constitutive relations in eqs. (10) and (11) are inconsistent with the results
of Jensen:2011xb ; Jensen:2012jh  by a factor of 2 whenever
$\tilde{\alpha}_{1}$ shows up. To derive the correct 2+1 equilibrium
constitutive relations, we must start with a 3+1 system with a generating
functional that is not analytic at small $B^{\mu}$, so that $p=p(T,\mu,B)$.
Then a similar analysis leads instead to the identification

$p_{2+1}(T,\mu)=p(T,\mu,0)\,,\quad\tilde{\alpha}_{1}=\,p_{,B}(T,\mu,0)\,,\quad\tilde{\alpha}_{2}=B\,M_{5}(T,\mu,0)\,.$
(119)

The thermodynamic transport coefficients $\tilde{\alpha}_{i}$ can now be
nonzero. Note that $\tilde{\alpha}_{2}\neq 0$ requires $M_{5}$ to diverge at
small magnetic field, which is allowed since we didn’t assume $W_{s}$ was
analytic at small $B^{\mu}$. The resulting equilibrium constitutive relations
would then match precisely the results of Jensen:2011xb ; Jensen:2012jh  after
truncating fluctuations along the magnetic field and higher order terms in the
magnetic field.

A different way of dimensional reduction could be defined for hydrodynamics on
a (2+1)-dimensional hypersurface with normal vector $n^{\mu}$ pointing along
the magnetic field, in a (3+1)-dimensional material. Using this vector
$n^{\mu}$ to project all tensor structures onto the lower-dimensional surface,
one obtains a (2+1)-dimensional hydrodynamic description of such a
hypersurface Hernandez:2017xan .

We are interested in the Hall response
$(\tilde{\chi}_{E}+\tilde{\sigma})\tilde{E}^{\mu}$ and its (3+1)-dimensional
analog. Note that within the (2+1)-dimensional constitutive equations in
Landau versus thermodynamic frame the following relations hold Jensen:2012jh
:373737Note that the charge density is denoted by $\rho$ in Jensen:2011xb ;
Jensen:2012jh , whereas we have used $n$ in this work. For the purpose of this
comparison we simplify $\rho=n_{2+1}=n$. Similarly, the pressure is denoted
$P=p_{2+1}=p$. Note also that the coefficients in Jensen:2012jh  were related
as
$\tilde{\phi}_{1}=\tilde{\delta}_{1},\,\tilde{\phi}_{2}=\tilde{\gamma}_{1}$.

$\tilde{\chi}_{E}=\tilde{\phi}_{1}-\frac{n}{\epsilon+p}\tilde{\phi}_{2}\,.$
(120)

There is one obvious contribution to the Hall response in the current,
$J^{\mu}=\tilde{\sigma}_{\perp}\tilde{E}^{\mu}+...$, but there is another
contribution to the Hall response, coming from $\mathcal{Q}^{\mu}$, in the
frame invariant combination

$J^{\mu}-\frac{n}{\epsilon+p-{\chi_{B}}B^{2}}\mathcal{Q}^{\mu}\sim\tilde{E}^{\mu}\left(\tilde{\sigma}_{\perp}+2B\,p_{,B^{2},\mu}-\frac{n}{\epsilon+p-{\chi_{B}}B^{2}}(M_{5,\mu}-2p_{,B^{2}})\,B\right)\,.$
(121)

This should be Taylor-expanded in small $B$ in order to keep only terms linear
in derivatives for comparison. In the (2+1)-dimensional frame-invariant from
Jensen:2012jh , one finds

$J_{2+1}^{i}-\frac{n}{\epsilon+p}\mathcal{Q}_{2+1}^{i}\sim\tilde{E}^{i}\left(\tilde{\sigma}_{\perp}+\tilde{\phi}_{1}-\frac{n}{\epsilon+p}\tilde{\phi}_{2}\right)\,.$
(122)

This implies that our relations projected on (2+1) dimensions and taking
hydrodynamic frames into account give

$\tilde{\chi}_{E}=\tilde{\phi}_{1}-\frac{n}{\epsilon+p}\tilde{\phi}_{2}=2B\,n_{,B^{2}}+\frac{n}{\epsilon+p}(2p_{,B^{2}}-M_{5,\mu})\,B\,,$
(123)

where we have used that $n=p_{,\mu}$. Now this agrees with Jensen:2011xb ;
Jensen:2012jh  if we recall that in (2+1) dimensions $\frac{\partial
p}{\partial\Omega}=B\,M_{5}$, as is shown in Jensen:2011xb .383838There, the
coefficient equivalent to our $M_{5}$ is named $M_{\Omega}$.

##  3 Holography

Our goal is to investigate an explicit example of a charged plasma in quantum
field theory with a chiral anomaly at strong coupling and in presence of a
(strong) magnetic background field. In particular, we determine the transport
coefficients within that quantum field theory and show that most of the novel
transport coefficients discussed in section 2 are indeed non-vanishing.

For this purpose, we consider $\mathcal{N}=4$ Super-Yang-Mills (SYM) theory
with gauge group $SU(N_{c})$. The field content of $\mathcal{N}=4$ SYM theory
consists of the following: one vector, four left-handed Weyl fermions, and six
real scalar fields. All the matter content is in the adjoint representation of
the $SU(N_{c})$ gauge group. $\mathcal{N}=4$ SYM theory is invariant under a
global $SU(4)_{R}$ R-charge symmetry. In the following we consider a $U(1)$
subgroup of $SU(4)_{R}$, which we label $U(1)_{A}$. In $\mathcal{N}=4$ SYM
theory the associated global current is axial and has non-vanishing divergence
due to a chiral anomaly in the theory. Hence we refer to this as an axial
$U(1)_{A}$ current. The fermions and scalars are charged under the anomalous
$U(1)_{A}$ symmetry, while the vector field is uncharged. The anomalous
current $J^{\alpha}$ associated with the $U(1)_{A}$ symmetry is coupled to an
external axial gauge field $A_{\alpha}^{\text{ext}}$ via the interaction
term393939Note that indices referring to boundary field theory coordinates
$\\{t,\,x,\,y,\,z\\}$ are represented by Greek indices such as
$\alpha,\,\beta,\,\mu,\,\nu,\,\dots=0,\,1,\,2,\,3$.

$S=S_{\text{SYM}}+\int d^{4}x\,J^{\alpha}A_{\alpha}^{\text{ext}}\,.$ (124)

We are specifically interested in the effects of a constant external magnetic
field $B$ described by $F=\textrm{d}A^{\text{ext}}=B\,dx\wedge dy$. Moreover,
we allow for a nonzero chemical potential $\mu$ associated with the $U(1)_{A}$
symmetry. We may think of $\mu$ and $B$ as an axial chemical potential and a
magnetic field associated with an axial $U(1)_{A}$ symmetry, respectively.
This is an example for the $U(1)_{A}$ symmetry, associated gauge fields,
chemical potential, and currents discussed in section 2.

In order to obtain results for the charged plasma state within this strongly
coupled theory, we utilize the gauge/gravity correspondence in the large
$N_{c}$ limit and at large ’t Hooft coupling $\lambda$ limit Maldacena:1997re
(for textbooks see Ammon:2015wua ; Nastase ; Schalm ). Then we perform the
relevant calculations in the dual gravitational theory, or a consistent
truncation thereof, namely within Einstein-Maxwell-Chern-Simons theory subject
to an external magnetic field. The relevant solutions dual to the desired
equilibrium state at finite temperature, finite axial chemical potential, and
subject to an external magnetic field, are charged magnetic black branes
D'Hoker:2009mm ; Ammon:2016szz . The stability of these black branes was
investigated in Ammon:2017ded  by computing the quasinormal modes.404040Note
that the linear stability of the dual gravitation theory is not guaranteed in
the presence of a magnetic field. In fact, there are examples of holographic
models in which a magnetic field induces a linear instability, see e.g.
Ammon:2011je .

###  3.1 Holographic Setup

The holographically dual gravitational theory mentioned above is five-
dimensional Einstein-Maxwell-Chern-Simons theory with a negative cosmological
constant $\Lambda=-6/L^{2}$ and the $AdS_{5}$ radius $L$. This theory is
defined via414141Note that we refer to five-dimensional coordinate indices
with lower case Latin letters, such as $m,n=0,\,1,\,2,\,3,\,4$. Recall that we
use Greek indices such as $\mu,\nu=0,\dots,3$ for field theory vectors, as
well as tensors.

$S_{\textrm{grav}}=\frac{1}{2\kappa^{2}}\left[\,\int\limits_{\mathcal{M}}\\!d^{5}x\,\sqrt{-g}\left(R+\frac{12}{L^{2}}-\frac{1}{4}F_{mn}F^{mn}\right)-\frac{\gamma}{6}\int\limits_{\mathcal{M}}A\wedge
F\wedge F\right]\,,$ (125)

where $\mathcal{M}$ denotes the asymptotically $AdS_{5}$ spacetime, while
$\partial\mathcal{M}$ denotes its conformal boundary. Furthermore, $g_{mn}$ is
the five-dimensional metric and $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$
is the five-dimensional $U(1)_{A}$ field strength tensor. Moreover, the
parameter $\gamma$ is the Chern-Simons coupling which is related to the
anomaly coefficient $C$ introduced in section 2.1.2 by $C=-\gamma.$ Let us
specify the parameters $c_{1}$ and $c_{2}$ introduced in the consistent
generating functional (16): the coefficient $c_{1}$ is related to the mixed
gauge-gravitational anomaly and is sub-leading in the large $N_{c}$ limit.
Hence, in our case $c_{1}=0.$424242In order to mimic a nonzero coefficient
$c_{1}$ we may add by hand a mixed gauge-gravitational Chern-Simons term to
the action $S_{\textrm{grav}}$. See Landsteiner:2011iq  for more details.
Moreover, since the dual $\mathcal{N}=4$ supersymmetric field theory is CPT-
invariant, we also conclude that $c_{2}=0$.

For $\gamma=2/\sqrt{3}$, the action (125) is the bosonic part of minimal
gauged supergravity in five spacetime dimensions and hence it is a consistent
truncation of the most general class of type IIB supergravity in ten
dimensions or supergravity in eleven dimensions which are dual to
$\mathcal{N}=1$ superconformal field theories, see e.g. Buchel:2006gb ;
Gauntlett:2006ai ; Gauntlett:2007ma ; Colgain:2014pha . In this paper however,
we restrict ourselves not just to this particular value of $\gamma$ but rather
treat $\gamma$ as a free parameter and study the transport coefficients as a
function thereof. In particular, we investigate the cases $\gamma=0$ and
$\gamma=2/\sqrt{3}$. The latter value, we refer to as supersymmetric.

The action (125) has to be amended by boundary terms Henningson:1998gx ;
Balasubramanian:1999re ; Taylor:2000xw ; Sahoo:2010sp  of the form

$S_{\text{bdy}}=\frac{1}{\kappa^{2}}\int\limits_{\partial\mathcal{M}}\\!d^{4}x\sqrt{-\hat{g}}\left(K-\frac{3}{L}-\frac{L}{4}\hat{R}+\ln\left(\frac{\varrho}{L}\right)\left(\frac{L}{8}F_{\mu\nu}F^{\mu\nu}-\frac{L^{3}}{8}\hat{R}_{\mu\nu}\hat{R}^{\mu\nu}+\frac{L^{3}}{24}\hat{R}^{2}\right)\right),$
(126)

where $\varrho$ is the radial coordinate of $AdS_{5}$ in the Poincaré slicing.
The metric $\hat{g}_{\mu\nu}$ is induced by $g_{mn}$ on the conformal boundary
of $AdS_{5}$, while the extrinsic curvature is given by

$K_{mn}=\mathcal{P}_{m}^{\ \,o}\,\mathcal{P}_{n}^{\
\,p}\,\nabla_{o}n_{p}\,,\qquad\mbox{with}\quad\mathcal{P}_{m}^{\
\,o}=\delta_{m}^{\ o}-n_{m}n^{o}\,.$ (127)

Here, $\nabla$ is the covariant derivative and $n_{m}$ are the components of
the outward pointing normal vector of the boundary $\partial\mathcal{M}.$
Moreover, $K$ is the trace of the extrinsic curvature with respect to the
metric at the boundary, $\hat{R}_{\mu\nu}$ denotes the Ricci tensor associated
with the metric $\hat{g}_{\mu\nu}$ and $\hat{R}$ is the corresponding Ricci
scalar. For simplicity we are going to choose $L=1$ and $2\,\kappa^{2}=16\pi
G_{5}=1$ from now on.

The equations of motion associated with the action (125) in terms of the Ricci
tensor $R_{mn}$, metric $g_{mn}$, and field strength $F_{mn}$ read

$R_{mn}=-4\,g_{mn}+\frac{1}{2}\left(F_{mo}\,F_{n}{}^{o}-\frac{1}{6}\,g_{mn}\,F_{op}F^{op}\right)$
(128)

from variation with respect to the metric, as well as

$\nabla_{m}F^{mn}+\frac{\gamma}{8\sqrt{-g}}\,\varepsilon^{nmopq}F_{mo}F_{pq}=0$
(129)

from variation with respect to the gauge field. Here, $\varepsilon^{mnopq}$ is
the totally antisymmetric Levi-Civita symbol in five spacetime dimensions with
$\varepsilon^{txyz\varrho}=1$.

We are interested in describing the charged plasma state in the presence of an
external (axial) magnetic background field, (axial) chemical potential and in
the presence of the chiral anomaly. The appropriate ansatz for the metric and
the field strength tensor $F$ in ingoing Eddington-Finkelstein coordinates
reads434343To keep the notation simple, we still use the variable $t$ for the
null ingoing Eddington-Finkelstein time.

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\varrho^{2}}\left[\left(-u(\varrho)+c(\varrho)^{2}\,w(\varrho)^{2}\right)\,dt^{2}-2\,dt\,d\varrho+2\,c(\varrho)\,w(\varrho)^{2}\,dz\,dt\right.$
(130)
$\displaystyle\left.+v(\varrho)^{2}\,\left(dx^{2}+dy^{2}\right)+w(\varrho)^{2}\,dz^{2}\right]\,,$
$\displaystyle F$ $\displaystyle=$ $\displaystyle
A_{t}^{\prime}(\varrho)\,d\varrho\wedge dt+B\,dx\wedge
dy+P^{\prime}(\varrho)\,d\varrho\wedge dz\,,$ (131)

where the horizon of the black brane is located at $\varrho=1$, while the
conformal boundary is located at $\varrho=0$. Moreover, prime denotes
derivatives with respect to the radial coordinate $\varrho$. The field
strength tensor (131) may be obtained from a gauge field $A$ of the form

$A=A_{t}(\varrho)\,dt+\frac{B}{2}\left(-y\,dx+x\,dy\right)+P(\varrho)\,dz\,.$
(132)

Note that the ansatz for the metric and the field strength tensor preserve the
$SO(2)$ rotational symmetry in the $(x,y)$-plane.

