# Higher-form symmetries, anomalous magnetohydrodynamics, and holography

Arpit Das<EMAIL_ADDRESS>Centre for Particle Theory, Department of
Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK
Nabil Iqbal<EMAIL_ADDRESS>Centre for Particle Theory, Department
of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK
Ruth Gregory<EMAIL_ADDRESS>Department of Physics, King’s College
London, The Strand, London, WC2R 2LS, UK

###### Abstract

In $U(1)$ Abelian gauge theory coupled to fermions, the non-conservation of
the axial current due to the chiral anomaly is given by a dynamical operator
$F_{\mu\nu}\tilde{F}^{\mu\nu}$ constructed from the field-strength tensor. We
attempt to describe this physics in a universal manner by casting this
operator in terms of the 2-form current for the 1-form symmetry associated
with magnetic flux conservation. We construct a holographic dual with this
symmetry breaking pattern and study some aspects of finite temperature
anomalous magnetohydrodynamics. We explicitly calculate the charge
susceptibility and the axial charge relaxation rate as a function of
temperature and magnetic field and compare to recent lattice results. At small
magnetic fields we find agreement with elementary hydrodynamics weakly coupled
to an electrodynamic sector, but we find deviations at larger fields.

###### Contents

  1. I Introduction
  2. II Hydrodynamic calculation of relaxation rate
  3. III Overview of holographic model
    1. III.1 Holographic bulk action
    2. III.2 Dualizing the action
      1. III.2.1 Poincaré dualization
      2. III.2.2 Gauge symmetries of $S_{5}$
      3. III.2.3 Inverse operation
  4. IV Finite temperature physics: zero frequency
    1. IV.1 Susceptibility
      1. IV.1.1 General solutions
      2. IV.1.2 Regular solution
  5. V Hydrodynamic limit
  6. VI Numerical results
    1. VI.1 Contributing equations of motion
    2. VI.2 Numerics
      1. VI.2.1 Logarithmic fall-off
      2. VI.2.2 Hydrodynamic mode
  7. VII Conclusion
  8. A Conventions
  9. B Inverse operation
  10. C $\zeta\to 0$ and hypergeometric differential equation
  11. D Implementation of numerics
  12. E Non-hydrodynamic (gapped) modes and quasinormal mode table

##  I Introduction

In this work we will discuss the finite temperature physics of a
magnetohydrodynamic _chiral_ plasma, i.e. an electrodynamic plasma with an
axial $U(1)_{A}$ current $j_{A}$ that is afflicted by an Adler-Bell-Jackiw
anomaly:

$\partial_{\mu}j^{\mu}_{A}=-\frac{1}{16\pi^{2}}\epsilon^{\mu\nu\rho\sigma}f_{\mu\nu}f_{\rho\sigma}\
.$ (1)

Here it is understood that this expression arises in a theory of dynamical
electromagnetism, and $f_{\mu\nu}$ is the field strength of this fluctuating
gauge field. We stress that the non-conservation of the axial current is given
by a dynamical operator. While the analysis presented here will be from a more
formal, holographic, perspective, the system has clear phenomenological
interest, with applications to baryon number violation figueroa1 ;
Figueroa:2019jsi ; PhysRevD.30.2212 ; KUZMIN198536 ; PhysRevD.43.2027 ,
primordial magnetic fields PhysRevD.78.074033 ; PhysRevLett.108.031301 ;
PhysRevLett.79.1193 , magnetised baryogenesis PhysRevD.57.2186 ;
PhysRevD.94.063501 ; PhysRevD.94.123509 , and Dirac and Weyl semi-metals in
condensed matter systems (for a review see Landsteiner:2016led ).

As described in figueroa1 , a quantity of interest in $U(1)$ anomalous
processes is the relaxation rate of the chiral charge density $j_{A}^{0}$.
This is what we seek to compute in this work. In our opinion a fully universal
hydrodynamic treatment of this problem has not yet been given; indeed it is
somewhat unclear whether one should exist.

We begin by carefully stating the problem and distinguishing it from the large
existing literature on anomalous hydrodynamics. To orient ourselves, it is
helpful to first imagine a weakly-coupled realization of the physics that we
are interested in. Consider the following Lagrangian describing a massless
Dirac fermion coupled to dynamical electromagnetism with photon $a$:

$S[a,\psi]=\int
d^{4}x\left(-\frac{1}{4e^{2}}f^{2}+\bar{\psi}\left(\not{\partial}-i\not{a}\right)\psi\right)$
(2)

We will be interested in placing this system at finite temperature and
understanding the hydrodynamic description.

What are the global symmetries of this system? It has a 1-form
$U(1)^{(1)}$-symmetry associated with the conservation of magnetic flux:

$\partial_{\mu}J^{\mu\nu}=0\qquad
J^{\mu\nu}\equiv{{\frac{1}{2}}}\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}$ (3)

In this work we assume the reader is familiar with higher-form symmetries
Gaiotto:2014kfa  and we will often denote a $p$-form $U(1)$-symmetry by
$U(1)^{(p)}$. A recent review of applications of higher-form symmetries can be
found in McGreevy:2022oyu .

The symmetry associated with vector phase rotations of the $\psi$ field
$\psi\to e^{i\alpha}\psi$ is gauged and does not correspond to a global
symmetry. Classically, the theory appears to have a conventional (i.e. 0-form)
$U(1)^{(0)}_{A}$-symmetry associated with $\psi\to e^{i\alpha\gamma^{5}}\psi$;
however at the quantum mechanical level the conservation of the associated
current is broken by the Adler-Bell-Jackiw anomaly (1). Note that the right-
hand side of this expression is an operator, as $f$ is a fluctuating dynamical
field. This should be contrasted with the case of a ’t-Hooft anomaly, where
the right-hand side of a current-conservation equation involves a fixed
external source that can be set to zero.

We are interested in understanding the realization of the symmetries at finite
temperature. This is the domain of hydrodynamics, which describes how
conserved quantities relax towards thermal equilibrium. Hydrodynamics in the
presence of a ’t-Hooft anomaly is a rich and well-studied field Son:2009tf ;
Neiman:2010zi  (see also Landsteiner:2016led ). The situation with the anomaly
above is somewhat different. As the right-hand side is a fluctuating operator,
there is no longer a strictly universal sense in which the axial current is
conserved. Thus it appears that the only true global symmetry of the system is
the 1-form symmetry (3). A naive application of the conventional formalism of
hydrodynamics applied to this system would then only involve a study of the
1-form symmetry in thermal equilibrium. Such an analysis was performed in
Grozdanov:2016tdf , where it was shown that the resulting framework is
essentially a reformulation of conventional relativistic magnetohydrodynamics,
i.e. a description of an electrodynamic plasma (a holographic description of
this plasma from this point of view was given in Hofman:2017vwr ;
Grozdanov:2017kyl ). The only conserved quantity here is the usual magnetic
flux, and this description of course makes no reference to the axial current
whatsoever.

Nevertheless, to us this situation seems somewhat unsatisfactory; after all,
from an applied viewpoint, it seems clear that the finite-temperature dynamics
of (2) has a rich and physically relevant phenomenology. This physics is
usually accessed by coupling the equations of (ungauged) hydrodynamics with a
’t Hooft anomaly to weakly coupled electrodynamics “by hand” Yamamoto:2016xtu
; PhysRevLett.108.031301 ; in particular see PhysRevD.100.065023  which
constructs a hydrodynamic theory in a formal expansion in the anomaly
coefficient. These constructions are not fully universal, and the domain of
validity of the resulting theories is not entirely clear. In particular,
recent work on the lattice figueroa1 ; Figueroa:2019jsi  that computes the
charge relaxation rate $\Gamma_{A}$ shows a disagreement with the predictions
of the above hydrodynamic theories, suggesting that short-distance
fluctuations play an important role that is not captured by the non-universal
theories above.

It is difficult to come up with a universal hydrodynamic theory for this
model. In fact, towards the end of our analysis, we will see that we have
reasons to believe that such a universal hydrodynamic description might not
exist for such a set up. For now we will explore this problem in a new way, by
using holographic duality to explore aspects of the finite-temperature
dynamics of a system in the same universality class as the weakly coupled
theory described above. To construct our holographic dual theory, we must
first carefully understand the symmetries in a manner that is independent of
the description. One route to understand this is to note that the right-hand
side of the expression above can be written in terms of the current
$J^{\mu\nu}$ for the 1-form symmetry:

$\partial_{\mu}j^{\mu}_{A}=k\,\epsilon_{\mu\nu\rho\sigma}J^{\mu\nu}J^{\rho\sigma}\qquad
k\equiv\frac{1}{16\pi^{2}}$ (4)

We may now try to describe the dynamics of a system with a conserved 2-form
current $J^{\mu\nu}$ and a 1-form axial current $j^{\mu}_{A}$ that satisfies
the above non-conservation equation; this may be thought of as a kind of
intertwining of the (genuine) 1-form symmetry and the (broken by the anomaly)
$0$-form symmetry. (To our knowledge, this particular intertwining does not
appear to have a precise universal characterization in the field theory
literature; however see further discussion in the conclusion).111Note added:
after the first version of this paper was released on the arXiv, Choi:2022jqy
; Cordova:2022ieu  appeared: these works present a precise field-theoretical
characterization of the ABJ anomaly in terms of non-invertible symmetries.

In this work we will first construct a holographic model that possesses the
above symmetries. We will then perform a preliminary investigation of the
resulting holographic system. In particular, we study the system in the
presence of a background magnetic field. We will explicitly compute the charge
relaxation rate in this model and compare it both to elementary hydrodynamics
with weakly coupled electromagnetism and to recent lattice results; we will
find agreement with hydrodynamics at low magnetic fields, but disagreement at
large magnetic fields; this suggests that UV fluctuations are important for a
quantitative determination of this relaxation rate.

A short outline of the rest of this paper follows. In Section II we review a
simple hydrodynamic discussion of the charge relaxation rate. In Section III
we introduce the holographic model that we will study in the remainder of the
paper. In Section IV we place this model at finite temperature and study some
aspects of static response (i.e. the analogue of the charge susceptibility).
In Sections V and VI we study finite-frequency response (both analytically at
small frequencies and numerically) and we conclude with a brief discussion in
Section VII.

##  II Hydrodynamic calculation of relaxation rate

We begin our study by defining the relaxation rate that we will compute and
using elementary physical arguments to understand what may control it; at the
end we will compare the resulting physics to our holographic construction.

We first review the usual hydrodynamic computation of this charge relaxation
rate. This is done in the usual framework of “chiral MHD”. As described above,
this means we assume a certain anomalous contribution to the dynamical
electric current and couple it perturbatively to an MHD sector. We
particularly highlight PhysRevD.100.065023 , where the authors perform a
hydrodynamic study where the anomaly coefficient $k$ is treated
perturbatively; they compute the chiral charge relaxation rate $\Gamma_{A}$
for small $k$. In this limit, they find that, $\Gamma_{A}\sim k^{2}\,B^{2}$
with $B$ being the magnetic field.

We review a similar calculation below: this is physically instructive but as
discussed above is not truly universal. This calculation is a very slight
generalization of the one presented in figueroa1 .

In our notation the anomaly takes the form

$\displaystyle\partial_{\mu}j^{\mu}_{A}=-2kF_{\mu\nu}\tilde{F}^{\mu\nu}=k\epsilon_{\mu\nu\rho\sigma}J^{\mu\nu}J^{\rho\sigma},$
(5)

where $j^{\mu}_{A}$ is the chiral current, $\tilde{F}_{\mu\nu}$ is the Hodge
dual of the field strength $F_{\mu\nu}$. (In the case of a single Dirac
fermion studied in figueroa1  we have $2k=-\frac{e^{2}}{8\pi^{2}}$, where $e$
is the electromagnetic coupling.)

For the homogeneous case (see figueroa1 ), we have $\vec{j}_{A}=0$ and thus
the anomaly equation becomes (for $j_{A}^{0}\equiv\rho=\chi\mu_{A}$),

$\displaystyle\chi\frac{d\mu_{A}}{dt}=-\frac{8k}{V}\int\ d^{3}x\
\vec{E}\cdot\vec{B},\ \ \ \ (\text{as},\
F_{\mu\nu}\tilde{F}^{\mu\nu}=4\vec{E}\cdot\vec{B})$ (6)

where $\mu_{A}$ is the space-independent axial chemical potential, $\rho$ the
axial charge density, and $\chi$ the axial charge susceptibility, which in
principle could depend on the temperature and background magnetic field.

Thus, we have the chiral relaxation rate as,

$\displaystyle\frac{d\mu_{A}}{dt}=-\frac{8k}{\chi V}\int\ d^{3}x\
\vec{E}\cdot\vec{B},$ (7)

Following figueroa1  we give below the chiral MHD equations as,

$\displaystyle\frac{\partial\vec{B}}{\partial
t}=\nabla\times\vec{E},\qquad\frac{\partial\vec{E}}{\partial
t}+\nabla\times\vec{B}=-\sigma\vec{E}+4k\mu_{A}\vec{B}.$ (8)

where $\sigma$ is the electric conductivity of the plasma, and we have assumed
that the density of electric charge is zero, and the plasma has zero velocity.
The last term in $k\mu_{A}B$ is the contribution from the chiral magnetic
effect. This system of equations is complemented by the anomaly equation above
(7).

