DCPT-21/11

Dissipation in Holographic Superfluids

Aristomenis Donos1, Polydoros Kailidis1 and Christiana Pantelidou2

1 Centre for Particle Theory and Department of Mathematical Sciences,
Durham University, Durham, DH1 3LE, U.K.
2 School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

Abstract

> We study dissipation in holographic superfluids at finite temperature and
> zero chemical potential. The zero overlap with the heat current allows us to
> isolate the physics of the conserved current corresponding to the broken
> global $U(1)$. By using analytic techniques we write constitutive relations
> including the first non-trivial dissipative terms. The corresponding
> transport coefficients are determined in terms of thermodynamic quantities
> and the black hole horizon data. By analysing their behaviour close to the
> phase transition we show explicitly the breakdown of the hydrodynamic
> expansion. Finally, we study the pseudo-Goldstone mode that emerges upon
> introducing a perturbative symmetry breaking source and we determine its
> resonant frequency and decay rate.

##  1 Introduction

Holography provides a powerful framework to study large classes of strongly
coupled systems at finite temperature. One of its most useful applications
concerns the study of phase transitions and broken phases of thermodynamic
systems. Moreover, the duality allows us to study large-$N$ systems at all
energy scales. This includes the hydrodynamic limit of finite temperature
systems which is expected to be universal.

In this paper we will be interested in the long wavelength excitations of the
Goldstone mode that emerges in superfluid phases of matter which spontaneously
break a global $U(1)$ symmetry. One of our aims is to include the first non-
trivial dissipative effects which will result to a finite decay rate for the
expected “second sound”. The second aim of this paper is to introduce
perturbative sources which explicitly break the global symmetry and write down
the effective theory of the resulting pseudo-Goldstone mode.

One important aspect of the phases we will consider is that they will be at
zero chemical potential and electric charge. This will allow us to isolate the
long wavelength dynamics of the condensate avoiding its mixing with the other
hydrodynamic degrees of freedom of the system.

Holography has been extensively used to construct and study superfluid phases
after their original discovery [1, 2, 3]. From the bulk point of view, the
boundary theory global $U(1)$ is mapped to a gauged $U(1)$ symmetry acting on
a complex scalar field. The vector field that gauges the $U(1)$ in the bulk is
the gravitational dual of the conserved field theory Noether current operator.
Below a critical temperature $T_{c}$ the charged scalar develops a
perturbative instability yielding a new branch of broken phase black holes.

The effective theory of the Goldstone mode in the limit of long wavelengths is
a topic which has been explored before. The starting point is the non-
dissipative fluids which were first described in the framework of the Landau-
Tisza theory [4, 5]. Dissipative effects at finite temperature were
subsequently studied in the literature starting from [6, 7] and more recently
in [8], in the context of a systematic classification111See also [9] for more
recent developments in the framework of superfluids..

Due to the absence of chemical potential and the presence of rotational and
time reversal symmetry, the only information we will need to extract from our
holographic theories is two transport coefficients. We will be able to express
these in terms of horizon data related to finite temperature static
configurations. This, is in addition to the charge and current
susceptibilities which are fixed at the level of thermodynamics and don’t
capture dissipative dynamics. A particularly useful tool in extracting these
coefficients will be the symplectic current for gravity [10]. At a practical
level, this can be seen as the generalisation of the Wronskian for ordinary
differential equations. However, it is powerful enough to be applicable in
situations where the spacetime is inhomogeneous. Given the explicit
expressions for our transport coefficients, we will be able to study our
system near the phase transition. As we will see, the speed of second sound
will go to zero as we approach the transition and the mode will become
diffusive. However, its diffusion constant is not continuously connected to
that of the incoherent current pole which exists in the normal phase.

Apart from the purely spontaneous case, we will also consider a scenario in
which the global $U(1)$ is an almost exact symmetry222So far, the interplay of
weak explicit and spontaneous breaking of symmetries has been mainly
considered in the context of translations in e.g. [11, 12, 13]. For the case
of global symmetries in the bulk see [14, 12].. To implement the explicit
breaking of the $U(1)$ in a controlled manner, we will introduce a
perturbative static source for our complex operator that condenses at low
temperatures. As a result, the corresponding current becomes only partially
conserved and the propagating pseudo-Goldstone mode acquires a mass as well as
a finite gap. Our analysis is based on a modified Ward identity for the now
almost conserved electric current and is therefore applicable beyond the
details of our specific model.

In section 2 we discuss the class of holographic models we will employ in
order to realise our scenario along with some relevant aspects of their
thermodynamics. In section 3 we construct the theory of hydrodynamics which
captures the dynamics of our Goldstone mode along with the technical aspects
of our computation. We conclude the section with the computation of the
Green’s functions and the Kubo formulae for the two transport coefficients
that enter our description. In section 4 we introduce perturbative static
sources which break the global symmetry in a controlled manner as well as
spacetime dependent ones for the complex scalar operator. We use the sourced
Ward identity for the currents to extract the retarded Green’s functions of
the system. From the poles we extract the dispersion relations of second sound
pseudo-Goldstone mode. Section 5 is devoted to numerical checks of some of the
analytic results we obtain in the previous sections.

##  2 Setup

In this section we will introduce our holographic superfluids. Our main
results on the hydrodynamics of the superfluidity Goldstone mode are
independent of the specific scenario in which we will realise the neutral
superfluid phase transition. For concreteness, we will consider a holographic
CFT which is deformed by introducing a source $\phi_{(s)}$ for the neutral
relevant operator $\mathcal{O}_{\phi}$ with bulk dual $\phi$. Below a critical
temperature $T_{c}$, which is set by the scale $\phi_{(s)}$, a boundary
operator $\mathcal{O}_{\psi}$ which is charged under a global $U(1)$ symmetry
condenses. From the bulk point of view, this condensation leads to black holes
with non-zero profile for the dual bulk scalar $\psi$.

To realise this setup in holography, we consider a gravitational theory in the
bulk consisting of a metric, a neutral scalar $\phi$ and a complex scalar
$\psi$ which is charged under a local $U(1)$ symmetry and a Maxwell field
$A_{\mu}$ gauging the symmetry. The local symmetry in the bulk corresponds to
a global $U(1)$ symmetry on the boundary theory and the gauge field $A_{\mu}$
is dual to the corresponding Noether current operator
$\hat{\mathcal{J}}^{\mu}$.