The metric functions $u(\varrho),v(\varrho),w(\varrho)$ and $c(\varrho)$ are
chosen such that the spacetime is asymptotically $AdS_{5}$ with a flat
Minkowski metric $\hat{g}$ of the conformal boundary which is located at
$\varrho=0$. In particular, we set $u(0)=v(0)=w(0)=1$ and $c(0)=0$. Moreover,
the leading component of $A_{t}$ is identified with the axial chemical
potential $\mu$, while there is no explicit source for $P(\varrho)$. The
latter choice sets the source for the field theory current $J_{z}$ to zero.

Moreover, we impose the conditions $A_{t}(1)=0$ and $c(1)=u(1)=0$ at the
horizon which is located at $\varrho=1$. The condition on $c(1)$ and $u(1)$
prevents a conical singularity at the horizon in the Euclideanized metric. The
subleading coefficient of $u(\varrho)$ at the horizon is related to the
temperature

$T=\frac{|u^{\prime}(1)|}{4\pi}\,.$ (133)

The explicit form of functions close to the conformal boundary and near the
horizon are shown in appendix B.

In order to find the functions
$u(\varrho),\,c(\varrho),\,w(\varrho),\,v(\varrho),\,A_{t}(\varrho)$, and
$P(\varrho)$ subject to the boundary conditions specified above we use
spectral methods to solve the ordinary differential equations (for more
details see Ammon:2016szz ). We can also read off thermodynamic quantities
such as the density of the partition function in the grand canonical ensemble
$\Omega$, the entropy density $s$ as well as the one-point function of the
energy-momentum tensor $T_{\mu\nu}$, the covariant current $J^{\mu}_{cov}$ and
the consistent current $J^{\mu}_{cons}$ from the numerical solution.

The form of the expectation value of the energy-momentum tensor depends on the
chosen action (125) and on its boundary terms (126). Given these boundary
terms, which corresponds to a particular choice of the renormalization scheme,
the energy-momentum tensor may be extracted from Balasubramanian:1999re
444444If we do not set $2\kappa^{2}=1$, the expressions for the energy-
momentum tensor and the current have to be multiplied by $2\kappa^{2}.$ In
case of $\mathcal{N}=4$ SYM with gauge group $SU(N)$, this overall prefactor
is given by $N^{2}/(8\pi^{2})$ in terms of field theory quantities.

$\displaystyle\left\langle
T_{\mu\nu}\right\rangle=\lim\limits_{\varrho\rightarrow
0}\frac{1}{\varrho^{2}}$ $\displaystyle\Big{(}$
$\displaystyle-2K_{\mu\nu}+2(K-3)\,g_{\mu\nu}+\ln(\varrho)\left(F_{\mu}^{\
\alpha}F_{\nu\alpha}-\frac{1}{4}\,g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\right)$
(134)
$\displaystyle+\hat{R}_{\mu\nu}-\frac{1}{2}\hat{R}\hat{g}_{\mu\nu}+8\ln(\varrho)\,h^{(4)}_{\mu\nu}\Big{)}\,$

with

$h^{(4)}_{\mu\nu}=\frac{1}{8}\hat{R}_{\mu\nu\rho\sigma}\hat{R}^{\rho\sigma}+\hat{R}\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}\hat{R}-\frac{1}{16}\hat{\nabla}^{2}\hat{R}_{\mu\nu}-\frac{1}{24}\hat{R}\hat{R}_{\mu\nu}+\frac{1}{96}\left(\hat{\nabla}^{2}\hat{R}+\hat{R}^{2}-3\hat{R}_{\rho\sigma}\hat{R}^{\rho\sigma}\right)\hat{g}_{\mu\nu}$
(135)

Here, $\hat{g}_{\mu\nu}$ is the metric on the conformal boundary of $AdS_{5}$;
$\hat{\nabla}_{\mu}$, $\hat{R}_{\mu\nu\rho\sigma}$, $\hat{R}_{\mu\nu}$ and
$\hat{R}$ are the covariant derivative, the Riemann curvature tensor, the
Ricci tensor and the Ricci scalar of the boundary metric $\hat{g}$. Also,
$h^{(4)}_{\mu\nu}$ is proportional to the Bach tensor, which spoils the power
series expansion of the metric in the Fefferman-Graham expansion FG ;
Fefferman:2007rka  for $d=4$, introducing a term that is logarithmic in the
radial coordinate Alvarez:2002jy . Moreover, $K_{\mu\nu}$ is the (projected)
extrinsic curvature (127).

Let us now turn to the expectation value of the consistent and covariant form
of the current $\left\langle J^{\mu}\right\rangle.$ The consistent current is
given in terms of a variation of the consistent generating functional
$W_{cons}$ with respect to the boundary gauge field $A^{\textrm{ext}}_{\mu}$
(see eq. (124) for the definition of the external gauge field). The consistent
generating functional is identified with the action (125) – including its
boundary terms (126) – evaluated on-shell. Note that this proposal is adequate
since both, $W_{cons}$ as well as the gravitational action $S_{\textrm{grav}}$
transforms in the same way under an infinitesimal $U(1)$ gauge transformation,
namely as specified by equation (13). Hence, the expectation value of the
consistent current $J_{cons}^{\mu}$ is given by D'Hoker:2009bc ;
Landsteiner:2011iq ; Gynther:2010ed ; Landsteiner:2016led ; Landsteiner:2011tf
; Landsteiner:2013aba

$\left\langle J_{cons}^{\mu}\right\rangle=\lim\limits_{\varrho\rightarrow
0}\left(\sqrt{-g}n_{a}g^{a\nu}F_{\nu\sigma}g^{\sigma\mu}+\frac{\gamma}{6}\epsilon^{\alpha\beta\gamma\mu}A_{\alpha}F_{\beta\gamma}+\ln\varrho\,\sqrt{-\hat{g}}\,\hat{\nabla}_{\nu}F^{\nu\mu}\right)\,,$
(136)

where $n_{a}$ is the unit normal vector orthogonal to the AdS-boundary.

Let us turn to the expectation value of the covariant form of the current
$J_{cov}^{\mu}$. The recipe is to drop the term in (136) which arises from the
Chern-Simons term, i.e. the term
$\frac{\gamma}{6}\epsilon^{\alpha\beta\gamma\mu}A_{\alpha}F_{\beta\gamma}$.
This is also in accordance with the proposal put forward in the previous
paragraph: the covariant current is given in terms of a variation of the
covariant generating functional $W_{cov}$ which differs from $W_{cons}$ by a
Chern-Simons term – see eq. (22). This is chosen in such a way that $W_{cov}$
is gauge invariant. In terms of the dual gravitational theory this means that
we have to drop the Chern-Simons term from the action $S_{\textrm{grav}}$
before evaluating $S_{\textrm{grav}}$ and $S_{\textrm{bdy}}$ on-
shell.454545However, this does not imply that the Chern-Simons parameter is
not relevant for computing $\left\langle J_{cov}^{\mu}\right\rangle$. We still
have to solve the equation (129) for the gauge fields even though we have
droped the Chern Simons term to determine the generating functional $W_{cov}.$
Note that this also fits nicely with the identification $C=-\gamma$ which we
stated earlier.

In summary, the expectation value of the covariant current reads

$\left\langle J_{cov}^{\mu}\right\rangle=\lim\limits_{\varrho\rightarrow
0}\left(\sqrt{-g}n_{a}g^{a\nu}F_{\nu\sigma}g^{\sigma\mu}+\ln\varrho\,\sqrt{-\hat{g}}\,\hat{\nabla}_{\nu}F^{\nu\mu}\right)\,.$
(137)

In appendix B we evaluate the expectation value of the energy momentum tensor
and the covariant current for the charged magnetic brane considered here and
relate them to coefficients in the near-boundary expansion. A word of caution
in order: components of the one-point functions, e.g. the energy density
$\epsilon=\langle T_{tt}\rangle,$ as well as the pressure $p$ may be scheme-
dependent. Adding finite counter terms in $S_{\text{bdy}}$ (see eq. (126))
will change the renormalization scheme. In fact, here we use $1/L$ as our
renormalization scale.464646Note that due to setting $L=1$ the renormalization
scale is not transparent in the explicit expressions. Other physically
significant choices of the renormalization scale are discussed in
Fuini:2015hba .

Besides thermodynamics, we are also interested in thermodynamic and
hydrodynamic transport coefficients given in terms of Kubo formulas in section
2.2.4. In order to compute the corresponding correlation functions, we
consider fluctuations to linear order in the metric,
$h_{mn}(\varrho,x^{\mu}),$ and of the gauge field, $a_{m}(\varrho,x^{\mu})$,
on top of the background discussed above. We perform a Fourier transformation
along the field theory coordinates $x^{\mu}$ and solve the corresponding
equations of motion (128) and (129) for the fluctuations
$\tilde{h}_{mn}(\varrho,k^{\mu})$ and
$\tilde{a}_{m}(\varrho,k^{\mu})$.474747To keep the notation simple, we
suppress the $k$-dependence of $\tilde{h}_{mn}(\varrho,k^{\mu})$ and
$\tilde{a}_{m}(\varrho,k^{\mu})$ and neglect the tilde in the following.

Since we either determine correlation functions at zero momentum, or we choose
the momentum to be aligned with the magnetic field, (i.e. along the $z$-axis)
we may classify the fluctuations according to the unbroken SO(2) symmetry
corresponding to rotations around the $z$-axis (for more details see e.g.
Ammon:2017ded ). In order to consider only the physical modes of the system,
we have to fix the gauge freedom. To do so, we choose a modified radial gauge
in which $a_{\varrho}=0$ and $h_{m\varrho}=0$ for $m\neq t$, as well as
$h_{t\varrho}=1/2\,h_{tt}.$484848Note these unusual gauge. However, setting
$h_{t\varrho}$ to zero does not allow us to specify arbitrary sources for
$h_{\mu\nu}(\varrho=0)$ at the conformal boundary.

To determine the thermodynamic/transport coefficients we proceed as follows:
we perturb the system by switching on a source term, e.g.
$h_{\alpha\beta}(0)=\delta g_{\alpha\beta}$ and then study its linear response
$\delta\langle T_{\mu\nu}\rangle$ and $\delta\langle J_{cov}^{\mu}\rangle$;
using eq. (46) we may read off the correlators
$G_{T^{\mu\nu}T^{\alpha\beta}}(\omega,{\bf k})$ and
$G_{J_{cov}^{\mu}T^{\alpha\beta}}(\omega,{\bf k}).$ Likewise we may allow for
non-vanishing sources $a_{\alpha}(0)$ from which we deduce
$G_{J_{cov}^{\mu}J_{cons}^{\alpha}}(\omega,{\bf k})$ (see eq. (63)). The naive
way to compute correlators of the form $\lim\limits_{k\rightarrow
0}\frac{1}{k}G_{\mathcal{O}\tilde{\mathcal{O}}}(\omega=0,k\hat{\bf{z}})$ and
$\lim\limits_{\omega\rightarrow
0}\frac{1}{\omega}G_{\mathcal{O}\tilde{\mathcal{O}}}(\omega,\bm{k}=\bm{0})$ is
to evaluate the Green’s function at a small value of $k$ (or a small value of
$\omega$), and then divide by that small value of $k$ (or $\omega$). Not
surprisingly, this leads to large numerical errors. In appendix B we outline a
method which just determines the linear coefficient in a small $k$ (or
$\omega$) expansion of the corresponding Green’s function.

###  3.2 Results: Thermodynamics and thermodynamic transport coefficients

#### Partition function in the grand canonical ensemble and magnetization

First we determine thermodynamic quantities of the dual field theory such as
$\Omega(T,\mu,B)$ being the density of the partition function in the grand
canonical ensemble which in turn gives the pressure $p=-\Omega.$ In fact, we
study the system as a function of the dimensionless chemical potential
$\tilde{\mu}=\mu/T$ and the dimensionless magnetic field $\tilde{B}=B/T^{2}.$
From here on out we will refer to these two quantities simply as the chemical
potential $\tilde{\mu}$ and the magnetic field $\tilde{B}$ for brevity.
Moreover, any observable discussed in the following sections will be rescaled
by powers of temperature such that it is dimensionless. For example, the
dimensionless density of the grand canonical potential $\Omega$ is given by
$\Omega(T,\mu,B)/T^{4}$.

In figure 1 we display the dimensionless pressure $p/T^{4}$ as a function of
the magnetic field $\tilde{B}$ for different values of the chemical potential
$\tilde{\mu}.$ Since there is no major qualitative difference between the
cases $\gamma=0$ and $\gamma=2/\sqrt{3}$ we only display the latter one. As a
cross-check for our numerics we checked that the – from the gravitational
point of view non-trivial – relation $p=\langle T_{zz}\rangle$ holds to high
accuracy.494949See Ammon:2012qs  or a discussion concerning thermodynamic
potentials in the presence of magnetic fields and their relation to components
of the energy-momentum tensor.

Next, we determine the magnetization $M$ which is defined by
$M=-\left(\frac{\partial\Omega}{\partial B}\right)_{\mu,T}$ where $\mu$ and
$T$ are kept constant. Note that we can express the dimensionless
magnetization $M/T^{2}$ by
$\partial\\!\left(p/T^{4}\right)\\!/\partial\tilde{B}|_{\tilde{\mu}}$ where we
keep $\tilde{\mu}$ constant.505050In the following we abbreviate the
derivative by $\left(p/T^{4}\right)_{,\tilde{B}}$. We refer the reader to
appendix B.4 for more details. The results are shown in figure 2. The left
figure shows the dimensionless magnetization for the supersymmetric case while
the right figure displays the case $\gamma=0.$ In both cases, the
magnetization is a symmetric function of $\tilde{\mu}$. Figure 2 shows three
different cases. Some remarks are in order: Firstly, the magnetization
vanishes for $\tilde{B}\rightarrow 0$. In fact, the red curve in the left and
right panel of figure 2 corresponds to the smallest value of $\tilde{B}$ which
we investigated, namely $\tilde{B}=0.05$. In both cases, the magnitude of the
magnetization nearly vanishes (however, zooming into the red curve we see a
behavior similar to the one with larger values of $\tilde{B}$). Secondly, the
magnetization is an even function of the chemical potential $\tilde{\mu}$.
Note that in the case $\gamma=0$ the magnetization increases as a function of
$|\tilde{\mu}|,$ while for the supersymmetric value of $\gamma$, the
magnetization as a function of $|\tilde{\mu}|$ first decreases and then again
increases. Our numerics do not allow an exact statement, however, figure 2
suggests that in both cases the magnetization approaches zero for large values
of $\tilde{\mu}$. Finally, the values for the dimensionless magnetization
agree for both cases of $\gamma$ at zero chemical potential which is an
important cross-check for our numerics. This is because at zero chemical
potential our solutions reduce to magnetic black branes, which can be shown to
receive no contribution from the Chern-Simons term, neither through the on-
shell action, nor through the equations of motion.