Now if we neglect the time-derivative of $\vec{E}$ in (8) we can express
$\vec{E}$ in terms of $\vec{B}$ using (8). Then, we get for long-range
fluctuations of the gauge fields (that is $\nabla\times\vec{B}\to 0$),

$\displaystyle\frac{d\mu_{A}}{dt}$ $\displaystyle=-\frac{8k}{\sigma\chi
V}\int\ d^{3}x\
\left(4k\mu_{A}\vec{B}\right)\cdot\vec{B}=-\frac{32k^{2}}{\sigma\chi}\mu_{A}B^{2}\equiv-\Gamma_{A}\mu_{A},$
(9)

where in the last equality $\Gamma_{A}$ is defined as the rate of chirality
non-conservation in the presence of an external homogeneous magnetic field
$\vec{B}$. The solution of Eq.(9) goes as,

$\displaystyle\mu_{A}(t)=e^{-\Gamma_{A}t}\mu_{A,0}\ \ \ \ \ (\text{where}\
\mu_{A,0}\ \text{is an integration constant})$ (10)

From Eq.(9) we see that,

$\displaystyle\Gamma_{A}=\frac{32k^{2}B^{2}}{\sigma\chi}.$ (11)

i.e. the relaxation rate is quadratic in the magnetic field. We will compare
this elementary discussion to an explicit holographic calculation later.

We note that in figueroa1 ; Figueroa:2019jsi , the authors perform a numerical
lattice computation in determining the chiral charge relaxation rate
$\Gamma_{A}$. They also found it to be quadratic in the magnetic field $B$. As
mentioned above, it was observed that the pre-factor in $\Gamma_{A}\sim B^{2}$
is approximately 10 times that of the theoretical predictions of the same pre-
factor from hydrodynamics. Calculating this pre-factor in a strongly coupled
yet solvable holographic model and comparing it to the existing literature
serves as a pragmatic motivation for this study.

##  III Overview of holographic model

In this section, we will present a bulk holographic theory which realizes the
pattern of symmetry non-conservation in (4). We will begin by presenting the
bulk action and demonstrating the deformed Ward identity; in the next section,
we will describe how we arrived at this theory from dualizing a different bulk
action. (For a summary of our conventions and notation see Appendix A.)

###  III.1 Holographic bulk action

We desire a bulk theory with the following properties: it should have a bulk
massless 2-form $B_{MN}$; as explained in detail in Hofman:2017vwr , this is
associated with a global 1-form symmetry in the boundary, as $B_{MN}$ is dual
to the boundary 2-form current $J^{\mu\nu}$. The action should also have a
vector field which we call $E_{M}$. $E_{M}$ is dual to a vector operator
representing the axial current $j_{A}^{\mu}$ on the boundary; this vector
operator should be understood as the axial current, and it is not conserved.
Thus, $E_{\mu}$ should not enjoy a bulk gauge symmetry. (We will see in a
later section that $E_{\mu}$ is of the form $A_{\mu}-\partial_{\mu}\phi$,
where $A_{\mu}$ and $\phi$ enjoy (bulk) gauge symmetries in such a way that
$E_{\mu}$ is gauge-invariant). However, the divergence of $j_{A}^{\mu}$ on the
boundary is not completely unconstrained; rather its divergence should be
related to the a double-trace operator of the 2-form current $J_{\mu\nu}$ by
the following anomaly equation

$\partial_{\mu}j^{\mu}_{A}=k\,\epsilon_{\mu\nu\rho\sigma}J^{\mu\nu}J^{\rho\sigma}$
(12)

where $k$ is a parameter that should enter the bulk action.

We now present a bulk action which satisfies the above properties:

$\displaystyle S[E,B]=\int d^{5}x\sqrt{-g}\ $
$\displaystyle\left[-\frac{1}{4}G^{2}-\frac{1}{12}H^{2}+k^{2}\left(E\cdot
H\right)^{2}-\frac{k}{12}\epsilon_{PQRMN}H^{PQR}E_{L}H^{LMN}\right].$ (13)

Here $G=dE$ and $H=dB$ are the field strengths of $E$ and $B$ respectively,
and we have defined $H^{2}=H^{PQR}H_{PQR}$, $\left(E\cdot
H\right)^{2}=E_{L}H^{LMN}E^{P}H_{PMN}$. The theory has an invariance under a
1-form gauge symmetry:

$B\to B+d\Lambda$ (14)

with $\Lambda$ an arbitrary 1-form. $E$ clearly enjoys no explicit gauge
symmetry; note however that the “mass” terms for $E$ have a specific
structure, involving couplings to $H$ that are parameterized by a single
coupling $k$. We will show that this structure encodes the anomaly (12). This
action should be understood as being correct to order
${{\mathcal{O}}}(E^{2})$; as we will show, the anomaly structure (12) above is
only correctly represented to that order. Below we will also present an
algorithm that can be used to obtain an action that is correct to all orders
in $E$, though we will not require it for our purposes.

Motivated by studies pertaining to similar anomalies, a bulk action involving
anomaly-inspired mass terms for gauge fields was studied in Jimenez-
Alba:2014iia  (see Eq.(30) in Jimenez-Alba:2014iia ). There, the anomaly is
thought to be sourced by a dynamical non-Abelian gauge field, which does not
have an associated 1-form symmetry. In our case, the dynamical gauge field is
Abelian, and thus the non-conservation of the current is precisely related to
a 2-form current with universal dynamics; thus our action takes a more
constrained form (and describes somewhat different physics) compared to that
in Jimenez-Alba:2014iia .222We note that Jimenez-Alba:2014iia  also consider
external vector Abelian magnetic fields; however these fields act as a source
and are non-dynamical.

The variation of the action above with respect to $B$ results in the following
equations of motion:

$\displaystyle-\frac{1}{2}\partial_{L}\left(\sqrt{-g}H^{LMN}\right)-\frac{k}{4}\partial_{L}\left[\sqrt{-g}\
\epsilon^{LMNQR}E^{P}H_{PQR}\right]$
$\displaystyle-\frac{k}{12}\partial_{L}\left[\sqrt{-g}\
H_{PQR}\left(E^{L}\epsilon^{PQRMN}+E^{M}\epsilon^{PQRNL}+E^{N}\epsilon^{PQRLM}\right)\right]=0.$
(15)

and similarly we have the following equations for the variation with respect
to $E$:

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{M}\left(\sqrt{-g}G^{ML}\right)=-2k^{2}H^{LQR}E^{P}H_{PQR}+\frac{k}{12}\epsilon_{PQRMN}H^{PQR}H^{LMN}.$
(16)

Note that if $k=0$ these equations of motion decouple into free Maxwell
equations for $E$ and $B$ respectively. Here we work only to linearized order
in $E$.

We would now like to interpret this bulk physics holographically. We begin
with the 2-form $B$. As usual, we may construct the boundary 2-form current
$J^{\mu\nu}$ by varying the action with respect to the boundary value of
$B_{\mu\nu}$, i.e.

$J^{\mu\nu}(x)=\ \frac{\delta S}{\delta B_{\mu\nu}\left(\infty\right)}$ (17)

The on-shell variation of the boundary value of the action may be reduced to a
variation with respect to the radial derivative $\partial_{r}B_{\mu\nu}$, and
we thus find

$\displaystyle J^{\rho\sigma}(x)$ $\displaystyle=\lim_{r\to\infty}\frac{\delta
S}{\delta\left(\partial_{r}\left(B_{\rho\sigma}\right)\right)}$ (18)
$\displaystyle=\lim_{r\to\infty}\sqrt{-g}\left[-\frac{1}{2}H^{r\rho\sigma}-\frac{k}{4}\epsilon^{r\rho\sigma\mu\nu}E^{\alpha}H_{\alpha\mu\nu}-\frac{k}{12}\left[H_{\alpha\beta\gamma}\\{E^{r}\epsilon^{\alpha\beta\gamma\rho\sigma}+E^{\rho}\epsilon^{\alpha\beta\gamma\sigma
r}+E^{\sigma}\epsilon^{\alpha\beta\gamma r\rho}\\}\right]\right]$ (19)

Here the first term is standard Hofman:2017vwr ; the others arise from the
physics associated with the anomaly. We have omitted terms of order
${{\mathcal{O}}}(E^{2})$; this is because in this work we will study only the
linearized equations of motion of $E$.

Note that the radial component of the bulk wave equation (15) for $B_{2}$
ensures that we have

$\partial_{\mu}J^{\mu\nu}(x)=0$ (20)

i.e. that the 2-form current is conserved.

We now turn to $E$. We may construct the dual current as in (17):

$j^{\mu}_{A}=\frac{\delta S}{\delta
E_{\mu}\left(\infty\right)}=-\sqrt{-g}G^{r\mu}\,(r\to\infty).$ (21)

The last expression is standard for the boundary operator dual to a vector
field.

Let us now understand the non-conservation of $j^{\mu}_{A}$, i.e. let us
derive (12) holographically at linear order in ${{\mathcal{O}}}(E)$. From the
above expression for the 2-form current at the boundary let us now
compute333Note, the boundary metric is flat. Thus we have the following
expression relating boundary and bulk Levi-Civita tensors,
$\epsilon_{rabcd}=\sqrt{-g}\,\epsilon_{abcd}$. $k\ J\wedge_{4}J$:

$\left[-\frac{k}{4}\epsilon_{\alpha\beta\mu\nu}J^{\alpha\beta}J^{\mu\nu}\right]=\left(\sqrt{-g}\right)^{2}\
\left(\frac{k^{2}}{2}\frac{E^{\alpha}H_{\alpha\mu\nu}H^{r\mu\nu}}{\sqrt{-g}}-\frac{k}{16}\epsilon_{\alpha\beta\mu\nu}H^{r\alpha\beta}H^{r\mu\nu}\right)$
(22)

(Note the explicit appearance of factors of the determinant of the metric;
this arises from the fact that from the point of view of the bulk
$J^{{\alpha}\beta}$ not a tensor, as can be seen from its definition in (18)).
Now let us consider $L=r$ in (16),

$2k^{2}H^{r\mu\nu}E^{\alpha}H_{\alpha\mu\nu}-\frac{k}{4}\sqrt{-g}\epsilon_{\alpha\beta\mu\nu}H^{r\alpha\beta}H^{r\mu\nu}=-\partial_{\sigma}G^{\sigma
r},\ \ \ \ \ (\sigma\neq r)$ (23)

Usually in AdS/CFT this component of the bulk Maxwell equations of motion is a
radial constraint that enforces the conservation of the current $j^{\mu}_{A}$;
here we see that the current is instead not conserved. Using (22) we see that
it is equal to

$\partial_{\sigma}j^{\sigma}_{A}=\partial_{\sigma}\left(\sqrt{-g}G^{\sigma
r}\right)=k\epsilon_{\alpha\beta\mu\nu}J^{\alpha\beta}J^{\mu\nu}$ (24)

i.e. equivalent to (12) as desired.

This shows that this holographic theory is in the correct universality class,
by which we mean that it correctly links the non-conservation of the 0-form
axial current with a bilinear constructed from the 2-form current
$J^{\mu\nu}$. The fact that $E$ has no gauge symmetry at all in the bulk is
dual to the fact that its non-conservation is given in terms of a dynamical
operator that cannot be turned off. We note also that the intermediate steps
are somewhat complicated and rely on the detailed structure of the action
(13). The reader who is willing to take this action as a given and is
interested only in results can now skip to the next section, where we compute
the holographic observables of interest.

In the remainder of this section we describe how we construct this action
through bulk Poincaré duality.

###  III.2 Dualizing the action

Our approach to constructing the bulk action is essentially the bulk dual of
the operation of “gauging a global $U(1)$ symmetry”; i.e. we begin by
considering the very well-studied bulk action Jimenez-Alba:2014iia ;
Gynther:2010ed ; Jimenez-Alba:2014pea 444We also found the exposition in
ammon2016holographic ; grieninger2017holographic  useful. for a theory with
two 0-form global symmetries $U(1)_{A}\times U(1)_{V}$ with a mixed ’t Hooft
anomaly between them:

$\displaystyle S_{5}\
[A_{1},V_{1}]=\int_{\mathcal{M}^{5}}\left(-\frac{1}{2}F_{2}\wedge\star
F_{2}-\frac{1}{2}G_{2}\wedge\star G_{2}-kA_{1}\wedge F_{2}\wedge
F_{2}-\frac{k}{3}A_{1}\wedge G_{2}\wedge G_{2}\right).$ (25)

Here $A_{1}$ and $V_{1}$ are two 1-form potentials which are holographically
dual to the $0$-form axial and vector currents respectively, and
$F_{2}=dV_{1}$ and $G_{2}=dA_{1}$. The action is invariant (up to a boundary
term) under the following gauge symmetries:

$A_{1}\to A_{1}+d\Lambda_{0}\qquad V_{1}\to V_{1}+d\lambda_{0}.$ (26)

The boundary variation of the above gauge-transformation is nonzero, and it is
well-understood Gynther:2010ed  that this means that the dual field theory has
a ’t Hooft anomaly for the axial current:

$\partial_{\mu}j^{\mu}_{A}=k\epsilon^{\mu\nu\rho\sigma}\left(F_{\mu\nu}F_{\rho\sigma}+\frac{1}{3}G_{\mu\nu}G_{\rho\sigma}\right)$
(27)

where $F$ and $G$ are the field strengths of the fixed external sources for
the vector and axial currents respectively.