The system is described by the bulk action

$\displaystyle S_{\mathrm{bulk}}=\int
d^{4}x\,\sqrt{-g}\,\left(R-V(\phi,|\psi|^{2})-\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi-(D_{\mu}\psi)(D^{\mu}\psi)^{\ast}-\frac{1}{4}\tau(\phi,|\psi|^{2})\,F^{\mu\nu}F_{\mu\nu}\right)\,,$
(2.1)

with the covariant derivative $D_{\mu}\psi=\nabla_{\mu}\psi+iqA_{\mu}\,\psi$
and the field strength $F=dA$. We will be mostly interested in configurations
which are dual to superfluids and therefore spontaneously break the $U(1)$
symmetry. These backgrounds will have complex scalar with non-trivial modulus
and for such configurations it will be beneficial to perform the field
redefinition $\psi=\rho\,e^{i\theta}$. This brings the action to the
equivalent form

$\displaystyle S=\int
d^{4}x\,\sqrt{-g}\,\left(R-V(\phi,\rho^{2})-\frac{1}{2}(\partial\phi)^{2}-(\partial\rho)^{2}-\rho^{2}\,(\partial\theta+qA)^{2}-\frac{1}{4}\tau(\phi,\rho^{2})\,F^{\mu\nu}F_{\mu\nu}\right)\,.$
(2.2)

The corresponding equations of motion are

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}V-\frac{\tau}{2}\left(F_{\mu\rho}F_{\nu}{}^{\rho}-\frac{1}{4}g_{\mu\nu}\,F^{2}\right)\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle-\frac{1}{2}\partial_{\mu}\phi\,\partial_{\nu}\phi-\partial_{\mu}\rho\,\partial_{\nu}\rho-\rho^{2}(\partial_{\mu}\theta+qA_{\mu})\,(\partial_{\nu}\theta+qA_{\nu})=$
$\displaystyle 0$
$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi-\partial_{\phi}V-\frac{1}{4}\partial_{\phi}\tau\,F^{2}=$
$\displaystyle 0$
$\displaystyle\nabla_{\mu}\nabla^{\mu}\rho-\partial_{\rho^{2}}V\,\rho-\frac{1}{4}\partial_{\rho^{2}}\tau\,\rho\,F^{2}-\rho\,(\partial\theta+qA)^{2}=$
$\displaystyle 0$
$\displaystyle\nabla_{\mu}\left(\rho^{2}\left(\nabla^{\mu}\theta+q\,A^{\mu}\right)\right)=$
$\displaystyle 0$
$\displaystyle\nabla_{\mu}(\tau\,F^{\mu\nu})-2q\,\rho^{2}\,(\nabla^{\nu}\theta+q\,A^{\nu})=$
$\displaystyle 0\,.$ (2.3)

In order to implement our holographic scenario, we will assume that for small
$\phi$ and $\psi$ the potential and the gauge coupling admit the expansions

$\displaystyle V$
$\displaystyle\approx-6+\frac{1}{2}m_{\phi}^{2}\,\phi^{2}+m_{\psi}^{2}\,|\psi|^{2}+\cdots$
$\displaystyle\tau$ $\displaystyle\approx 1+\cdots\,.$ (2.4)

Under these conditions, our theory admits the geometry of $AdS_{4}$ as
solution with metric

$\displaystyle
ds^{2}=r^{2}\left(-dt^{2}+dx^{2}+dy^{2}\right)+\frac{dr^{2}}{r^{2}}\,$ (2.5)

and trivial scalars and gauge field. Given the expansion (2), the bulk scalars
$\phi$ and $\psi$ correspond to operators $\mathcal{O}_{\phi}$ and
$\mathcal{O}_{\psi}$ whose dimensions $\Delta_{\phi}$ and $\Delta_{\psi}$ are
fixed by the mass terms according to
$\Delta_{\phi}(\Delta_{\phi}-3)=m_{\phi}^{2}$ and
$\Delta_{\psi}(\Delta_{\psi}-3)=m_{\psi}^{2}$.

Without loss of generality, a suitable ansatz which captures all the necessary
ingredients is

$\displaystyle ds^{2}$
$\displaystyle=-U(r)\,dt^{2}+\frac{dr^{2}}{U(r)}+e^{2g(r)}\,\left(dx^{2}+dy^{2}\right)\,,$
$\displaystyle\phi$ $\displaystyle=\phi(r),\qquad\rho=\rho(r),\qquad A=0\,.$
(2.6)

This ansatz leads to a non-linear system of ODEs for the radial functions that
appear in it. Notice that the above choice of coordinates does not fully fix
the radial coordinate $r$ which we are still free to shift by an arbitrary
constant. We will choose to fix this freedom by requiring that the horizon of
Hawking temperature $T$ is located at $r=0$. This allows us to write the near
horizon expansion

$\displaystyle U(r)\approx\,4\pi T\,r+\mathcal{O}(r^{2}),\qquad g(r)\approx
g^{(0)}+\mathcal{O}(r)\,,$
$\displaystyle\phi(r)\approx\phi^{(0)}+\mathcal{O}(r),\qquad\rho(r)\approx\rho^{(0)}+\mathcal{O}(r)\,.$
(2.7)

Ultimately, the backgrounds we wish to consider correspond to thermal states
of the theory deformed by a source of the relevant operator
$\mathcal{O}_{\phi}$ and a spontaneous VEV for the charged operator
$\mathcal{O}_{\psi}$. Given these considerations, the appropriate near
conformal boundary expansions at $r\to\infty$ are

