}\nabla{\cdot}u+\eta_{2}b^{\alpha}b^{\beta}\nabla_{\alpha}u_{\beta}-c_{14}b{\cdot}V\right)$
$\displaystyle\ \ \
\,-\tilde{\eta}_{\perp}\tilde{\sigma}^{\mu\nu}_{\perp}-\tilde{\eta}_{\parallel}(b^{\mu}\tilde{\Sigma}^{\nu}+b^{\nu}\tilde{\Sigma}^{\mu})+c_{15}(b^{\mu}V_{\perp}^{\nu}+b^{\nu}V^{\mu})+c_{17}(b^{\mu}\tilde{V}^{\nu}+b^{\nu}\tilde{V}^{\mu})\,,$
(43b)

where
$\sigma^{\mu\nu}_{\perp}\equiv{\textstyle{\frac{1}{2}}}\left(\mathbb{B}^{\mu\alpha}\mathbb{B}^{\nu\beta}+\mathbb{B}^{\nu\alpha}\mathbb{B}^{\mu\beta}-\mathbb{B}^{\mu\nu}\mathbb{B}^{\alpha\beta}\right)\sigma_{\alpha\beta}$
is the part of the shear tensor transverse to the magnetic field, and
$\tilde{\sigma}_{\perp}^{\mu\nu}=\frac{1}{2}\left(\epsilon^{\mu\lambda\alpha\beta}u_{\lambda}b_{\alpha}\sigma_{\perp\beta}^{\
\ \
\nu}+\epsilon^{\nu\lambda\alpha\beta}u_{\lambda}b_{\alpha}\sigma_{\perp\beta}^{\
\ \ \mu}\right)$. We used the projection orthogonal to the magnetic field and
the fluid velocity $\mathbb{B}^{\mu\nu}=\Delta^{\mu\nu}-b^{\mu}b^{\nu}$. The
current is given by

$\displaystyle{\cal N}$ $\displaystyle={\cal N}_{\rm eq.}\,,$ (44a)
$\displaystyle{\cal J}^{\mu}$ $\displaystyle={\cal J}^{\mu}_{\rm
eq.}+\sigma_{\perp}V^{\mu}_{\perp}+\tilde{\sigma}_{\perp}\,\tilde{V}^{\mu}+b^{\mu}(\sigma_{||}b{\cdot}V+c_{4}\nabla{\cdot}u+c_{5}b^{\alpha}b^{\beta}\nabla_{\alpha}u_{\beta})+c_{8}\Sigma^{\mu}+c_{10}\tilde{\Sigma}^{\mu}\,.$
(44b)

The magnetic polarization vector is given in (12) and the polarization vector
is $p_{\mu}=M_{4}B_{\mu}$.

####  2.2.2 Hydrodynamic correlation functions

We can find the two point correlation functions of energy-momentum and
conserved currents by varying the one-point functions given by the
constitutive relations in the presence of external sources with respect to the
external sources. To do this, we solve the hydrodynamic equations in the
presence of plane wave external source perturbations $\delta A,\delta g$
(proportional to $\exp(-i\omega t+i\mathbf{k\cdot x})$) to find $\delta
T[A,g],\delta\mu[A,g],\delta u^{\mu}[A,g]$, then vary the resulting on-shell
expressions $T^{\mu\nu}_{\rm on-shell}[A,g]$ and $J^{\mu}_{\rm on-shell}[A,g]$
with respect to $g_{\mu\nu}$ and $A_{\mu}$ to find the retarded hydrodynamic
correlation functions

$\displaystyle
G^{R}_{T^{\mu\nu}T^{\alpha\beta}}=\frac{2}{\sqrt{-g}}\frac{\delta}{\delta
g_{\alpha\beta}}\left(\sqrt{-g}\,T^{\mu\nu}_{\textrm{on-
shell}}[A,g]\right)\,,$ $\displaystyle
G^{R}_{J^{\mu}T^{\alpha\beta}}=\frac{2}{\sqrt{-g}}\frac{\delta}{\delta
g_{\alpha\beta}}\left(\sqrt{-g}\,J^{\mu}_{\textrm{on-shell}}[A,g]\right)\,,$
(45a) $\displaystyle G^{R}_{T^{\mu\nu}J^{\alpha}}=\frac{\delta}{\delta
A_{\alpha}}T^{\mu\nu}_{\textrm{on-shell}}[A,g]\,,$ $\displaystyle
G^{R}_{J^{\mu}J^{\alpha}}=\frac{\delta}{\delta A_{\alpha}}J^{\mu}_{\textrm{on-
shell}}[A,g]\,,$ (45b)

where the source perturbations $\delta g$ and $\delta A$ are set to zero after
the variation. The above expressions are to be understood as

$\delta(\sqrt{-g}\,T^{\mu\nu}_{\textrm{on-
shell}})={\textstyle{\frac{1}{2}}}\sqrt{-g}\,G^{R}_{T^{\mu\nu}T^{\alpha\beta}}\,\delta
g_{\alpha\beta}(\omega,{\bf k})\,,$ (46)

etc. This provides a direct method to evaluate the retarded functions, and
allows both to find constrains due to the Onsager relations and to derive Kubo
formulas for transport coefficients.

####  2.2.3 Symmetry constraints and Onsager relations

Time reversal covariance adds additional constraints to the transport
coefficients, called the Onsager relations Onsager1 ; Onsager2 . We consider a
state characterized by a density matrix $\rho(\chi)$ and an anti-unitary
operator $\Theta$ such that

$\Theta^{-1}\rho(\chi)\Theta=\rho(-\chi)\,,$ (47)

where $\chi$ are some $\Theta$ symmetry breaking parameters of the state
associated with the density matrix $\rho(\chi)$. Recall that the expectation
values in this state are given by

$\langle{\cal O}\rangle_{\rho(\chi)}={\rm Tr}\left(\rho(\chi){\cal
O}\right)\,,$ (48)

and the retarded two point correlation functions are given by

$G^{R}_{\varphi_{a}\varphi_{b}}(t,\mathbf{x};\chi)=i\theta(t)\langle[\varphi_{a}(t,\mathbf{x}),\varphi_{b}(0,\mathbf{0})]\rangle_{\rho(\chi)}\,.$
(49)

Now, for states that are homogeneous in space-time (i.e. with space-time
translation invariance), the transformation properties of $\rho(\chi)$ under
$\Theta$ leads to

$G^{R}_{\varphi_{a}\varphi_{b}}(t,\mathbf{x};\chi)=\eta_{\varphi_{a}}\eta_{\varphi_{b}}G^{R}_{\varphi_{b}^{\dagger}\varphi_{a}^{\dagger}}(t,-\mathbf{x};-\chi)\,,$
(50)

where $\eta_{\varphi_{a}}$ is the $\Theta$ eigenvalue of $\varphi_{a}$ and
similarly for $\varphi_{b}$.

This relation can be translated to the Fourier basis correlators

$G^{R}_{\varphi_{a}\varphi_{b}}(\omega,\mathbf{k};\chi)=\int
d^{4}x\,e^{-i\omega
t+i\mathbf{k}{\cdot}\mathbf{x}}G^{R}_{\varphi_{a}\varphi_{b}}(t,\mathbf{x};\chi)\,,$
(51)

where we find

$G^{R}_{\varphi_{a}\varphi_{b}}(\omega,\mathbf{k};\chi)=\eta_{\varphi_{a}}\eta_{\varphi_{b}}G^{R}_{\varphi_{b}^{\dagger}\varphi_{a}^{\dagger}}(\omega,-\mathbf{k};-\chi)\,.$
(52)

To derive the Onsager relations in our system we have the option of using
$\Theta={\cal T}$, $\chi=B_{0}$ or $\Theta={\cal PT}$, $\chi=\mu$. The Onsager
relations are derived by using (52) on two point functions of energy-momentum
and currents.

A similar argument using the unitary parity operator ${\cal P}$ also gives the
constraint

$G^{R}_{\varphi_{a}\varphi_{b}}(\omega,\mathbf{k};\chi)=\epsilon_{\varphi_{a}}\epsilon_{\varphi_{b}}G^{R}_{\varphi_{a}\varphi_{b}}(\omega,-\mathbf{k};-\chi)\,,$
(53)

where $\epsilon_{\varphi_{a}}$ is the ${\cal P}$ eigenvalue of $\varphi_{a}$
and in this case $\chi=(B_{0},\mu)$. We will refer to the constraints derived
from eq. (53) as the parity constraints. These constraints are the same that
can be derived from considering the parity eigenvalue of the terms in the
constitutive relations: $\tilde{\eta}_{\perp}$, $\tilde{\eta}_{\parallel}$,
$\tilde{\sigma}_{\perp}$, $c_{3}$, $c_{4}$, $c_{5}$, $c_{8}$, $c_{14}$ and
$c_{15}$ are odd functions of the chemical potential, while
$\sigma_{\parallel}$, $\rho_{\perp}$, $\eta_{\perp}$, $\eta_{\parallel}$,
$\zeta_{1}$, $\zeta_{2}$, $\eta_{1}$, $\eta_{2}$, $c_{10}$ and $c_{17}$ are
even functions of the chemical potential.

####  2.2.4 Hydrodynamic Kubo formulas for systems in strong magnetic fields

The Kubo fomulas for the non-equilibrium transport coefficients can be found
by evaluating the zero spatial momentum, low frequency limit of the retarded
functions in flat space-time. For parity preserving systems coupled to strong
vector magnetic fields, only the viscosities ($\zeta_{1}$, $\zeta_{2}$,
$\eta_{1}$, $\eta_{2}$, $\eta_{\parallel}$, $\tilde{\eta}_{\parallel}$,
$\eta_{\perp}$ and $\tilde{\eta}_{\perp}$) and the conductivities
($\sigma_{\parallel}$, $\sigma_{\perp}$ and $\tilde{\sigma}_{\perp}$) appear
in the constitutive relations. The two-point function of the longitudinal
current $J^{z}$ gives the longitudinal conductivity,191919Note that we drop
the superscript “$R$” for all retarded Green’s functions from here on in order
to declutter the notation.