We now want to study a field theory where we have “gauged” the global symmetry
$U(1)_{V}$. In this operation the boundary gauge field $V_{1}$ will become
dynamical, and we will thus lose the 0-form symmetry $U(1)_{V}$. However we
expect to obtain a new 1-form symmetry (and 2-form current) associated with
the conserved magnetic flux in our new $U(1)$ gauge theory; in holographic
duality, we thus expect to obtain a new 2-form bulk field which we call
$B_{2}$.

It is thus very natural to expect that the bulk operation equivalent to
“gauging” on the boundary is to perform a bulk Poincaré duality on the bulk
1-form potential $V_{1}$, replacing it with a 2-form $B_{2}$. Similar
operations have a long history in AdS/CFT, and may be viewed as a higher-form
generalization of Witten:2003ya ; see DeWolfe:2020uzb ; Iqbal:2021tui  for
recent work in a similar holographic context.

We now describe the dualization process below, which proceeds essentially as
it would in flat space. This is a well-posed operation, but to the best of our
knowledge the details are not present in the literature for a non-linear
action of the form (25) even in flat space. We will see some interesting
wrinkles arising from the presence of the mixed Chern-Simons term.

####  III.2.1 Poincaré dualization

We follow the usual algorithm to dualize $V_{1}$, as can be found e.g. in
Appendix B of Polchinski:1998rr  (see also berman1988 ). The action does not
depend on $V_{1}$ directly, but only on its field strength $F_{2}$; it is thus
possible to treat $F_{2}$ as the dynamical variable rather than $V_{1}$.
However we then need to impose its closure $dF_{2}=0$ through the use of a
Lagrange multiplier $B_{2}$.

We construct the parent action $S_{5p}$ by adding the Lagrange multiplier
term’s action ($S_{c}$) to the action $S_{5}$. We get for $S_{5p}$,555In the
action $S_{5p}$, $F_{2}$ is now a dynamical field.

$\displaystyle S_{5p}\ [A_{1},F_{2},B_{2}]=$
$\displaystyle\underbrace{\int_{\mathcal{M}^{5}}-\frac{1}{2}F_{2}\wedge\star
F_{2}-\frac{1}{2}G_{2}\wedge\star G_{2}-kA_{1}\wedge F_{2}\wedge
F_{2}-\frac{k}{3}A_{1}\wedge G_{2}\wedge G_{2}}_{S_{5}}$
$\displaystyle+\underbrace{\int_{\mathcal{M}^{5}}dB_{2}\wedge F_{2}}_{S_{c}},$
(28)

where,

$S_{c}=-\int_{\mathcal{M}^{5}}dB_{2}\wedge
F_{2}=\int_{\mathcal{M}^{5}}B_{2}\wedge dF_{2}-d(B_{2}\wedge F_{2})$ (29)

Then, $\delta_{B_{2}}S_{c}=0$ gives $dF_{2}=0$ (closure of $F_{2}$). Now
imposing the equation of motion $\delta_{F_{2}}S_{5p}=0$ yields,

$\displaystyle\star$ $\displaystyle F_{2}=dB_{2}-2k\ (A_{1}\wedge F_{2}),$
(30)

The standard procedure is to now solve for $dB_{2}$ as a function of $F_{2}$
and eliminate the latter from the action entirely.

####  III.2.2 Gauge symmetries of $S_{5}$

Before doing so, we discuss the symmetries: note that the realization of the
0-form gauge symmetry associated with $A_{1}$ has changed, as $F_{2}$ is now
closed only on-shell. In the action as given in (28), consider instead the
following gauge transformations,

$\displaystyle A_{1}\to A_{1}+d\Lambda_{0},\qquad B_{2}\to
B_{2}+2k\,\Lambda_{0}\,F_{2}\ .$ (31)

It is easy to show that with the above gauge transformations, the action
$S_{5p}$ as given in (28) is gauge-invariant but the equation of motion (30)
is not (off-shell). It fails to be gauge-invariant by a term
$2k\Lambda_{0}\,dF_{2}$; since $F_{2}$ is an independent dynamical field now,
it is not necessarily a closed 2-form unless we impose $B_{2}$’s equations of
motion. This may appear problematic: the action is gauge-invariant under the
gauge transformations but the equations of motion are not off-shell gauge-
invariant under the same gauge transformations. We shall remedy this below by
introducing a new auxiliary field $\phi_{0}$. The equations of motion of
$\phi_{0}$ shall serve as a constraint (which we refer to as a gauge-invariant
constraint) for imposing the closure of $F\wedge dF_{2}$.

Alternatively, we can introduce $\phi_{0}$ by the following argument. Taking a
step back, let us first consider variations of the action $S_{5}$ as given in
Eq.(25) w.r.t. the gauge transformations as given in Eq.(26). We have,

$\displaystyle\delta S_{5}=2k\,\Lambda_{0}F_{2}\wedge
dF_{2}\,+\,\frac{2k}{3}\,\Lambda_{0}\,G_{2}\wedge dG_{2}\,+\,\text{boundary
terms},$ (32)

where the first two terms on the LHS vanish owing to the fact that both
$F_{2}$ and $G_{2}$ in $S_{5}$ are closed 2-forms. This then ensures the
invariance of $S_{5}$ (up to boundary terms) under gauge transformations (26).
However, in the dualized form $S_{5p}$, $F_{2}$ is an arbitrary 2-form and not
necessarily closed. So, due to the non-closure of $F_{2}$ we now may or may
not have $F_{2}\wedge dF_{2}=0$ (off-shell). Therefore, in addition to
imposing the closure of $F_{2}$ by a Lagrange multiplier $B_{2}$ we have to
add to $S_{5p}$ another Lagrange multiplier to impose the constraint
$F_{2}\wedge dF_{2}=0$ (gauge-invariant constraint). From the degree of the
term $F_{2}\wedge dF_{2}$, it is clear that the Lagrange multiplier in this
case would be a 0-form, say $\phi_{0}$. Furthermore, as $S_{5p}$ has to remain
gauge-invariant under $A_{1}\to A_{1}+d\Lambda_{0}$, $\phi_{0}$ has to be a
gauge field with its own gauge transformation given as,
$\phi_{0}\to\phi_{0}+\Lambda_{0}$ (by construction). Then, we have $S_{5p}$
as,

$\displaystyle S_{5p}=$
$\displaystyle\int_{\mathcal{M}^{5}}-\frac{1}{2}F_{2}\wedge\star
F_{2}-\frac{1}{2}G_{2}\wedge\star G_{2}-kA_{1}\wedge F_{2}\wedge
F_{2}-\frac{k}{3}A_{1}\wedge G_{2}\wedge G_{2}+dB_{2}\wedge F_{2}$
$\displaystyle-\int_{\mathcal{M}^{5}}2k\ \phi_{0}\,F_{2}\wedge dF_{2},$ (33)

which can be re-written as,

$\displaystyle S_{5p}=\int_{\mathcal{M}^{5}}-\frac{1}{2}F_{2}\wedge\star
F_{2}-\frac{1}{2}G_{2}\wedge\star G_{2}-k(A_{1}-d\phi_{0})\wedge F_{2}\wedge
F_{2}-\frac{k}{3}A_{1}\wedge G_{2}\wedge G_{2}+dB_{2}\wedge F_{2},$ (34)

with $E_{1}\equiv A_{1}-d\phi_{0}$ being a vector field. Note that the
emergence of the gauge-invariant field $E_{1}$ is precisely the structure
anticipated earlier, which we now see emerges naturally when demanding off-
shell gauge-invariance. We also note that the equations of motion of
$\phi_{0}$ are redundant – they follow automatically from the equations of
$A_{1}$.666Another way to understand $\phi_{0}$ is that $e^{i\phi_{0}}$ is an
operator that is charged under the bulk $0$-form “instanton” current
$\star_{5}F\wedge F$; in a conventional formalism where $F=dA$ this current is
conserved identically, but in this formalism its conservation must be enforced
by $\phi_{0}$’s equations of motion.

Now let us give below the gauge transformations of $A_{1}$ and $\phi_{0}$,

$\displaystyle A_{1}\to
A_{1}+d\Lambda_{0},\qquad\phi_{0}\to\phi_{0}+\Lambda_{0}.$ (35)

With these gauge transformations the $F_{2}$ equation of motion,
$F_{2}=-\star\left[dB_{2}-2k\ (E_{1}\wedge F_{2})\right]$, remains gauge-
invariant even off-shell, and everything is consistent with the Poincaré
dualization procedure. Clearly, $S_{5p}$ is also invariant under (35) up to
boundary terms.

The conclusion of the above discussion is that one should be careful while
performing Poincaré dualization, as at times one may be required to impose a
gauge-invariant constraint along with the usual closure constraint.

####  III.2.3 Inverse operation

Let us now proceed to eliminate $F_{2}$ from the action. We can now invert
(30) to get $F_{2}$ in terms of $E_{1}$ and $B_{2}$ as below. See Appendix B
for details of this calculation.

$\displaystyle F_{MN}$
$\displaystyle=-\frac{\tilde{c}_{1}}{6}\epsilon_{PQRMN}H^{PQR}+2\tilde{c}_{1}k\
E^{P}H_{PMN}+\frac{4}{3}\tilde{c}_{1}k^{2}\
H^{PQR}E^{L}\epsilon_{PQRL[M}E_{N]}.$ (36)

where $H_{PQR}=\partial_{P}B_{QR}+\partial_{Q}B_{RP}+\partial_{R}B_{PQ}\
(\text{as},\ H_{3}=dB_{2})$ and $\tilde{c}_{1}\equiv\frac{1}{1+4k^{2}E^{2}}$.

Note that, if in $S_{5}$ we have $k\to 0$, $F_{2}$’s equations of motion
become $F_{2}=-\star dB_{2}$ and if we take $k\to 0$ in the above equation we
get $F_{2}=-\star dB_{2}$. $F_{MN}$ above has been written in such a way so
that it is manifestly anti-symmetric.

Now let us substitute Eq.(36) into Eq.(34) to obtain below the full non-linear
bulk action,

$\displaystyle S_{5p}=\int_{\mathcal{M}^{5}}\sqrt{-g}\,d^{5}x\,$
$\displaystyle\left[\left\\{-\frac{H^{2}}{12}-\frac{k}{12}\epsilon_{PQRMN}H^{PQR}E_{D}H^{DMN}+k^{2}\left((E\cdot
H)^{2}-\frac{2}{3}E^{2}H^{2}\right)\right.\right.$
$\displaystyle\left.-k^{3}\epsilon_{PQRMN}E^{P}E_{L}H^{LQR}E_{J}H^{JMN}+4k^{4}\left(E^{2}(E\cdot
H)^{2}-\frac{1}{3}E^{4}H^{2}\right)\right\\}\tilde{c}_{1}^{2}$
$\displaystyle+\left.\left\\{\frac{k}{12}\epsilon^{PQRMN}E_{P}G_{QR}G_{MN}-\frac{1}{4}G^{2}\right\\}\right]$
(37)

Note that truncating the above action to $\mathcal{O}(E^{2})$ results in the
quadratic action in Eq.(13) above, which we will use for the remainder of this
study, in which we consider only small fluctuations about equilibrium.

Now we give below the full 2-form current $J^{\mu\nu}$ obtained from the
action above Eq.(III.2.3),

$\displaystyle J^{\rho\sigma}$ $\displaystyle=\lim_{r\to\infty}\frac{\delta
S_{5p}}{\delta\left(\partial_{r}B_{\rho\sigma}\right)}$
$\displaystyle=\lim_{r\to\infty}\sqrt{-g}\left[-\frac{H^{r\rho\sigma}}{2}-\frac{k}{4}\epsilon^{r\rho\sigma\mu\nu}E^{\eta}H_{\eta\mu\nu}-\frac{k}{12}H_{\alpha\beta\gamma}\left(E^{r}\epsilon^{\alpha\beta\gamma\rho\sigma}+E^{\rho}\epsilon^{\alpha\beta\gamma\sigma
r}+E^{\sigma}\epsilon^{\alpha\beta\gamma r\rho}\right)\right.$
$\displaystyle\left.+2\left(k^{2}+4k^{4}E^{2}\right)E_{\alpha}\left(E^{r}H^{\alpha\rho\sigma}+E^{\rho}H^{\alpha\sigma
r}+E^{\sigma}H^{\alpha
r\rho}\right)-4\left(1+2k^{2}E^{2}\right)k^{2}E^{2}H^{r\rho\sigma}\right.$
$\displaystyle\left.-2k^{3}E_{\alpha}E^{\kappa}H_{\kappa\beta\gamma}\left(E^{r}\epsilon^{\alpha\beta\gamma\rho\sigma}+E^{\rho}\epsilon^{\alpha\beta\gamma\sigma
r}+E^{\sigma}\epsilon^{\alpha\beta\gamma
r\rho}\right)\right]\tilde{c}_{1}^{2}.$ (38)

Now one can, in principle, check that the anomaly structure of Eq.(12) can be
obtained from the above 2-form current by performing an order-by-order (in
$k$) comparison of coefficients on both sides of Eq.(12). We have checked this
explicitly to $\mathcal{O}(k^{3})$.