$\displaystyle U(r)\approx(r+R)^{2}+\cdots+g_{(v)}\,(r+R)^{-1}+\cdots\,,$
$\displaystyle g(r)\approx\ln(r+R)+\dots\,,$
$\displaystyle\phi(r)\approx\phi_{(s)}\,(r+R)^{\Delta_{\phi}-3}+\cdots+\phi_{(v)}\,(r+R)^{-\Delta_{\phi}}+\cdots\,,$
$\displaystyle\rho(r)\approx\rho_{(s)}\,(r+R)^{\Delta_{\psi}-3}+\cdots\rho_{(v)}\,(r+R)^{-\Delta_{\psi}}+\cdots\,,$
(2.8)

where we have the leading behaviour as well as the various constants of
integration for our system. Given that we are looking for phases where
$\mathcal{O}_{\psi}$ takes a VEV spontaneously, we will either set the sources
$\rho_{(s)}$ equal to zero or, as we will do in section 4, keep them as
perturbative deformations. The VEVs of the scalar operators then are fixed by
$\phi_{(v)}$ and $\rho_{(v)}$333In general, holographic renormalisation can
involve the sources $\rho_{(s)}$ as part of the VEV of our scalar operators.
However, we consider phases in which our operator takes a VEV spontaneously
and we will naturally be working in the limit
$\rho_{(s)}\ll\rho_{(v)}^{(3-\Delta_{\psi})/\Delta_{\psi}}$. In this
situation, the VEV will be approximately fixed by $\rho_{(v)}$.. The constant
shift $R$ is an artefact of the way we chose to fix our radial coordinate by
placing the horizon at $r=0$.

At this point, it is worth understanding the asymptotics of the phase field
$\theta$. In the absence of (or for perturbatively small) source $\rho_{(s)}$
a perturbation of $\delta\theta$ admits the asymptotic expansion

$\displaystyle\delta\theta(r)\approx
r^{2\Delta_{\psi}-3}\,\delta\theta_{(s)}+\cdots+\delta\theta_{(v)}+\cdots\,.$
(2.9)

In the complete absence of a source for $\psi$, the relevant bit for us is the
constant term in $r$, which parametrises the phase of the order parameter.
Indeed, for the VEV of the condensed operator we will have that
$\langle\mathcal{O}_{\psi}\rangle\propto\rho_{(v)}\,e^{i\,\theta_{(v)}}$. From
the boundary point of view, we can simply change the global phase without any
cost in energy. The fact that we can freely rotate the VEV
$\langle\mathcal{O}_{\psi}\rangle$ without changing the energy of our system
reflects the existence of a Goldstone mode in the boundary theory. This
suggests that the dynamics of the Goldstone mode is captured by the VEV of the
operator

$\displaystyle\mathcal{O}_{Y}=\frac{1}{2i\,\left|\langle\mathcal{O}_{\psi}\rangle_{b}\right|}\,(\langle\mathcal{O}_{\bar{\psi}}\rangle_{b}\,\mathcal{O}_{\psi}-\langle\mathcal{O}_{\psi}\rangle_{b}\,\mathcal{O}_{\bar{\psi}})\,,$
(2.10)

where $\langle\mathcal{O}_{\psi}\rangle$ is the thermal state VEV. In the
absence of a source for $\psi$, for the fluctuations we can write

$\displaystyle\delta\langle\mathcal{O}_{Y}\rangle=\langle\mathcal{O}_{\psi}\rangle\,\delta\theta_{(v)}\,.$
(2.11)

In section 4 of this paper will be interested in hydrodynamic perturbations in
which the source $\delta\theta_{(s)}$ will be much smaller than the scale of
$\delta\theta_{(v)}$. Given this information, in this language the
perturbative source for the scalar will be $\delta
s_{\psi}=i(\rho_{(v)}\,\delta\theta_{(s)}+\delta\rho_{(s)}\,\delta\theta_{(v)})$
while the VEV will be given by
$\langle\mathcal{O}_{\psi}\rangle\propto(2\Delta_{\psi}-3)\,\rho_{(v)}\,(1+i\delta\theta_{(v)})$
after a perturbation. Therefore the source of the operator $\mathcal{O}_{Y}$
will be simply
$s_{Y}=(\rho_{(v)}\,\delta\theta_{(s)}+\delta\rho_{(s)}\,\delta\theta_{(v)})$
while for its VEV we can write equation (2.11).

A central point of interest to our paper is the fluctuations of the bulk gauge
field $A$ around the background black holes of equation (2). From the
equations of motion (2), we see that fluctuations of the gauge field $A_{\mu}$
and phase field $\theta$ are captured by the last two equations. In fact,
after defining the one-form field $B_{\mu}=\partial_{\mu}\theta+q\,A_{\mu}$,
we can simply consider the equation of motion of a massive vector field in the
bulk

$\displaystyle\nabla_{\mu}(\tau\,W^{\mu\nu})-2q^{2}\rho^{2}\,B^{\nu}=0\,,$
(2.12)

where we have defined the two form $W=dB$. The penultimate equation of (2) now
reads

$\displaystyle\nabla_{\mu}\left(\rho^{2}\,B^{\mu}\right)=0\,,$ (2.13)

and is simply a consequence of taking the divergence of equation (2.12). For
this reason we will only need to consider equation (2.12). Close to the
conformal boundary, in the absence of background sources for complex scalar,
the 1-form field $B_{\nu}$ admits the expansion

$\displaystyle
B_{\alpha}=\partial_{\alpha}\theta_{(s)}\,r^{2\Delta_{\psi}-3}+\cdots+v_{\alpha}+\cdots+\frac{q\,j_{\alpha}}{r}+\cdots\,,$
(2.14)

where $v_{\alpha}=\partial_{\alpha}\theta_{(v)}+q\,\mu_{\alpha}$ is a gauge
invariant combination of the superfluid velocity
$\partial_{\alpha}\theta_{(v)}$ and the source $\mu_{\alpha}$ for the $U(1)$
current. Moreover, as we will explain later the constants of integration
$j_{\alpha}$ satisfy the Ward identity for the currents which are given by
equation (2.20). Note that in the thermal states we are interested in we have
$\mu_{\alpha}=0$; we will only consider non-zero $\mu_{\alpha}$ when computing
the thermodynamic susceptibilities.