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{z}J^{z}}(\omega,{\bf k}{=}0)=\sigma_{\parallel}+\cdots\,,$ (54a) in
the limit of first setting ${\bf k}=0$, and then taking $\omega\to 0$. In what
follows, we take this limit in all the Kubo relations for hydrodynamic
transport coefficients. The ellipsis denote terms that vanish for $B_{0}\ll
T_{0}^{2}$ or when $M_{1}=M_{3}=M_{4}=0$. The Kubo formulas for the transverse
conductivities simplify when written in terms of the transverse resistivities.
We define the $2\times 2$ conductivity matrix in the plane transverse to ${\bf
B}_{0}$ as
$\sigma_{ab}\equiv\sigma_{\perp}\delta_{ab}+\left(\frac{n_{0}}{|{\bf
B}_{0}|}+\tilde{\sigma}_{\perp}\right)\epsilon_{ab}$, and the corresponding
resistivity matrix as
$\rho_{ab}\equiv(\sigma^{-1})_{ab}=\rho_{\perp}\delta_{ab}+\tilde{\rho}_{\perp}\,\epsilon_{ab}$,
which defines $\rho_{\perp}$ and $\tilde{\rho}_{\perp}$. Using these
definitions, the two-point functions of the transverse currents $J^{x}$,
$J^{y}$ give the transverse resistivities,
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{x}J^{x}}(\omega,{\bf
k}{=}0)=\omega^{2}\rho_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{4}}\,,$
(54b) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{x}J^{y}}(\omega,{\bf
k}{=}0)=\frac{n_{0}}{B_{0}}-\omega^{2}\tilde{\rho}_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{4}}\,{\rm
sign}(B_{0})\,,$ (54c)

Alternatively, the transverse resistivities can be found from correlation
functions of momentum density,

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{tx}T^{tx}}(\omega,{\bf
k}{=}0)=\rho_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{2}}\,,$
(55a) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{tx}T^{ty}}(\omega,{\bf k}{=}0)=-\tilde{\rho}_{\perp}{\rm
sign}(B_{0})\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{2}}\,.$ (55b)

where $O_{3}=\frac{1}{2}(T^{xx}-T^{yy})$. The “bulk” viscosities may be
expressed as

$\displaystyle{\textstyle{\frac{1}{\omega}}}\delta_{ij}{\rm
Im}\,G_{T^{ij}O_{1}}(\omega,{\bf k}{=}0)=3\zeta_{1}+\cdots\,,$ (56a)
$\displaystyle{\textstyle{\frac{1}{3\omega}}}\delta_{ij}\delta_{kl}\,{\rm
Im}\,G_{T^{ij}T^{kl}}(\omega,{\bf k}{=}0)=3\zeta_{1}+\zeta_{2}+\cdots\,,$
(56b) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{O_{1}O_{1}}(\omega,{\bf
k}{=}0)=\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1}+\cdots\,,$ (56c)
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{O_{2}O_{2}}(\omega,{\bf k}{=}0)=2\eta_{2}+\cdots\,,$ (56d) where
$O_{1}=\frac{1}{2}(T^{xx}+T^{yy})$, and
$O_{2}=T^{zz}-\frac{1}{2}(T^{xx}+T^{yy})$. The $\delta^{ij}$ is the projector
onto the spatial coordinates, i.e. $i=x,y,z$. The ellipsis denote terms that
vanish when $M_{1}=M_{3}=M_{4}=0$, or when $B_{0}\ll T_{0}^{2}$. The shear
viscosities are given by202020For parity preserving systems the $c_{i}$
coefficients vanish and the Kubo formulas are identical to those in
Hernandez:2017mch .

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{xy}T^{xy}}(\omega,{\bf k}{=}0)=\eta_{\perp}\,,$ (56e)
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{xy}O_{3}}(\omega,{\bf k}{=}0)=\tilde{\eta}_{\perp}\,{\rm
sign}(B_{0})\,,$ (56f) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{xz}T^{xz}}(\omega,{\bf
k}{=}0)=\eta_{\parallel}+(c_{8}c_{15}-c_{10}\bar{c}_{17})\rho_{\perp}-(c_{8}\bar{c}_{17}+c_{10}c_{15})\tilde{\rho}_{\perp}\,,$
(56g) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{yz}T^{xz}}(\omega,{\bf
k}{=}0)=\left(\tilde{\eta}_{\parallel}+(c_{8}\bar{c}_{17}+c_{10}c_{15})\rho_{\perp}+(c_{8}c_{15}-c_{10}\bar{c}_{17})\tilde{\rho}_{\perp}\right){\rm
sign}(B_{0})\,,$ (56h)

Using relation (52) yields the Onsager relations for the parity preserving
transport coefficients

$3\zeta_{2}-6\eta_{1}-2\eta_{2}=0\,.$ (57)

In addition, the parity constraints coming from relation (53) imply that the
tilded transport coefficients $\tilde{\rho}_{\perp}$, $\tilde{\eta}_{\perp}$
and $\tilde{\eta}_{\parallel}$ are odd functions of the chemical potential,
while the untilded $\sigma_{\parallel}$, $\rho_{\perp}$, $\eta_{\perp}$,
$\eta_{\parallel}$, $\zeta_{1}$, $\zeta_{2}$, $\eta_{1}$ and $\eta_{2}$ are
even functions of the chemical potential.212121These behaviours can be derived
using ${\cal C}$ instead of ${\cal P}$ for vector gauge fields instead of
axial gauge fields.

The Kubo formulas for the parity violating non-equilibrium coefficients
appearing in (42) are given by

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{tx}T^{xz}}(\omega,{\bf
k}{=}0)=-\frac{w_{0}-{M_{5}}_{,\mu}B_{0}^{2}}{B_{0}}(c_{8}\tilde{\rho}_{\perp}+c_{10}\rho_{\perp})\,,$
(58a) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{tx}T^{yz}}(\omega,{\bf
k}{=}0)=-\frac{w_{0}-{M_{5}}_{,\mu}B_{0}^{2}}{|B_{0}|}(c_{8}\rho_{\perp}-c_{10}\tilde{\rho}_{\perp})\,,$
(58b) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{xz}T^{tx}}(\omega,{\bf
k}{=}0)=\frac{w_{0}}{B_{0}}(c_{15}\tilde{\rho}_{\perp}+\bar{c}_{17}\rho_{\perp})\,,$
(58c) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{yz}T^{tx}}(\omega,{\bf
k}{=}0)=-\frac{w_{0}}{|B_{0}|}(c_{15}\rho_{\perp}-\bar{c}_{17}\tilde{\rho}_{\perp})\,,$
(58d) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{z}O_{1}}(\omega,{\bf k}{=}0)=-c_{4}\,{\rm sign}(B_{0})+\cdots\,,$
(58e) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{z}O_{2}}(\omega,{\bf k}{=}0)=-c_{5}\,{\rm sign}(B_{0})+\cdots\,$
(58f) $\displaystyle{\textstyle{\frac{1}{\omega}}}\delta_{ij}{\rm
Im}\,G_{T^{ij}J^{z}}(\omega,{\bf k}{=}0)=3c_{3}\,{\rm sign}(B_{0})+\cdots\,,$
(58g) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{O_{2}J^{z}}(\omega,{\bf k}{=}0)=2c_{14}\,{\rm sign}(B_{0})+\cdots\,,$
(58h)

where once again the terms in the ellipsis vanish for $M_{1}=M_{3}=M_{4}=0$ or
$B_{0}\ll T^{2}_{0}$. The rest of the Kubo formulas in the previous section
(54) (55) and (2.2.4) remain valid when the gauge fields are axial. As
mentioned in section 2.2.3, the Onsager relations give constraints on the
transport coefficients. The constraints on the parity-violating coefficients
can be derived using $\Theta={\cal T}$ in (52). Here, ${\cal T}$ refers to
time-reversal. In addition to (57), these are

$c_{3}=-c_{4}-{\textstyle{\frac{1}{3}}}c_{5}\,,\quad
c_{14}=-{\textstyle{\frac{1}{2}}}c_{5}\,,\quad
c_{15}=-\frac{w_{0}-{M_{5}}_{,\mu}B^{2}_{0}}{w_{0}}c_{8}\,,\quad\bar{c}_{17}=-\frac{w_{0}-{M_{5}}_{,\mu}B^{2}_{0}}{w_{0}}c_{10}\,.$
(59)

In addition, the parity constraints (53) imply that $c_{10}$ and $c_{17}$ are
even functions of the chemical potential, while $c_{3}$, $c_{4}$, $c_{5}$,
$\bar{c}_{8}$, $c_{14}$ and $c_{15}$ are odd functions of the chemical
potential.

For a microscopic theory with a chiral anomaly, the Kubo formulas for the
parallel shear viscosities are slightly modified

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{xz}T^{xz}}(\omega,{\bf
k}{=}0)=\eta_{\parallel}+(c_{8}c_{15}-\tilde{c}_{10}\tilde{c}_{17})\rho_{\perp}-(c_{8}\tilde{c}_{17}+\tilde{c}_{10}c_{15})\tilde{\rho}_{\perp}\,,$
(60a) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{yz}T^{xz}}(\omega,{\bf
k}{=}0)=\left(\tilde{\eta}_{\parallel}+(c_{8}\tilde{c}_{17}+\tilde{c}_{10}c_{15})\rho_{\perp}+(c_{8}c_{15}-\tilde{c}_{10}\tilde{c}_{17})\tilde{\rho}_{\perp}\right){\rm
sign}(B_{0})\,,$ (60b)

where

$\tilde{c}_{10}=c_{10}-\xi_{TB}\,,\quad\tilde{c}_{17}=\bar{c}_{17}+\xi_{TB}=c_{17}+B_{0}^{2}M_{2,\mu}+\xi_{TB}\,.$

Recall that $\xi_{TB}=\frac{1}{2}C\mu^{2}+c_{1}T^{2}+2c_{2}T\mu$. The Kubo
formulas for parity violating non-equilibrium coefficients that are modified
are222222To isolate $c_{8}$ and $\tilde{c}_{10}$, we invert eqs. (62a) and
(60a), then, using eqs. (55) we find  $\displaystyle\tilde{c}_{10}$
$\displaystyle=-\frac{w_{0}}{B_{0}}\frac{{\rm Im}G_{T^{tx}T^{tx}}\,{\rm
Im}\,G_{T^{tx}T^{xz}}+{\rm Im}\,G_{T^{tx}T^{ty}}\,{\rm
Im}\,G_{T^{tx}T^{yz}}}{\left({\rm Im}\,G_{T^{tx}T^{tx}}\right)^{2}+\left({\rm
Im}\,G_{T^{tx}T^{ty}}\right)^{2}}\,,$ (61) $\displaystyle c_{8}$
$\displaystyle=\frac{w_{0}}{|B_{0}|}\frac{{\rm Im}\,G_{T^{tx}T^{ty}}\,{\rm
Im}\,G_{T^{tx}T^{xz}}-{\rm Im}\,G_{T^{tx}T^{tx}}\,{\rm
Im}\,G_{T^{tx}T^{yz}}}{\left({\rm Im}\,G_{T^{tx}T^{tx}}\right)^{2}+\left({\rm
Im}\,G_{T^{tx}T^{ty}}\right)^{2}}\,.$ We may isolate $c_{15}$ and $c_{17}$ in
a similar way.