##  IV Finite temperature physics: zero frequency

With the holographic action in hand, we will now study the plasma that is
obtained from the realization of these symmetries at finite temperature. To
heat up our system, we consider the background metric given by the usual
planar black brane background

$ds^{2}=r^{2}\left(-f(r)dt^{2}+d\vec{x}^{2}\right)+\frac{dr^{2}}{r^{2}f(r)}\
,$ (39)

where $f(r)=1-\left(\frac{r_{h}}{r}\right)^{4}$ and where we are working in
units where the AdS radius $R=1$. The Hawking temperature of the black brane
is $T=\frac{r_{h}}{\pi}$. We are interested in the physics in the presence of
a background magnetic field in the $z$ direction, i.e. a configuration where
$\langle J^{tz}\rangle=-\frac{b}{2}$. From the holographic dictionary (19), we
see that this means that $H^{rtz}\neq 0$; solving the equations of motion (15)
we see that the background profile is:

$\displaystyle H^{rtz}=\frac{b}{r^{3}},\qquad H_{rtz}=-\frac{b}{r},\qquad
E=0.$ (40)

Note that, here we are working in the so called probe limit – where we neglect
the backreaction of the magnetic field onto the geometry. In other words, we
assume the above background profile (Eq.(40)) doesn’t affect the metric
components given in Eq.(39). This is justified in the high-temperature limit;
at low temperatures this backreaction cannot be neglected, and one would
replace the background with a magnetic brane solution DHoker:2009mmn ;
DHoker:2012rlj . Now we will begin our study by computing the static axial
charge susceptibility $\chi$ in the presence of the background magnetic field.
For a theory with a conserved charge, the susceptibility can be defined as:

$\langle j_{A}^{t}\rangle=\chi\mu_{A}$ (41)

where $\mu_{A}$ is the axial chemical potential. In the case of a non-
conserved axial current the precise definition of the axial chemical potential
as a dynamical hydrodynamic variable is somewhat more subtle (see e.g.
Gynther:2010ed ), but in thermal equilibrium it can be understood as the value
of the axial source $A_{t}$, which coincides with $E_{t}$ when all fields are
static.

###  IV.1 Susceptibility

In this section we shall consider the low frequency limit of Eq.(16) and
compute the axial charge susceptibility in this model. From Eq.(16) we find,

$\displaystyle\delta E_{r}=\frac{(i\omega r^{2})\partial_{r}\left(\delta
E_{t}\right)+kbf\partial_{r}\left(\delta
B_{xy}\right)}{r^{2}\omega^{2}-4b^{2}k^{2}f},\ \ \ \ \ \ \text{(with}\ L=r\
\text{in Eq.(\ref{Eeoms}))}$ (42) $\displaystyle\partial_{r}^{2}\left(\delta
E_{t}\right)+i\omega\partial_{r}\left(\delta
E_{r}\right)+\left(\frac{3}{r}\right)\partial_{r}\left(\delta
E_{t}\right)+\left(\frac{3}{r}\right)i\omega\delta
E_{r}=\frac{4b^{2}k^{2}}{r^{6}f}\delta E_{t}-\frac{kbi\omega}{r^{6}f}\delta
B_{xy},$ $\displaystyle\text{(with}\ L=t\ \text{in Eq.(\ref{Eeoms}))}$ (43)

Now plugging $\delta E_{r}$ from Eq.(42) in Eq.(43) and taking the $\omega\to
0$ limit we obtain777Note that we could have obtained Eq.(44) by directly
taking $\omega\to 0$ limit in Eq.(43). This is because in $\omega\to 0$ limit
$\delta E_{r}$ and $\delta E_{t}$ decouple.,

$\displaystyle\partial_{r}^{2}\left(\delta
E_{t}\right)+\left(\frac{3}{r}\right)\partial_{r}\left(\delta
E_{t}\right)-\frac{4b^{2}k^{2}}{r^{6}f}\delta E_{t}=0\ .$ (44)

####  IV.1.1 General solutions

Let us define the following dimensionless combination of temperature and
background magnetic field for later convenience:

$\displaystyle\zeta(b/T^{2})\equiv\frac{\sqrt{r_{h}^{4}-4b^{2}k^{2}}}{r_{h}^{2}}=\left(\frac{b}{T^{2}\pi^{2}}\right)\sqrt{\frac{\pi^{4}}{\left(b^{2}/T^{4}\right)}-4k^{2}}.$
(45)

In the small magnetic field limit (with $T$ fixed), that is $b\to 0$, we have,

$\displaystyle\zeta(b/T^{2})\underset{b\to
0}{\to}1-\frac{2k^{2}}{\pi^{4}}\left(\frac{b}{T^{2}}\right)^{2}-\frac{2k^{4}}{\pi^{8}}\left(\frac{b}{T^{2}}\right)^{4}+\mathcal{O}\left(\left(\frac{b}{T}\right)^{6}\right)$
(46)

In the large magnetic field limit (with $T$ fixed), that is $b\to\infty$, we
have,

$\displaystyle\zeta(b/T^{2})\underset{b\to\infty}{\to}\frac{2ik}{\pi^{2}}\left(\frac{b}{T^{2}}\right)-\frac{i\pi^{2}}{4k}\left(\frac{b}{T^{2}}\right)^{-1}-\frac{i\pi^{6}}{64k^{3}}\left(\frac{b}{T^{2}}\right)^{-3}+\mathcal{O}\left(\left(\frac{b}{T}\right)^{-5}\right)$
(47)

Notice from the definition of $\zeta$ (in Eq.(45)), it appears that $\zeta=0$
could be a point of non-analyticity for the susceptibility $\chi(\zeta)$; we
shall show below that $\chi$ is actually a function of $\zeta^{2}$ and not of
$\zeta$ and hence is analytic at $\zeta=0$.

Solving Eq.(44) analytically we find the general solution as,

$\displaystyle\delta E_{t}(r)_{gen}$ $\displaystyle=d_{1}\ r_{h}^{1+\zeta}\
r^{-1-\zeta}\
{}_{2}F_{1}\left(-\frac{1}{4}-\frac{\zeta}{4},\frac{1}{4}-\frac{\zeta}{4};1-\frac{\zeta}{2};\frac{r^{4}}{r_{h}^{4}}\right)$
$\displaystyle+d_{2}\ r_{h}^{1-\zeta}\ r^{-1+\zeta}\
{}_{2}F_{1}\left(-\frac{1}{4}+\frac{\zeta}{4},\frac{1}{4}+\frac{\zeta}{4};1+\frac{\zeta}{2};\frac{r^{4}}{r_{h}^{4}}\right),$
(48)

where $d_{1}$ and $d_{2}$ are integration constants. From Eq.(48) we will fix
them such that $\delta E_{t}(r)$ is regular near the horizon (or in the
interior). The boundary condition we seek is $\delta E_{t}(r=r_{h})_{gen}=0$.
We define

$\displaystyle\left.\delta E_{t}(r,r_{h},b,k,d_{3})\right|_{p}$
$\displaystyle\equiv r_{h}^{1+\zeta}\ r^{-1-\zeta}\
{}_{2}F_{1}\left(-\frac{1}{4}-\frac{\zeta}{4},\frac{1}{4}-\frac{\zeta}{4};1-\frac{\zeta}{2};\frac{r^{4}}{r_{h}^{4}}\right)$
$\displaystyle+d_{3}\ r_{h}^{1-\zeta}\ r^{-1+\zeta}\
{}_{2}F_{1}\left(-\frac{1}{4}+\frac{\zeta}{4},\frac{1}{4}+\frac{\zeta}{4};1+\frac{\zeta}{2};\frac{r^{4}}{r_{h}^{4}}\right),$
(49) (such that $\delta(E_{t})_{gen}=d_{1}\ \left.\delta
E_{t}(r,r_{h},b,k,d_{3})\right|_{p}$ and $d_{3}:=d_{2}/d_{1}$)

Then we evaluate $\left.\delta E_{t}(r=r_{h},r_{h},b,k,d_{3})\right|_{p}$ at
the horizon and find,

$\displaystyle\left.\delta
E_{t}(r=r_{h},r_{h},b,k,d_{3})\right|_{p}=\frac{\Gamma\left(1-\frac{\zeta}{2}\right)}{\Gamma\left(\frac{3}{4}-\frac{\zeta}{4}\right)\Gamma\left(\frac{5}{4}-\frac{\zeta}{4}\right)}+\frac{d_{3}\
\Gamma\left(1+\frac{\zeta}{2}\right)}{\Gamma\left(\frac{3}{4}+\frac{\zeta}{4}\right)\Gamma\left(\frac{5}{4}+\frac{\zeta}{4}\right)}$
(50)

We further fix $d_{3}$ as

$\displaystyle
d_{3}(r_{h},b,k)=-\frac{\Gamma\left(1-\frac{\zeta}{2}\right)\Gamma\left(\frac{3}{4}+\frac{\zeta}{4}\right)\Gamma\left(\frac{5}{4}+\frac{\zeta}{4}\right)}{\Gamma\left(1+\frac{\zeta}{2}\right)\Gamma\left(\frac{3}{4}-\frac{\zeta}{4}\right)\Gamma\left(\frac{5}{4}-\frac{\zeta}{4}\right)}$
(51)

Note that, $d_{3}$ is chosen such a way that $\left.\delta
E_{t}(r=r_{h},r_{h},b,k,d_{3})\right|_{p}=0$ or in other words, $\delta
E_{t}(r=r_{h})_{gen}=0$888Note that, $\left.\delta
E_{t}(r=r_{h},r_{h},b,k,d_{3})\right|_{p}$ is a ‘particular’ solution (hence
the notation $\left.\delta E_{t}\right|_{p}$) which satisfies the boundary
condition $\delta E_{t}(r=r_{h})_{gen}=0$..

####  IV.1.2 Regular solution

After $d_{3}$ is fixed as above we obtain the following regular solution,

$\displaystyle\delta E_{t}(r)=\ $ $\displaystyle r_{h}^{1-\zeta}\
r^{-1-\zeta}\ \Gamma\left(1-\frac{\zeta}{2}\right)\left[r_{h}^{2\zeta}\
{}_{2}\tilde{F}_{1}\left(-\frac{1}{4}-\frac{\zeta}{4},\frac{1}{4}-\frac{\zeta}{4};1-\frac{\zeta}{2};\frac{r^{4}}{r_{h}^{4}}\right)-\right.$
$\displaystyle\left.2^{-\zeta}\ r^{2\zeta}\
\frac{\Gamma\left(\frac{3}{2}+\frac{\zeta}{2}\right)}{\Gamma\left(\frac{3}{2}-\frac{\zeta}{2}\right)}\
{}_{2}\tilde{F}_{1}\left(-\frac{1}{4}+\frac{\zeta}{4},\frac{1}{4}+\frac{\zeta}{4};1+\frac{\zeta}{2};\frac{r^{4}}{r_{h}^{4}}\right)\right],$
(52)

where ${}_{2}\tilde{F}_{1}$ is the regularized hypergeometric function.