###  2.1 Thermodynamics

An important ingredient in studying the thermodynamics and the linear response
of our system is the free energy density $w$. In order to compute it, we first
need to regularise our bulk action (2.1) by introducing appropriate
couterterms which render the total action finite and the boundary value
problem well defined. These form a boundary action which is defined on a
hypersurface $\partial M$ of constant radial coordinate $r$ near the conformal
boundary. An appropriate choice of boundary action includes the terms

$\displaystyle S_{bdr}=$ $\displaystyle-\int_{\partial
M}d^{3}x\,\sqrt{-\gamma}\,\left(-2K+4+R_{bdr}\right)$
$\displaystyle\quad-\frac{1}{2}\int_{\partial
M}d^{3}x\,\sqrt{-\gamma}\,[(3-\Delta_{\phi})\phi^{2}-\frac{1}{2\Delta_{\phi}-5}\,\partial_{a}\phi\partial^{a}\phi]$
$\displaystyle\quad-\int_{\partial
M}d^{3}x\,\sqrt{-\gamma}\,[(3-\Delta_{\psi})|\psi|^{2}-\frac{1}{2\Delta_{\psi}-5}\,D_{a}\psi
D^{a}\psi^{\ast}]+\cdots\,,$ (2.15)

with $\gamma_{\alpha\beta}$ the induced metric on $\partial M$, $R_{bdr}$ the
associated Ricci scalar and $K$ the extrinsic curvature scalar of the constant
$r$ hypersurface. In fact, for the case with $\Delta_{\phi}=\Delta_{\psi}=2$
on which we focus in our numerics section 5, these are all the terms needed.

In order to compute the free energy we need to evaluate the regularised action
$S_{tot}=S_{bulk}+S_{bdr}$ on the Euclidean version of our backgrounds (2)
with $t=-i\,\tau$. Since we are dealing with an infinite field theory system,
this is an infinite quantity and we should be discussing densities instead.
After dropping the integration over the boundary spatial non-compact
coordinates, this yields the density $I_{tot}$ which is related to the free
energy density $w_{FE}$ via $w_{FE}=TI_{tot}$ with $T$ the Hawking temperature
of our black brane. Notice that, in the absence of sources, the terms which
involve the complex scalar $\psi$ in (2.1) will not contribute to the
thermodynamics of our system. However, these are needed in order to extract
the VEV of the complex operator $\mathcal{O}_{\psi}$.

In order to compute the on-shell value of the Euclidean bulk action
$I_{bulk}$, it is useful to write the integrand as a total derivative. After
exploiting the existence of the Killing vector $\partial_{t}$ and using a
Komar type of argument, we can write

$\displaystyle
I_{bulk}=\frac{1}{T}\int_{0}^{\infty}dr\,\left(e^{2g}\,U^{\prime}\right)^{\prime}\,.$
(2.16)

Being in the grand canonical ensemble and at zero background chemical
potential, the free energy density of our system is

$\displaystyle w_{FE}=\epsilon-T\,s,$ (2.17)

where $\epsilon$ is the energy density and $s$ is the Bekenstein-Hawking
entropy. This can be expressed in terms of the horizon data (2) as

$\displaystyle s=4\pi\,e^{2\,g^{(0)}}\,.$ (2.18)

In the case of e.g. $\Delta_{\phi}=\Delta_{\psi}=2$ the bulk on-shell action
(2.16) combines with the boundary action (2.1) to yield

$\displaystyle w_{FE}=-2g_{(v)}-\phi_{(s)}\,\phi_{(v)}-T\,s\,.$ (2.19)

Even though we have not included a source $\mu_{\alpha}$ for the current, this
is certainly an important part of thermodynanics for a superfluid. The
response of the system to an external source includes a non-trivial
expectation value for the conserved $U(1)$ current

$\displaystyle J^{\alpha}=\lim_{r\to\infty}\sqrt{-g}\,\tau\,F^{\alpha
r}=\lim_{r\to\infty}\sqrt{-g}\,\tau\,W^{\alpha r}\,.$ (2.20)

For our purposes, we will need this information in the context of linear
response.

In addition to the free energy $w$, we are also interested in the
thermodynamic susceptibilities of our system. In order to compute them, we
consider the static perturbations $\delta B^{(t)}$ and $\delta B^{(x)}$ with
zero superfluid velocity. In terms of the asymptotics (2.14), we have $\delta
v_{\nu}=q\,\delta^{t}_{\nu}\,\delta\mu_{t}$ and $\delta
v_{\nu}=q\,\delta^{x}_{\nu}\,\delta\mu_{x}$. Close to the conformal boundary
these perturbations will admit the expansion

$\displaystyle\delta B^{(t)}_{t}$
$\displaystyle=q\,\delta\mu_{t}+q\,\frac{\delta j_{t}}{r+R}+\cdots\,,$
$\displaystyle\delta B^{(x)}_{x}$
$\displaystyle=q\,\delta\mu_{x}+q\,\frac{\delta j_{x}}{r+R}+\cdots\,.$ (2.21)

In terms of the thermodynamic susceptibilities $\chi_{QQ}$ and $\chi_{JJ}$ we
must have $\delta j_{t}=-\chi_{QQ}\,\delta\mu_{t}$444The charge density is
$\delta\rho=-\delta j_{t}$. and $\delta j_{x}=-\chi_{JJ}\,\delta\mu_{x}$.
Close to the event horizon, regularity demands the expansion

$\displaystyle\delta B^{(t)}_{t}$
$\displaystyle=q\,\delta\mu_{t}\,a_{t}^{(0)}\,r+\mathcal{O}(r^{2})\,,$
$\displaystyle\delta B^{(x)}_{x}$
$\displaystyle=q\,\delta\mu_{x}\,a_{x}^{(0)}+\mathcal{O}(r)\,.$ (2.22)

The quantities $a_{t}^{(0)}$ and $a_{x}^{(0)}$ are certainly part of the
thermodynamics of a superfluid. Later, we will see that together with the
horizon data of (3.2) they will determine the transport coefficients of our
superfluid when we express the current in terms of the superfluid velocity
$\partial_{\alpha}\theta_{(v)}$.

Later in the paper, we will also want to understand our hydrodynamic expansion
close to the phase transition. For the case where the transition is second
order, close to the critical temperature $T_{c}$ we must have
$\rho\propto\sqrt{T_{c}-T}$ corresponding to a VEV which scales like
$\langle\mathcal{O}_{\psi}\rangle\propto\sqrt{T_{c}-T}$. This will let us see
the important fact that the Goldstone mode does not connect continuously to
the diffusive mode of the incoherent current which exists above $T_{c}$. More
interestingly, we will see that the source for this discontinuity is that our
hydrodynamic expansion breaks down close to the critical temperature making
the radius of convergence infinitesimally small.