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{tx}T^{xz}}(\omega,{\bf
k}{=}0)=-\frac{w_{0}-{M_{5}}_{,\mu}B_{0}^{2}}{B_{0}}(c_{8}\tilde{\rho}_{\perp}+\tilde{c}_{10}\rho_{\perp})\,,$
(62a) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{tx}T^{yz}}(\omega,{\bf
k}{=}0)=-\frac{w_{0}-{M_{5}}_{,\mu}B_{0}^{2}}{|B_{0}|}(c_{8}\rho_{\perp}-\tilde{c}_{10}\tilde{\rho}_{\perp})\,,$
(62b) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{xz}T^{tx}}(\omega,{\bf
k}{=}0)=\frac{w_{0}}{B_{0}}(c_{15}\tilde{\rho}_{\perp}+\tilde{c}_{17}\rho_{\perp})\,,$
(62c) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{T^{yz}T^{tx}}(\omega,{\bf
k}{=}0)=-\frac{w_{0}}{|B_{0}|}(c_{15}\rho_{\perp}-\tilde{c}_{17}\tilde{\rho}_{\perp})\,.$
(62d)

In addition, we need to specify what currents we use in the correlation
functions. From the on-shell expressions $T^{\mu\nu}_{\rm on-shell}[A,g]\,,\
J^{\mu}_{cov,\,\rm on-shell}[g,A]$, every gauge field variation introduces a
consistent current, that is

$G_{T^{\mu\nu}J_{cons}^{\alpha}}=\frac{\delta}{\delta
A_{\alpha}}T^{\mu\nu}_{\textrm{on-shell}}[A,g]\,,\quad
G_{J_{cov}^{\mu}J_{cons}^{\alpha}}=\frac{\delta}{\delta
A_{\alpha}}J^{\mu}_{cov,\,\textrm{on-shell}}[A,g]\,.$ (63)

Let us rewrite (54) and the rest of (58) with the explicit labels for these
currents232323We use here the covariant-consistent correlation functions for
our Kubo formulas. In appendix A.1, we write the Kubo formulas in terms of the
consistent-consistent correlation functions instead.

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{z}_{cov}J^{z}_{cons}}(\omega,{\bf
k}{=}0)=\sigma_{\parallel}+\cdots\,,$ (64a)
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cov}^{x}J^{x}_{cons}}(\omega,{\bf
k}{=}0)=\omega^{2}\rho_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{4}}\,,$
(64b) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cov}^{x}J_{cons}^{y}}(\omega,{\bf
k}{=}0)=\frac{n_{0}}{B_{0}}-\omega^{2}\tilde{\rho}_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{4}}\,{\rm
sign}(B_{0})\,,$ (64c) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cov}^{z}O_{1}}(\omega,{\bf k}{=}0)=-c_{4}\,{\rm
sign}(B_{0})+\cdots\,,$ (64d) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cov}^{z}O_{2}}(\omega,{\bf k}{=}0)=-c_{5}\,{\rm
sign}(B_{0})+\cdots\,$ (64e)
$\displaystyle{\textstyle{\frac{1}{\omega}}}\delta_{ij}{\rm
Im}\,G_{T^{ij}J_{cons}^{z}}(\omega,{\bf k}{=}0)=3c_{3}\,{\rm
sign}(B_{0})+\cdots\,,$ (64f) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{O_{2}J_{cons}^{z}}(\omega,{\bf k}{=}0)=2c_{14}\,{\rm
sign}(B_{0})+\cdots\,,$ (64g)

where $O_{1}$ and $O_{2}$ are defined below (2.2.4). The terms omitted vanish
for $B_{0}\ll T_{0}^{2}$ or when $M_{1}=M_{3}=M_{4}=0$. Note that the Bardeen-
Zumino polynomial $J^{\mu}_{BZ}$ is proportional to $1/\sqrt{-g}$ so that
$\sqrt{-g}J^{\mu}_{BZ}$ is independent of the metric and therefore
$G_{J_{cov}^{\mu}T^{\nu\rho}}=G_{J_{cons}^{\mu}T^{\nu\rho}}$. This is
important for using time reversal covariance to derive the Onsager constraints
by (52). The modified Onsager relations are

$c_{3}=-c_{4}-{\textstyle{\frac{1}{3}}}c_{5}\,,\quad
c_{14}=-{\textstyle{\frac{1}{2}}}c_{5}\,,\quad
c_{15}=-\frac{w_{0}-{M_{5}}_{,\mu}B^{2}_{0}}{w_{0}}c_{8}\,,\quad\tilde{c}_{17}=-\frac{w_{0}-{M_{5}}_{,\mu}B^{2}_{0}}{w_{0}}\tilde{c}_{10}\,.$
(65)

####  2.2.5 A comment on frequency-dependent transport coefficients

Note that we may also compute frequency-dependent transport coefficients and
find Kubo relations for them. A common example are the AC electric
conductivities defined in electrodynamics. Generally, one can define any
frequency-dependent thermodynamic transport coefficient, $\kappa$ as

$\kappa_{\text{thermo}}(\omega)=\lim\limits_{k\to
0}\frac{G(\omega,k)-G(\omega,k=0)}{-ik}\,,$ (66)

with the Green’s function for the appropriate operator. Similarly, any
frequency-dependent hydrodynamic transport coefficient can be defined as

$\kappa_{\text{hydro}}(\omega)=\frac{G(\omega,k=0)}{-i\omega}\,\qquad\text{(no
limit on $\omega$ is implied)}\,.$ (67)

In this work, however, we are not going to consider such frequency-dependent
transport coefficients, and instead leave this as a future task.

###  2.3 Entropy constraints

To find constraints on the transport coefficients, one method is to impose a
local version of the second law of thermodynamics: the existence of a local
entropy current with positive semi-definite divergence for every non-
equilibrium configuration consistent with the hydrodynamic equations. As was
shown in Bhattacharyya:2013lha ; Bhattacharyya:2014bha 242424 This was
demonstrated in the example of (2+1)-dimensional parity-violating
hydrodynamics to first order in derivatives before Jensen:2011xb . , the
constraints on transport coefficients derived from the entropy current are the
same as those derived from the equilibrium generating functional, plus the
inequality constraints on dissipative transport coefficients. We take the
entropy current to be

$S^{\mu}=S^{\mu}_{\rm canon}+S^{\mu}_{\rm eq.}\,,$

where the canonical part of the entropy current is

$S^{\mu}_{\rm canon}=\frac{1}{T}\left(pu^{\mu}-T^{\mu\nu}_{A}u_{\nu}-\mu
J_{cov}^{\mu}\right)\,,$ (68)

and $S^{\mu}_{\rm eq.}$ is found from the equilibrium partition function, as
described in Bhattacharyya:2013lha ; Bhattacharyya:2014bha . The constraints
on transport coefficients follow by demanding $\nabla_{\\!\mu}S^{\mu}\geqslant
0$. Using the hydrodynamic equations (15), the divergence of the modified
canonical entropy current is

$\nabla_{\\!\mu}S^{\mu}_{\rm
canon}=\nabla_{\\!\mu}\left(\frac{p}{T}u^{\mu}\right)-T_{A}^{\mu\nu}\nabla_{\\!\mu}\frac{u_{\nu}}{T}+J_{cov}^{\mu}\left(\frac{E_{\mu}}{T}-\partial_{\mu}\frac{\mu}{T}\right)-\frac{\mu}{T}CB{\cdot}E\,.$

The $S^{\mu}_{\rm eq.}$ part of the entropy current is explicitly built to
cancel out the part of $\nabla_{\\!\mu}S^{\mu}_{\rm canon}$ that arises from
the equilibrium terms in the constitutive relations, i.e. the terms in
$T^{\mu\nu}$ and $J^{\mu}$ derived from the equilibrium generating functional.
These include the anomalous term $\frac{\mu}{T}CB{\cdot}E$. We thus focus on
non-equilibrium terms, and write the thermodynamic frame constitutive
relations as $T_{A}^{\mu\nu}=T^{\mu\nu}_{\rm eq.}+T^{\mu\nu}_{\textrm{non-
eq.}}$ and $J^{\mu}_{cov}=J^{\mu}_{\rm eq.}+J^{\mu}_{\textrm{non-eq.}}$. The
divergence of the entropy current is then

$\displaystyle\nabla_{\\!\mu}S^{\mu}$
$\displaystyle=\frac{1}{T}J^{\mu}_{\textrm{non-
eq.}}\left(E_{\mu}-T\partial_{\mu}\frac{\mu}{T}\right)-T^{\mu\nu}_{\textrm{non-
eq.}}\nabla_{\\!\mu}\frac{u_{\nu}}{T}$
$\displaystyle=\frac{1}{T}\left(\ell^{\mu}_{\perp\textrm{non-
eq.}}+\frac{B^{\mu}}{B}\ell_{\textrm{non-
eq.}}\right)V_{\mu}-\frac{1}{T}f_{\textrm{non-
eq.}}\nabla{\cdot}u-\frac{1}{2T}t^{\mu\nu}_{\textrm{non-
eq.}}\sigma_{\mu\nu}\,.$

Using the constitutive relations (43), (44), this leads to

$\displaystyle T\nabla_{\\!\mu}S^{\mu}$
$\displaystyle={\textstyle{\frac{1}{2}}}\eta_{\perp}(\sigma^{\mu\nu}_{\perp})^{2}+\sigma_{\perp}V_{\perp}^{2}+\eta_{\parallel}\Sigma^{2}+\left(c_{8}-c_{15}\right)\Sigma{\cdot}V_{\perp}$
$\displaystyle+(\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1})S_{3}^{2}+2\eta_{2}S_{4}^{2}+\sigma_{\parallel}S_{5}^{2}+(2\eta_{1}+\zeta_{2}-{\textstyle{\frac{2}{3}}}\eta_{2})S_{3}S_{4}$
$\displaystyle+\left(c_{4}-c_{3}+{\textstyle{\frac{2}{3}}}c_{14}\right)S_{3}S_{5}+\left(c_{5}-2c_{14}\right)S_{4}S_{5}\,,$
(69)

where $S_{3}\equiv\nabla{\cdot}u$, $S_{4}\equiv
b^{\mu}b^{\nu}\nabla_{\\!\mu}u_{\nu}$ and $S_{5}\equiv b{\cdot}V$. Demanding
$\nabla_{\\!\mu}S^{\mu}\geqslant 0$ now gives $\eta_{\perp}\geqslant 0$
together with the condition that the quadratic forms made out of $V_{\perp}$,
and $\Sigma$ and $S_{3}$, $S_{4}$ and $S_{5}$ are non-negative, which implies