Figure 1: The dimensionless pressure $p/T^{4}$ as a function of the magnetic
field $\tilde{B}=B/T^{2}$ for $\gamma=2/\sqrt{3}$ and three different values
of $\tilde{\mu}$, namely $\tilde{\mu}=0$ (red), $\tilde{\mu}=4$ (blue),
$\tilde{\mu}=10$ (purple). Inset: Difference between $p$ and $\langle
T_{zz}\rangle$ which serves as a check of the numerics.

Figure 2: The dimensionless magnetization $(p/T^{4})_{,\tilde{B}}$ as a
function of $\tilde{\mu}=\mu/T$ for $\gamma=2/\sqrt{3}$ (left figure) and
$\gamma=0$ (right figure). The different curves correspond to magnetic fields
$B/T^{2}=\\{0.05,12.5,30\\}$ (red, blue, purple).

Next, we determine the thermodynamic transport coefficients $M_{i}$ with
$i=1,\dots,5$ in this holographic model using the Kubo formulas (30) and (32).
From the correlation functions within the Kubo formulas using dimensional
analysis we then read off that $M_{1,3,5}$ have dimension of temperature (or
chemical potential), while $M_{2}$ has dimension of inverse temperature.
Finally, $M_{4}$ is dimensionless.

#### Magneto-vortical susceptibility $M_{5}$

First, we study the thermodynamic coefficient $M_{5}$ which is also known as
magneto-vortical susceptibility and has energy dimension one. We consider the
dimensionless combination $M_{5}/T$. We numerically confirm that in both cases
for $\gamma$, the magneto-vortical susceptibility $M_{5}/T$ is non-vanishing
for nonzero magnetic field and/or nonzero chemical potential. Moreover, as
expected and as evident from Figure 3, $M_{5}/T$ is an anti-symmetric function
of the chemical potential $\tilde{\mu}$.

Figure 3: The dimensionless magneto-vortical susceptibility $M_{5}/T$ as a
function of the chemical potential $\mu/T$ for $\gamma=2/\sqrt{3}$ (left
subfigure) and $\gamma=0$ (right subfigure). The different curves correspond
to magnetic fields $B/T^{2}=\\{0.05,12.5,30\\}$ (red, blue, purple).

Note that the left and right panel of Figure 3 correspond to the case
$\gamma=2/\sqrt{3}$ (left panel) and to the case $\gamma=0$ (right panel),
respectively. We first discuss the case $\gamma=0.$ For vanishing magnetic
field $\tilde{B}=0$ we find a linear relation between $M_{5}/T$ and
$\tilde{\mu}$ (at least for the values of the axial chemical potential
displayed) of the form $M_{5}/T=-\tilde{\mu}/2$ which is in agreement with
Bu:2019qmd . However for finite magnetic fields, $M_{5}/T$ is no longer
directly proportional to $\mu/T$ as evident from the inset in the lower left
corner. In fact, we numerically find that

$\frac{\partial(M_{5}/T)}{\partial\tilde{\mu}}\Big{|}_{\tilde{\mu}=0}=-\frac{1}{2}+c\tilde{B}^{2}$
(138)

where $c$ is a positive constant. If we evaluate
$\partial(M_{5}/T)/\partial\tilde{\mu}$ at non-vanishing (but small) chemical
potential $\tilde{\mu}$ there are additional corrections to the right hand
side of eq. (138).

In case of $\gamma=2/\sqrt{3}$ the thermodynamic coefficient $M_{5}/T$
transitions through zero at a finite value of $\tilde{\mu}$ as opposed to the
case $\gamma=0$ where $M_{5}/T$ is decreasing monotonically with increasing
chemical potential. In particular, for $\tilde{B}=0$, the relation
$M_{5}/T=-\tilde{\mu}\,T/2$ does not longer hold for finite values of the
chemical potential $\tilde{\mu}$. This can be seen from the inset in the lower
left corner of the right panel of figure 3. For finite but small magnetic
fields, there will be again corrections To be more precise we numerically
verified the relation (138) for sufficiently small $\tilde{B}$. However, the
constant $c$ is negative in contrast to the case in which $\gamma=0$.

#### Thermodynamic coefficient $M_{2}$

Let us turn to the thermodynamic coefficient $M_{2}$ which has inverse energy
dimension, hence we consider the dimensionless combination $M_{2}\cdot T$. We
here determine this transport coefficient for the first time in holography. In
fact, $M_{2}$ vanishes in the holographic Einstein-Maxwell model, i.e. for the
case $\gamma=0$. Hence, in figure 4 we only present results for
$\gamma=2/\sqrt{3}$.

Figure 4: The dimensionless thermodynamic susceptibility $M_{2}T$ as a
function of the chemical potential $\mu/T$ for $\gamma=2/\sqrt{3}$. The
different curves correspond to magnetic fields $B/T^{2}=\\{0.05,12.5,30\\}$
(red, blue, purple).

As expected, $M_{2}T$ is an odd function of the chemical potential
$\tilde{\mu}.$ For small chemical potential $\tilde{\mu}<0.25$ we
approximately find a linear proportionality between $M_{2}T$ and $\tilde{\mu}$
as evident from the inset in the lower right corner of figure 4, implying
$M_{2}T=a\,\tilde{\mu}$, where the coefficient $a$ depends on $\tilde{B}$. In
fact, the coefficient $a$ is given by $a\approx 0.024$ for vanishing magnetic
field which is not related to the chiral anomaly coefficient $C=-\gamma$ in an
obvious way. Finally, the value of $M_{2}T$ seems to not be sensitive to
$\tilde{B}$ for chemical potentials of order $\tilde{\mu}=7$ or larger – at
least for reasonably small magnetic fields $\tilde{B}$ displayed in figure 4.

#### Remaining thermodynamic coefficient $M_{1},M_{3}$ and $M_{4}$

Next, we turn our attention to the remaining thermodynamic transport
coefficients $M_{1},M_{3}$ and $M_{4}.$ As discussed in the section on
hydrodynamics there are no Kubo formulas available so far which give us
directly one of the thermodynamic transport coefficients $M_{i}$ with
$i=1,3,4.$ However, we can evaluate the Kubo relations (32) to numerically
determine $M_{4,\tilde{\mu}},$ $(M_{1}/T)_{,\tilde{\mu}}$ and
$(M_{3}/T)_{,\tilde{\mu}}$ i.e. the derivative of the dimensionless
thermodynamic coefficient $M_{i}$ with respect to the chemical potential
$\tilde{\mu}.$ It turns out that all these three derivatives vanish
numerically. Following the logic of section 2.1.4 we conclude that
$M_{4}(\tilde{\mu},\tilde{B})=0$ in our holographic model. A similar
conclusion cannot be made for the thermodynamic transport coefficients $M_{1}$
and $M_{3}$. $M_{3}$ is zero due to conformal invariance of theory.

###  3.3 Results: Dissipationless transport coefficients

In this section we determine the dissipationless transport coefficients in our
holographic model which are the Hall conductivity $\tilde{\sigma}_{\perp}$,
the Hall viscosities $\tilde{\eta}_{\perp}$ and $\tilde{\eta}_{\parallel}$ as
well as the coefficient $c_{10}$ (or alternatively $c_{17}$) which may be
interpreted as a shear-induced Hall conductivity.

We first turn our attention to the Hall viscosity $\tilde{\eta}_{\perp}$ which
may be computed by the Kubo formula (56f). We find that this transport
coefficient vanishes in our holographic model. In fact this is obvious from
the gravity side since the relevant fluctuations of the metric, namely
$h_{xy}$ and $h_{xx}-h_{yy}$, decouple.

#### Hall conductivity $\tilde{\sigma}_{\perp}$

Next, we determine numerically the Hall conductivity $\tilde{\sigma}_{\perp}$.
To do so we first employ the Kubo formulas (55) for $\rho_{\perp}$ and
$\tilde{\rho}_{\perp}$ and then invert the resistivity matrix to obtain the
conductivity matrix which contains the Hall conductivity
$\tilde{\sigma}_{\perp}$. The Hall conductivity $\tilde{\sigma}_{\perp}$ has
dimension of temperature and hence we consider the dimensionless Hall
conductivity $\tilde{\sigma}_{\perp}/T$. Its behaviour for fixed chemical
potential as a function of the magnetic field is displayed in figures 5 and 6.
The Hall conductivity is nonzero even for vanishing chiral anomaly coefficient
$C=-\gamma$ as evident from the right panels of those figures. Moreover, the
Hall conductivity is only nonzero for nonzero chemical potential and nonzero
magnetic field.

Figure 5: The dimensionless Hall conductivity $\tilde{\sigma}_{\perp}/T$ as a
function of a small magnetic field $\tilde{B}$. The left panel shows the case
$\gamma=2/\sqrt{3}$, while the right panel displays $\gamma=0$. The different
curves correspond to chemical potentials
$\tilde{\mu}\in\\{0.016,0.253,1.224,2.078,3.404,5.105,6.521,7.5\\}$. Note that
the specific values of $\tilde{\mu}$ are part of a Chebychev grid in
$\tilde{\mu}$ needed to compute the thermodynamic derivatives to high
precision. For further reference see appendix B.4.

We first focus on the behavior of the dimensionless Hall conductivity as a
function of the magnetic field for fixed chemical potential. As evident from
figure 5 the dimensionless Hall conductivity $\tilde{\sigma}_{\perp}/T$ is
linear in $\tilde{B}$ for small magnetic fields $\tilde{B}$ and for fixed
$\tilde{\mu}$, i.e. $\tilde{\sigma}_{\perp}/T\approx
a(\tilde{\mu})\,\tilde{B}.$ The proportionality constant $a(\tilde{\mu})$
displays an interesting behavior as a function of $\tilde{\mu}$ for both
values of the chiral anomaly coefficient $C=-\gamma=0$ and
$\gamma=2/\sqrt{3}$: first it monotonically increases with increasing chemical
potential $\tilde{\mu}$, then turns around and decreases. Finally, the
proportionality constant turns negative for even larger values of
$\tilde{\mu}$.

Figure 6: The dimensionless Hall conductivity $\tilde{\sigma}_{\perp}/T$ as a
function of a large dimensionless magnetic field $\tilde{B}$. The left panel
shows the case $\gamma=2/\sqrt{3}$, while the right panel displays $\gamma=0$.

In addition, the behavior of the dimensionless Hall conductivity for large
values of the magnetic field $\tilde{B}$ at fixed chemical potential
$\tilde{\mu}$ is displayed in figure 6. The dimensionless Hall conductivity
approaches a $\tilde{\mu}$-dependent value for large magnetic fields.
Moreover, we may investigate the ratio $\tilde{\sigma}_{\perp}/\sigma_{\perp}$
of the Hall conductivity and the perpendicular conductivity $\sigma_{\perp}$
(the latter one will be discussed in the next subsection) as a function of the
magnetic field. For fixed chemical potential $\tilde{\mu},$ the ratio
$\tilde{\sigma}_{\perp}/\sigma_{\perp}$ is directly proportional to the
magnetic field. This linear relationship persists even for very large values
of the magnetic field. The proportionality constant depends on $\tilde{\mu}$
and shows as a function of $\tilde{\mu}$ a behavior similar to that of the
proportionality constant $a(\tilde{\mu})$ of the previous paragraph.
Specifically, the similarities are that both proportionality constants vanish
at $\tilde{\mu}=0$, both increase monotonically with increasing
$\tilde{\mu}>0$, reach a maximum value, then decrease monotonically and then
turn negative within the range covered by our data.

#### Hall viscosity $\tilde{\eta}_{\parallel}$ and shear-induced Hall
conductivity $c_{10}$

Finally, we investigate the Hall viscosity $\tilde{\eta}_{\parallel}$ and the
coefficient $c_{10}$ which may be interpreted as a shear-induced Hall
conductivity. Note that both transport coefficients turn out to be zero in
case of vanishing chiral anomaly and hence these are novel transport
coefficients which are determined here for the first time within a holographic
model.

Figure 7: The dissipationless transport coefficients
$\tilde{\eta}_{\parallel}/T^{3}$ and $c_{10}/T^{2}$ in presence of the chiral
anomaly $\gamma=2/\sqrt{3}$. Without chiral anomaly, both transport
coefficients are zero. The green curve with $\tilde{\mu}=3.404$ is hidden
under the blue curve.

Figures 7 and 8 display dimensionless versions of the Hall viscosity
$\tilde{\eta}_{\parallel}$ and of $c_{10}$ as a function of the magnetic field
for fixed chemical potential. Note that the Hall viscosity
$\tilde{\eta}_{\parallel}$ has units of temperature $T^{3}$, and hence
$\tilde{\eta}_{\parallel}/T^{3}$ is dimensionless.515151We may instead plot
the dimensionless quantity $\tilde{\eta}_{\parallel}/s$ where $s$ is the
entropy density instead as it is usually done for viscosities. Moreover,
$c_{10}$ has the same units as $T^{2}$ implying that $c_{10}/T^{2}$ is
dimensionless.

As evident from figures 7 and 8 both dimensionless quantities
$\tilde{\eta}_{\parallel}/s$ and $c_{10}/T^{2}$ are only nonzero in the
presence of both a non-vanishing magnetic field and a non-vanishing chemical
potential. Moreover, note that both quantities are also positive (at least for
the magnetic fields and chemical potentials investigated in this paper) even
though these quantities are not constrained by the entropy positivity argument
from hydrodynamics.

We first investigate the behaviour of the Hall viscosity and the novel shear-
induced Hall conductivity for small magnetic fields $\tilde{B}$. As indicated
in figure 7 the dimensionless quantities $\tilde{\eta}_{\parallel}/T^{3}$ and
$c_{10}/T^{2}$ show the following scaling laws with the magnetic field for
fixed chemical potential:

$\tilde{\eta}_{\parallel}/T^{3}\sim\tilde{B}^{3}\,,\qquad\textrm{and}\qquad
c_{10}/T^{2}\sim\tilde{B}^{2}\,.$ (139)

We defer the plots with the fit curves superimposed, figure 16, to the
appendix. The behavior of the corresponding $\tilde{\mu}$-dependent
proportionality constants are not very illuminating; both proportionality
constants monotonically increase for small chemical potential $\tilde{\mu}$,
while they monotonically decrease for larger values of $\tilde{\mu}$.