By integrating the bulk equations of motion (2.12) we obtain the relations

$\displaystyle\delta\rho$
$\displaystyle=e^{2g^{(0)}}\tau^{(0)}\delta\mu_{t}\,a_{t}^{(0)}+2q\,\int_{0}^{\infty}dr\,\rho^{2}\frac{e^{2g}}{U}\delta
B_{t}^{(t)}\,,$ $\displaystyle\delta j_{x}$
$\displaystyle=-2q\,\int_{0}^{\infty}dr\,\rho^{2}\,\delta B_{x}^{(x)}\,.$
(2.23)

As we can see, the right hand side of the above equations still involves the
perturbation of the one-form field. Close to the phase transitions these
equations yield

$\displaystyle\chi_{QQ}$
$\displaystyle=e^{2g^{(0)}}\tau^{(0)}\,a_{t}^{(0)}+\mathcal{O}\left(T_{c}-T\right)\,,$
$\displaystyle\chi_{JJ}$
$\displaystyle=2q^{2}\,\int_{0}^{\infty}dr\,\rho^{2}+\mathcal{O}\left((T_{c}-T)^{2}\right)\,.$
(2.24)

This is indeed the behaviour we would have expected close to a superfluid
phase transition with the charge susceptibility $\chi_{QQ}$ remaining finite
and the current susceptibility $\chi_{JJ}$ approaching zero.

##  3 Linear Response

In this section, we will study long wavelength fluctuations of the $U(1)$
current of our system. The thermal state above the critical temperature is
simply an electrically neutral hot plasma corresponding to a CFT that has been
deformed by a relevant operator while preserving Poincare invariance. In the
absence of electric charge, the electric current fluctuations are dominated by
a diffusive mode in the hydrodynamic limit [15, 16, 17]. The existence of this
mode in a neutral plasma is due to the existence of an incoherent current in
strongly coupled CFTs [18, 19].

Below the critical temperature the hydrodynamics of the electric current will
be dominated by the Goldstone mode associated to the spontaneous breaking of
the global $U(1)$ of the boundary theory. In section 3.1 we will introduce the
symplectic current in the bulk which will be the main technical tool we will
use for our computations. In section 3.2 we will construct the hydrodynamic
perturbations which couple to the Goldstone mode of our boundary theory. This
will allow us to write constitutive relations for the expectation value of the
current operator in terms of the phase of our order parameter. Using these and
current conservation we will extract the dispersion relation of the second
sound mode in terms of thermodynamic susceptibilities and two transport
coefficients which are expressed in terms of horizon quantities. Finally, in
section 3.3 we will include sources for the current in our analysis. We will
give expressions for the retarded Green’s functions of the system and Kubo
formulae for the transport coefficients of our superfluid.

###  3.1 Symplectic Current

A powerful tool we will use is the symplectic current for the bulk theory.
This will allow us to deal with the one-form field fluctuations in an elegant
way. Its benefit lies in the fact that it is a conserved current in the bulk
and will allow us to relate boundary quantities to horizon ones. As we will
see, this logic will allow us to express the boundary electric current in
terms of derivatives of the phase of our order parameter, the superfluid
velocity, and the sources.

In order to introduce, it we imagine that we have two perturbations $\delta
B^{<1>}_{\mu}$ and $\delta B^{<2>}_{\mu}$ which solve our bulk equation
(2.12). We can then construct the vector density

$\displaystyle P^{\mu}=\sqrt{-g}\,\tau\,\left(\delta
B^{<1>}_{\nu}\delta(W^{<2>})^{\nu\mu}-\delta
B^{<2>}_{\nu}\delta(W^{<1>})^{\nu\mu}\right)\,,$ (3.1)

which can be shown to be divergence free,

$\displaystyle\partial_{\mu}P^{\mu}=0\,.$ (3.2)

This current is not a consequence of a conservation law as it is based on two
independent solutions. It merely comes from the fact that the equations of
motion of our classical theory in the bulk are derived from a local classical
action. In the hydrodynamic limit, the dominant term in (3.2) will be the one
involving the radial derivative which will let us relate the symplectic
current on the black brane horizon to the symplectic current on the conformal
boundary boundary.

This is especially helpful when one of the two solutions we are using in the
construction in the construction of (3.1) is considered to be known. For our
purposes, the role of the known solution will be played by the static
perturbations $\delta B^{(t)}_{\mu}$ and $\delta B^{(x)}_{\mu}$ we discussed
in section 2.1. At a philosophical level, we will need to consider as known
all the static black holes which can be used to describe the thermodynamics of
our system. That would include the black holes corresponding to finite
chemical potential as well as persistent supercurrents. The reason is that, as
we will see, the susceptibilities $\chi_{QQ}$ and $\chi_{JJ}$ will play an
important role even though we will be studying transport in thermal states of
zero chemical potential and external vector field. For the purpose of
extracting those, the knowledge of the static perturbations $\delta
B^{(t)}_{\mu}$ and $\delta B^{(x)}_{\mu}$ is sufficient.

###  3.2 Second sound

In this section we would like to construct the bulk perturbation corresponding
to the Goldstone mode of the boundary theory: the second sound. As we
discussed in section 2.1, the Goldstone mode involves the phase of our order
parameter and therefore the associated conserved current due to the global
symmetry on the boundary. In the absence of electric charge, the current
decouples from the stress tensor of the theory within linear response. This
will allow us to clearly isolate the dynamics of the Goldstone mode from the
rest of the hydrodynamics modes of our system.

From the bulk point of view we will only need to examine perturbations of the
one-form field $B_{\mu}$. The spacetime translational symmetries allow us to
study Fourier modes. Thus, in order to study perturbations in the hydrodynamic
limit, we will consider long wavelength excitations of the form

$\displaystyle\delta B_{\mu}(t,x;r)=e^{-i\omega(t+S(r))+i\varepsilon
k\,x}\,\delta b_{\mu}(r)\,dx^{\mu}\,,$ (3.3)

for some small number $\varepsilon$. The function $S(r)$ is chosen so that it
drops faster that $\mathcal{O}(1/r^{3})$ close to the conformal boundary and
it therefore doesn’t interfere with holographic renormalisation. However,
close to the horizon, it is chosen so that it approaches $S(r)\to\frac{1}{4\pi
T}\ln r$ and the combination $t+S(r)$ is regular and ingoing. Note that we
picked the momentum $k$ to point in the direction $x$ without loss of
generality given that the background is isotropic.