Now we examine Eq.(52) in the vanishing magnetic field limit that is for
$b=0$,

$\displaystyle\delta E_{t}(r)\underset{b\to
0}{\to}-1+\frac{r_{h}^{2}}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{2}}\right).$
(53)

Now let us look at the boundary expansion of Eq.(52) $\left(\text{up to}\
\mathcal{O}\left(\frac{1}{r^{2}}\right)\right))$,

$\displaystyle\delta E_{t}(r)\underset{r\to\infty}{\to}\ $
$\displaystyle\left(-\frac{1}{r_{h}^{4}}\right)^{\frac{1}{4}-\frac{\zeta}{4}}\
r_{h}^{1-\zeta}\left(\left(-\frac{1}{r_{h}^{4}}\right)^{\frac{\zeta}{2}}\
r_{h}^{2\zeta}-\tan\left(\left(\frac{1}{4}+\frac{\zeta}{4}\right)\pi\right)\right)\left(\frac{\sqrt{\pi}\
\Gamma\left(1-\frac{\zeta}{2}\right)}{\Gamma\left(\frac{1}{4}-\frac{\zeta}{4}\right)\Gamma\left(\frac{5}{4}-\frac{\zeta}{4}\right)}\right)$
$\displaystyle+$
$\displaystyle\frac{2}{r^{2}}\left(-\frac{1}{r_{h}^{4}}\right)^{\frac{3}{4}-\frac{\zeta}{4}}\
r_{h}^{5-\zeta}\left(\left(-\frac{1}{r_{h}^{4}}\right)^{\frac{\zeta}{2}}\
r_{h}^{2\zeta}-\cot\left(\left(\frac{1}{4}+\frac{\zeta}{4}\right)\pi\right)\right)\left(\frac{\sqrt{\pi}\
\Gamma\left(1-\frac{\zeta}{2}\right)}{\Gamma\left(-\frac{1}{4}-\frac{\zeta}{4}\right)\Gamma\left(\frac{3}{4}-\frac{\zeta}{4}\right)}\right).$
(54)

Notice that the above expression is of the form, $A+Br^{-2}$. From this we can
find the charge susceptibility as $\chi=-\frac{2B}{A}$ (using the definition
of the current from (21)).

$\displaystyle\chi(r_{h},b)=\ $
$\displaystyle-2r_{h}^{2}\left(\frac{\zeta^{2}-1}{16\pi^{2}}\cos\left(\frac{\zeta\pi}{2}\right)\Gamma^{2}\left(\frac{1-\zeta}{4}\right)\Gamma^{2}\left(\frac{1+\zeta}{4}\right)\right)=\
-2r_{h}^{2}\ g(\zeta)\equiv\ -2T^{2}\,\tilde{g}(b/T^{2}),$ (55)

where999We have used $\Gamma(1+x)=x\Gamma(x)$ and
$\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin\left(\pi x\right)}$ to obtain Eq.(55)
from Eq.(54).
$\tilde{g}(b/T^{2})=\pi^{2}g(\zeta)\equiv\frac{\zeta^{2}-1}{16}\cos\left(\frac{\zeta\pi}{2}\right)\Gamma^{2}\left(\frac{1-\zeta}{4}\right)\Gamma^{2}\left(\frac{1+\zeta}{4}\right)$.
Note that $g(\zeta)$ is manifestly an even function of $\zeta$. Hence,
$g(\zeta)$ is analytic as a function of $\zeta^{2}$ and not $\zeta$ which we
wanted to show (see (45)), and $g(\zeta)$ is analytic at $\zeta=0$ (see
Appendix C for further details on the $\zeta=0$ case).

Note that $\lim\limits_{\zeta\to\pm 1}g(\zeta)=-1$. Furthermore, the vanishing
magnetic field limit is $b=0$ which corresponds to $\zeta=\pm 1$ (from the
definition of $\zeta$). Hence, we find that in the vanishing magnetic field
limit, $\chi(r_{h},b=0)=2r_{h}^{2}$, which is the usual charge susceptibility
for the conventional black brane, and matches with what we would have gotten
from computing the susceptibility from Eq.(53)).

In the field-theoretical study figueroa1 , the corresponding susceptibility
was taken to be the free fermion result at zero magnetic field
$\chi_{s}=\frac{1}{6}\,T^{2}$. Let us contrast this to the susceptibility
obtained above in Eq.(55) from holography. A key difference is that the
proportionality factor relating $\chi$ and $T^{2}$ is no longer a constant but
a function of $b/T^{2}$, namely $\tilde{g}(b/T^{2})$. (Presumably a similar
effect would exists in a perturbative approach, where one would simply
consider the effects of Landau levels on the charge susceptibility).

A plot of $\chi/T^{2}$ as a function of $kb/T^{2}$ is shown in Figure 1.

Figure 1: $\chi/T^{2}$ as a function of $kb/T^{2}$

##  V Hydrodynamic limit

We now solve the bulk equations of motion in a small frequency limit; we will
see an analogue of the chiral magnetic effect appear in this limit, and we
will also reproduce from the bulk certain aspects of the hydrodynamic
calculation in Section II.

Let us begin by noting from (19) that the equation of motion for the 2-form
$B_{2}$ can be written as

$\nabla_{P}{{\mathcal{H}}}^{PQR}=0$ (56)

where the 3-form ${{\mathcal{H}}}_{3}$ is defined as

${{\mathcal{H}}}^{PQR}\equiv
H^{PQR}\,+\,\frac{k}{2}\left[\epsilon^{PQRMN}E^{L}H_{LMN}\right]\,+\,\frac{k}{6}\left[H_{LMN}\left(E^{P}\epsilon^{LMNQR}+E^{Q}\epsilon^{LMNRP}+E^{R}\epsilon^{LMNPQ}\right)\right]$
(57)

with $H_{3}=dB_{2}$. This general form – i.e. that the equation of motion can
be written as the divergence of a 3-form ${{\mathcal{H}}}_{3}$ – follows from
the fact that the action is a function of $dB_{2}$ alone.

We are now interested in solving these equations of motion in a hydrodynamic
limit, i.e. with $\frac{\omega}{T}\to 0$. When taking a small frequency limit
in AdS/CFT, it is useful to use the formalism of the membrane paradigm
Iqbal:2008by . The usual infalling membrane boundary condition as applied to
the modified field-strength ${{\mathcal{H}}}$ results in the following
condition at the black hole horizon:

$\sqrt{-g}{{\mathcal{H}}}^{rxy}(r_{h})={{\mathcal{H}}}_{txy}(r_{h})\Sigma(r_{h})\qquad\Sigma(r)\equiv\sqrt{\frac{-g}{-g_{rr}g_{tt}}}g^{xx}g^{yy}$
(58)

See Hofman:2017vwr  for an application of these techniques to a minimally
coupled 2-form $B_{2}$. Importantly, it is shown there that the quantity
$\Sigma(r_{h})$ can be understood as (twice)101010The factor of $2$ arises
from a different normalization of the bulk action from Hofman:2017vwr . the
conventional electric resistivity $2\rho$.

We now study the consequences of this boundary condition for fluctuations
about a background field configuration where $H_{rtz}\neq 0$ as in (40). We
will study a configuration where the nonzero components of the fluctuations
are ${{\mathcal{H}}}_{rxy},{{\mathcal{H}}}_{txy}$, $E_{t}$ and $E_{r}$.

We begin by writing out:

${{\mathcal{H}}}_{txy}=H_{txy}+2k\sqrt{-g}E_{t}H^{rtz}$ (59)

(where we have used an orientation in which $\epsilon_{txyrz}<0$). The
boundary condition (58) thus implies that at the horizon we have

$\sqrt{-g}{{\mathcal{H}}}^{rxy}(r_{h})=\Sigma(r_{h})\left(H_{txy}+2k\sqrt{-g}E_{t}H^{rtz}\right)\bigg{|}_{r=r_{h}}$
(60)

We would now like to propagate this information to the boundary, where it can
be given an interpretation in the field theory. The equations of motion in the
low-frequency limit take the form

$\partial_{r}\left(\sqrt{-g}{{\mathcal{H}}}^{rxy}\right)=0\qquad\partial_{r}H_{txy}=0$
(61)

where the former is the $xy$ component of the diagonal equation of motion (56)
and the latter is the Bianchi identity associated with $H_{3}=dB_{2}$. Thus we
can evaluate the expression above at the AdS boundary:

$\sqrt{-g}{{\mathcal{H}}}^{rxy}(\infty)=\Sigma(r_{h})\left(H_{txy}(\infty)+2k\sqrt{-g}E_{t}(r_{h})H^{rtz}(r_{h})\right)$
(62)

Now we note that
$J^{xy}=-\lim_{r\to\infty}{{\frac{1}{2}}}\sqrt{-g}{{\mathcal{H}}}^{rxy}$, and
we thus find that

$J^{xy}=-\Sigma(r_{h}){{\frac{1}{2}}}\left(H_{txy}(\infty)+2k\sqrt{-g}E_{t}(r_{h})H^{rtz}(r_{h})\right)$
(63)

Here $J^{xy}$ may be understood as the electric field in the $z$ direction
$\mathcal{E}_{z}$; as explained before,
$J^{tz}=-{{\frac{1}{2}}}\sqrt{-g}H^{rtz}$ is the background magnetic field.
Now $H_{txy}$ is the applied source; let us set it to zero. We then find the
following expression:

$J^{xy}=2k\Sigma(r_{h})E_{t}(r_{h})J^{tz}$ (64)

Let us now pause to dissect this result. This appears reminiscent of the
chiral magnetic effect; in the conventional language of electric and magnetic
fields as in (3), $J^{xy}$ is proportional to the electric field in the $z$
direction; thus we see that in the absence of external sources, there is an
electric field parallel to the applied magnetic field, where the constant of
proportionality is $2k\Sigma(r_{h})E_{t}(r_{h})$. In comparing to field
theory, we note that $\Sigma(r_{h})=2\rho$, i.e. the conventional electric
resistivity in this theory.

Of course, if we are at precisely zero frequency, then we are required to have
that $E_{t}(r_{h})=0$. This is completely consistent with the known physics of
the chiral magnetic effect, which states that the equilibrium value of the
chiral magnetic effect for the consistent vector current is zero
Landsteiner:2016led .111111Here – as explained in detail in
Landsteiner:2016led  – one must be careful about the distinction between the
consistent and covariant currents; the covariant chiral magnetic effect is not
zero, but the consistent one receives contributions both from the axial
chemical potential and the value of the axial gauge field source, which
precisely cancel in equilibrium. This is thus an unexciting but expected
result.

Now we should note however that in this work we are interested in small
fluctuations around equilibrium. If $\omega\neq 0$, then it is no longer
required that $E_{t}(r_{h})=0$ (indeed, in the conventional case of a massless
gauge field, this quantity is no longer even gauge invariant). Let us instead
allow $E_{t}(r)\neq 0$ and use the small frequency analysis above to compute
the relaxation rate of the axial charge. Here we will make contact with the
hydrodynamic calculation above, and we will thus study a situation where
$E_{t}(\infty)=0$, i.e. there is no axial source applied.

We first use (24) to write down

$\partial_{t}j_{A}^{t}=8kJ^{xy}J^{tz}$ (65)

This equation holds at all $r$, (indeed, from above, all of the expressions in
it are radially constant), and so we can evaluate it at the boundary to find:

$-i\omega j^{t}_{A}=16k^{2}\Sigma(r_{h})(J^{tz})^{2}E_{t}(r_{h})$ (66)

Solving this for $\omega$ we find:

$\omega=16i\left(\frac{E_{t}(r_{h})}{j^{t}_{A}}\right)\Sigma(r_{h})\left(kJ^{tz}\right)^{2}$
(67)

Thus we see that there is a diffusion pole, where the coefficient of the pole
varies as the magnetic field squared. We stress that the approximation made
was $\omega\to 0$; from above we see that this also requires that $kJ^{tz}\to
0$. Away from that limit, we expect to see deviations from the quadratic
expression above.

Let us also examine the pre-factor of the expression in the limit $kJ^{tz}\to
0$. We see that the ratio of $E_{t}(r_{h})$ and $j^{t}_{A}$ appears. We can
evaluate this from Eq.(53) (with the appropriate boundary conditions:
$E_{t}(r_{h})\neq 0$ and $E_{t}(\infty)=0$) and Eq.(21) to get,

$\displaystyle\frac{E_{t}(r_{h})}{j^{t}_{A}}\underset{r\to\infty}{\to}\,-\frac{1}{2r_{h}^{2}}\,\equiv\,-\chi^{-1}.$
(68)

We thus find (evaluating $\Sigma(r_{h})=\frac{1}{r_{h}}$ from (39)):

$\omega=-2{ik^{2}}\,\left(\frac{b^{2}}{r_{h}^{3}}\right).$ (69)

We should compare it to the expectation from elementary hydrodynamics given in
(11). Putting in the holographic expressions for the field-theoretical
quantities in (11) in the small magnetic field limit, using
$\sigma=\frac{1}{\rho}=2r_{h}$, and $\chi=2r_{h}^{2}$, and $B=\frac{b}{2}$, we
find that (11) becomes $\Gamma_{5,n}=\frac{2k^{2}b^{2}}{r_{h}^{3}}$, i.e.
precisely the same as the pole exhibited above. This agreement is not
surprising; indeed it can be seen that the derivation above parallels in the
bulk the hydrodynamic calculation leading to (11).

However, here we see the limitations of the hydrodynamic calculation – in
particular, we see explicitly in this holographic model that the analytic
calculation is expected to break down if $kJ^{tz}$ is not small; i.e. the
result above is valid only in the small $b$ limit. In the next section we
explicitly compute the same relaxation rate numerically and compare with
lattice results.

##  VI Numerical results

In this section we calculate the quasi-normal modes of our system using
standard holographic techniques.