Figure 8: The dissipationless transport coefficients
$\tilde{\eta}_{\parallel}/T^{3}$ and $c_{10}/T^{2}$ in presence of the chiral
anomaly $\gamma=2/\sqrt{3}$. Without chiral anomaly, both transport
coefficients are zero.

Figure 8 displays the dissipationless Hall viscosity and shear-induced Hall
conductivity for fixed chemical potential and large magnetic fields. In
particular, we find that the dissipationless Hall viscosity is linear in
$\tilde{B}$ as shown in the left panel of figure 8. In fact, we find that the
proportionality constant between $\tilde{\eta}_{\parallel}/T^{3}$ and
$\tilde{B}$ is linear in $\tilde{\mu}$, hence giving us the following
universal result for the holographic model considered here:

$\frac{\tilde{\eta}_{\parallel}}{T^{3}}\approx
0.305\,\tilde{\mu}\,\tilde{B}\,.$ (140)

Within the full parameter range covered by our numerical data, the above
approximation is valid for sufficiently large $\tilde{\mu}\tilde{B}$. The
shear-induced Hall conductivity for large magnetic fields $\tilde{B}$ is shown
in the right panel of figure 8. It is also tempting to speculate that the
novel dimensionless shear-induced Hall conductivity, $c_{10}/T^{2}$,
approaches a value for large magnetic fields which is independent of the
chemical potential.

###  3.4 Results: Dissipative hydrodynamic transport coefficients

In this section we investigate the dissipative transport coefficients in our
holographic model which may be grouped into

  * •

components of the shear viscosity tensor, namely $\eta_{\perp}$,
$\eta_{\parallel}$, $\eta_{1}$ and $\eta_{2}$, as well as the components of
the bulk viscosity tensor, namely $\zeta_{1}$ and $\zeta_{2}$,

  * •

dissipative components of the conductivity tensor, namely the longitudinal and
perpendicular conductivities $\sigma_{\parallel}$ and $\sigma_{\perp}$,

  * •

novel transport coefficients $c_{4}$ and $c_{5}$, as well as $c_{8}$ which in
our model are only nonzero in the presence of a nonzero chiral anomaly
coefficient $C=-\gamma\neq 0$, magnetic field $B\neq 0$, and chemical
potential $\mu\neq 0$. (In our model $c_{3}=0$ for any values of
$\gamma,\,B,\,\mu$.)

#### Viscosities

We first focus on the various components of the shear viscosity tensor. The
perpendicular shear viscosity $\eta_{\perp}$ satisfies
$\eta_{\perp}/s=1/(4\pi)$. Note that both $\eta_{\perp}/T^{2}$ and $s/T^{2}$
change as functions of the magnetic field. Remarkably, their functional
dependence on $B$ is identical and cancels in the ratio $\eta_{\perp}/s$. This
can be shown analytically even in the presence of the chiral anomaly as well
as in the presence of $B$ and $\mu$, following Critelli:2014kra .

Figure 9:  Parallel shear viscosity as function of $\tilde{B}$ for fixed
values of $\tilde{\mu}$. The dashed red line indicates the well-known result
for $\eta_{\perp}/s$, namely $1/(4\pi).$ Left: $\gamma=2/\sqrt{3}$. For this
supersymmetric value of $\gamma$, the ratio of parallel shear viscosity
increases with $\tilde{B}$ for $\tilde{\mu}\geq 5$ within the range of
parameters displayed here. See figure 10 for a refined statement. Right:
$\gamma=0$.

The parallel shear viscosity $\eta_{\parallel}/s$ depends non-trivially on the
magnetic field and the chemical potential. The dimensionless parallel shear
viscosity $\eta_{\parallel}/s$ is shown in figure 9 for small values of the
magnetic field $\tilde{B}$ and in figure 10 for medium and large magnetic
fields. It is evident from both figures that $\eta_{\parallel}$ is positive as
implied by the hydrodynamic stability analysis. Moreover, for vanishing
magnetic field, $\eta_{\parallel}/s$ takes the value $1/(4\pi)$ as expected.
Let us first investigate the behavior for small magnetic fields $\tilde{B}$
displayed in figure 9. The dimensionless ratio $\eta_{\parallel}/s$ deviates
from $1/(4\pi)$ quadratically, i.e.

$\frac{\eta_{\parallel}}{s}\sim\frac{1}{4\pi}-c(\tilde{\mu})\tilde{B}^{2}\,,$
(141)

where $c(\tilde{\mu})$ is a model-dependent coefficient. In fact,
$c(\tilde{\mu})$ is positive for vanishing chiral anomaly and monotonically
decreases with increasing $\tilde{\mu}$; in contrast, for $\gamma=2/\sqrt{3}$
the coefficient $c(\tilde{\mu})$ turns negative for $\tilde{\mu}>5$. In other
words, for those values of $\tilde{\mu}$ the ratio $\eta_{\parallel}/s$
increases for small and intermediate magnetic fields $\tilde{B}$, as evident
from the right panel of figure 10. However, for large magnetic fields
$\tilde{B}$ the dimensionless parallel shear viscosity $\eta_{\parallel}/s$
decreases monotonically to zero. This is true for both values of the chiral
anomaly.

Figure 10: Ratio of parallel shear viscosity and entropy density as function
of $\tilde{B}$ with $\gamma=2/\sqrt{3}$; in the left plot the color scheme
distinct from right since we use a different set of $\tilde{\mu}$. We observe
that $\eta_{\|}/s$ increases initially for $\tilde{\mu}\geq 5$. This behavior
is only present with the chiral anomaly. Eventually, all curves tend to zero
as displayed in the right figure.

We also investigated whether we can predict the transport coefficient
$\eta_{\parallel}/s$ by using horizon data of the charge magnetic black brane.
In fact, extending the analysis of Critelli:2014kra  we were able to show that
this is indeed the case for vanishing chiral anomaly coefficient and in the
presence of a magnetic field and a chemical potential. However, in the case of
non-vanishing chiral anomaly coefficient we were not able to find an horizon
formula for $\eta_{\parallel}/s$.

Figure 11: Dimensionless ratio of bulk viscosity $\eta_{2}$ and entropy
density for fixed chemical potentials $\tilde{\mu}$. The red dashed line
indicates the value for $\eta_{2}/s$ at zero magnetic field, namely
$3/(8\pi).$ Note the behavior similar to $\eta_{\parallel}/s$, namely that for
$\gamma=2/\sqrt{3}$ there is a distinctly different behavior from the case
$\gamma=0$: for the largest three displayed chemical potentials, i.e. for
$\tilde{\mu}\geq 5$, the value of the transport coefficient $\eta_{2}/s$
increases with $\tilde{B}$, while it always decreases for $\gamma=0$.

Next, we turn to the dissipative components $\eta_{1}$ and $\eta_{1}$ of the
shear viscosity tensor. It suffices to display only $\eta_{2}$ since the bulk
viscosity $\zeta_{2}$ vanishes in our holographic model. Hence, the transport
coefficients $\eta_{1}$ and $\eta_{2}$ are related by
$\eta_{2}=-\frac{1}{3}\eta_{1}$ due to the Onsager relation (65). The behavior
of the dimensionless ratio $\eta_{2}/s$ as a function of the magnetic field
$\tilde{B}$ is depicted in figure 11.525252Note that we do not display the
transport coefficient $\eta_{2}/s$ for large magnetic fields since we expect
contributions from the thermodynamic transport coefficients $M_{1},M_{3}$ and
$M_{4}$ which we cannot determine in our holographic model. As expected, the
ratio $\eta_{2}/s=3/(8\pi)$ for vanishing magnetic field and quadratically
deviates from that value for small magnetic fields up to $\tilde{B}=1$. We
obtain scaling law similar to the one for $\eta_{\parallel}/s,$ see equation
(141). The analogous coefficient $c(\tilde{\mu})$ qualitatively shows the same
behavior as in the case of $\eta_{\parallel}/s:$ it monotonically decreases
for $\gamma=0$ but stays positive; in contrast for $\gamma=2/\sqrt{3}$ it is
not monotonic for the whole range of chemical potentials $\tilde{\mu}$ and
even turns negative for intermediate values of $\tilde{\mu}.$

Finally, we numerically checked that the bulk viscosities $\zeta_{1}$ and
$\zeta_{2}$ vanish in our model.

#### Conductivities

Next, we investigate the dissipative components of the conductivity tensor,
namely the longitudinal conductivity $\sigma_{\parallel}$ and the
perpendicular component $\sigma_{\perp}$. Both quantities have units of
temperature, implying that $\sigma_{\perp}/T$ and $\sigma_{\parallel}/T$ are
dimensionless. Both dimensionless quantities are almost insensitive to the
magnetic field and to $\gamma$. In fact, for both values of the chiral anomaly
coefficient and within the range covered by our numerical data, that is
$\tilde{\mu}$ between $0$ and $7.5$ $\sigma_{\parallel}/T$ has a relative
variation of order $10^{-3}$ for $\tilde{B}<1.$

According to our numerical data, $\sigma_{\perp}/\sigma_{||}$ changes only
slightly (at most by $10^{-2}$) in the range $\tilde{B}=0.0,...,1.0$ where our
Kubo formula is valid; see derivation in section 2. Due to systematic and
numerical errors for this particular ratio it is not possible to consider this
change significant. In figure 12 we depict the ratio
$\sigma_{\perp}/\sigma_{\parallel}$ which is a measure of the anisotropy of
the system. The left panel shows the case $\gamma=2/\sqrt{3}$. Note that in
this case the ratio is always larger than one. In contrast, in the case of
vanishing chiral anomaly which is shown in the right panel of figure 12 the
ratio may be also less than one, depending on the value of the chemical
potential. In both cases, i.e. for $\gamma=0$ and for $\gamma=2/\sqrt{3}$ the
ratio $\sigma_{\perp}/\sigma_{\parallel}$ deviates from one quadratically in
$\tilde{B},$ i.e.

$\frac{\sigma_{\perp}}{\sigma_{\parallel}}=1+\tilde{c}(\tilde{\mu})\,\tilde{B}^{2}\,,$
(142)

which holds up to magnetic fields of order $\tilde{B}=1.$

Figure 12:  The ratio of longitudinal and perpendicular conductivity
$\sigma_{\perp}/\sigma_{||}$ as function of $\tilde{\mu}$ at fixed $\tilde{B}$
for $\gamma=2/\sqrt{3}$ (left) and $\gamma=0$ (right). Each color corresponds
to a fixed value of $\tilde{B}$ given by $\tilde{B}=\\{1/30,1/3,2/3,1\\}$
(red, blue, purple, dark green).

#### Novel transport coefficients $c_{3}$, $c_{4}$ and $c_{5}$ as well as
$c_{8}$

Figure 13: The transport coefficient $c_{4}/T^{3}$ in presence of the chiral
anomaly $\gamma=2/\sqrt{3}$. Without chiral anomaly or for vanishing chemical
potential, the transport coefficients is zero.

The novel transport coefficients $c_{4}$, $c_{5}$ as well as $c_{8}$ are
nonzero in our holographic model. However, they vanish if either the anomaly
is absent, i.e. if $\gamma=0$, or if the magnetic field vanishes, or if the
chemical potential vanishes. In other words, our holographic calculation is
the first to reveal these transport coefficients, because they need $\mu\neq
0$, $B\neq 0$, and the chiral anomaly coefficient $C=-\gamma\neq 0$
simultaneously. Our numerical data shows that $c_{3}$ vanishes in our model.
This is expected due to the conformal invariance of theory. The vanishing of
$c_{3}$ implies that $c_{4}=-c_{5}/3$ due to the Onsager relation (65). The
two relations $c_{4}=-\frac{1}{3}c_{5}$ and $\eta_{2}=-\frac{1}{3}\eta_{1}$
explain why the bound $\det M\geq 0$ with $M$ given by eq. (71) is saturated
(within numerical accuracy $\det M\approx 10^{-6}$). The Onsager relation for
$c_{4}$ and $c_{5}$ is satisfied up to errors of order $10^{-16}$ at worst
($\tilde{B}=1$, and $\tilde{\mu}=2.078$ corresponding to the dark green
curve). We hence plot only the dimensionless quantity $c_{4}/T^{3}$, see
figure 13. It turns out that $c_{4}$ is negative which in turn implies that
$c_{5}$ is positive. Furthermore, we find $c_{4}/T^{3}\sim
d(\tilde{\mu})\tilde{B}$, where $d(\tilde{\mu})$ is negative. Note that
$d(\tilde{\mu})$ first decreases as a function of the chemical potential
$\tilde{\mu}$ and then increases again. The dimensionless quantity
$c_{8}/T^{2}$ as a function of $\tilde{B}$ is depicted in figure 14. Note that
$c_{8}/T^{2}$ is proportional to $\tilde{B}$ for small $\tilde{B}$, i.e. the
slope changes its sign. Finally, as observed in the other transport
coefficients above, $c_{8}/T^{2}$ seems to go to zero for large magnetic
fields.

We make the observation that many of the dissipative transport coefficients
vanish at large magnetic fields. This may be interpreted as dissipation being
suppressed at large magnetic fields, as indicated for example by $\eta_{||}$,
see figure 11. A subset of the current authors has previously observed this
behavior in the sound attenuation coefficient and the shear diffusion
coefficient Ammon:2016fru ; Ammon:2017ded  computed in the holographic model
we also consider here.

Figure 14: The transport coefficient $c_{8}/T^{3}$ for small and large
magnetic fields in presence of the chiral anomaly $\gamma=2/\sqrt{3}$. Without
chiral anomaly or for vanishing chemical potential, the transport coefficients
is zero.

##  4 Discussion

In this paper we have used effective field theory methods in order to
construct a hydrodynamic description of chiral charged relativistic fluids
subject to a strong external magnetic field. In particular, hydrodynamic
constitutive relations are generated for the energy-momentum tensor,
consistent, and covariant (axial) charge currents. All the independent
transport coefficients appearing in the constitutive equations are summarized
and classified in tables 3 (thermodynamic), 4 (hydrodynamic, non-dissipative),
and 5 (hydrodynamic, dissipative). A discussion and physical interpretation of
these transport coefficients is provided in section 2.5.1. Instead of
repeating the discussions already given above, in the following we highlight
novel phenomena and interesting observations. Novel coefficients and transport
effects arise due to the combination of magnetic field, axial charge, and
chiral anomaly. These three quantities break time-reversal symmetry,
rotational symmetry from $O(3)$ down to $O(2)$, parity symmetry, and chiral
symmetry. This symmetry breaking has a drastic impact on the effective field
theory description, leading to odd transport effects.