An important ingredient in extracting the second sound is the absence of
boundary sources. In general, the asymptotic expansion close to the conformal
boundary is

$\displaystyle\delta
b_{\alpha}=\delta\hat{v}_{\alpha}+q\,\frac{\delta\hat{j}_{\alpha}}{r}+\cdots\,.$
(3.4)

The leading terms are a gauge invariant combination of the external one-form
source and the superfluid velocity. Absence of sources is equivalent to
demanding

$\displaystyle\delta\hat{v}_{t}=-i\omega\,\delta\hat{c},\qquad\delta\hat{v}_{x}=i\varepsilon
k\,\delta\hat{c}\,,$ (3.5)

so that $v_{\alpha}=\partial_{\alpha}\theta_{(v)}$ in the notation of equation
(2.14). Moreover, charge conservation implies that

$\displaystyle\partial_{\alpha}\delta
j^{\alpha}=0\Rightarrow\omega\delta\hat{j}_{t}+\varepsilon
k\delta\hat{j}_{x}=0\,.$ (3.6)

At this point it will be illuminating to count constants of integration. This
is crucial in order to see that we are doing the right thing in terms of
boundary conditions since we are after specific quasi-normal modes. The
equation of motion (2.12) yields two second order equations for $\delta b_{t}$
and $\delta b_{x}$ while $\delta b_{r}$ can be solved algebraically. In order
to find a unique solution we will therefore need to fix four constants.

Close to the conformal boundary we have the constant $\delta\hat{c}$, while
$\delta\hat{j}_{t}$ and $\delta\hat{j}_{t}$ give one more constant since they
have to satisfy the constraint (3.6). Close to the horizon, in-falling
boundary conditions fix the expansion

$\displaystyle\delta b_{\alpha}$ $\displaystyle=\delta
b_{\alpha}^{(0)}+r\,\delta b_{\alpha}^{(1)}+\cdots\,,$ $\displaystyle\delta
b_{r}$ $\displaystyle=\frac{1}{4\pi Tr}\delta b_{t}^{(0)}+\delta
b_{r}^{(1)}+\cdots\,.$ (3.7)

We therefore have an extra two constants of integration coming from the near
horizon expansion, giving an overall total of 4 constants. However, we are
only solving a linear system of homogeneous equations. Since we are not
introducing any sources, we can use the scaling symmetry of the problem to set
any of the above constants of integration to one leaving with only three. The
fourth constant is precisely the frequency $\omega$ which will ultimately
become a function of the wavenumber $k$ that we are free to choose. This
procedure will fix the dispersion relation for $\omega(\epsilon k)$ for the
quasi-normal modes.

For our purposes however, we are after a particular mode which is hydrodynamic
and has $\omega=0$ for $\varepsilon=0$. This simply corresponds to the trivial
solution for the one form field. This allows us to consider the hydrodynamic
expansion

$\displaystyle\omega$
$\displaystyle=\varepsilon\,\omega_{[1]}+\varepsilon^{2}\,\omega_{[2]}+\cdots\,,$
$\displaystyle\delta\hat{j}_{\mu}$
$\displaystyle=\varepsilon\,\delta\hat{j}_{\mu[1]}+\varepsilon^{2}\,\delta\hat{j}_{\mu[2]}+\cdots\,.$
(3.8)

For the radial function in the ansatz (3.3) we will write the expansion

$\displaystyle\delta b_{t}(r)$
$\displaystyle=\varepsilon\,\delta\hat{B}^{(t)}_{t}(r)+\varepsilon^{2}\,\delta
B^{(2)}_{t}(r)+\cdots\,,$ $\displaystyle\delta b_{x}(r)$
$\displaystyle=\varepsilon\,\delta\hat{B}^{(x)}_{x}(r)+\varepsilon^{2}\,\delta
B^{(2)}_{x}(r)+\cdots\,,$ $\displaystyle\delta b_{r}(r)$
$\displaystyle=\varepsilon^{2}\,\delta B^{(2)}_{r}(r)+\cdots\,,$ (3.9)

where $\delta\hat{B}^{(t)}_{t}(r)$ and $\delta\hat{B}^{(x)}_{x}(r)$ are
precisely the thermodynamic perturbations we discussed in equation (2.14) with

$\displaystyle\delta\mu_{t}=-\frac{i\omega_{[1]}}{q}\delta\hat{c},\qquad\delta\mu_{x}=\frac{ik}{q}\delta\hat{c}\,.$
(3.10)

For the subleading terms we impose the near conformal boundary expansion

$\displaystyle\delta B_{t}^{(n)}$
$\displaystyle=-i\omega_{[n]}\delta\hat{c}+q\frac{\delta\hat{j}_{t[n]}}{r}+\cdots\,,$
$\displaystyle\delta B_{x}^{(n)}$
$\displaystyle=q\frac{\delta\hat{j}_{x[n]}}{r}+\cdots\,,$ (3.11)

which is simply a reorganisation of the perturbation with boundary conditions
as in equation (3.5). Close to the horizon, we impose in-falling boundary
conditions. This gives us the expansion,

$\displaystyle\delta B_{t}^{(n)}$ $\displaystyle=\delta B^{(n)(0)}_{t}+\delta
B^{(n)(1)}_{t}\,r+\cdots\,,$ $\displaystyle\delta B_{r}^{(n)}$
$\displaystyle=\frac{\delta B^{(n)(0)}_{t}}{4\pi Tr}+\delta
B^{(n)(0)}_{r}+\cdots\,,$ $\displaystyle\delta B_{x}^{(n)}$
$\displaystyle=\delta B^{(n)(0)}_{x}+\delta B^{(n)(1)}_{x}\,r+\cdots\,.$
(3.12)

At leading order in the $\varepsilon$ expansion the components of the
conserved current are

$\displaystyle
q\,\delta\hat{j}_{t[1]}=i\omega_{[1]}\chi_{QQ}\,\delta\hat{c},\qquad
q\,\delta\hat{j}_{x[1]}=-ik\chi_{JJ}\,\delta\hat{c}\,,$ (3.13)

which follows from the definition of the charge and current susceptibilities.
At that order, we see that equation (3.6) gives

$\displaystyle\omega_{[1]}\delta\hat{j}_{t[1]}+k\delta\hat{j}_{x[1]}=0\Rightarrow\omega_{[1]}^{2}\,\chi_{QQ}-k^{2}\,\chi_{JJ}=0$
$\displaystyle\Rightarrow\omega_{[1]}=\pm\,c_{s}\,k\,,$ (3.14)

with $c_{s}^{2}=\chi_{JJ}/\chi_{QQ}$ being the universal speed of second
sound.