###  VI.1 Contributing equations of motion

From here on we shall work in ingoing Eddington-Finkelstein coordinates
($r,v,x,y,z$) rather than the Schwarzschild coordinates used above (see
Appendix A for a brief review of the coordinate system). First let us give
some useful expressions,

$\displaystyle E^{r}=(E_{v}+r^{2}fE_{r}),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
E^{v}=E_{r},$ $\displaystyle
H^{rxy}=\frac{f}{r^{2}}\partial_{r}\left(B_{xy}\right)-\frac{i\omega}{r^{4}}B_{xy},\
\ \ \ \ H^{vxy}=\frac{1}{r^{4}}\partial_{r}\left(B_{xy}\right).$

We will study finite frequency fluctuations about the equilibrium solution
(40). Consider $H\to H_{0}+\delta H$ and $E\to E_{0}+\delta E$ with $E_{0}=0$
and $H_{0}=H^{rvz}$, which are the background solutions Eq.(40) (in the
ingoing coordinates). $H_{0}$ and $E_{0}$ are the background fields and
$\delta H$ and $\delta E$ are the fluctuations of the fields that we are
interested in.

Before proceeding to solve Eq.(15) let us first note that we are interested in
wave-like solutions for $\delta B$ and $\delta E$ of the form $e^{-i\omega
v}$. So, we are considering the corresponding wave vector to be of the form
$k^{\mu}=(\omega,\vec{k})=(\omega,0)$, i.e. the only non-zero momentum is in
the time direction. Furthermore, since the magnetic field is in the
$z$-direction, the little group for fluctuations is $SO(2)$: the group of
rotations in the $x-y$ plane. Therefore, we have the following contributing
equations of motion in the relevant channels for the fluctuations: in the
antisymmetric tensor channel (i.e. from the $(\rho\sigma=xy)$ components of
Eq.(15)),

$\displaystyle-\left[\partial_{v}\left(\sqrt{-g}\ \delta
H^{vxy}\right)+\partial_{r}\left(\sqrt{-g}\ \delta H^{rxy}\right)\right]$
$\displaystyle-$ $\displaystyle 2k\ \left[\partial_{v}\left(\sqrt{-g}\ \delta
E^{v}\epsilon^{rvzxy}H_{rvz}\right)+\partial_{r}\left(\sqrt{-g}\ \delta
E^{r}\epsilon^{rvzxy}H_{rvz}\right)\right]=0.\ \ \ \ (\rho\sigma=xy)$ (70)

Similarly, from Eq.(16) we have

$\displaystyle 4k^{2}H^{2}\ \delta E^{v}-k\epsilon_{rvzxy}H^{rvz}\ \delta
H^{vxy}=-\frac{1}{\sqrt{-g}}\partial_{r}\left(\sqrt{-g}\ \delta
G^{rv}\right),$ (71) $\displaystyle 4k^{2}H^{2}\ \delta
E^{r}-k\epsilon_{rvzxy}H^{rvz}\ \delta
H^{rxy}=-\frac{1}{\sqrt{-g}}\partial_{v}\left(\sqrt{-g}\ \delta
G^{vr}\right).$ (72)

In the above contributing equations of motion, we have not considered terms
with $\partial_{x},\ \partial_{y},\ \partial_{z}$ as we have set spatial
momenta to vanish. The vector channel involving $\delta E^{x}$, $\delta
H^{xyz}$ decouples and we will not consider it.

In these coordinates, the background solutions (40) become,

$\displaystyle H^{rvz}=\frac{b}{r^{3}},\qquad H_{rvz}=-\frac{b}{r}.$ (73)

Now we solve for $\delta E^{r}$ in Eq.(72) to obtain,

$\displaystyle\delta E_{r}=\frac{i\omega r^{4}\ \partial_{r}(\delta
E_{v})+4k^{2}b^{2}\delta E_{v}+kbr^{2}f\ \partial_{r}(\delta
B_{xy})-kbi\omega\delta B_{xy}}{\tilde{\gamma}},$ (74)

where $\tilde{\gamma}:=r^{4}\omega^{2}-4b^{2}k^{2}r^{2}f$.

Next we give Eqs.(70) and (71) in terms of the fields with all indices
downstairs,

$\displaystyle\partial_{r}^{2}\left(\delta
B_{xy}\right)\left[-rf\right]+\partial_{r}\left(\delta
B_{xy}\right)\left[-f-\frac{4r_{h}^{4}}{r^{4}}+\frac{2i\omega}{r}\right]+\delta
B_{xy}\left[\frac{-i\omega}{r^{2}}\right]$ $\displaystyle-$
$\displaystyle\partial_{r}\left(\delta
E_{v}\right)\left[\frac{2kb}{r}\right]+\delta
E_{v}\left[\frac{2kb}{r^{2}}\right]$ $\displaystyle-$
$\displaystyle\partial_{r}\left(\delta E_{r}\right)\left[2kbrf\right]+\delta
E_{r}\left[-2kbf-\frac{8kbr_{h}^{4}}{r^{4}}+\frac{2kbi\omega}{r}\right]=0,$
(75) $\displaystyle\frac{-4b^{2}k^{2}}{r^{4}}\delta
E_{r}-\frac{kb}{r^{4}}\partial_{r}\left(\delta
B_{xy}\right)=\partial_{r}^{2}\left(\delta
E_{v}\right)+\left(\frac{3}{r}\right)\partial_{r}\left(\delta
E_{v}\right)+i\omega\partial_{r}\left(\delta
E_{r}\right)+\left(\frac{3}{r}\right)i\omega\delta E_{r}.$ (76)

If we substitute $\delta E_{r}$ from Eq.(74) into Eqs.(75) and (76), then we
obtain two coupled ODEs – these are somewhat lengthy so we do not present them
explicitly, but we solve them numerically below.

###  VI.2 Numerics

Now we shall solve Eq.(75) and Eq.(76) numerically using a mid-point
shooting121212In mid-point shooting method we numerically integrate the
boundary solution from boundary and horizon solution from horizon and adjust
these till they meet somewhere in the middle and this adjustment yields the
QNM. method (see e.g. 2019  for a discussion of the shooting method, with some
previous applications to quasinormal modes in PhysRevD.62.024027 ;
PhysRevD.66.124013 ). Below we present some details of the boundary
conditions; the reader interested only in the results can feel free to skip to
the next section.

####  VI.2.1 Logarithmic fall-off

$B_{xy}$ has a logarithmic fall-off near the boundary, associated with the
fact that the double-trace deformation associated with $J^{2}$ on the boundary
is marginally (ir)relevant Hofman:2017vwr . As explained in detail in that
work, the correct boundary condition at the UV cut-off $u=u_{\Lambda}$ takes
the form:

$\displaystyle B_{xy}\left(u_{\Lambda}\right)-\frac{J}{\kappa}=0,$ (77)

where $J=u\partial_{u}B_{xy}$ and $\kappa$ the double-trace coupling for
$J^{2}$. The form of $B_{xy}$ we take is,

$\displaystyle B_{xy}(u)=d_{0}+\sum\limits_{j}d_{j}\
u^{j}+\ln(u)\left[d_{0}^{{}^{\prime}}+\sum\limits_{j}d_{j}^{{}^{\prime}}\
u^{j}\right]$ (78)

where the $d_{i}$ are expansion coefficients. Using (77) we then find:

$d_{0}+\sum\limits_{j}d_{j}\
u^{j}+\ln(u_{\Lambda})\left[d_{0}^{{}^{\prime}}+\sum\limits_{j}d_{j}^{{}^{\prime}}\
u^{j}\right]\\\ -\frac{1}{\kappa}\left[\sum\limits_{j}jd_{j}\
u^{j}+d_{0}^{{}^{\prime}}+\sum\limits_{j}d_{j}^{{}^{\prime}}\
u^{j}+\sum\limits_{j}\ j\ u^{j}\ln(z)d_{j}^{{}^{\prime}}\right]=0.$

Now at $u=u_{\Lambda}$, (VI.2.1) becomes (for $u\to 0$),

$\displaystyle
d_{0}+\ln(u_{\Lambda})d_{0}^{{}^{\prime}}-\frac{d_{0}^{{}^{\prime}}}{\kappa}=0$
$\displaystyle
d_{0}=d_{0}^{{}^{\prime}}\left[\ln\left(e^{\frac{1}{\kappa}}/u_{\Lambda}\right)\right].$
(79)

Thus, at the boundary ($u=0$) we obtain $B_{xy}$ as,

$\displaystyle
B_{xy}(u)=d_{0}^{{}^{\prime}}\ln\left(ue^{\frac{1}{\kappa}}/u_{\Lambda}\right)+\sum\limits_{j}d_{j}\
u^{j}+\left(\ln(u)\right)\sum\limits_{j}d_{j}^{{}^{\prime}}\ u^{j},$
$\displaystyle
B_{xy}(u)=d_{0}^{{}^{\prime}}\ln\left(u/u^{*}\right)+\sum\limits_{j}d_{j}\
u^{j}+\left(\ln(u)\right)\sum\limits_{j}d_{j}^{{}^{\prime}}\ u^{j}.$ (80)

where $u^{*}=u_{\Lambda}e^{-\frac{1}{\kappa}}$ is an RG-invariant combination
of the double-trace coupling and the UV cutoff; this is the analogue of the
Landau pole in regular QED, and by dimensional transmutation all physical
results can depend on this alone (see Hofman:2017vwr ; Faulkner:2012gt  for a
discussion in the holographic context).

Now since $B_{xy}$ at the boundary has a logarithmic fall-off, the coupled
nature of the equations of motion as given in Eq.(75) and Eq.(76) imply that
$E_{v}$ has the following form at the boundary:

$\displaystyle E_{v}(u)=u^{2}\left(c_{0}+\sum\limits_{j}c_{j}\
u^{j}\right)+\ln(u)\left(\sum\limits_{j}{cc}_{j}\ u^{j}\right).$ (81)

The logarithm appearing in this boundary condition appears to follow from the
fact that the axial current $j^{\mu}_{A}$ mixes with the 1-form current
$J^{\mu\nu}$.

Next we present the numerical results; see Appendix D for further details on
the numerical implementation.

####  VI.2.2 Hydrodynamic mode

The lowest quasinormal mode $\omega_{l}=-i\Gamma_{A}$ approaches the origin as
$b$ approaches zero. For small $b$ (i.e. for $0<b\leq 3$) it varies with $b$
as

$\displaystyle\Gamma_{A}(r_{h},b)=0.048\
\left(\frac{b^{2}}{r_{h}^{3}}\right).$ (82)

where the prefactor was obtained from a numerical fit, and where we have
restored $r_{h}$ on dimensional grounds.

Next we move onto some higher values of $b$ to show that away from a small
neighbourhood of $b=0$; the quadratic $b^{2}$ behaviour of no longer captures
the full dependence and we obtain a more complicated function of $b$. We
consider $0<b\leq 6.5$ and obtain the behaviour shown in Fig.(2),

Figure 2: $\Gamma_{A}$ vs $b$ with $k=0.15$ and $r_{h}=1$ (blue curve is
numerics, orange curve is quadratic fit)

where the orange curve is as given in Eq.(82). Note that both the curves above
match till about $b\approx 3$.

Now let us compare the above result obtained from numerics to the result
obtained in Section V using the membrane paradigm formalism (in the
$kJ^{tz}\to 0$ limit) in Eq.(69). Note that if we put $k=0.15$ in Eq.(69) then
we find,

$\displaystyle\Gamma_{A}(r_{h},b)=0.045\
\left(\frac{b^{2}}{r_{h}^{3}}\right).$ (83)

in approximate agreement (within 6 %) with the small-frequency limit of the
numerics. As mentioned in that section, the membrane paradigm analysis also
agrees with elementary hydrodynamics arguments arising from treating
electrodynamics perturbatively; thus we conclude that at small $b$ the the
conventional chiral MHD approach from weakly gauged electrodynamics is valid.

However, from Fig.(2) we notice that for $b\gg 1$, the functional dependence
of $\Gamma_{A}$ on $b$ is no longer quadratic, and is a non-trivial function
of $b$. This function now appears to depend on UV physics, and is not simply
determined by other thermodynamic quantities such as the susceptibility
$\chi$.

For example, we can try to improve the hydrodynamic result for $\Gamma_{A}$ in
(11) with the holographically determined susceptibility in (55). The resulting
plot as a function of the magnetic field $B=\frac{b}{2}$ is shown in Figure 3.

Figure 3: $\Gamma^{\mathrm{improved}}_{A}$ from (11) (i.e. chiral MHD with
weakly coupled electrodynamics) as a function of $b$ with $k=0.15$ and
$r_{h}=1$; note it does not capture the non-trivial dependence on $b$ seen in
the numerical results of Figure 2

.

It appears barely different from the quadratic dependence as $\chi$ does not
depend strongly on $b$. Indeed if we expand this improved hydrodynamic attempt
at approximating $\Gamma^{\mathrm{improved}}_{A}$ as a series in $b$ we get,

$\displaystyle\Gamma_{A}^{\mathrm{improved}}(b)=2\,k^{2}\left(\frac{b^{2}}{r_{h}^{3}}\right)+\mathcal{O}(b^{4})=0.045\,b^{2}+\mathcal{O}(b^{4}).$
(84)

where in the second equality above we have put $k=0.15$ and $r_{h}=1$ (for a
comparison with the numerical parameters) and the coefficient of
$\mathcal{O}(b^{2})$ is $100$ times that of the coefficient of
$\mathcal{O}(b^{4})$. Thus the dependence of the charge susceptibility on $b$
is insufficient to account for the non-trivial dependence of $\Gamma_{A}(b)$,
and it appears to not be determined by thermodynamic data.