Remarkably, one novel non-dissipative hydrodynamic transport coefficient is
found: the shear-induced Hall conductivity $c_{10}$. This transport effect
falls into the same category as the known Hall viscosity
$\tilde{\sigma}_{\perp}$ and the transverse Hall viscosity
$\tilde{\eta}_{\perp}$. This is because all three of them describe time-
dependent but dissipationless transport effects. There exists no longitudinal
Hall conductivity.

In addition, three novel dissipative transport coefficients are found: the
shear-induced conductivity $c_{8}$, as well as the expansion-induced
longitudinal conductivities $c_{4}$ and $c_{5}$. Even the well-known charge
conductivity and resistivity highlight the drastic changes to the fluid
description. In standard hydrodynamic Kubo formulas the charge conductivity is
the transport coefficient appearing in the lowest order of the current-current
correlator. However, here the lowest transport coefficient is the charge
resistivity, see e.g. eq. (54b). The four novel coefficients, $c_{4}$,
$c_{5}$, $c_{8}$ and $c_{10}$ can be nonzero in a strong magnetic field if
either a chiral anomaly or an axial chemical potential is breaking the parity
symmetry.

Kubo relations are derived for 8 independent thermodynamic ($M_{2},\,M_{5}$,
$\xi$, $\xi_{B}$, $\xi_{T}$, $M_{1}$, $M_{3}$, $M_{4}$), for 4 independent
non-dissipative hydrodynamic ($\tilde{\eta}_{\perp}$, $\tilde{\eta}_{||}$,
$c_{10}$, $\tilde{\sigma}_{\perp}$), and for 10 independent dissipative
hydrodynamic ($\eta_{\perp}$, $\eta_{||}$, $c_{8}$, $\sigma_{\perp}$,
$\sigma_{||}$, $\eta_{1}$, $\eta_{2}$, $\zeta_{1}$, $c_{4}$, $c_{5}$)
transport coefficients. For the explicit formulas refer to the equations
listed in the third column in tables 3 (thermodynamic), 4 (hydrodynamic, non-
dissipative), and 5 (hydrodynamic, dissipative), respectively. In order to
derive the thermodynamic transport effects, a field theory generating
functional for time-independent $n$-point functions was constructed, see eq.
(16). Five Onsager relations, (57) and (65), are found and can be used for
cross-checks in explicit computations of the 19 possible hydrodynamic
transport coefficients. Kubo relations for the dependent hydrodynamic
transport coefficients ($c_{3}$, $c_{14}$, $c_{15}$, $c_{17}$, $\zeta_{2}$) in
a parity-violating microscopic theory (one with a chiral anomaly), are
provided in eq. (64a), (62c), and (62d). The coefficients
$c_{3},\,c_{4},\,c_{5}$ have counterparts in a theory which does not feature
an anomaly but which breaks parity through the axial chemical potential, those
are given in (59) and (58).

Most prominently, our work discusses strong magnetic fields. The magnetic
field breaks rotation symmetry leading to an anisotropic equilibrium state (in
the rest frame of the fluid). Let us summarize here which transport effects
newly arise at strong magnetic fields of order $\mathcal{O}(\partial^{0})$,
versus the effects appearing already in states with weak magnetic fields of
order $\mathcal{O}(\partial)$. Consider the list of all transport coefficients
provided in tables 3, 4, and 5. First, some of the first order transport
coefficients each split into two distinct coefficients, one for transport
perpendicular and the other parallel to the magnetic field:
$\eta_{||},\,\eta_{\perp},\,\tilde{\eta}_{||},\,\tilde{\eta}_{\perp},\,\sigma_{||},\,\sigma_{\perp}$
(or equivalently $\rho_{||},\,\rho_{\perp}$), $\tilde{\sigma}_{||}$, and
$\tilde{\sigma}_{\perp}$. At weak magnetic field all
$\alpha_{||}=\alpha_{\perp}$, where $\alpha$ stands for any transport
coefficient within this group, and all tilded coefficients vanish. Second, at
strong magnetic field new tensor structures in the constitutive relations give
rise to $\eta_{1},\,\eta_{2},\,\zeta_{1},\,\zeta_{2}$. At weak magnetic fields
$2/3\,\eta_{2}=-2\eta_{1}=\eta$, $\zeta_{1}=\zeta$, and $\zeta_{2}=0$. Third,
two of the chiral conductivities, namely $\xi_{B},\,\xi_{TB}$, are promoted
from first order in derivative effects at weak magnetic fields to zeroth order
effects at strong magnetic fields. Fourth, all of the $M_{n}$ vanish at weak
magnetic field, as seen from the $s_{n}$ in table 1. Finally, all $c_{i}$ with
$i\geq 3$ vanish at weak magnetic field, as can be deduced from eq. (42).

Interestingly, we find three incarnations of the Nernst effect. Strikingly, we
confirm a previous claim Chernodub:2017jcp , that the standard Nernst
coefficient, see (112), is proportional to the conformal anomaly, which in
turn is then related to the magnetic susceptibility $\chi_{B}$. In addition,
we find a magnetic version of the Nernst effect, see (93), with the magnetic
Nernst coefficient related to the perpendicular magnetic vorticity
susceptibility $M_{2}$, and a momentum Nernst effect, see (111), with momentum
Nernst coefficient related to the magneto-vortical susceptibility $M_{5}$.

As a proof of existence and consistency of this hydrodynamic description of
chiral fluids we provide an explicit computation in a holographic model. This
model is the well-known $\mathcal{N}=4$ Super-Yang-Mills (SYM) theory
minimally coupled to an external axial $U(1)_{A}$ gauge field. In order to
probe a charged fluid in a magnetic field, we consider perturbations around
charged magnetic black brane solution. A Chern-Simons term yields the desired
anomalous (chiral) charge current. We find values for 25 of the transport
coefficients, for which we verify the 5 Onsager relations (57) and (65). For
the remaining two, $M_{1}$ and $M_{3}$ we find only their derivatives as the
hydrodynamic theory gives only Kubo relations for those derivatives. However,
as stated in section 2, $M_{3}=0$ in a conformal field theory. Of the 25
computed coefficients 20 are nonzero. In our model the gravitational anomaly
is absent, hence all effects associated with $c_{1}$, see eq. (17), vanish in
this model. The magneto-acceleration susceptibility $M_{3}$ vanishes due to
conformal invariance. Also the transverse Hall viscosity
$\tilde{\eta}_{\perp}$ vanishes, however, without any reason obvious to us.

Due to conformal invariance the bulk viscosities $\zeta_{1}$, and $\zeta_{2}$
vanish. In addition, $c_{3}$ is zero.

As discussed in the introduction, we have restricted our attention to an axial
$U(1)_{A}$ symmetry in this work. In Nature, however, an interplay between
axial and vector symmetries is generic. In the standard model of particle
physics, for example, currents of left- and right-handed fermions may be
combined into axial and vector currents. Thus, it is natural to extend the
hydrodynamic description to the combination, $U(1)_{V}\times U(1)_{A}$. As
usual, the vector and axial currents can be related to the current of left-
and right-handed particles by $J^{\mu}_{V/A}=(J^{\mu}_{L}\pm
J^{\mu}_{R})/2$.535353It would also be interesting to have two U(1) gauge
fields in the gravity theory Jimenez-Alba:2014pea , i.e. introducing an axial
and a conserved vector current in the dual field theory. This is relevant for
testing some predictions for chiral magnetic waves and for Weyl semimetals and
their surface states, see Landsteiner:2015lsa ; Landsteiner:2015pdh ;
Copetti:2016ewq ; Ammon:2016mwa ; Grignani:2016wyz .  A different combination
of currents may be interesting in the context of the $\Lambda$-hyperon
polarization effect recently discovered in heavy ion collisions STAR:2017ckg :
one may consider the helicity current instead of the axial current
Ambrus:2019khr ; Ambrus:2020oiw . This would require disentangling the axial
current carried by particles from that carried by anti-particles, which is the
definition of the helicity current.

A next logical step is to restrict the vector gauge fields to satisfy
Maxwell’s equations and allow them to interact dynamically with the fluid. The
resulting effective field theory may be labelled magnetohydrodynamics.
However, previous constructions excluding anomalies Grozdanov:2016ala ;
Grozdanov:2016tdf ; Kovtun:2018dvd ; Grozdanov:2017kyl  indicate that such a
description goes well beyond what is known as textbook
magnetohydrodynamics.545454Important steps towards incorporating the effects
of chiral anomalies into magnetohydrodynamics have been taken in
Boyarsky:2015faa . Although the setting with dynamical gauge fields
Grozdanov:2017kyl  is distinct from the one we consider here, some of the Kubo
relations associated with energy momentum tensor correlators agree with ours.
In order to understand this observation, recall that the constitutive
equations for hydrodynamics coupled to external electromagnetic fields can be
interpreted as those of hydrodynamics coupled to dynamical gauge fields
Hernandez:2017mch .

Over the past decade progress has been made towards a generating functional
yielding non-equilibrium contributions to the constitutive relations and more
generally to the $n$-point functions Jensen:2012jh ; Crossley:2015evo ;
Haehl:2015pja ; Haehl:2015foa ; Haehl:2016pec ; Haehl:2018lcu ;
Glorioso:2018wxw . This progress will eventually establish hydrodynamics as an
effective field theory. Our work provides a step towards this goal by encoding
the equilibrium response in our generating functional (16). Extending it to
include non-equilibrium transport effects would be accomplishing that goal. In
order to understand the fluid response to time-dependent or spatially
modulated sources, frequency and wave length dependence of the transport
coefficients should be studied, which can be computed with our methods. In the
holographic context frequency- and wavelength-dependent transport coefficients
have been computed before Amado:2011zx ; Hoyos2013 . One may also venture to
extend hydrodynamics in strong electromagnetic fields to descriptions of
fluids far-from-equilibrium. Recent developments applying insights from
resurgence are promising in this regard Romatschke:2017vte ; Heller:2013oxa ;
Heller:2015dha ; Heller:2020anv ; Heller:2020uuy . For an alternate expansion,
linearizing in hydrodynamic fields, see Bu:2016oba ; Bu:2016vum ; Bu:2018drd ;
Bu:2018psl ; Bu:2019mow .

An application of our fluid description is the hydrodynamic modelling of heavy
ion collisions Romatschke:2017ejr , as well as the hydrodynamic description of
astrophysical fluids in strong magnetic fields RezzollaHydroTextbook ;
Joyce:1997uy ; Boyarsky:2011uy ; Tashiro:2012mf ; Brandenburg:2017rcb  (see
Boyarsky:2020ani ; Boyarsky:2020cyk  for recent papers), and the description
of condensed matter materials and cold atom gases subjected to magnetic
fields, see Fujita:2012fp ; An:2020tkn ; Hoyos:2020ywe ; Baggioli:2020edn  for
holographic models in this context. While these systems are subject to a
strong _dynamical_ vector magnetic field, when the backreaction of the fluid
on the electromagnetic field can be neglected, the description in terms of
external magnetic fields as outlined in this paper may be useful. For example
this is the case in (quantum) Hall effect experiments, ultracold atom gases in
strong magnetic field traps, and possibly intermediate time phase of heavy-
ion-collisions where the magnetic field changes from growing to decaying.

Moreover, external axial gauge fields are of importance in the context of Weyl
semimetals. Elastic deformations therein couple to the fermionic low-energy
excitations with different signs, giving rise to effective axial gauge fields
Cortijo_2015 ; Cortijo:2016wnf . Such deformations naturally arise at the
surfaces of Weyl semimetals, inducing localised axial magnetic fields in their
vicinity. In fact, the surface states (so-called Fermi arcs) may be understood
from this perspective in terms of lowest Landau levels generated by those
axial magnetic fields (see e.g. Chernodub:2013kya ; Ammon:2016mwa ).

In quantum chromodynamics (QCD) the chiral magnetic effect and more generally
QCD thermodynamics in the presence of an external magnetic field has been
investigated using lattice simulations Buividovich:2009wi ; Buividovich:2010tn
; Bali:2011qj ; Bali:2012zg ; Bali:2014kia ; Astrakhantsev:2019zkr . In
particular, a result from lattice gauge theory gives motivation to the study
of holographic models as a surrogate for QCD with strong magnetic fields: the
magnetoresponse of $\mathcal{N}=4$ SYM as computed in the same holographic
dual we used in this work has been found to have universal similarities to the
magnetoresponse of QCD Endrodi:2018ikq . The Kubo relations for thermodynamic
transport coefficients and constitutive relations derived in the present paper
facilitate a more detailed comparison of the equilibrium response in QCD and
SYM subject to external magnetic fields.

The time-evolution of (electro)magnetic fields in relativistic heavy-ion-
collisions has also been considered Voronyuk:2011jd ; Inghirami:2019mkc . It
would be interesting to compare the hydrodynamic transport in our holographic
model to the results from these other approaches. Even an analysis of the data
from the other approaches using our Kubo relations may lead to interesting
results in and out of equilibrium.

Especially with regard to condensed matter physics, we predict additional
effects in the Hall response of materials. Hall effects of quantum or
classical nature have been studied in condensed matter physics since Hall’s
discovery in 1879.

For example, Hall viscosity Avron:1995fg ; Avron:1997  is a dissipationless
transport coefficient of relevance to topological states of matter, e.g.
fractional quantum Hall systems, see for example Wen:2017 ; Hoyos:2014pba . In
(2+1) dimensions Hall viscosity in a hydrodynamic context has been found to be
given by the angular momentum density in condensed matter Read:2008rn ;
Read:2010epa , and in holographic systems, e.g. in application to $p$-wave
superfluids Son:2013xra . In fractional quantum Hall systems the Hall
viscosity is quantized and may be used to detect topological phase transitions
Hoyos:2014pba .

However, the Hall viscosities which we find in (3+1) dimensions prepared in an
anisotropic state are novel. We predict two distinct Hall viscosities, one
associated with the plane parallel to a strong magnetic field, the other with
a plane perpendicular to it with generically distinct values. It would be very
interesting to understand how the anomalous Hall effect Nagosa2010  may fit
into our hydrodynamic description. We assume that it would be measured in our
Hall conductivity $\tilde{\sigma}_{\perp}$. Potentially analogous effects may
appear in the other Hall coefficients. Finally, the novel dissipationless
coefficient $c_{10}$ is to be interpreted as a Hall response of a charge
current to the shear of a fluid in a plane in the fluid creates a charge
current perpendicular to that plane. This considerably extends the
possibilities for Hall physics to be studied in the future and promises new
technological applications.