$\displaystyle\sigma_{\perp}\geqslant 0\,,\quad\eta_{\parallel}\geqslant
0\,\quad\eta_{2}\geqslant
0\,,\quad\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1}\geqslant
0\,,\quad\sigma_{\parallel}\geqslant 0\,,$ (70)
$\displaystyle\sigma_{\perp}\eta_{\parallel}\geqslant{\textstyle{\frac{1}{4}}}\left(c_{8}-c_{15}\right)^{2}\,,\quad
2\eta_{2}(\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1})\geqslant{\textstyle{\frac{1}{4}}}(2\eta_{1}+\zeta_{2}-{\textstyle{\frac{2}{3}}}\eta_{2})^{2}\,,$
$\displaystyle\sigma_{\parallel}(\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1})\geqslant{\textstyle{\frac{1}{4}}}(c_{4}-c_{3}+{\textstyle{\frac{2}{3}}}c_{14})^{2}\,,\quad
2\eta_{2}\sigma_{\parallel}\geqslant{\textstyle{\frac{1}{4}}}(c_{5}-2c_{14})^{2}\,,$
$\displaystyle{\rm det}(M)\geqslant 0\,,$

where

$M=\begin{bmatrix}\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1}&\eta_{1}+{\textstyle{\frac{1}{2}}}\zeta_{2}-{\textstyle{\frac{1}{3}}}\eta_{2}&{\textstyle{\frac{1}{2}}}c_{4}-{\textstyle{\frac{1}{2}}}c_{3}+{\textstyle{\frac{1}{3}}}c_{14}\\\
\eta_{1}+{\textstyle{\frac{1}{2}}}\zeta_{2}-{\textstyle{\frac{1}{3}}}\eta_{2}&2\eta_{2}&{\textstyle{\frac{1}{2}}}c_{5}-c_{14}\\\
{\textstyle{\frac{1}{2}}}c_{4}-{\textstyle{\frac{1}{2}}}c_{3}+{\textstyle{\frac{1}{3}}}c_{14}&{\textstyle{\frac{1}{2}}}c_{5}-c_{14}&\sigma_{\parallel}\end{bmatrix}\,.$
(71)

The coefficients $\tilde{\eta}_{\perp}$, $\tilde{\eta}_{\parallel}$,
$\tilde{\sigma}$, $c_{10}$ and $c_{17}$ do not contribute to entropy
production, and are not constrained by the above analysis. Thus,
$\tilde{\eta}_{\perp}$, $\tilde{\eta}_{\parallel}$, $\tilde{\sigma}$, $c_{10}$
and $c_{17}$ are non-equilibrium non-dissipative coefficients. Note that using
the Onsager relations (57) and (65) these constraints reduce to the linear
constraints

$\eta_{\perp}\geqslant 0\,,\quad\sigma_{\perp}\geqslant
0\,,\quad\eta_{\parallel}\geqslant 0\,\quad\eta_{2}\geqslant
0\,,\quad\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1}\geqslant
0\,,\quad\sigma_{\parallel}\geqslant 0\,,$ (72)

the quadratic constraints

$\displaystyle\sigma_{\perp}\eta_{\parallel}\geqslant(1-\frac{M_{5,\mu}B_{0}^{2}}{2})^{2}c_{8}^{2}\,,\quad\quad
2\eta_{2}(\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1})\geqslant
4\eta_{1}^{2}\,,$ (73)
$\displaystyle\sigma_{\parallel}(\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1})\geqslant
c_{4}^{2}\,,\quad\quad\quad\quad\quad 2\eta_{2}\sigma_{\parallel}\geqslant
c_{5}^{2}\,,$

and the qubic constraint

${\rm det}(M)\geqslant 0$ (74)

where now

$M=\begin{bmatrix}\zeta_{1}-{\textstyle{\frac{2}{3}}}\eta_{1}&2\eta_{1}&c_{4}\\\
2\eta_{1}&2\eta_{2}&c_{5}\\\ c_{4}&c_{5}&\sigma_{\parallel}\end{bmatrix}.$

###  2.4 Eigenmodes

From the hydrodynamic equations (15) together with the constitutive relations
(43), (44), (18a), (28), one can study the eigenmodes of small oscillations
about the thermal equilibrium state. We begin by including only the anomaly
induced transport coefficients from the parity violating sector, i.e.
$\xi,\,\xi_{B},\,\xi_{T},\,\xi_{TB}$, and set
$c_{3}=c_{4}=c_{5}=c_{8}=c_{10}=c_{14}=c_{15}=c_{17}=0$.252525For $c_{i}\neq
0$, there are some corrections to the hydrodynamic dispersion relations. We
also keep the $CPT$ violating constant $c_{2}=0$, and begin by ignoring the
$c_{1}$ term that is related to the gravitational anomaly coefficient. At the
end of the section we comment on the changes due to keeping $c_{1}\neq 0$. We
set the external sources to zero, and linearize the hydrodynamic equations
near the flat-space equilibrium state with constant $T=T_{0}$, $\mu=\mu_{0}$,
$u^{\alpha}=(1,{\bf 0})$, and $B^{\alpha}=(0,0,0,B_{0})$. Taking the
fluctuating hydrodynamic variables proportional to $\exp(-i\omega t+i{\bf
k}{\cdot}{\bf x})$, the source-free system admits five eigenmodes, two gapped
($\omega({\bf k}{\to}0)\neq 0$), and three gapless ($\omega({\bf
k}{\to}0)=0$). The frequencies of the gapped eigenmodes are262626All
dispersion relations in this section are exact to the order in momentum shown.
There is one potential exception which we discuss separately below.

$\displaystyle\omega=\pm\frac{B_{0}^{2}}{w_{0}}\sigma_{12}-\frac{iB_{0}^{2}}{w_{0}}\sigma_{11}+v_{gap\pm}k\,\cos\theta-
iD_{c}(\theta)k^{2}\,,$ (75)

where $w_{0}\equiv\epsilon_{0}+p_{0}$ is the equilibrium enthalpy density, and
we have taken ${\chi_{B}}B_{0}^{2}\ll w_{0}$, ${{M_{5}}}_{,\mu}B_{0}^{2}\ll
w_{0}$ in the hydrodynamic regime $B_{0}\ll T_{0}^{2}$. The $2\times 2$
conductivity matrix in the plane transverse to ${\bf B}_{0}$ was defined in
section 2.2.4. We repeat it here for the reader’s convenience:
$\sigma_{ab}\equiv\sigma_{\perp}\delta_{ab}+\left(\frac{n_{0}}{|{\bf
B}_{0}|}+\tilde{\sigma}\right)\epsilon_{ab}$. The corresponding resistivity
matrix is
$\rho_{ab}\equiv(\sigma^{-1})_{ab}=\rho_{\perp}\delta_{ab}+\tilde{\rho}_{\perp}\,\epsilon_{ab}$,
which defines $\rho_{\perp}$ and $\tilde{\rho}_{\perp}$. Stability of these
eigenmodes requires $\sigma_{11}=\sigma_{\perp}>0$, which is a direct
consequence of the entropy production argument (70). The analogous mode in 2+1
dimensional hydrodynamics was christened the hydrodynamic cyclotron mode in
Hartnoll:2007ih , which also explored its implications for transport near two-
dimensional quantum critical points. The gapped mode velocity

$v_{gap\pm}=\frac{B^{2}_{0}C\mu_{0}^{3}}{3w_{0}^{2}}(\sigma_{12}\pm
i\sigma_{11})$ (76)

is unique to systems in the presence of anomalies.

The coefficient $D_{c}(\theta)$ in the cyclotron mode eigenfrequency (75) at
small $B_{0}$ is272727Note that the Hall viscosities and the Hall conductivity
show up in this coefficient and not in any of the others. Only the Hall
conductivity contributes to the gap in eq. (76), while both Hall viscosities
and the Hall conductivity contribute to the diffusion coefficient (77).

$\displaystyle D_{c}(\theta)$
$\displaystyle=\left(\frac{3\zeta_{1}{+}6\eta_{\perp}{-}2\eta_{1}+6i\tilde{\eta}_{\perp}}{6w_{0}}+\frac{(n_{0}^{2}\chi_{11}{-}w_{0}^{2}\chi_{33})w_{0}}{2n_{0}^{2}\det(\chi)}(\sigma_{11}\mp
i\tilde{\sigma})\pm\frac{iv_{s}^{2}w_{0}}{2n_{0}B_{0}}\right)\sin^{2}\theta$
(77) $\displaystyle+\left(\frac{\eta_{\perp}}{w_{0}}\pm
i\left(\frac{\tilde{\eta}_{\parallel}}{w_{0}}+\frac{C\mu_{0}^{3}}{3}\frac{v_{gap\,\pm}}{w_{0}}\right)\right)\cos^{2}\theta+O(B_{0}^{2})\,,$

where $\theta$ is the angle between $\mathbf{k}$ and $\mathbf{B}_{0}$. The
nonzero elements of the $3\times 3$ susceptibility matrix are
$\chi_{11}=T(\partial\epsilon/\partial T)_{\mu/T}$,
$\chi_{13}=\chi_{31}=(\partial\epsilon/\partial\mu)_{T}$, $\chi_{33}=(\partial
n/\partial\mu)_{T}$, and $\chi_{22}=w_{0}$, with derivatives evaluated at
constant $B^{2}$ in equilibrium. The speed of sound $v_{s}$ in eq. (77) is
given by

$v_{s}^{2}=\frac{n_{0}^{2}\chi_{11}+w_{0}^{2}\chi_{33}-2n_{0}w_{0}\chi_{13}}{\det(\chi)}\,.$
(78)

For the gapped modes, the limits $\theta\to 0$ and $\mathbf{k}\to 0$ as well
as $\theta\to\pi/2$ and $\mathbf{k}\to 0$ commute.