Finally, in our numerical investigation we also observed many non-hydrodynamic
gapped modes. These do not seem to have model-independent relevance, but we we
discuss them in Appendix E.

##  VII Conclusion

In this work we have discussed a holographic model that is in the same
universality class as a Dirac fermion coupled to QED at finite temperature,
i.e. where axial charge is non-conserved due to an anomaly with a dynamical
operator (involving a topological density constructed from the 2-form current
associated with magnetic flux conservation) on the right hand side. We
described the bulk dualization process by which we constructed the holographic
model and computed some basic observables.

Perhaps our most significant result was explicit computation of the axial
charge relaxation rate $\Gamma_{A}$ in the presence of a background magnetic
field; we found that due to the anomaly the axial charge density $j^{t}_{A}$
is not conserved, and instead obeys an equation of the form $j^{t}_{A}\sim
e^{-\Gamma_{A}t}$, where $\Gamma_{A}$ was numerically found through solving
the bulk equations of motion. It is a nontrivial function of the background
magnetic field $B$ and the temperature $T$, and can be seen in Figure 2. Note
that at small magnetic field this relaxation rate is quadratic in the field
$B$; indeed as the pole approaches the origin the pre-factor may be computed
analytically from the small frequency limit of the bulk equations of motion.
This pre-factor can also be obtained from elementary magneto-hydrodynamic
arguments that essentially treat the anomaly coefficient perturbatively, as
reviewed in Section II. The resulting pre-factor agrees with our holographic
computation, and we thus find that at small magnetic fields:

$\Gamma^{\mathrm{hol}}_{A}(B\to 0)\approx\gamma^{\mathrm{MHD}}B^{2}$ (85)

At larger magnetic fields this is a non-trivial function of $B$ and can only
be obtained from a full holographic treatment.

We now discuss the previous literature on this result. A lattice study of a
field-theoretical model in the same universality class was recently performed
in figueroa1 ; Figueroa:2019jsi ; in particular, figueroa1  studied the
diffusion of the operator $F\wedge F$ (which can be related to the charge
relaxation rate by a fluctuation-dissipation argument), and Figueroa:2019jsi
directly measured $\Gamma_{A}$. Those works numerically observe an expression
of the form:

$\Gamma^{\mathrm{lattice}}_{A}(B)\approx\gamma^{\mathrm{lattice}}B^{2},$ (86)

Interestingly, those works found that
$\gamma^{\mathrm{lattice}}/\gamma^{\mathrm{MHD}}\approx 10$, i.e. that the
pre-factor obtained from the lattice differs from the hydrodynamic estimate by
an order of magnitude Figueroa:2019jsi . In those works it was argued that
this means that short-distance physics that is not taken into account in the
hydrodynamic analysis is important. Interestingly, this is not what we find
from our (UV-complete) holographic calculation; instead our holographic result
precisely coincides with the hydrodynamic result at small magnetic fields,
differing from it only at larger fields when the magnetic field _itself_
probes UV scales.

It is interesting to speculate on the cause of this discrepancy between the
lattice and holography. An ingredient entering into the computation of
$\gamma^{\mathrm{MHD}}$ is the resistivity of the electromagnetic sector; in
holography it is very easy to see how this enters into the calculation and
separate it from the anomalous dynamics but in a purely field-theoretical
treatment it seems possible that uncertainties in this conductivity – a
notoriously complicated quantity to calculate from first principles – could
cloud this analysis, as was already suggested in Figueroa:2019jsi . It would
be interesting to perform further tests of this hypothesis, perhaps by
computing more observables from holography and the lattice and comparing them
further.

If we take our results at face value, it suggests that for this observable, a
hydrodynamic treatment of conventional MHD (treating the anomaly as a
perturbation) is sufficient at weak magnetic fields, though it differs
quantitatively from the true result at stronger fields.131313This is
philosophically aligned with previous results Jensen:2013vta  that argue that
various transport coefficients are generically renormalized if the gauge
fields sourcing the anomaly are dynamical.

There are many directions for future research. Our bulk action Eq.(III.2.3)
permits the explicit study of a strongly interacting system in the same
universality class as the chiral plasma. It would be very interesting to
understand other phenomena, e.g. if the instabilities (due to non-vanishing
$\vec{k}$) PhysRevD.100.065023  exist in this model, or the study of chiral
magnetic waves Kharzeev:2010gd .

From a field-theoretical point of view, it would be very interesting to go
further than our holographic considerations and construct a true effective
hydrodynamic theory for this system. Indeed this was one of the motivations
for our construction of the holographic action (III.2.3), though it is
sufficiently complicated that it does not shed much immediate light on how (or
whether) an effective description could be computed. Indeed, the prospect of
such an analysis is clouded by the fact that we are not aware of a completely
universal field-theoretical description of the anomaly (1); conventional lore
would tell us the axial symmetry is simply completely broken, though as we
have argued this appears to miss important physics associated with the fact
that it is broken not by a generic operator but rather by a topological
density $J\wedge J$ constructed from the current of a 1-form symmetry. Indeed,
along these lines recent work describes a novel higher group structure that is
present when the axial symmetry is spontaneously broken Brennan:2020ehu ;
Hidaka:2020iaz ; Hidaka:2020izy . It would be very interesting to understand
whether such an analysis could be extended to the phase when the axial
symmetry is unbroken or realized at finite temperature, as a first step
towards constructing a hydrodynamic EFT.

Note added: Shortly after the first version of this paper appeared on the
arXiv, two papers Choi:2022jqy ; Cordova:2022ieu  appeared where a precise
characterization of the ABJ anomaly is given in terms of non-invertible
symmetries. It would be very interesting to use this refined understanding to
construct a hydrodynamic theory.

## Acknowledgments

This work was supported in part by the STFC Consolidated Grant ST/P000371/1
(NI & RG). We would like to express our gratitude to Daniele Dorigoni, Tin
Sulejmanpasic, Stefano Cremonesi, Iñaki García Etxebarria and Aristomenis
Donos for valuable discussions on gauge transformations, monopoles and
hydrodynamics. We would like to thank Umut Gursoy, Diego Hofman, Kieran
Macfarlane, Napat Poovuttikul, Zheng Liang Lim, Gabriel Arenas-Henriquez,
Richie Dadhley and T.V. Karthik for useful discussions on related issues.

##  Appendix A Conventions

We work in natural units with $c=\hbar=1$ and our metric signature is mostly
plus: $(-,+,+,\cdots\cdots,+)$.

For boundary indices we have $\mu\nu\rho\sigma\lambda\cdots$ and for bulk
indices we have $MNPQR\cdots$. We have for the epsilon symbol,
$\tilde{\epsilon}_{0123\cdots}=+1$ and for the volume form,
$\epsilon=+\sqrt{-g}\ d^{D}x$, in $D$ spacetime dimensions.

In ingoing EF coordinates ($r,v,x,y,z$) we have,

$\displaystyle
ds^{2}_{5}=-r^{2}f(r)dv^{2}+2dvdr+r^{2}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(87) $\displaystyle\sqrt{-g}=r^{3},$ (88)
$\displaystyle\epsilon_{rvzxy}=r^{3}\tilde{\epsilon}_{rvzxy}=r^{3},$ (89)
$\displaystyle\epsilon^{rvzxy}=-r^{-3},$ (90)

where $f\equiv f(r):=\left(1-\frac{r_{h}^{4}}{r^{4}}\right)$ and
$\partial_{r}f(r)=\frac{4r_{h}^{4}}{r^{5}}$.

We also record a few useful identities relating differential forms to their
components, starting with:

$A_{p}\wedge\star
A_{p}=\frac{1}{p!}A_{\mu_{1}\cdots\mu_{p}}A^{\mu_{2}\cdots\mu_{p}}\epsilon,$
(91)

Integrating we find,

$\int_{{{\mathcal{M}}}_{D}}A_{p}\wedge\star
A_{p}=\frac{1}{p!}\int_{{{\mathcal{M}}}_{D}}d^{D}x\sqrt{-g}A_{\mu_{1}\cdots\mu_{p}}A^{\mu_{1}\cdots\mu_{p}}.$
(92)

We now also sometimes use expressions of the following form,

$\int_{{{\mathcal{M}}}_{5}}H_{3}\wedge
F_{2}=\frac{(-1)^{s}}{2!3!}\int_{{{\mathcal{M}}}_{5}}d^{5}x\sqrt{-g}\epsilon^{\mu\nu\rho\alpha\beta}H_{\mu\nu\rho}F_{\alpha\beta}$
(93)

where $s$ is the number of minus signs in the metric.

##  Appendix B Inverse operation

In terms of tensor-index notation, Eq.(30) is,

$\displaystyle F_{MN}=-\frac{1}{2}\epsilon_{PQRMN}\ (\partial^{P}B^{QR})+k\
\epsilon_{PQRMN}\ A^{P}F^{QR},$ $\displaystyle\text{leading to},\ $
$\displaystyle\left[\delta^{Q}_{M}\delta^{R}_{N}-k\ \epsilon_{IJKMN}\
g^{JQ}g^{KR}A^{I}\right]F_{QR}=-\frac{1}{2}\epsilon_{IJKMN}\
\partial^{I}(B^{JK}),$ (94) $\displaystyle\text{leading to},\ $
$\displaystyle\mathcal{O}^{QR}_{\ \ MN}\ F_{QR}=-\frac{1}{2}\epsilon_{IJKMN}\
\partial^{I}(B^{JK}),$ (95)

where $\mathcal{O}^{QR}_{\ \ MN}\ :=\ \delta^{Q}_{M}\delta^{R}_{N}-k\
\epsilon_{IJKMN}\ g^{JQ}g^{KR}A^{I}$.

Now the task is to invert $\mathcal{O}^{QR}_{\ \ MN}$ to obtain
$({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ such that $\mathcal{O}^{QR}_{\ \ MN}\
({\mathcal{O}^{-1}})^{MN}_{\ \ LK}=\delta^{Q}_{L}\delta^{R}_{K}$.141414Note
that $\delta^{Q}_{L}\delta^{R}_{K}F_{QR}=F_{LK}$. So, in the space of $2$
forms $\delta^{Q}_{L}\delta^{R}_{K}$ is the identity operator. Then,
$({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ would enable us to write $F_{2}$ in terms
of $B_{2}$ and $A_{1}$ and their derivatives which is what we are after.

Note that we are not assuming that $\mathcal{O}^{QR}_{\ \ MN}$ or
$({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ is anti-symmetric at this point. However
ultimately they will be contracted with $F_{QR}$ (for instance, see Eq.(94))
and their symmetric parts would cancel out and things will turn out to be
consistent.

Let us consider the most general $({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ possible
(arranged in ascending powers of $A_{1}$) and then we shall demand it to be
$\mathcal{O}^{QR}_{\ \ MN}$’s inverse. The most general expression for
$({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ is,

$\displaystyle({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ $\displaystyle=c_{0}\
\delta^{M}_{L}\delta^{N}_{K}\ +\ \bar{c}_{0}\ \delta^{M}_{K}\delta^{N}_{L}\ +\
bk\ \epsilon_{PIJLK}\ A^{P}\ g^{IM}g^{JN}\ +\
c_{2}k^{2}\delta^{M}_{L}A^{N}A_{K}\ +\
\bar{c}_{2}k^{2}\delta^{N}_{L}A^{M}A_{K}$ $\displaystyle+\
c_{2}^{{}^{\prime}}k^{2}\delta^{N}_{K}A^{M}A_{L}\ +\
\tilde{c}_{2}k^{2}\delta^{M}_{K}A^{N}A_{L},$ (96)

where $c_{0},\ \bar{c}_{0},\ c_{1},\ c_{2},\ \bar{c}_{2},\
c_{2}^{{}^{\prime}},\ \tilde{c}_{2}$ are coefficients to be determined by
demanding that $({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ is the inverse of
$\mathcal{O}^{QR}_{\ \ MN}$ (their subscript is numbered as per the powers of
$A_{1}$ they are coefficients of). Note that we cannot have any more powers of
$A_{1}$ in the above expression (in the sense of anti-symmetric indices of
$A_{1}$) as when they would be contracted with $\epsilon_{PIJQR}$ they would
cancel. Note that, every term other than terms whose coefficients are $c_{0}$
and $\tilde{c}_{0}$ have to come with some powers of $k$ otherwise on $k\to 0$
limit they would not give the proper inverse of the $\left.\mathcal{O}^{QR}_{\
\ \ MN}\right|_{k\to 0}=\delta^{Q}_{M}\delta^{R}_{N}$ as they would survive
the $k\to 0$ limit.