One promising testing ground are Weyl- or Dirac-semimetals Nielsen:1983 ;
Cortijo_2015 ; Cortijo:2016wnf ; Pikulin:2016 ; Taguchi:2016 ; Breunig:2019kjd
; Zhang:2018 . A quantum Hall effective action for the anisotropic Dirac
semimetal is discussed in Hoyos:2020ywe . Experiments with Weyl semimetals
report the observation of chiral transport effects in presence of magnetic
fields 2014arXiv1412.6543L ; 2015arXiv150606577S ; 2015arXiv150407698Z ;
2015arXiv150600924W ; 2015arXiv150603190Y . In these experiments, the relevant
observable is the negative magnetoresistance. However, note that negative
magnetoresistance cannot unambigously be related to the presence of a chiral
anomaly (see e.g. the holographic computations in Baumgartner:2017kme ;
Ammon:2009jt ).555555 It is not yet clear if Weyl- or Dirac-semimetals are
governed by strong correlations. Thus an exploration from the strongly-
interacting perspective is rather interesting. The interplay between external
magnetic fields and a chiral anomaly within holographic models of Weyl-
semimetals is discussed in Landsteiner:2015lsa ; Landsteiner:2015pdh ;
Copetti:2016ewq ; Ammon:2016mwa ; Grignani:2016wyz ; Kharzeev:2016mvi .
Hydrodynamic behavior has been measured most reliably or has been
theoretically argued for in (2+1)-dimensional materials Berdyugin162 ;
Molenkamp:1994 ; Moll1061  where the parity anomaly leads to anomalous
transport effects DiSante:2019zrd ; Tutschku:2020rjq ; Tutschku:2020drw ;
parityAnomalyQHEWuerzburgPRL ; Haldane:1988zza . Our discussion of
(2+1)-dimensional hydrodynamics in section 2.5.2 will help relating our
(3+1)-dimensional hydrodynamic transport effects to these lower dimensional
experiments and theoretical descriptions.

All these examples, taken from fields as different as particle physics and
condensed matter physics, illustrate the importance of a more detailed
understanding of the response of (non)relativistic fluids in the presence of
electromagnetic fields and anomalies, towards which this work is taking a
step.

coefficient | name | Kubo formulas | $\mathcal{C}$ | $\mathcal{P}$ | $\mathcal{T}$
---|---|---|---|---|---
Thermodynamic $\left(\lim\limits_{\mathbf{k}\to 0}\lim\limits_{\omega\to
0}\right)$, non-dissipative
momentum diffusion sector
${M_{2}}$ | perp. magnetic vorticity susceptibility |  $T^{xz}T^{yz}$ (30) | + | - | +
${M_{5}}$ | magneto-vortical susceptibility |  $T^{tx}T^{yz}$ (30,31) | + | - | +
$\xi=\xi_{TB}$ | chiral vortical conductivity |  $J^{x}T^{ty}$ (38,39) | + | + | +
$\xi_{B}$ | chiral magnetic conductivity |  $J^{x}J^{y}$ (38,39) | + | - | +
$\xi_{T}$ | chiral vortical heat conductivity |  $T^{tx}T^{ty}$ (38,39) | + | - | +
scalar sector
${M_{1}}$ | magneto-thermal susceptibility |  $J^{t}T^{xx}$ (32) | + | + | -
${M_{3}}$ | magneto-acceleration susceptibility |  $J^{t}T^{tt}$ (32) | + | + | -
${M_{4}}$ | magneto-electric susceptibility |  $J^{t}J^{t}$ (32) | + | - | -
Table 3:  Independent thermodynamic transport coefficients in a charged chiral
thermal plasma subjected to a strong $U(1)_{A}$ magnetic field. The column
“Kubo formulas” points to the equations containing the relevant Kubo formulas.
In that column, the operator combinations, e.g. $J_{x}T_{ty}$ indicate the
correlation function appearing in the Kubo formula for the spatial momentum to
be aligned with the magnetic field. The susceptibilities $\chi_{11}$,
$\chi_{13}$, $\chi_{33}$ and $\chi_{B}$ are not counted here as transport
coefficients because they are thermodynamic derivatives of pressure. Here,
“perp.” denotes “perpendicular”.  Non-dissipative Hydrodynamic
$\left(\lim\limits_{\omega\to 0}\lim\limits_{\mathbf{k}\to 0}\right)$
---
coefficient | name | Kubo formulas | $\mathcal{C}$ | $\mathcal{P}$ | $\mathcal{T}$
shear sector
${\tilde{\eta}_{\perp}}$ | transverse Hall viscosity | $T^{xy}(T^{xx}-T^{yy})\eqref{eq:tildeEtaPerpKubo}$ | + | - | +
momentum diffusion sector
$\boxed{c_{10}}\propto c_{17}$ | shear-induced Hall cond. |  $T^{tx}T^{xz},T^{tx}T^{yz}$ (61,62a,62b) | + | + | +
${\tilde{\eta}_{||}}$ | parallel Hall viscosity |  $T^{yz}T^{xz}$ (60b) | + | - | +
${\tilde{\sigma}_{\perp}}$ | Hall conductivity |  $J^{x}J^{x}$,$J^{x}J^{y}$ (55,54b,54c) | + | - | +
Table 4:  Independent non-dissipative hydrodynamic transport coefficients in a
charged chiral thermal plasma subjected to a strong $U(1)_{A}$ magnetic field.
Similar to table 3. Note that the Hall conductivity $\tilde{\sigma}_{\perp}$
is dissipationless, however, it is computed from the resistivity matrix
including $\rho$ and $\tilde{\rho}$, which are both dissipative. Here, “cond.”
denotes “conductivity”. Boxed quantities are novel in this paper.
dissipative, hydrodynamic $\left(\lim\limits_{\omega\to
0}\lim\limits_{\mathbf{k}\to 0}\right)$
---
coefficient | name | Kubo formulas | $\mathcal{C}$ | $\mathcal{P}$ | $\mathcal{T}$
shear sector
$\eta_{\perp}$ | perp. shear viscosity |  $T^{xy}T^{xy}$ (2.2.4) | + | + | -
momentum diffusion sector
${\eta_{||}}$ | parallel shear viscosity |  $T^{xz}T^{xz}$ (60a) | + | + | -
$\boxed{c_{8}}\propto c_{15}$ | shear-induced conductivity |  $T^{tx}T^{xz}$, $T^{tx}T^{yz}$ (58) | + | + | +
${\rho_{\perp}}$ | perp. resistivity |  $J^{x}J^{x}$ (55) | + | + | -
$\sigma_{||}$ | long. conductivity |  $J^{z}J^{z}$ (54a) | + | + | -
$\sigma_{\perp}$ | perp. conductivity | $\rho_{ab}\equiv(\sigma^{-1})_{ab}=\rho_{\perp}\delta_{ab}+\tilde{\rho}_{\perp}\,\epsilon_{ab}$ | + | + | -
scalar sector
$\eta_{1}$ | bulk viscosity |  $\mathcal{O}_{1}\mathcal{O}_{1}$ (56c) | + | + | -
$\eta_{2}$ | bulk viscosity |  $\mathcal{O}_{2}\mathcal{O}_{2}$ (56d) | + | + | -
$\zeta_{1}$ | bulk viscosity |  $T^{ij}(T^{xx}+T^{yy})$(56a) | + | + | -
$\zeta_{2}$ | bulk viscosity | $3\zeta_{2}-6\eta_{1}=2\eta_{2}$ | + | + | -
$\boxed{c_{4}}$ | expan.-induced long. cond. |  $J^{x}T^{xx}$ (58) | + | - | -
$\boxed{c_{5}}\propto c_{14}$ | expan.-induced long. cond. |  $J^{z}T^{zz}$ (58) | + | - | -
$c_{3}$ |  | $c_{5}=-3(c_{3}+c_{4})$ | + | - | -
Table 5:  Independent dissipative hydrodynamic transport coefficients in a
charged chiral thermal plasma subjected to a strong $U(1)_{A}$ magnetic field.
Similar to table 4. Abbreviations: perpendicular (perp.), longitudinal
(long.), conductivity (cond.), expansion (expan.). Gray shaded rows display
relation of dependent transport coefficients to the set of independent
transport coefficients. Boxed quantities are novel in this paper.

###### Acknowledgements.

We would like to thank C. Cartwright, A. Jain, P. Kovtun, K. Landsteiner, A.
Shukla, L. Yaffe for helpful discussions, as well K. Landsteiner for detailed
remarks on a draft of the paper. MA is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Grant
No.406235073 within the Heisenberg program. SG is in part supported by the
‘Atracción de Talento’ program (2017-T1/TIC-5258, Comunidad de Madrid) and
through the grants SEV-2016-0597 and PGC2018-095976-B-C21. SG furthermore
gratefully acknowledges financial support by the Fulbright Visiting Scholar
Program which is sponsored by the US Department of State and the German-
American Fulbright Commission and the DAAD. JH is supported in part by the
Natural Sciences and Engineering Research Council of Canada through a
Postgraduate Doctoral Scholarship. This work was supported, in part, by the U.
S. Department of Energy grant DE-SC-0012447. Research at Perimeter Institute
is supported in part by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Colleges and Universities. SG would like to
thank the University of Victoria for their hospitality.

##  Appendix A Details of the hydrodynamic calculation

quantity | ${C}$ | ${P}$ | ${T}$
---|---|---|---
$t$ | + | + | -
$x^{i}$ | + | - | +
$\varrho$ | + | + | +
$T,h_{tt},T^{tt}$ | + | + | +
$\mu_{A},A_{t},J^{t}$ | + | - | +
$\mu_{V},V_{t},J_{V}^{t}$ | - | + | +
$A_{i},J^{i}$ | + | + | -
$V_{i},J_{V}^{i}$ | - | - | -
$A_{\varrho}$ | + | - | -
$V_{\varrho}$ | - | + | -
$u^{i},h_{ti},T^{ti}$ | + | - | -
$h_{ij},T^{ij}$ | + | + | +
$B^{i}$ | + | - | -
$B_{V}^{i}$ | - | + | -
$E^{i}$ | + | + | +
$E_{V}^{i}$ | - | - | +
$dx^{\mu}\wedge dx^{\nu}\wedge dx^{\rho}\wedge dx^{\sigma}\wedge dx^{\kappa}$ | + | - | -
$\int\limits_{i}^{f}A\wedge F\wedge F$ | + | + | +
$\int\limits_{i}^{f}V\wedge F_{V}\wedge F_{V}$ | - | - | +
$u^{t}$ | + | + | +
generating functionals $W_{s},\,W_{cons},\,W_{cov}$ (axial $U(1)_{A}$) | + | + | +
Table 6:  Transformation properties of the following quantities under charge-
parity ${C}$, parity ${P}$, and time-reversal ${T}$: field theory coordinates
$x^{\mu}=t,\,x,\,y,\,z$, field theory axial $U(1)_{A}$ (or vector $U(1)_{V}$)
current $J^{\mu}$ (or $J^{\mu}_{V}$) and energy-momentum tensor $T^{\mu\nu}$,
hydrodynamic variables $T,\,u^{\mu},\,\mu$ (or $\mu_{V}$) and sources
$h_{\mu\nu}$, $A_{\mu}$ (or $V_{\mu}$), as well as the extra spatial direction
$r$ and Chern-Simons term, both appearing in the anomaly inflow formulation in
section 2. In some entries we have split the time $t$ from the spatial
components in the field theory directions, labeled by $i=x,\,y,\,z$, and the
$\varrho$-component. Note that we consider here an axial gauge field $A_{\mu}$
which has transformation properties distinct from that of a vector gauge
field. We here refer to the parity ${P}$ with respect to the field theory
directions $x,\,y,\,z$. The integral boundaries are indicated as initial, $i$,
and final, $f$.

###  A.1 Consistent-consistent Kubo Formulas

The Kubo formulas in sections 2.1 and 2.2.4 were written in terms of
covariant-consistent correlation functions. The consistent-consistent
connected correlation functions are the same as the covariant-covariant
connected correlation functions since

$\displaystyle\langle J^{\mu}_{cov}J^{\nu}_{cov}\rangle$ $\displaystyle=$
$\displaystyle\langle(J^{\mu}_{cons}+J^{\mu}_{BZ})(J^{\nu}_{cons}+J^{\nu}_{BZ})\rangle$
(143) $\displaystyle=$ $\displaystyle\langle
J^{\mu}_{cons}J^{\nu}_{cons}\rangle+J^{\mu}_{BZ}\langle
J^{\nu}_{cons}\rangle+\langle J^{\mu}_{cons}\rangle
J^{\nu}_{BZ}+J^{\mu}_{BZ}J^{\nu}_{BZ}\,,$

and the last three terms are disconnected. The consistent-consistent connected
correlation functions differ from the covariant-consistent connected
correlation functions (63) by the variation of the Bardeen-Zumino polynomial

$G_{J_{cons}^{\mu}J_{cons}^{\nu}}=G_{J_{cov}^{\mu}J_{cons}^{\nu}}+\frac{\delta
J^{\mu}_{BZ}}{\delta A_{\nu}}\,.$ (144)

Alternatively, the static consistent-consistent correlation functions can be
found by varying the 3+1 dimensional generating functional (16)

$G_{J_{cons}^{\mu}J_{cons}^{\nu}}(\omega=0,\mathbf{k})=\frac{\delta^{2}W_{cons}}{\delta
A_{\mu}(\mathbf{k})\delta A_{\nu}(-\mathbf{k})}\,.$ (145)

The Kubo formula for $\xi_{B}$ in terms of static consistent-consistent
correlation functions is then

$\displaystyle\langle
J^{x}_{cons}(\mathbf{k})J^{z}_{cons}(-\mathbf{k})\rangle=-ik_{y}(\xi_{B}-{\textstyle{\frac{1}{3}}}CA_{t})\,,$
(146a) for fluctuations perpendicular to the magnetic field and
$\displaystyle\langle
J^{x}_{cons}(\mathbf{k})J^{y}_{cons}(-\mathbf{k})\rangle=-ik_{z}(\xi_{B}-{\textstyle{\frac{1}{3}}}CA_{t})\,,$
(146b)

for fluctuations parallel to the magnetic field. These thermodynamic formulas
are unchanged by the presence of strong magnetic fields, and agree with the
expressions found in Amado:2011zx ; Jensen:2013vta . We can also rewrite the
current-current Kubo formulas in (64a) in terms of the consistent currents

$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J^{z}_{cons}J^{z}_{cons}}(\omega,{\bf
k}{=}0)=\sigma_{\parallel}+\cdots\,,$ (147a)
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cons}^{x}J^{x}_{cons}}(\omega,{\bf
k}{=}0)=\omega^{2}\rho_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{4}}\,,$
(147b) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cons}^{x}J_{cons}^{y}}(\omega,{\bf
k}{=}0)=\frac{n_{0}}{B_{0}}-\omega^{2}\tilde{\rho}_{\perp}\frac{w_{0}(w_{0}-{M_{5}}_{,\mu}B_{0}^{2})}{B_{0}^{4}}\,{\rm
sign}(B_{0})\,,$ (147c) $\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cons}^{z}O_{1}}(\omega,{\bf k}{=}0)=-c_{4}\,{\rm
sign}(B_{0})+\cdots\,,$ (147d)
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{J_{cons}^{z}O_{2}}(\omega,{\bf k}{=}0)=-c_{5}\,{\rm
sign}(B_{0})+\cdots\,$ (147e)
$\displaystyle{\textstyle{\frac{1}{\omega}}}\delta_{ij}{\rm
Im}\,G_{T^{ij}J_{cons}^{z}}(\omega,{\bf k}{=}0)=3c_{3}\,{\rm
sign}(B_{0})+\cdots\,,$ (147f)
$\displaystyle{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{O_{2}J_{cons}^{z}}(\omega,{\bf k}{=}0)=2c_{14}\,{\rm
sign}(B_{0})+\cdots\,,$ (147g)

where $O_{2}$ is defined below equation (2.2.4). The terms omitted vanish for
$B_{0}\ll T_{0}^{2}$ or when $M_{1}=M_{3}=M_{4}=0$.