For momenta ${\bf k}\parallel{\bf B}_{0}$, the three gapless eigenmodes are
the two “sound” waves, and the chiral magnetic wave Kharzeev:2010gd . The
eigenfrequencies in the small momentum limit are

$\displaystyle\omega=kv_{\pm}-i\frac{\Gamma_{\parallel\pm}}{2}k^{2}\,,$ (79a)
$\displaystyle\omega=kv_{0}-iD_{\parallel}k^{2}\,.$ (79b)

The velocities $v_{0}\,,\ v_{+}$ and $v_{-}$ can be expressed in terms of the
speed of sound as well as the following expressions

$\alpha=\frac{(s_{0}T_{0})^{2}}{{\rm
det}(\chi)}\,,\quad\gamma=\frac{\mu_{0}^{2}}{w_{0}}-2n_{0}\mu_{0}\frac{\chi_{11}-\mu_{0}\chi_{13}}{{\rm
det}(\chi)}+w_{0}\frac{\chi_{11}-\mu_{0}\chi_{33}}{{\rm det}(\chi)}\,,$ (80)

from which we find

$\displaystyle v_{0}$ $\displaystyle=B_{0}C\frac{\alpha}{v_{s}^{2}}+\cdots\,,$
(81) $\displaystyle v_{\pm}$ $\displaystyle=\pm v_{s}+B_{0}C\frac{\gamma
v_{s}^{2}-\alpha}{2v_{s}^{2}}+\cdots\,,$

where we have omitted terms of order $B_{0}^{2}C^{2}$ and higher.

The damping coefficient is

$\displaystyle\Gamma_{s,\parallel}$
$\displaystyle=\frac{3\zeta_{1}+10\eta_{1}+6\eta_{2}}{3W_{0}}+\frac{v_{s}^{2}\chi_{11}-w_{0}}{\det(\chi)}\frac{w_{0}}{v_{s}^{2}}\sigma_{\parallel}$
(82)
$\displaystyle+CB_{0}\left(\Sigma_{\parallel}\sigma_{\parallel}+\Sigma_{\perp}\sigma_{\perp}\right)+\cdots\,,$

where

$\frac{w_{0}}{W_{0}}=1+{\cal O}\left(CB_{0}\right)\,.$ (83)

The ${\cal O}(CB_{0})$ part of $\frac{1}{W_{0}}$ as well as the ${\cal O}(1)$
part of $\Sigma_{\perp}$ can be found in appendix A.2. We have omitted higher
order terms in $CB_{0}$ in eq. (82).

The longitudinal diffusion constant is

$D_{\parallel}=\frac{w_{0}^{2}\sigma_{\parallel}}{v_{s}^{2}\det(\chi)}+{\cal
O}\left(B_{0}^{2}C^{2}\right)\,,$ (84)

where once again, the ${\cal O}\left(C^{2}B_{0}^{2}\right)$ terms can be found
in appendix A.2. The positivity of the diffusion constant implies
$\sigma_{\parallel}>0$.

For modes propagating at an angle $\theta\neq\pi/2$ with respect to ${\bf
B}_{0}$, the velocities (and damping coefficients) of the “sound” waves and
the chiral magnetic wave depend on the angle of propagation. For a fixed value
of $\theta$, the small-momentum eigenfrequencies are
$\omega=kv_{\pm}\cos\theta-\frac{i}{2}\Gamma_{s}(\theta)k^{2}$, and
$\omega=kv_{0}\cos\theta-iD(\theta)k^{2}$, where

$\displaystyle D(\theta)$ $\displaystyle=$ $\displaystyle
D_{\parallel}\cos^{2}\theta+\left(\frac{n_{0}^{2}\,w_{0}^{2}\,\rho_{\perp}}{B_{0}^{2}v_{s}^{2}\det(\chi)^{2}}+{\cal
O}(B_{0}C)\right)\sin^{2}\theta\,,$ (85) $\displaystyle\Gamma_{s}(\theta)$
$\displaystyle=$
$\displaystyle\Gamma_{s,\parallel}\cos^{2}\theta+\left(\frac{\eta_{\parallel}}{w_{0}}+\frac{(n_{0}\chi_{13}-w_{0}\chi_{33})^{2}w_{0}^{3}}{B_{0}^{2}\,\det(\chi)^{2}}\rho_{\perp}+{\cal
O}(B_{0}C)\right)\sin^{2}\theta\,.$ (86)

The limits $\theta\to\pi/2$ and $k\to 0$ in the gapless eigenfrequencies do
not commute. For momenta ${\bf k}\perp{\bf B}_{0}$, the three gapless
eigenmodes include two diffusive modes, and one “subdiffusive” mode with a
quartic dispersion relation,282828One might worry that the quartic relation in
eq. (87b) could be affected by ${\cal O}\left(\partial^{2}\right)$ terms in
the constitutive relations. However, we verified that the only term that could
modify eq. (87b) is a term in $\nabla_{\mu}J^{\mu}\sim k_{y}^{3}$. But
$\nabla_{\mu}J^{\mu}$ is a scalar equation and there are no scalar terms which
contain $k_{y}^{3}$.

$\displaystyle\omega=-iD_{\perp\,\pm}k^{2}\,,$ (87a)
$\displaystyle\omega=-i\frac{\eta_{\perp}k^{4}}{B_{0}^{2}\,\chi_{33}}\,.$
(87b)

The transverse diffusion constant is given by

$\displaystyle
D_{\perp\pm}=D_{\perp}\pm\sqrt{D_{\perp}^{2}-\frac{w_{0}^{2}\chi_{33}\eta_{\parallel}\rho_{\perp}}{B_{0}^{2}\left(\det(\chi)-B_{0}^{2}C^{2}T_{0}^{2}\mu_{0}^{2}\frac{ds}{dT}\right)}}\,,$
(88)

where

$\displaystyle D_{\perp}=$
$\displaystyle\frac{w_{0}^{4}\chi_{33}+B_{0}^{2}C^{2}\mu_{0}^{2}\left(w_{0}^{2}\mu_{0}T_{0}\frac{dn}{dT}+{\textstyle{\frac{1}{4}}}\mu_{0}^{2}\det(\chi)-w_{0}^{3}\right)}{w_{0}B_{0}^{2}\left(\det(\chi)-B_{0}^{2}C^{2}T_{0}^{2}\mu_{0}^{2}\frac{ds}{dT}\right)}\frac{\rho_{\perp}}{2}$
(89)
$\displaystyle+\frac{\det(\chi)}{w_{0}\left(\det(\chi)-B_{0}^{2}C^{2}T_{0}^{2}\mu_{0}^{2}\frac{ds}{dT}\right)}\frac{\eta_{\parallel}}{2}\,.$

again using ${{M_{5}}}_{,\mu}B_{0}^{2}\ll w_{0}$. Stability of the equilibrium
state requires $\eta_{\perp}>0$, $\eta_{\parallel}>0$, $\rho_{\perp}>0$, which
is ensured by the entropy production argument (70).

The $c_{1}$ constant related to the gravitational anomaly modifies the
dispersion relations in a similar way than the $C$ gauge anomaly coefficient.
For example, the combination in the denominators in the transverse diffusion
constants $D_{\perp\pm}$ in eq. (87) with definitions given in eqs. (88) and
(89) get modified

$T_{0}^{2}\mu_{0}^{2}C^{2}\frac{ds}{dT}\to
T_{0}^{2}\mu_{0}^{2}C^{2}\frac{ds}{dT}+4c_{1}C\frac{dn}{dT}T_{0}^{3}\mu_{0}-4c_{1}^{2}T_{0}^{4}\chi_{33}\,,$
(90)

and the terms in the gap velocity $v_{gap}$ and the cyclotron frequency
$D_{c}(\theta)$ in eq. (75) which include the gauge anomaly coefficient appear
in the following combination

$\frac{1}{3}C\mu_{0}^{3}\to\frac{1}{3}C\mu_{0}^{3}+2c_{1}T_{0}^{2}\mu_{0}=\xi_{T}\,,$
(91)

with $\xi_{T}$ in eq. (19).

Lastly, we mention that taking $C=c_{1}=0$, and therefore
$\xi=\xi_{B}=\xi_{T}=\xi_{TB}=0$, the results of this section agree with those
found in section 3.5 of Hernandez:2017mch . In turn, those results reduce to
the standard results for $B_{0}\to 0$. Including the other coefficients
$c_{n}$ complicates this eigenmode analysis considerably, and is a excellent
direction for future investigation.

The equations (77), (87b), (88), (82), (84), (85) may be regarded as Einstein
relations. They are relating several transport coefficients to each other, in
analogy to the simple examples of the shear diffusion
$D_{\eta}=\eta/(\epsilon+P)$, the charge diffusion $D_{\sigma}=\sigma/\chi$
(with the charge susceptibility $\chi$), and sound attenuation
$\Gamma=(\zeta+4\eta/3)/(\epsilon+P)$ (with bulk viscosity $\zeta$) in the
uncharged isotropic system.

###  2.5 Interpretation of transport coefficients

With the systematic approach applied in this section, 22 independent transport
coefficients have been identified. Some of them have a standard interpretation
with a new twist, some are novel and will be given a first interpretation
here.292929The pressure $p$ in our formulation acts as a generating functional
for equilibrium $n$-point functions Jensen:2011xb ; Jensen:2012jh ;
Banerjee:2012iz ; Kovtun:2016lfw . It is not counted as a transport
coefficient. Susceptibilities such as the $U(1)_{A}$ charge susceptibility
$\chi_{33}$ are derivatives of the pressure and are not counted as individual
transport coefficients.

####  2.5.1 Discussion of all transport coefficients

If not specified otherwise, in the examples here we assume a flat metric
$\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$, and the equilibrium fluid velocity
$u^{\mu}=(1,0,0,0)$.

##### The perpendicular magnetic vorticity susceptibility $\bm{M_{2}}$.

In order to interpret $M_{2}$, we may consider how it arises in various
expressions originating from the generating functional.

First, one may interpret $M_{2}$ with the help of the vorticity of the
magnetic field, $\Omega_{B}^{\mu}$. This quantity appears in the most
prominent terms in the equilibrium constitutive equations (10), which contain
$M_{2}$, namely

$\mathcal{E}_{\text{eq}}\sim\mathcal{P}_{\text{eq}}\sim
M_{2}B{\cdot}\Omega_{B}\,.$ (92)

Here, in analogy to the vorticity of the fluid velocity,
$\Omega^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\nabla_{\rho}u_{\sigma}$, we
define the vorticity of the magnetic field303030Both, $u$ and $B$ are vector
fields, however, $u$ is a dynamical field, while $B$ is a source, i.e. an
external field. as
$\Omega_{B}^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}\nabla_{\rho}B_{\sigma}$.
In this sense, $M_{2}$ measures the response of energy or pressure to the
vorticity of $B$ parallel to $B$. In other words, $M_{2}$ measures the
response to that component of the curl of the magnetic field, which is
perpendicular to both the magnetic field itself and to the fluid velocity.
Example: Thermodynamic (time-independent): Consider a time-independent
uncharged equilibrium state with an inhomogeneous anisotropic background
magnetic field $\bm{B}=(B_{x}(y),B_{y}(x),B_{z})$. Here, $B_{z}$ is a constant
in space and time, however, $B_{x}$ depends on $y$, and $B_{y}$ depends on
$x$. This leads to $T^{00}\sim
M_{2}\epsilon^{\mu\nu\rho\sigma}u_{\mu}B_{\nu}\nabla_{\rho}B_{\sigma}\sim
M_{2}\epsilon^{tzyx}u_{t}B_{z}\nabla_{y}B_{x}(y)\sim-
M_{2}B_{z}\partial_{i}\times B_{j}\sim-M_{2}B_{z}\partial_{y}B_{x}(y)$. So in
such an equilibrium state, the energy density (and pressure) receive a
contribution from that part of the curl of the magnetic field, which is
perpendicular to itself. $M_{2}$ measures how large that contribution is.