To specify $\omega_{[2]}$, we will need to fix the currents
$\delta\hat{j}_{\alpha[2]}$. In order to do this we will employ the
conservation (3.2) of the symplectic current (3.1). As we explained in section
3.1, the symplectic current is particularly useful when we compare the
solution we are after to a solution we already know. For this reason we will
consider it for two different choices of pairs of perturbations. For both
cases, we will choose $\delta B^{<1>}$ to be the perturbation in (3.3).

Our first choice for $\delta B^{<2>}$ is the static perturbation $\delta
B^{(x)}$ of section 2.1. This will allow us to specify the component
$\delta\hat{j}_{x}$. Expanding the symplectic current in $\varepsilon$ we have

$\displaystyle P^{\mu}=e^{-i\omega(t+S(r))+i\varepsilon
k\,x}\,\left(\varepsilon^{2}\,p^{\mu}_{(2)}+\varepsilon^{3}\,p^{\mu}_{(3)}+\cdots\right)\,.$
(3.15)

One would expect that the leading term would be $\mathcal{O}(\varepsilon)$
since our perturbation $\delta B^{<1>}$ starts at order $\varepsilon$ and
$\delta B^{<2>}$ starts at zeroth order. However, the leading term in the
$\varepsilon$-expansion of $\delta B^{<1>}$ in equation (3.2) is proportional
to $\delta B^{<2>}$ and the would-be leading term in the symplectic current
turns out to vanish.

More explicitly we have

$\displaystyle p^{t}_{(2)}$ $\displaystyle=\frac{i\tau}{U}\delta
B^{(x)}_{x}\,\left(\omega_{[1]}\delta\hat{B}^{(x)}_{x}+k\delta\hat{B}^{(t)}_{t}\right)\,,$
$\displaystyle p^{r}_{(2)}$ $\displaystyle=-U\tau\,\left(\delta
B^{(2)}_{x}\,\partial_{r}\delta B^{(x)}_{x}+\delta
B^{(x)}_{x}\,\left(i\omega_{[1]}S^{\prime}\,\delta\hat{B}^{(x)}_{x}-\partial_{r}\delta
B^{(2)}_{x}\right)\right)\,,$ $\displaystyle p^{x}_{(2)}$
$\displaystyle=U\tau\,\delta B_{r}^{(2)}\,\partial_{r}\delta B^{(x)}_{x}\,.$
(3.16)

We now consider the divergence (3.2) and keep only terms up to
$\mathcal{O}(\varepsilon^{2})$ giving

$\displaystyle\partial_{r}p_{(2)}^{r}=0\,.$ (3.17)

Integrating this equation from the horizon up to the conformal boundary, we
obtain the relation

$\displaystyle
q\,\delta\hat{j}_{x[2]}=-\omega_{[1]}k\,\sigma_{d}\,\delta\hat{c}\,,$ (3.18)

where we have defined 555For this transport coefficient, a similar result was
obtained in [20]. Here we clarify the significance of the quantities that
appear in this equation as data related to the geometries dual to the thermal
state after the inclusion of the superfluid velocity in the ensemble
variables.

$\displaystyle\sigma_{d}=\tau^{(0)}\,(a_{x}^{(0)})^{2}\,.$ (3.19)

Choosing now $\delta B^{<2>}$ to be $\delta B^{(t)}$, we obtain

$\displaystyle p^{t}_{(2)}$ $\displaystyle=-e^{2g}\tau\,\delta
B^{(2)}_{r}\,\partial_{r}\delta B^{(t)}_{t}\,,$ $\displaystyle p^{r}_{(2)}$
$\displaystyle=e^{2g}\tau\,\left(\delta B^{(2)}_{t}\,\partial_{r}\delta
B^{(t)}_{t}+\delta
B^{(t)}_{t}\,\left(i\omega_{[1]}S^{\prime}\,\delta\hat{B}^{(t)}_{t}-\partial_{r}\delta
B^{(2)}_{t}\right)\right)\,,$ $\displaystyle p^{x}_{(2)}$
$\displaystyle=-\frac{i\tau}{U}\delta
B^{(t)}_{t}\,\left(\omega_{[1]}\,\delta\hat{B}^{(x)}_{x}+k\,\delta\hat{B}^{(t)}_{t}\right)\,.$
(3.20)

Keeping once again only terms up to order $\mathcal{O}(\varepsilon^{2})$ in
equation (3.2) and integrating from the horizon up to the conformal boundary,
we obtain

$\displaystyle
q\,\delta\hat{j}_{t[2]}=i\omega_{[2]}\,\chi_{QQ}\,\delta\hat{c}+e^{2g^{(0)}}\tau^{(0)}\,\delta
B_{t}^{(2)(0)}\,a_{t}^{(0)}\,.$ (3.21)

We have used the near horizon expansion (3.2) and we need to specify the
constant of integration $\delta B_{t}^{(2)(0)}$. We can do this by considering
the near horizon limit of the radial component of the equation of motion
(2.12) along with the expansion (3.2). This fixes

$\displaystyle\delta
B^{(2)(0)}_{t}=-\frac{\omega_{[1]}^{2}}{2q^{2}(\rho^{(0)})^{2}}a_{t}^{(0)}\tau^{(0)}\,\delta\hat{c}\,,$
(3.22)

yielding

$\displaystyle
q\,\delta\hat{j}_{t[2]}=i\omega_{[2]}\,\chi_{QQ}\,\delta\hat{c}-e^{2g^{(0)}}\,\frac{(\tau^{(0)}a_{t}^{(0)})^{2}}{2q^{2}(\rho^{(0)})^{2}}\omega_{[1]}^{2}\,\delta\hat{c}\,.$
(3.23)