###  A.2 Eigenmodes

In this appendix, we give more details about the gapless eigenmodes found in
section 2.4.

The velocities $v_{0}\,,\ v_{+}$ and $v_{-}$ appearing in eq. (79) are the
solutions to the cubic equation

$a_{3}v^{3}+a_{2}v^{2}+a_{1}v+a_{0}=0\,,$ (148)

where

$\displaystyle a_{0}=-B_{0}C\frac{(s_{0}T_{0})^{2}}{\det(\chi)}\,,$
$\displaystyle
a_{1}=v_{s}^{2}-\frac{(B_{0}C\mu_{0}T_{0})^{2}}{\det(\chi)}\frac{ds}{dT}\,,$
$\displaystyle
a_{2}=B_{0}C\left(\frac{\mu_{0}^{2}}{w_{0}}-2n_{0}\mu_{0}\frac{\chi_{11}-\mu_{0}\chi_{13}}{{\rm
det}(\chi)}+w_{0}\frac{\chi_{11}-\mu_{0}^{2}\chi_{33}}{{\rm
det}(\chi)}\right)\,,$ $\displaystyle
a_{3}=-1+\frac{(B_{0}C\mu_{0}T_{0})^{2}}{\det(\chi)}\frac{ds}{dT}\,,$

and the speed of sound expressed in terms of the elements of the
susceptibility matrix is given by (78)

These can be solved perturbatively in $B_{0}C$. We first label

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

from which we find

$\displaystyle v_{0}$ $\displaystyle=B_{0}C\frac{\alpha}{v_{s}^{2}}+\cdots\,,$
(149) $\displaystyle v_{\pm}$ $\displaystyle=\pm v_{s}+B_{0}C\frac{\gamma
v_{s}^{2}-\alpha}{2v_{s}^{2}}\mp B_{0}^{2}C^{2}\frac{3\alpha^{2}-2\alpha\gamma
v_{s}^{2}+v_{s}^{4}(4\beta(1-v_{s}^{2})-\gamma^{2})}{8v_{s}^{5}}+\cdots\,,$

where we have omitted terms of order $B_{0}^{3}C^{3}$ and higher.

The damping coefficient in eq. (82) has the following corrections

$\displaystyle\frac{w_{0}}{W_{0}}$
$\displaystyle=1+CB_{0}\frac{\chi_{11}w_{0}+2n_{0}\mu_{0}(\mu_{0}\chi_{13}-\chi_{11})+\mu_{0}^{2}\left((3\det(\chi)-\chi_{13}^{2}w_{0})/\chi_{11}-\det(\chi)/w_{0}\right)-(s_{0}T_{0}/v_{s})^{2}}{v_{s}\det(\chi)}\,,$
(150)
$\displaystyle\quad\quad\quad\Sigma_{\perp}=\frac{8\mu_{0}w_{0}^{2}(\mu_{0}\chi_{13}-\chi_{11})(w_{0}\chi_{13}-n_{0}\chi_{11})}{v_{s}(n_{0}\chi_{11}-2w_{0}\chi_{13})^{2}\det(\chi)}\,,$

and $\Sigma_{\parallel}$ is a lengthy function of the susceptibilities, other
thermodynamic derivatives of the pressure, the chemical potential and the
temperature. We have omitted higher order terms in $CB_{0}$ in eq. (82).

The leading correction to the longitudinal diffusion constant in eq. (84) is

$D_{\parallel}=\frac{w_{0}^{2}\sigma_{\parallel}}{v_{s}^{2}\det(\chi)}+B_{0}^{2}C^{2}\left(\left(\frac{ds}{d\mu}\right)_{p}\frac{s_{0}^{4}T_{0}^{6}}{3v_{s}^{6}{\rm
det}(\chi)^{3}}(3\zeta_{1}+10\eta_{1}+6\eta_{2})+F\sigma_{\parallel}\right)+\cdots\,,$

where $\left(\frac{ds}{d\mu}\right)_{p}$ is the derivative keeping pressure
and magnetic field fixed, and

$\displaystyle F$
$\displaystyle=\frac{s_{0}^{4}\,T_{0}^{6}\,w_{0}}{v_{s}^{8}\,{\rm
det}(\chi)^{4}}\Bigg{[}2\left(\frac{dn}{d\mu}\right)_{p}^{2}s_{0}^{2}\mu_{0}^{2}+2\left(\frac{ds}{d\mu}\right)_{p}^{3}n_{0}T_{0}\mu_{0}^{2}$
$\displaystyle+\left(\frac{ds}{d\mu}\right)_{p}^{2}\left(3s_{0}^{2}T_{0}^{2}-2\left(\frac{dn}{d\mu}\right)_{p}s_{0}T_{0}\mu_{0}^{2}-\mu_{0}^{2}n_{0}\left(n_{0}-2\left(\frac{dn}{d\mu}\right)_{p}\mu_{0}\right)\right)$
$\displaystyle+2\left(\frac{dn}{d\mu}\right)_{p}\left(\frac{ds}{d\mu}\right)_{p}s_{0}\mu_{0}\left(3s_{0}T_{0}+\mu_{0}\left(n_{0}-\left(\frac{dn}{d\mu}\right)_{p}\mu_{0}\right)\right)\Bigg{]}\,.$

The positivity of the diffusion constant implies $\sigma_{\parallel}>0$.

###  A.3 A comment on magnetic susceptibilities

In this appendix, we comment on the choice of naming ${\chi_{B}}=2p_{,B^{2}}$
as magnetic susceptibility. The thermodynamic function ${\chi_{B}}$ appears in
the constitutive relation for the magnetic polarization in front of $B^{\mu}$.
To be more precise, we can define $\delta
W=\frac{1}{2}\int\sqrt{-g}M^{\mu\nu}\delta
F_{\mu\nu}=\int\sqrt{-g}\left(m^{\mu}\delta B_{\mu}+p^{\mu}\delta
E_{\mu}\right)$ (see Kovtun:2016lfw ). The polarization tensor $M_{\mu\nu}$ is
related to the polarization and magnetization vectors $p_{\mu}$ and $m_{\mu}$
by

$M_{\mu\nu}=p_{\mu}u_{\nu}-p_{\nu}u_{\mu}-\epsilon_{\mu\nu\rho\sigma}u^{\rho}m^{\sigma}\,.$
(151)

Varying the generating functional $W=\\!\int\sqrt{-g}{\cal
F}(T,\mu,B^{2},E^{2},B{\cdot}E,B{\cdot}a,B{\cdot}\Omega,E{\cdot}a,E{\cdot}\Omega,\cdots)$
we find the constitutive relations for $p^{\mu}$ and $m^{\mu}$

$\displaystyle m^{\mu}=2{\cal F}_{,B^{2}}B^{\mu}+{\cal
F}_{,B{\cdot}E}E^{\mu}+{\cal F}_{,B{\cdot}a}a^{\mu}+{\cal
F}_{,B{\cdot}\Omega}\Omega^{\mu}+\cdots\,,$ (152) $\displaystyle
p^{\mu}=2{\cal F}_{,E^{2}}E^{\mu}+{\cal F}_{,B{\cdot}E}B^{\mu}+{\cal
F}_{,E{\cdot}a}a^{\mu}+{\cal F}_{,E{\cdot}\Omega}\Omega^{\mu}+\cdots\,.$

Now, if we focus on the zeroth order constitutive relations and with strong
magnetic fields so that $B^{\mu}={\cal O}(1)$ while
$E^{\mu}=a^{\mu}=\Omega^{\mu}={\cal O}(\partial)$, we simply find

$m^{\mu}=2p_{,B^{2}}B^{\mu}+\cdots\,,\quad p^{\mu}=\cdots\,,$ (153)

that is, the thermodynamic function relating $B^{\mu}$ to $m^{\mu}$ is simply
$2p_{,B^{2}}$. When written in terms of magnetic field strength
$H^{\mu}=B^{\mu}-m^{\mu}$, the relation reads

$m^{\mu}=\frac{2p_{,B^{2}}}{1-2p_{,B^{2}}}H^{\mu}\,.$ (154)

However, if we are interested in seeing how the magnetization changes as we
change the magnetic field $\delta m^{\mu}=m^{\mu\nu}\delta B_{\nu}$, we find

$m^{\mu\nu}=4p_{,B^{2}B^{2}}B^{\mu}B^{\nu}+2p_{,B^{2}}g^{\mu\nu}\,,$ (155)

where, in particular, the term aligned with $B^{\mu}$ is

$m^{\mu\nu}B_{\mu}B_{\nu}=4p_{,B^{2}B^{2}}B^{4}+2p_{,B^{2}}B^{2}=p_{,BB}B^{2}$
(156)

where $B=\sqrt{B^{2}}$. This is often quoted as the magnetic susceptibility at
finite magnetic field.

Note that the change in magnetization orthogonal to the existing magnetic
field is given by $2p_{,B^{2}}$. That is, for

$\delta B^{\mu}=\delta B^{\mu}_{\parallel}+\delta B^{\mu}_{\perp}\,,$ (157)

we have the following change in the magnetization

$\delta m^{\mu}=p_{,BB}\,\delta B^{\mu}_{\parallel}+2p_{,B^{2}}\delta
B^{\mu}_{\perp}\,.$ (158)

This suggests we could call $2p_{,B^{2}}$ the perpendicular magnetic
susceptibility and $p_{,BB}$ the parallel magnetic susceptibility.

##  Appendix B Details of the holographic calculation

In this section, we present more details of the holographic calculation. We
also outline the numerical procedure used to construct the numerical solutions
in this work and compute the transport coefficients.

###  B.1 Expansion close to horizon and conformal boundary

Here we include the near-boundary and near-horizon expansions for the
functions defined in the metric and gauge field ansatz, eq. (130) and eq.
(131), used in the holographic model. The solution close to the conformal
boundary (i.e. for $\varrho=0$) reads

$\displaystyle u(\varrho)=1+\varrho^{4}\left[u_{4}+{\cal
O}(\varrho^{2})\right]+\varrho^{4}\ln(\varrho)\left[\frac{B^{2}}{6}+{\cal
O}(\varrho^{2})\right]\,,$ $\displaystyle
v(\varrho)=1+\varrho^{4}\left[-\frac{w_{4}}{2}+{\cal
O}(\varrho^{2})\right]+\varrho^{4}\ln(\varrho)\left[-\frac{B^{2}}{24}+{\cal
O}(\varrho^{2})\right]\,,$ $\displaystyle
w(\varrho)=1+\varrho^{4}\left[w_{4}+{\cal
O}(\varrho^{2})\right]+\varrho^{4}\ln(\varrho)\left[\frac{B^{2}}{12}+{\cal
O}(\varrho^{2})\right]\,,$ $\displaystyle
c(\varrho)=\varrho^{4}\left[c_{4}+{\cal
O}(\varrho^{2})\right]+\varrho^{8}\ln(\varrho)\left[-\frac{B^{2}}{12}{c_{4}}+{\cal
O}(\varrho^{2})\right]\,,$ $\displaystyle
A_{t}(\varrho)=\mu-\frac{\rho}{2}\varrho^{2}-\frac{\gamma
Bp_{1}}{8}\varrho^{4}+{\cal O}(\varrho^{6})\,,$ $\displaystyle
P(\varrho)=\varrho^{2}\left(\frac{p_{1}}{2}+\frac{\gamma
B\rho}{8}\varrho^{2}+{\cal O}(\varrho^{4})\right)\,,$ (159)

where $u_{4},\,w_{4},\,c_{4},\,\rho,\,p_{1}$ are undetermined coefficients.
Here we have chosen to set the non-normalizable mode for $P(\varrho)$ to zero
at the conformal boundary. This choice is dual to switching off the source for
the current in $z$-direction on the boundary. Near the horizon at $\varrho=1$,
the expansion of the same functions reads

$\displaystyle u(\varrho)$ $\displaystyle=(1-\varrho)\left[\bar{u}_{1}+{\cal
O}(1-\varrho)\right],\qquad$ $\displaystyle c(\varrho)$
$\displaystyle=(1-\varrho)\left[\bar{c}_{1}+{\cal O}(1-\varrho)\right],$
$\displaystyle v(\varrho)$ $\displaystyle=\bar{v}_{0}+{\cal O}(1-\varrho),$
$\displaystyle A_{t}(\varrho)$
$\displaystyle=(1-\varrho)\left[\bar{A}_{t\,0}+{\cal O}(1-\varrho)\right],$
$\displaystyle w(\varrho)$ $\displaystyle=\bar{w}_{0}+{\cal O}(1-\varrho),$
$\displaystyle P(\varrho)$ $\displaystyle=\bar{P}_{0}+{\cal O}(1-\varrho)\,,$
(160)

where $\bar{u}_{1},\,\bar{c}_{1},\,\bar{w}_{0},\,\bar{v}_{0},\,\bar{A}_{t\,0}$
and $\bar{P}_{0}$ are undetermined coefficients. At the horizon, we choose the
coefficients of order $(1-\varrho)^{0}$ in $u(\varrho)$, $c(\varrho)$, and
$A_{t}(\varrho)$ to vanish.

###  B.2 Summary of thermodynamic details of the charged magnetic black brane

In the following we display how to extract the expectation value of the
energy-momentum tensor and of the covariant and consistent currents from the
metric and gauge field solutions.