Alternatively, $M_{2}$ also appears in the magnetization measuring response to
the curl of the magnetic field perpendicular to the fluid flow (in the fluid
rest frame), or as response to the temperature gradient perpendicular to the
magnetic field and the fluid flow (in the fluid rest frame):

$\mathfrak{m}^{\mu}\sim
M_{2}(2\Omega^{\mu}_{B}-\epsilon^{\mu\nu\kappa\sigma}u_{\nu}B_{\kappa}\partial_{\sigma}T)\,.$
(93)

The second term may be interpreted as a magentic version of the Nernst effect
as both are a response to the same tensor structure
$\epsilon^{\mu\nu\kappa\sigma}u_{\nu}B_{\kappa}\partial_{\sigma}T$. However,
here the response occurs in the magnetization, whereas the original Nernst
effect has a response in the electric field. One may think of $-M_{2}$ as a
magnetization Nernst coefficient.

Second, one may interpret $M_{2}$ in terms of the Poynting vector. Consider a
setup with $B=B_{0}\hat{z}$, $E=E_{0}\hat{y}$, $T=T_{0}$ and $\mu=E_{0}y$ on
$|y|\leq R$ where $E_{0}={\cal O}(\partial)$. Then there is a nonzero shear
term $T^{zx}=-M_{2,\mu}B^{z}\epsilon^{xtyz}u_{t}E_{y}B_{z}\sim
M_{2,\mu}B_{0}^{2}E_{0}$ due to $M_{2}$. That is, $M_{2,\mu}$ measures the
response of the shear tensor to an external Poynting vector
$S^{\mu}=\epsilon^{\mu\nu\rho\sigma}u_{\nu}E_{\rho}B_{\sigma}$ in the plane
spanned by the magnetic field $B^{\mu}$ and the Poynting vector $S^{\mu}$. In
the same setup, $M_{2,\mu}$ also gives the response of the magnetization to
the external Poynting vector $\mathfrak{m}^{\mu}\sim-M_{2,\mu}S^{\mu}$.

In a theory which microscopically preserves parity invariance, $M_{2}$ can be
non-vanishing in states in which parity is broken through an axial
($U(1)_{A}$) chemical potential.313131Although a vector ($U(1)_{V}$) magnetic
field breaks parity, the only scalar that can be formed from it to appear in
the equilibrium generating function is the parity even $B^{2}$. It can also be
nonzero if parity is broken microscopically through a chiral anomaly.

##### The magneto-thermal susceptibility $\bm{M_{1}}$.

It appears in the constitutive relation of the equilibrium energy momentum
tensor, (10), as a response of the energy density to the gradient of the
dimensionless ratio $B^{2}/T^{4}$. This gradient is parallel to the magnetic
field

$\mathcal{E}_{\text{eq}}\sim
M_{1}B^{\mu}\partial_{\mu}\left(\frac{B^{2}}{T^{4}}\right)\,.$ (94)

Example: Consider a spatially modulated magnetic field in the $z$-direction,
$B_{z}$, which is constant in time but depends on the $z$-coordinate, e.g.
$B_{z}(z)=B_{0}\,\text{sin}(kz)$ with the wave vector of the modulation, $k$.
Assume the temperature is constant. This leads to a spatially modulated energy
density $T^{00}\sim
M_{1}B_{z}(z)(\partial_{z}B_{z}(z)^{2})/T^{4}=M_{1}\frac{2kB_{0}^{3}}{T^{4}}\text{sin}^{2}(kz)\,\text{cos}(kz)$.
Alternatively, the temperature can be spatially modulated. A pure time-
modulation would not lead to any response, because the gradient
$\partial_{\mu}$ needs to be aligned with the (spatial) magnetic field $B$.

Alternatively, $M_{1}$ measures the response of the magnetization to the part
of the gradient of $B^{2}/T^{4}$ which is perpendicular to the fluid velocity

$\mathfrak{m}^{\mu}\sim
M_{1}\Delta^{\mu\nu}\partial_{\nu}\frac{B^{2}}{T^{4}}\,.$ (95)

##### The magneto-acceleration susceptibility $\bm{M_{3}}$.

It multiplies $B{\cdot}a$ in the generating functional, eq. (2). In the energy
momentum constitutive relation, (10), one finds the term

$\mathcal{E}_{\text{eq}}\sim\mathcal{P}_{\text{eq}}\sim
M_{3,B^{2}}\,B{\cdot}a\,.$ (96)

Hence, the thermodynamic derivative of $M_{3}$ with respect to $B^{2}$
measures the response of equilibrium energy and pressure to a magnetic field
aligned with the acceleration of the fluid in any of the spatial directions.
In the magnetization, $M_{3}$ appears directly measuring the response to the
acceleration of the fluid

$\mathfrak{m}^{\mu}\sim M_{3}\,a^{\mu}\,.$ (97)

Example: A fluid which is accelerated in the $x$-direction gets magnetized
along that direction. Its magnetization is proportional to the acceleration.

The magneto-acceleration susceptibility $M_{3}$ vanishes in conformal field
theories regardless of the state breaking conformal invariance.

##### The magneto-electric susceptibility $\bm{M_{4}}$.

This susceptibility has been discussed previously with both electric and
magnetic field being strong ($B\sim\mathcal{O}(1)$) Kovtun:2016lfw .323232In
Kovtun:2016lfw  the magneto-electric susceptibility $M_{4}$ was named
$\chi_{EB}$. There and in our case, this susceptibility multiplies $B{\cdot}E$
in the generating functional, eq. (2). For our case

$\mathcal{E}_{\text{eq}}\sim M_{4,T}\,B{\cdot}E\,,\qquad{\cal P}_{\rm eq}\sim
M_{4,B^{2}}\,B{\cdot}E\,.$ (98)

The magneto-electric susceptibility $M_{4}$ measures the response of the
energy density or pressure to an electric field projected onto the direction
of the magnetic field. Hence this response is proportional to
$|B|\,|E|\text{cos}\,\theta$ with the angle, $\theta$, between the two fields.
In the magnetization, $M_{4}$ measures the response to an electric field

$\mathfrak{m}^{\mu}\sim M_{4}\,E^{\mu}\,,$ (99)

and the only leading contribution to the polarization is given by

$\mathfrak{p}^{\mu}=M_{4}\,B^{\mu}\,.$ (100)

These two last equations highlight that the term $M_{4}B{\cdot}E$ generates a
response (polarization and magnetization) symmetric under exchange of $B$ and
$E$.

##### Novel “expansion-induced longitudinal conductivities” $\bm{c_{4}}$ and
$\bm{c_{5}}$.

These transport coefficients appear in the constitutive relation for the
current

$\mathcal{J}^{\mu}\sim
b^{\mu}(c_{4}\nabla{\cdot}u+c_{5}b^{\alpha}b^{\beta}\nabla_{\alpha}u_{\beta})\,.$
(101)

There is similarity between these two terms and the longitudinal viscosity
terms contributing to the energy-momentum tensor $T^{\mu\nu}\sim
b^{\langle\mu\nu\rangle}(\eta_{1}\nabla\cdot
u+\eta_{2}b^{\alpha}b^{\beta}\nabla_{\alpha}u_{\beta})$ in eq. (43). The
latter is a symmetric traceless two-tensor contribution aligned with the
magnetic field, and eq. (101) is a current aligned with the magnetic field. We
refer to $c_{4}$ and $c_{5}$ as conductivities instead of viscosities since
they appear in the charge current. In this sense they are longitudinal
conductivities in analogy to $\eta_{1},\,\eta_{2}$ being longitudinal
viscosities. Both, $c_{4}$ and $\eta_{1}$ measure the response of the
respective currents to divergence of the velocity field, $\nabla{\cdot}u$. In
a similar way, the coefficient $c_{5}$ appears in analogy to $\eta_{2}$ as a
response to the gradient of the velocity field along the magnetic field
projected onto the magnetic field direction,
$b^{\mu}b^{\nu}\nabla_{\mu}u_{\nu}$. The latter can be thought of as an
expansion of the fluid along the magnetic field.

##### Novel shear-induced conductivity $\bm{c_{8}}$ and shear-induced Hall
conductivity $\bm{c_{10}}$ (the latter is dissipationless).