Now demanding, $\mathcal{O}^{QR}_{\ \ MN}\ ({\mathcal{O}^{-1}})^{MN}_{\ \
LK}=\delta^{Q}_{L}\delta^{R}_{K}$ we get the following equation,

$\displaystyle(c_{0}+2c_{1}k^{2}A^{2})\delta^{Q}_{L}\delta^{R}_{K}\ +\
(\bar{c}_{0}-2c_{1}k^{2}A^{2})\delta^{Q}_{K}\delta^{R}_{L}\ +\
(c_{1}-c_{0}+\bar{c}_{0})k\epsilon_{PIJLK}A^{P}\ g^{IQ}g^{JR}$
$\displaystyle+\ (c_{2}-2c_{1})k^{2}\delta^{Q}_{L}A^{R}A_{K}\ +\
(\bar{c}_{2}+2c_{1})k^{2}\delta^{R}_{L}A^{Q}A_{K}\ +\
(c_{2}^{{}^{\prime}}-2c_{1})k^{2}\delta^{R}_{K}A^{Q}A_{L}$ $\displaystyle+\
(\tilde{c}_{2}+2c_{1})k^{2}\delta^{Q}_{K}A^{R}A_{L}=\delta^{Q}_{L}\delta^{R}_{K}$

So the above equation is satisfied if,

$\displaystyle c_{1}=\frac{1}{1+4k^{2}A^{2}},\
c_{0}=\frac{1+2k^{2}A^{2}}{1+4k^{2}A^{2}},\
\bar{c}_{0}=\frac{2k^{2}A^{2}}{1+4k^{2}A^{2}},$ (97) $\displaystyle
c_{2}=c_{2}^{{}^{\prime}}=2c_{1}=\frac{2}{1+4k^{2}A^{2}},\
\bar{c}_{2}=\tilde{c}_{2}=-2c_{1}=\frac{-2}{1+4k^{2}A^{2}}.$ (98)

One can readily check that with the above values for the coefficients
$({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ as given in Eq.(96) is the inverse of
$\mathcal{O}^{QR}_{\ \ MN}$ (and also gives the correct inverse in the $k\to
0$ limit when $c_{0}\to 1$ and only the term with $c_{0}$ as the coefficient
survives).

Next we multiply, Eq.(95) with $({\mathcal{O}^{-1}})^{MN}_{\ \ LK}$ to get
Eq.(36),

For completeness here we express the differential forms that make up $S_{5p}$
in terms of their components (up to quadratic orders in $E$),

$\displaystyle-\frac{1}{2}F_{2}\wedge\star
F_{2}\rightarrow\left[\frac{H^{2}}{12}-3k^{2}\left(E\cdot
H\right)^{2}+\frac{2}{3}k^{2}H^{2}E^{2}+\frac{k}{6}\epsilon_{PQRLK}H^{PQR}E_{M}H^{MLK}\right]\tilde{c}_{1}^{2},$
(99) $\displaystyle+H_{3}\wedge
F_{2}\rightarrow\left[-\frac{H^{2}}{6}-\frac{4}{3}k^{2}H^{2}E^{2}-\frac{k}{6}\epsilon_{PQRLK}H^{PQR}E_{M}H^{MLK}+2k^{2}\left(E\cdot
H\right)^{2}\right]\tilde{c}_{1}^{2},$ (100) $\displaystyle-kE_{1}\wedge
F_{2}\wedge F_{2}\rightarrow\left[2k^{2}\left(E\cdot
H\right)^{2}-\frac{k}{12}\epsilon_{PQRLK}H^{PQR}E_{M}H^{MLK}\right]\tilde{c}_{1}^{2},$
(101) $\displaystyle-\frac{1}{2}G_{2}\wedge\star
G_{2}\rightarrow-\frac{1}{4}G^{2},$ (102)

and when expanded in powers of small $k$,
$\tilde{c}_{1}^{2}=1-8\tilde{k^{2}}E^{2}+\mathcal{O}(k^{4})$.

##  Appendix C $\zeta\to 0$ and hypergeometric differential equation

Note that if we naively put $\zeta=0$ in (48), then we do not get two linearly
independent solutions at $\zeta=0$. This is related to the structure of the
Riemann differential equation about the point $\zeta=0$. Here we give the
general solution to Eq.(44) in the limit of $\zeta\to 0$151515For more
information see section 15.10 of daalhuis1 .,

$\displaystyle\left.\delta E_{t}(r)_{gen}\right|_{\zeta\to 0}$
$\displaystyle=\frac{r_{h}}{r}\left[m_{1}\
{}_{2}F_{1}\left(-\frac{1}{4},\frac{1}{4};1;\frac{r^{4}}{r_{h}^{4}}\right)+m_{2}\
\left\\{{}_{2}F_{1}\left(-\frac{1}{4},\frac{1}{4};1;\frac{r^{4}}{r_{h}^{4}}\right)\ln\left(\frac{r^{4}}{r_{h}^{4}}\right)\right.\right.$
$\displaystyle\left.\left.+\sum\limits_{i=0}^{\infty}\frac{\left(a\right)_{i}\left(b\right)_{i}}{\left(c\right)_{i}i!}\left(\frac{r^{4}}{r_{h}^{4}}\right)^{i}\left(\psi\left(i-\frac{1}{4}\right)+\psi\left(i+\frac{1}{4}\right)-2\psi\left(1+i\right)\right)\right\\}\right],$
(103)

where $a=-\frac{1}{4},\ b=\frac{1}{4},\ c=1,\
\psi(x):=\frac{d}{dx}\ln(\Gamma(x))$ is the digamma function and
$\left(a\right)_{i}$ is the rising Pochhammer symbol defined as,

$\displaystyle\left(a\right)_{i}:=\begin{cases}1,&i=0\\\
a(a+1)\cdots\cdots(a+i-1),&i>0\end{cases}$

##  Appendix D Implementation of numerics

In this appendix we aim to find a searching condition for the numerical
implementation of our quasinormal modes. To do this we perform a simple linear
algebra exercise. Suppose we have $N$ fields $\Phi^{I}(r)$. Let
$\Phi_{a}^{I}(r)$ be a basis for the solutions that are ingoing at horizon
($a\in\\{1,\cdots\cdots,N\\}$). Let $\Phi_{\alpha}^{I}(r)$ be a basis for the
solutions that are outgoing at horizon ($\alpha\in\\{1,\cdots\cdots,N\\}$).
Let $r^{*}$ be the matching point. We wish to know when $\exists$ $C_{a}$ and
$D_{\alpha}$ such that,

$\displaystyle\sum\limits_{a}C_{a}\Phi_{a}^{I}(r^{*})=\sum\limits_{\alpha}D_{\alpha}\Phi_{\alpha}^{I}(r^{*}),$
(104)
$\displaystyle\sum\limits_{a}C_{a}\Phi_{a}^{{}^{\prime}I}(r^{*})=\sum\limits_{\alpha}D_{\alpha}\Phi_{\alpha}^{{}^{\prime}I}(r^{*}),$
(105)

which implies that there exists a solution that satisfies both sets of
boundary conditions.

We can frame the above as a linear algebra problem. Consider
$X_{a}^{A}:=\left\\{\Phi^{I}_{a}(r^{*}),\Phi^{{}^{\prime}I}_{a}(r^{*})\right\\}$
(with $A\in\\{1,\cdots\cdots,2N\\}$) and view $X^{A}_{a}$ as a subspace of
$\mathbb{R}^{2N}$. Similarly, consider
$Y_{\alpha}^{A}:=\left\\{\Phi^{I}_{\alpha}(r^{*}),\Phi^{{}^{\prime}I}_{\alpha}(r^{*})\right\\}$.
We wish to know when the subspaces $X_{a}^{A}$ and $Y_{\alpha}^{A}$ overlap.
This condition will then imply the existence of a solution that would satisfy
both sets of boundary conditions (104) and (105). Let us consider
$\left(Y^{\perp}\right)^{A}_{b}$ that satisfies,

$\displaystyle\sum\limits_{A}\left(Y^{\perp}\right)^{A}_{b}Y^{A}_{\alpha}=0,$
(106)

where $b\in\\{1,\cdots\cdots,N\\}$.

Then we want,

$\displaystyle\sum\limits_{A}\left(Y^{\perp}\right)^{A}_{b}X^{A}_{a}=0,$ (107)

to have a non-trivial solution. Eq.(107) will have a non-trivial solution if
and only if,

$\displaystyle\det\limits_{a,b}\left[\sum\limits_{A}\left(Y^{\perp}\right)^{A}_{b}X^{A}_{a}\right]=0.$
(108)

Eq.(108) is the QNM searching condition that we have been looking for. Next we
perform the numerics161616An order of convergence of $10^{-11}$ and lower has
been treated as zero in the numerics. The numerical results that are presented
have been verified (up to very slight variations) for the matching point in
the range $y_{m}\in[0.2,0.8]$. We have also dropped a few terms in the UV and
the IR expansions of the fields and have verified the robustness of the
numerical results. with the mid-point shooting method. For the matching point,
we have, $y_{m}=0.6$. The numerical parameters used are,

Tolerance ($t_{m}$) | Horizon radius ($r_{h}$) | UV cut-off ($u_{\Lambda}$)
---|---|---
$0.1$ | $1$ | $0.1$
Double-trace coupling ($\kappa$) | RG parameter ($u^{*}$) | Anomaly coefficient ($k$)
$-1/\ln(10)$ | $1$ | $0.15$

##  Appendix E Non-hydrodynamic (gapped) modes and quasinormal mode table

We observe from the numerics that, there exists a higher (generically) complex
non-hydrodynamic mode $\forall\ b\in[0.00001,20]$. This mode seems to be
independent of $b$ as it exists $\forall\ b\in[0.00001,20]$ and has the value,

$\displaystyle\Gamma^{\text{cplx}}_{\text{non-hydro,0}}=\pm 0.965-1.736i,$
(109)

However, $\forall\ b\in(0,15.3]$, $\Gamma^{\text{cplx}}_{\text{non-hydro,0}}$
is not the lowest QNM and for $\forall\ b\geq 15.4$,
$\Gamma^{\text{cplx}}_{\text{non-hydro,0}}$ becomes the lowest QNM.

We also give below some more generically complex non-hydro (gapped) modes
which exist $\forall\ b\in[0.00001,20]$,

$\displaystyle\Gamma^{\text{cplx}}_{\text{non-hydro,1}}=\pm 3.359-3.697i,$
(110) $\displaystyle\Gamma^{\text{cplx}}_{\text{non-hydro,2}}=\pm
5.334-5.663i,$ (111) $\displaystyle\Gamma^{\text{cplx}}_{\text{non-
hydro,3}}=\pm 7.175-6.797i,$ (112)
$\displaystyle\Gamma^{\text{cplx}}_{\text{non-hydro,4}}=\pm 8.96-8.604i,$
(113) $\displaystyle\Gamma^{\text{cplx}}_{\text{non-hydro,5}}=\pm
10.537-7.337i,$ (114) $\displaystyle\Gamma^{\text{cplx}}_{\text{non-
hydro,6}}=\pm 13.102-5.463i,$ (115)

Table 1: lowest QNM vs $b$ S.No | $b$ | $-i\Gamma_{A}$
---|---|---
0 | $10^{-5}$ | $-2.3721962\times 10^{-12}\ i$
1 | 0.1 | $-0.000450031\ i$
2 | 0.2 | $-0.0018005\ i$
3 | 0.3 | $-0.00405253\ i$
4 | 0.4 | $-0.00720806\ i$
5 | 0.5 | $-0.0112698\ i$
6 | 0.6 | $-0.0162414\ i$
7 | 0.7 | $-0.0221275\ i$
8 | 0.8 | $-0.0289337\ i$
9 | 0.9 | $-0.036667\ i$
10 | 1.0 | $-0.0453354\ i$
11 | 1.1 | $-0.0549485\ i$
12 | 1.2 | $-0.0655176\ i$
13 | 1.3 | $-0.0770557\ i$
14 | 1.4 | $-0.0895777\ i$
15 | 1.5 | $-0.103101\ i$
16 | 1.6 | $-0.117644\ i$
17 | 1.7 | $-0.13323\ i$
18 | 1.8 | $-0.149884\ i$
19 | 1.9 | $-0.167633\ i$
20 | 2.0 | $-0.186508\ i$
21 | 2.1 | $-0.206544\ i$
22 | 2.2 | $-0.227778\ i$
23 | 2.3 | $-0.250252\ i$
24 | 2.4 | $-0.27401\ i$
25 | 2.5 | $-0.299101\ i$
26 | 2.6 | $-0.325573\ i$
27 | 2.7 | $-0.353477\ i$
28 | 2.8 | $-0.382864\ i$
29 | 2.9 | $-0.41378\ i$
30 | 3.0 | $-0.446263\ i$
31 | 3.1 | $-0.480341\ i$
32 | 3.2 | $-0.516016\ i$
33 | 3.3 | $-0.553261\ i$

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