In order to compute the expectation value of the energy momentum tensor in
equilibrium for the charged magnetic brane we only have to evaluate the terms
in the first line565656The terms in the second line of (134) will be important
when computing correlation functions involving the energy-momentum tensor. of
(134). Given our ansatz for the metric (130) and for the gauge field (132) as
well as the boundary expansion (B.1), the expectation value of the energy
momentum tensor in equilibrium reads

$\langle
T^{\mu\nu}\rangle=\left(\begin{array}[]{cccc}-3\,u_{4}&0&0&-4\,c_{4}\\\
0&-\frac{B^{2}}{4}-u_{4}-4\,w_{4}&0&0\\\
0&0&-\frac{B^{2}}{4}-u_{4}-4\,w_{4}&0\\\
-4\,c_{4}&0&0&8\,w_{4}-u_{4}\end{array}\right)\,.$ (161)

Again, let us stress that the components of the energy-momentum tensor are
scheme-dependent. However, for any choice of the scheme, the trace of the
energy momentum tensor is $\langle T_{\mu}{}^{\mu}\rangle=-B^{2}/2$.

The expectation value of the covariant current $J_{\text{cov}}^{\mu}$ for the
charged magnetic black brane is given by

$\langle J_{cov}^{\mu}\rangle=\left(\rho,\,0,\,0,\,p_{1}\right)\,,$ (162)

where $\rho$ and $p_{1}$ are boundary coefficients defined in eq. (B.1). Note
that the equations of motion imply $p_{1}=-\gamma\mu B$ as shown in
Ammon:2016szz  and hence $\langle J^{z}_{cov}\rangle=-\gamma\mu B$ assuming
that $B$ is aligned along the (positive) $z$-axis.

In the following we will outline how to determine the other thermodynamic
quantities, such as the entropy $s$, the temperature $T$ and the (density of
the) grand canonical potential $\Omega.$ The entropy density $s$ is given by
the function $v(\varrho)$ and $w(\varrho)$ evaluated at the horizon
$\varrho=1$ using the expansion (B.1)

$s=4\pi\,v(1)^{2}\,w(1)=4\pi\,\bar{v}_{0}^{2}\,\bar{w}_{0}.$ (163)

while the temperature is given by

$T=\frac{|u^{\prime}(1)|}{4\pi}\,.$ (164)

Finally, the grand canonical potential or its density $\Omega$ is given by

$\Omega=\epsilon-sT-\mu\langle J_{cov}^{t}\rangle.$ (165)

This concludes the discussion concerning the thermodynamics of the charged
magnetic black brane.

###  B.3 Numerical details

####  B.3.1 Details for computing $\tilde{B}$ and $\tilde{\mu}$ derivatives of
thermodynamic coefficients

To extract the transport coefficients via Kubo formulas, we have to evaluate
thermodynamic derivatives with respect to $\tilde{B}$ and $\tilde{\mu}$,
respectively, while keeping the respective other quantity fixed. We compute
the derivatives with spectral derivative matrices, where we discretize the
parameter range in $\tilde{B}$ and $\tilde{\mu}$ in terms of a Chebychev grid.
From a numerical point of view, we can only control $\mu$ in terms of a
boundary condition on the temporal component of the gauge field and $B$ as
external parameter. To compute the background for a given $\tilde{B}$, we vary
$B$ in terms of an iterative solver until we find the desired value of
$\tilde{B}$ (while keeping $\tilde{\mu}$ fixed).

####  B.3.2 Spectral method for calculating the holographic Green’s functions
numerically

The numerical calculation of the transport coefficients in this paper are
based on a pseudo-spectral method (see Boyd00 ; Ammon:2016fru ;
Grieninger:2017jxz ; Grieninger:2020wsb ; Baggioli:2019abx  for a more
detailed introduction). In order to determine the thermo- and hydrodynamic
(transport) coefficients we have to (numerically) compute Green’s functions of
the form $\lim\limits_{\omega\rightarrow 0}{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{\mathcal{O}_{a}\mathcal{O}^{c}}(\omega,{\bf k}{=}0)$ and
$\lim\limits_{k_{z}\rightarrow 0}{\textstyle{\frac{1}{k_{z}}}}{\rm
Im}\,G_{\mathcal{O}_{a}\mathcal{O}^{c}}(\omega{=}0,k\,{\bf e}_{z})$ to very
high accuracy.

Green’s functions of the form $G_{\mathcal{O}_{a}\mathcal{O}^{c}}(\omega,{\bf
k})$ are determined by exploiting the relationship

$\delta\left\langle\mathcal{O}_{a}\right\rangle(\omega,\bm{k})=G_{\mathcal{O}_{a}\mathcal{O}^{c}}(\omega,\bm{k})\,\delta\phi_{c}(\omega,{\bf
k})\,,$ (166)

where $\delta\phi_{c}$ is the source dual to the operator $\mathcal{O}_{c}$
and $\delta\left\langle\mathcal{O}_{a}\right\rangle$ is the response to the
perturbation $\delta\phi_{c}.$

To directly compute two-point functions of the form
$\lim\limits_{\omega\rightarrow 0}{\textstyle{\frac{1}{\omega}}}{\rm
Im}\,G_{\mathcal{O}_{a}\mathcal{O}^{b}}(\omega,{\bf k}{=}0)$ and
$\lim\limits_{k_{z}\rightarrow 0}{\textstyle{\frac{1}{k}}}{\rm
Im}\,G_{\mathcal{O}_{a}\mathcal{O}^{b}}(\omega{=}0,k{\bf e}_{z})$ we found it
convenient to apply a three-step procedure which we explain in the following.
As outlined in chapter 3.1, the background of our gravity model is the
charged-magnetic black brane which we solve by means of a pseudo-spectral
method as described in Ammon:2016fru ; Grieninger:2017jxz ; Grieninger:2020wsb
; Ammon:2017ded . On top of that background we calculate the fluctuations to
linear order in $\omega$ and $k$ (were $k$ is the component along the
$z$-axis, _i.e._ , along the magnetic field), respectively, by doing an
expansion in terms of them which read for the metric fluctuations

$h_{mn}(\varrho,\omega)=h_{mn}^{(0)}(\varrho)+\omega\,h_{mn}^{(1)}(\varrho)\quad\text{and}\quad
h_{mn}(\varrho,k)=h_{mn}^{(0)}(\varrho)+k\,h_{mn}^{(1)}(\varrho),$ (167)

respectively. In order to compute a two-point functions of the form
${\textstyle{\frac{1}{\omega}}}{\rm Im}\,G_{T^{ab}T^{cd}}(\omega,{\bf
k}{=}0)$, for example, we have to source the fluctuation $h_{cd}^{(0)}$ and
read of the vacuum expectation value of the fluctuation $h_{ab}^{(1)}$. This
may be done by plugging the background solution into the fluctuation equations
of zeroth order in $\omega$ and then plugging the background solution and
zeroth order solution into the first order equations and solve for the
fluctuations of first order in $\omega$. For the metric fluctuations the near
boundary expansion is schematically of the form

$\displaystyle h_{mn}^{(0)}(\varrho)$
$\displaystyle=h_{mn,\bm{s}}^{(0)}+\varrho^{4}\,h_{mn,\bm{v}}^{(0)}+B^{2}h_{mn,\bm{s}}^{(0)}\varrho^{4}\log(\varrho)$
(168) $\displaystyle h_{mn}^{(1)}(\varrho)$
$\displaystyle=h_{mn,\bm{s}}^{(1)}+\ldots+\varrho^{4}\,h_{mn,\bm{v}}^{(1)}+(B\,h_{mn,\bm{s}}^{(0)}+B^{2}\,h_{mn,\bm{s}}^{(1)})\,\varrho^{4}\log(\varrho),$

where the prefactors $h_{mn,\bm{s}}^{(0)}$ and $h_{mn,\bm{s}}^{(1)}$ are
related to the source of the energy momentum tensor. Depending on the operator
$\mathcal{O}^{a}$ under considerations, we switch on the corresponding zeroth
order sources $h_{ab,\bm{s}}^{(0)}$. The first order sources
$h_{ab,\bm{s}}^{(1)}$ are always zero whereas the vacuum expectation value of
the operator $\mathcal{O}_{b}$ is encoded in the corresponding
$h_{ab,\bm{v}}^{(1)}$.

####  B.3.3 Convergence and numerical accuracy

In this subsection, we discuss the convergence and numerical precision of the
numerical procedure we used to compute the transport coefficients. The
asymptotic boundary expansions of the background fields (159) and the
fluctuations (168) contain logarithmic contributions proportional to the
magnetic field $B$. To improve the numerical accuracy close to the boundary
and simplify the process of reading of the expectation values, we introduce
auxiliary functions, schematically given by

$\displaystyle
h^{\text{old}}_{mn}=h_{mn,\,\bm{s}}+\varrho^{4}\,h^{\text{new}}_{mn}+c_{1}\,\varrho^{4}\,\log(\varrho)$
(169) $\displaystyle
a^{\text{old}}_{m}=a_{m,\,\bm{s}}+\varrho^{2}\,a^{\text{new}}_{m}+c_{2}\,\varrho^{2}\,\log(\varrho)$
(170)

where $c_{1},\,c_{2}$ is the coefficient of the logarithmic term proportional
to the expectation value. We found it convenient to apply these kind of
redefinition of the functions for all background fields and fluctuations.

Furthermore, to improve the numerical precision we shift the divergent
logarithmic coefficients at the boundary to higher powers in terms of the
radial coordinate by introducing the coordinate mapping
$\varrho\mapsto\varrho^{2}$ as introduced in Ammon:2016fru ;
Grieninger:2017jxz  and more extensively discussed in Grieninger:2020wsb . The
improvement of the numerical solution after applying this coordinate mapping
may be seen in figure 15, where we depict the convergence for the fluctuations
$h_{tx},\,h_{zx},\,a_{x}$ in the helicity-one sector without coordinate
mapping (left) and with coordinate mapping (right). We note that with
coordinate mapping, the coefficients fall off geometrically to machine
precision before they reach a plateau.

Figure 15: Convergence in the momentum diffusion sector without (left) and
with (right) the coordinate mapping $\varrho\mapsto\varrho^{2}$. The $c_{k}$
are the Chebychev coefficients of the numerical solution and $N$ is the number
of gridpoints of the Chebychev grid used to discretize the radial direction.
The parameters are $\gamma=2/\sqrt{3},\,\tilde{B}=109.658,\,\tilde{\mu}=7.5$
and the fluctuations are $h_{tx},\,h_{zx},\,a_{x}$ (red, green, black).

In section 3.3 we have mentioned a fit of our numerical data for $c_{10}$ and
$\tilde{\eta}_{||}$ to the form given in eq. (139). The fit is visualized here
in figure 16.

Figure 16:  Fit of our numerical data (dots) for $c_{10}$ and
$\tilde{\eta}_{||}$ to the form given in eq. (139) (solid lines).

###  B.4 Computing thermodynamic derivatives of dimensionless quantities

We compute dimensionless quantities of the (conformal) field theory using
holography. In particular, we use appropriate powers of the temperature $T$ to
introduce the dimensionless chemical potential $\tilde{\mu}=\mu/T$ and the
dimensionless magnetic field $\tilde{B}=B/T^{2}.$ In general, for a
(thermodynamic) quantity $\mathcal{O}$ with energy dimension $\alpha$ we may
introduce the dimensionless quantity
$\tilde{\mathcal{O}}=\mathcal{O}/T^{\alpha}.$ Assuming $\mathcal{O}$ is only a
function of $(T,\mu,B)$, the dimensionless quantity $\tilde{\mathcal{O}}$ can
only depend on the dimensionless quantities, i.e. $(\tilde{\mu},\tilde{B})$.
In particular this implies

$d\mathcal{O}(T,\mu,B)=\left(\frac{\partial\mathcal{O}}{\partial
T}\right)_{\mu,B}dT+\left(\frac{\partial\mathcal{O}}{\partial\mu}\right)_{T,B}d\mu+\left(\frac{\partial\mathcal{O}}{\partial
B}\right)_{T,\mu}dB\,,$ (171)

as well as

$d\mathcal{O}(T,\mu,B)=d\left(T^{\alpha}\,\tilde{\mathcal{O}}(\tilde{\mu},\tilde{B})\right)=\alpha
T^{\alpha-1}\tilde{\mathcal{O}}(\tilde{\mu},\tilde{B})\,dT+T^{\alpha}d\tilde{\mathcal{O}}(\tilde{\mu},\tilde{B})\,,$
(172)

with

$d\tilde{\mathcal{O}}(\tilde{\mu},\tilde{B})=\left(\frac{\partial\tilde{\mathcal{O}}}{\partial\tilde{\mu}}\right)_{\tilde{B}}d\tilde{\mu}+\left(\frac{\partial\tilde{\mathcal{O}}}{\partial\tilde{B}}\right)_{\tilde{\mu}}d\tilde{B}\,.$
(173)

Equating both expressions for $d\mathcal{O}$ and using
$d\tilde{\mu}=d\mu/T-\mu\,dT/T^{2}$ as well as
$d\tilde{B}=dB/T^{2}-2B\,dT/T^{3}$, we may relate derivatives of $\mathcal{O}$
with respect to $T$, $\mu$ and $B$ to derivatives of $\tilde{\mathcal{O}}$
with respect to $\tilde{\mu}$ and $\tilde{B}$ as follows

$\displaystyle\left(\frac{\partial\mathcal{O}}{\partial\mu}\right)_{T,B}$
$\displaystyle=$ $\displaystyle
T^{\alpha-1}\left(\frac{\partial\tilde{\mathcal{O}}}{\partial\tilde{\mu}}\right)_{\tilde{B}}\,,$
(174) $\displaystyle\left(\frac{\partial\mathcal{O}}{\partial
B}\right)_{T,\mu}$ $\displaystyle=$ $\displaystyle
T^{\alpha-2}\left(\frac{\partial\tilde{\mathcal{O}}}{\partial\tilde{B}}\right)_{\tilde{\mu}}\,,$
(175) $\displaystyle\left(\frac{\partial\mathcal{O}}{\partial
T}\right)_{\mu,B}$ $\displaystyle=$ $\displaystyle
T^{\alpha-1}\left(-\tilde{\mu}\,\left(\frac{\partial\tilde{\mathcal{O}}}{\partial\tilde{\mu}}\right)_{\tilde{B}}-2\tilde{B}\,\left(\frac{\partial\tilde{\mathcal{O}}}{\partial\tilde{B}}\right)_{\tilde{\mu}}+\alpha\,\tilde{\mathcal{O}}\right)\,.$
(176)

We may apply those expressions to the case of a grand canonical potential,
i.e. $\mathcal{O}(T,\mu,B)=\Omega(T,\mu,B)$ which for a four-dimensional
relativistic field theory has energy dimension four, i.e. $\alpha=4$. In this
case we obtain the usual relations for the charge density
$\rho=-\left(\frac{\partial\Omega}{\partial\mu}\right)_{T,B}$, the
magnetization $M=-\left(\frac{\partial\Omega}{\partial B}\right)_{T,\mu}$ and
the entropy density $s=-\left(\frac{\partial\Omega}{\partial
T}\right)_{\mu,B}$ in terms of their dimensionless counterparts.

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