Both transport coefficients measure the response to the shear within a
particular plane, in that sense both are shear-induced. But contrary to the
standard shear viscosities, $c_{8}$ and $c_{10}$ measure the response within
that plane in which the shear occurs. Hence, we refer to $c_{8}$ and $c_{10}$
as being transverse. In order to stress that both measure a response of the
current we refer to them as conductivities rather than viscosities. One
example for the interpretation of $c_{8}$ and $c_{10}$ can be based on the
constitutive relations containing:

$\displaystyle\mathcal{J}^{\mu}$ $\displaystyle=$
$\displaystyle\mathcal{J}_{\text{eq.}}^{\mu}+\dots+c_{8}\Sigma^{\mu}+c_{10}\tilde{\Sigma}^{\mu}\,,$
(102) $\displaystyle\Sigma^{\mu}$ $\displaystyle=$
$\displaystyle(\Delta^{\mu\nu}-b^{\mu}b^{\nu})\sigma_{\nu\rho}b^{\rho}\,,$
(103) $\displaystyle\tilde{\Sigma}^{\mu}$ $\displaystyle=$
$\displaystyle\epsilon^{\mu\nu\alpha\beta}u_{\nu}b_{\alpha}\Sigma_{\beta}\,.$
(104)

Example: Start by choice with the background magnetic field
$B^{\mu}=(0,0,0,B_{z})$, which implies $b^{\mu}=(0,0,0,1)$, which implies
$(\Delta^{\mu\nu}-b^{\mu}b^{\nu})=\text{diag}(0,1,1,0)$, which is the
projector onto the two directions perpendicular to the background magnetic
field, $B_{z}$, and the fluid velocity simultaneously. Working out eq. (103),
we find that $c_{8}$ measures the response of the spatial current components,
e.g. $\mathcal{J}_{x}$ ($\mathcal{J}_{y}$), to a shear of the fluid in the
plane of that current and the magnetic field, e.g. $(x,z)$-plane
($(y,z)$-plane) for the response in the $x$-direction ($y$-direction):

$\mathcal{J}_{x}\sim
c_{8}(\partial_{x}u_{z}+\partial_{z}u_{x})\,,\quad\mathcal{J}_{y}\sim
c_{8}(\partial_{y}u_{z}+\partial_{z}u_{y})\,,$ (105)

at linear order in derivatives. Equivalently, $c_{10}$ measures the (Hall-
like) response of the current $\mathcal{J}$ to a shear in the plane of the
magnetic field and perpendicular to the current response, e.g. $(y,z)$-plane
for the response in $x$-direction (and equivalently for the $y$-direction):

$\mathcal{J}_{x}\sim
c_{10}(\partial_{y}u_{z}+\partial_{z}u_{y})\,,\quad\mathcal{J}_{y}\sim
c_{10}(\partial_{x}u_{z}+\partial_{z}u_{x})\,.$ (106)

##### The transverse Hall viscosity $\bm{\tilde{\eta}_{\perp}}$, and the novel
longitudinal Hall viscosity $\bm{\tilde{\eta}_{||}}$.

Hall viscosities were first discovered in (2+1)-dimensional systems
Avron:1995fg ; Avron:1997 . The (3+1)-dimensional counterparts have been
discussed in Hernandez:2017mch . In (2+1)dimensions, the relevant term in the
energy momentum tensor constitutive relation takes a form which is our
$\tilde{\sigma}_{\perp}^{\mu\nu}$ projected onto the plane perpendicular to
the magnetic field $B^{\mu}$. When simplified, this reads
$T^{ij}\sim\eta_{H}(\epsilon^{ikl}u_{k}\sigma_{l}^{j}+\epsilon^{jkl}u_{k}\sigma_{l}^{i})$
with the (2+1)-dimensional traceless symmetric stress $\sigma_{l}^{j}$ defined
in analogy to our $\sigma_{\mu}^{\nu}$.

In our system, the transport in the plane perpendicular to the magnetic field
is associated with $\tilde{\eta}_{\perp}$. The relevant contribution to the
energy-momentum tensor constitutive equation, eq. (43), is given by

$T^{\mu\nu}\sim\tilde{\eta}_{\perp}\tilde{\sigma}_{\perp}^{\mu\nu}\,.$ (107)

One may imagine that in that plane the tensor structure giving rise to Hall
viscosity is simply that of a (2+1)-dimensional system, given in the previous
paragraph. It may be interpreted in the same way as in Avron:1995fg ;
Avron:1997 , namely as the response of the energy-momentum tensor’s diagonal
components to a shear in the plane perpendicular to the magnetic field.
Example: Considering the $(x,y)$-plane, one finds for example
$T^{xx}\sim\tilde{\eta}_{\perp}\sigma^{xy}$, if the magnetic field is chosen
along the $z$-direction.

On the contrary, the Hall viscosity in the plane along the magnetic field,
$\tilde{\eta}_{||}$, is novel. It measures the response of the energy-momentum
tensor off-diagonal components to a shear in the plane aligned with the
magnetic field and one of the other spatial directions as seen in the
constitutive relation, eq. (43),

$T^{\mu\nu}\sim\tilde{\eta}_{||}(b^{\mu}\tilde{\Sigma}^{\nu}+b^{\nu}\tilde{\Sigma}^{\mu})\,.$
(108)

Example: If the magnetic field is aligned with the $z$-direction, then we have
$T^{xz}\sim\tilde{\eta}_{||}\sigma^{yz}$.

##### The Hall conductivity $\bm{\tilde{\sigma}_{\perp}}$.

Hall transport is dissipationless and only occurs in the plane perpendicular
to the magnetic field. For example, consider a magnetic field along the
$z$-direction and an electric potential
$V^{x}=E^{x}-T\partial^{x}\frac{\mu}{T}$ along the $x$-direction. This
configuration induces a current in the $y$-direction proportional to
$\tilde{\sigma}_{\perp}$. An equivalent Hall response in the longitudinal
$(x,z)$\- or $(y,z)$-plane does not exist. Note that this can also be related
to the parity anomaly in the absence of any magnetic field from a
(2+1)-dimensional point of view as discussed below, see section 2.5.2.

The remaining 12 transport coefficients have been identified previously,
either at vanishing magnetic field, at vanishing charge density, and/or
without a chiral anomaly. Our study generalizes these previous results.

##### The magneto-vortical susceptibility $\bm{M_{5}}$.

This coefficient was first found, named, and interpreted in Hernandez:2017mch
.333333This coefficient in Hernandez:2017mch  is designated by $M_{\Omega}$.
It appears as the response of energy/pressure to the magnetic field along the
vorticity:

$T_{tt}\sim T_{zz}\sim M_{5}B{\cdot}\Omega\,.$ (109)

Heuristically, one may imagine vortices of charged fluid (due to nonzero
vorticity) distributed over the system. Each charged vortex acts like an
elementary magnet. Depending on its charge and orientation with respect to the
magnetic field, it either increases or decreases the energy of the fluid. The
fluid can respond like a diamagnet or paramagnet, depending on the sign of
$M_{5}$, which is a microscopic property of the system and has to be measured.
$M_{5}$ also measures the response of the magnetization to vorticity in the
fluid

$\mathfrak{m}^{\mu}\sim
M_{5}\Omega^{\mu}+M_{2}(2\epsilon^{\mu\nu\rho\sigma}u_{\nu}\nabla_{\rho}B_{\sigma}-\epsilon^{\mu\nu\kappa\sigma}u_{\nu}B_{\kappa}\partial_{\sigma}T)\,,$
(110)

and was interpreted to measure the angular momentum generated by an external
magnetic field due to non-vanishing surface currents (see discussion of
Hernandez:2017mch  for more details). Note that $M_{5}$ requires a nonzero
$\mu$ in order not to vanish, indicating that the fluid must be charged for
these effects to take place. These terms capture the intuitive effects of a
charged fluid with nonzero vorticity producing a magnetization, and a charged
fluid subject to a magnetic field acquiring some angular momentum in response.

Additionally, the magneto-vortical susceptibility induces a Nernst effect in
the energy current

${\cal Q}^{\mu}\sim\left(2M_{5}-TM_{5,T}-\mu
M_{5,\mu}\right)\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}T/T$
(111)

from which one can identify $2M_{5}/T-M_{5,T}-\mu M_{5,\mu}/T$ as a momentum
Nernst coefficient.

##### Chiral vortical, chiral magnetic, chiral thermal conductivities,
$\xi=\xi_{TB},\,\xi_{B},\,\xi_{T}$.

As expected, these dissipationless chiral conductivities are given
analytically as functions of thermodynamic quantities and the chiral anomaly
coefficient of the microscopic theory, see eq. (19). Therefore, we confirm
validity of these expressions in states with a strong magnetic field.

##### The Nernst effect.

The thermodynamic constitutive relations encode the Nernst effect in the
spatial current ${\cal J}^{\mu}$343434In deriving eq. (112), we separated the
electric field contribution from the temperature gradient contribution as a
source to the equilibrium current. That is, we chose
$T\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}\sim E^{\mu}$ and
$\Delta^{\mu\nu}\partial_{\nu}T$ as our independent first order equilibrium
scalars.

${\cal
J^{\mu}}\sim-\left(\chi_{B,T}+\mu\chi_{B,\mu}/T\right)\epsilon^{\mu\nu\rho\sigma}u_{\nu}B_{\rho}\partial_{\sigma}T\,.$
(112)

The Nernst coefficient can therefore be identified with
$-\chi_{B,T}-\mu\chi_{B,\mu}/T$. The magnetic susceptibility
$\chi_{B}=2\,p_{,B^{2}}$ was defined in section 2.1.3. Furthermore, it has
been shown that the conformal anomaly gives rise to a Nernst effect
Chernodub:2017jcp  with a Nernst coefficient proportional to the conformal
anomaly coefficient. This can be seen in our setup by considering that, in the
absence of a magnetic field, the conformal anomaly vanishes and thus
$T^{\mu}_{\mu}=4p-Tp{,T}-\mu p_{,\mu}-2\chi_{B}B^{2}=0$ at $B=0$. Taylor
expanding $T^{\mu}_{\mu}$ at small $B^{2}$ then gives the leading conformal
anomaly coefficient $c_{A}$

$T^{\mu}_{\mu}=c_{A}F_{\mu\nu}F^{\mu\nu}=2c_{A}B^{2}\,,$ (113)

where $c_{A}=-T\chi_{B,T}/2-\mu\chi_{B,\mu}/2$. This leads to the relation
$N_{Nernst}=2c_{A}/T$. This relation agrees with the result of
Chernodub:2017jcp  except for the numerical prefactor. The latter should
depend on which charges the fermions in the one-loop diagram carry, that
determines the conformal anomaly coefficient.

##### Very well known transport coefficients.

It should be noted first that while the transport coefficients to be discussed
here are well known, they have not yet been discussed for a strong magnetic
field associated with an axial $U(1)_{A}$. This aspect is novel in our work.

The remaining transport coefficients are all hydrodynamic. Due to the
anisotropy caused by the magnetic field, there are two shear viscosities,
$\eta_{\perp}$ for transport perpendicular, and $\eta_{||}$ for transport
longitudinal to the magnetic field.353535We will see in the holography section
that $\eta_{||}$ need not take on the value $s/(4\pi)$ because it does not
satisfy the equation of motion of a minimally coupled scalar in asymptotically
$AdS$ spacetime. See e.g. Rebhan:2009vc ; Huang:2011dc ; Critelli:2014kra ;
Hernandez:2017mch ; Grozdanov:2016tdf ; Grozdanov:2017kyl ; Garbiso:2020puw
for shear viscosities in anisotropic systems. The bulk viscosities
$\zeta_{1}$, $\zeta_{2}$, and $\eta_{1}$ (and the linearly dependent
$\eta_{2}$) have been discussed in Huang:2011dc ; Critelli:2014kra ;
Hernandez:2017mch ; Grozdanov:2016tdf ; Grozdanov:2017kyl . The same is true
for the perpendicular and longitudinal conductivities $\sigma_{\perp}$,
$\sigma_{||}$, and the associated resistivities.

##### Remarks on the origin of the transport effects:

  * •

The coefficient $M_{5}$ can be nonzero if the chemical potential is nonzero,
even if there is no anomaly.

  * •

We note that the susceptibilities $M_{1},\,M_{2},\,M_{3},\,M_{4}$ were already
considered in Hernandez:2017mch  for a vector magnetic field associated with a
$U(1)_{V}$. In that case, they have to vanish in a microscopic theory
preserving parity. However, in our case with an axial magnetic field present,
these coefficients can be nonzero even if the microscopic theory is parity
preserving. In other words, for the thermodynamic transport coefficients
$M_{1},\,M_{2},\,M_{3},\,M_{4}$ one source of parity-violation suffices in
order for them not to vanish. This parity violation may stem from a chiral
anomaly in the microscopic theory or alternately from an external axial
$U(1)_{A}$ chemical potential in a parity-preserving microscopic theory.
Hence, these coefficients are not exclusively caused by the anomaly.

  * •

While $M_{1,3,4}$ show up in constitutive equations multiplying a first order
scalar $s_{n}$, $M_{2}$ and $M_{5}$ stand out from the crowd as they multiply
other tensor structures at first order in derivatives. Consequently, $M_{2}$
and $M_{5}$ are the one of these transport coefficients which still contribute
to the constitutive equations if all $s_{1,2,3,4,5}=0$.

  * •

All transport coefficients $c_{i}$ and $\tilde{c}_{i}$ are nonzero only if the
system is chiral. This chirality can be caused by an anomaly, or a $U(1)_{A}$
chemical potential. Hence, these coefficients are not exclusively caused by
the anomaly.

####  2.5.2 Relation to hydrodynamics in 2+1 dimensional fluids

It is theoretically and experimentally motivated to consider slicing a
$(3+1)$-dimensional material into $(2+1)$-dimensional planes and suppressing
the interactions between such slices. For example, graphene or the high
temperature superconducting cuprates show a layered structure where transport
along the layers is different from transport perpendicular to the layers.

Parity-violating hydrodynamics in 2+1 dimensions has been constructed and
discussed in Jensen:2011xb . While there is no chiral anomaly, here the parity
anomaly manifests in the transport effects. In order to relate the transport
in that lower dimensional system, we may think of our magnetic field as
defining (2+1)-dimensional planes perpendicular to it. One can think of the
(3+1)-dimensional hydrodynamics described in this section as being projected
onto hypersurfaces perpendicular to the magnetic field. For the purpose of
constructing constitutive relations for hydrodynamic transport on those
(2+1)-dimensional hyperplanes, this simply means that we could project all
(3+1)-dimensional tensor structures onto those hyperplanes. For example, the
magnetic field may point along the $z$-direction. Then, itself is defined by
the projection of
$B^{\mu}=\frac{1}{2}\,\epsilon^{\mu\nu\rho\sigma}u_{\nu}F_{\rho\sigma}$ onto
the $z$-direction: $B_{z}=\frac{1}{2}\,\epsilon_{z}{}^{ijk}u_{i}F_{jk}$, where
$i,\,j,\,k\in\\{t,\,x,\,y\\}$. On the $(t,x,y)$-hyperplane $B_{z}$ transforms
like a pseudoscalar if $B^{\mu}$ is associated with a vector $U(1)_{V}$, and
like a scalar for an axial $U(1)_{A}$. Another example is the vorticity which
becomes a pseudoscalar
$\Omega_{z}=\epsilon_{z}{}^{ijk}u_{i}\partial_{j}u_{k}$. These are the
definitions of the (pseudo)scalar magnetic field and vorticity in
Jensen:2011xb . This projection procedure can also be applied to the other
tensor structures we used to construct the (3+1)-dimensional hydrodynamic
constitutive relations, eqs. (43) and (44). Of course, for a comparison, we
need to take into account that weak magnetic fields,
$B\sim\mathcal{O}(\partial)$ are considered in Jensen:2011xb . So similarities
will generally be more obvious in the non-equilibrium part of the constitutive
relations.

Among the equilibrium quantities, $s_{1},\,s_{2},\,s_{3},\,s_{4}$ have
derivatives or other vectors pointing in the $z$-direction, which have no
counterpart in (2+1) dimensions. However, $s_{5}$ has a trivial counterpart
$\Omega B$, with $B\sim\mathcal{O}(1)$, $\Omega\sim\mathcal{O}(\partial)$ in
our counting and $\Omega,B\sim\mathcal{O}(\partial)$ in the counting of
Jensen:2011xb . The generating functional may depend on the pseudoscalars $B$
and $\Omega$ discussed above, as well as on the temperature $T$.

It turns out that the perpendicular Hall viscosity of the (3+1)-dimensional
hydrodynamics after projection of constitutive relations eq. (43) is
identified with the Hall viscosity in the (2+1)-dimensional hydrodynamic
constitutive relation. To see this, consider that the term
$\tilde{\eta}_{\perp}\tilde{\sigma}_{\perp}^{\mu\nu}$ projected onto the
$(t,x,y)$-hyperplane becomes $\tilde{\eta}_{\perp}\tilde{\sigma}_{\perp}^{ij}$
in analogy to the same tensor structure defined in Jensen:2011xb . Similarly,
the perpendicular shear viscosity term $\eta_{\perp}\sigma^{\mu\nu}$ is
projected onto the shear viscosity term in (2+1) dimensions.

In the (2+1)-dimensional hydrodynamics the current constitutive relation
analogous to eq. (44) contains the Hall conductivity term
$(\tilde{\sigma}_{\perp}+\tilde{\chi}_{E})\epsilon^{ijk}u_{j}E_{k}$. The first
term can be thought of as a projection of
$\tilde{\sigma}_{\perp}\tilde{V}^{\mu}$. The thermodynamic transport
coefficient $\tilde{\chi}_{E}$ is entirely determined by thermodynamic
quantities, and it does not vanish at zero magnetic field. The
(3+1)-dimensional Hall conductivity also contains such a purely thermodynamic
contribution to the current, namely
$\epsilon^{\mu\nu\rho\sigma}u_{\nu}b_{\rho}E_{\sigma}$, which is first order
in derivatives but we may choose these to be only spatial derivatives. Note
that the parity anomaly leads to a Hall effect in absence of magnetic fields,
as had been realized early by Haldane Haldane:1988zza .

Projecting either of the longitudinal shear or longitudinal Hall viscosity
onto a $(t,x,y)$-hyperplane makes it vanish from the energy momentum tensor.
Heuristically, this is clear because there can not be any shear or Hall
response in the longitudinal $(x,z)$\- or $(y,z)$-planes if there are no such
planes in the (2+1)-dimensional system. We refer to such longitudinal
transport effects as out-of-plane transport from the perspective of
$(2+1)$-dimensional hyperplanes orthogonal to the anisotropy. The opposite to
that is the in-plane transport. It turns out that all of the novel363636These
are novel in that they are absent when the magnetic field is of linear or
higher order in derivatives. transport coefficients
$\tilde{\eta}_{||},\,\eta_{||}$, $\tilde{\sigma}_{||},\,\eta_{1},\,\eta_{2}$,
$\zeta_{2},\,c_{3},\,c_{4},\,c_{5},\,c_{8},\,c_{10},\,c_{14}$,
$c_{15},\,c_{17}$ describe out-of-plane transport effects as seen from the
projection of the constitutive relations eq. (43) and (44).

The (2+1)-dimensional constitutive relations in the thermodynamic frame are
given by

$\displaystyle T^{ij}_{(2+1)D}$ $\displaystyle=$
$\displaystyle(\epsilon_{0}+\tilde{\epsilon}_{1}B+\tilde{\epsilon}_{2}\Omega)u^{i}u^{j}+p_{0}\Delta^{ij}+\tilde{\phi}_{2}\tilde{E}^{(i}u^{j)}+\frac{\tilde{\epsilon}_{2}}{T}\epsilon^{(ikl}u_{k}\partial_{l}u^{j)}+\mathcal{O}(\partial^{2})\,,$
(114) $\displaystyle J^{i}_{(2+1)D}$ $\displaystyle=$
$\displaystyle(n_{0}+\tilde{\phi}_{1}B+\tilde{\phi}_{2}\Omega)u^{i}+\tilde{\phi}_{1}\tilde{E}^{i}+\frac{\tilde{\epsilon}_{1}}{T}\epsilon^{ijk}u_{j}\partial_{k}T+\mathcal{O}(\partial^{2})\,.$
(115)

The identification between the thermodynamic transport coefficients for
(2+1)-dimensional parity violating hydrodynamics and (3+1)-dimensional
hydrodynamics with strong magnetic fields can be easily done by comparing our
equilibrium constitutive relations to the results of Jensen:2012jh . Formally,
the comparison becomes straightforward by expanding the generating functional
in eq. (2) in small magnetic field and keeping only the terms which don’t
vanish under the assumption of no fluctuations parallel to the magnetic field

$W_{s}=\int
d^{4}x\sqrt{-g}\left(p(T,\mu,B^{2}=0)+p_{,B^{2}}(T,\mu,B^{2}=0)B^{2}+M_{5}(T,\mu,B^{2}=0)B^{\alpha}\Omega_{\alpha}+\cdots\right)\,.$
(116)

We can take this generating functional over a thin (2+1)-dimensional sheet
orthogonal to the magnetic field as the generating functional for
(2+1)-dimensional parity violating hydrodynamics and compare it to the
generating functional in Jensen:2012jh

$W_{2+1}=\int
d^{3}x\sqrt{-\gamma}\left(p_{2+1}(T,\mu)+\tilde{\alpha}_{1}B+\tilde{\alpha}_{2}\Omega+\cdots\right)\,,$
(117)

where $\Omega=B^{\mu}\Omega_{\mu}/B$ with $B=\sqrt{B^{2}}$. Comparing the two
expansions above leads to the identification of the thermodynamic transport
coefficients in the following way

$p_{2+1}(T,\mu)=p(T,\mu,0)\,,\quad\tilde{\alpha}_{1}=B\,p_{,B^{2}}(T,\mu,0)=0\,,\quad\tilde{\alpha}_{2}=B\,M_{5}(T,\mu,0)=0\,.$
(118)

Note that because of the derivative counting $B\sim{\cal O}(\partial)$, the
first order thermodynamic transport coefficients $\tilde{\alpha}_{i}$ must
vanish. This is a direct consequence of the assumption that the 3+1
dimensional system behaves analytically at small $B^{\mu}$, that is
$p=p(T,\mu,B^{2})$. Indeed, upon making these identifications, the