Moving on to next to leading order in equation (3.6) along with (3.23) and
(3.18), we have

$\displaystyle\omega_{[2]}\,\delta\hat{j}_{t[1]}+\omega_{[1]}\,\delta\hat{j}_{t[2]}+k\,\delta\hat{j}_{x[2]}$
$\displaystyle=0\Rightarrow$ $\displaystyle
2i\,\chi_{QQ}\,\omega_{[2]}-e^{2g^{(0)}}\,\frac{(\tau^{(0)}a_{t}^{(0)})^{2}}{2q^{2}(\rho^{(0)})^{2}}\omega_{[1]}^{2}-k^{2}\sigma_{d}$
$\displaystyle=0\Rightarrow$
$\displaystyle\omega_{[2]}=-i\frac{1}{2\chi_{QQ}}\left(e^{2g^{(0)}}\,\frac{(\tau^{(0)}a_{t}^{(0)})^{2}}{2q^{2}(\rho^{(0)})^{2}}\frac{\chi_{JJ}}{\chi_{QQ}}+\sigma_{d}\right)\,k^{2}$
$\displaystyle\,.$ (3.24)

Equations (3.13), (3.18) and (3.23) can be considered as the expansion of the
constitutive relations for the conserved current. By introducing the Fourier
transform $\delta c$ of $\delta\hat{c}$, we can write them in the form

$\displaystyle q\,\delta j_{t}$ $\displaystyle=-\chi_{QQ}\,\partial_{t}\delta
c+\Xi\,\partial_{t}^{2}\delta c\,,$ $\displaystyle q\,\delta j_{x}$
$\displaystyle=-\chi_{JJ}\,\partial_{x}\delta
c-\sigma_{d}\,\partial_{x}\partial_{t}\delta c\,,$ (3.25)

where we have defined

$\displaystyle\Xi=e^{2g^{(0)}}\,\frac{(\tau^{(0)}a_{t}^{(0)})^{2}}{2q^{2}(\rho^{(0)})^{2}}\,.$
(3.26)

In addition to the constitutive relations for the currents, the VEV of our
charged scalar operator is

$\displaystyle\langle\mathcal{O}_{\psi}\rangle=\langle\mathcal{O}_{\psi}\rangle_{b}\,(1+i\,\delta
c)\,,$ (3.27)

at leading order in the $\varepsilon$ expansion. Here, we have introduced the
VEV of the scalar in the thermal state
$\langle\mathcal{O}_{\psi}\rangle_{b}=(2\Delta-3)\,\rho_{(v)}$.

Using this notation, the dispersion relation for our sound mode is

$\displaystyle\omega=\pm\,c_{s}\,k-\frac{i}{2\chi_{QQ}^{2}}\left(\chi_{JJ}\,\Xi+\chi_{QQ}\,\sigma_{d}\right)\,k^{2}\,.$
(3.28)

Note that these constitutive relations are based on the two thermodynamic
susceptibilities $\chi_{QQ}$ and $\chi_{JJ}$ as well as the two transport
coefficients $\sigma_{d}$ and $\Xi$. In the next section we will introduce
sources for the external vector field which will allow us to compute the
retarded Green’s functions of the current. As a result, we will manage to
write Kubo formulae for our transport coefficients $\sigma_{d}$ and $\Xi$.

Before moving on, we would like to examine the behaviour of our hydrodynamics
close to the critical temperature $T_{c}$. As one can see from the
constitutive relations (3.3), the coefficients that enter in our hydrodynamic
expansion include the susceptibilities $\chi_{JJ}$ and $\chi_{QQ}$. The
behaviour of these coefficients close to $T_{c}$ was given in equation (2.1).
The first derivative with respect to the temperature of both of them exhibits
the expected discontinuity close to the transition with the former approaching
zero. From that, we can easily see that the speed of second sound fixing the
linear part of the dispersion relation(3.2) behaves like
$c_{s}\approx(T_{c}-T)^{1/2}$ close to the phase transition.

The dissipative part of the constitutive relations (3.2) is determined by the
transport coefficients $\sigma_{d}$ and $\Xi$. The holographic expressions
that we obtained for these are given in equations (3.19) and (3.26)
respectively. The behaviour of $\sigma_{d}$ is fairly simple to understand as
$\tau^{(0)}$ goes to a fixed value as we increase temperature and
$a_{x}^{(0)}$ is simply unity at $T_{c}$. On the other hand, equation (3.26)
is telling us that $\Xi$ blows up close to the transition since
$\rho^{(0)}\propto(T_{c}-T)^{1/2}$ and everything else remains finite in our
holographic expression. We interpret this fact as the breakdown of the
hydrodynamic expansion making the radius of convergence smaller and smaller as
we approach phase transition. One would have expected such a breakdown since
the phase $\delta c$ becomes a not well defined field close to the transition.
This is easy to see directly from the definition of $\mathcal{O}_{Y}$ itself
in equation (2.10). Note that there is another mode in the theory which
becomes gapless close to the transition which is associated with the modulus
of the complex VEV $\langle\mathcal{O}_{\psi}\rangle$ [21]. However, this mode
completely decouples from the system we are examining as it is captured by the
bulk field $\rho$ in our parametrisation.

Despite this fact, the speed of sound and the attenuation in the dispersion
relation (3.28) remain finite since, according to equation (2.1), the current-
current scalars behave like $\chi_{JJ}\approx(T_{c}-T)$. We can easily see
that close to the transition the dispersion relation (3.2) becomes

$\displaystyle\omega\approx-\frac{i}{2}\left(\frac{4\pi}{s}\,\int_{0}^{\infty}dr\,\frac{\rho^{2}}{(\rho^{(0)})^{2}}+\frac{\sigma_{d}}{\chi_{QQ}}\right)\,k^{2}\,,$
(3.29)

which remains finite as we take the $T\to T_{c}$ limit. Note that this
dispersion relation holds for $k\ll T_{c}-T$.

Above the phase transition where our system is in its normal phase, the $U(1)$
current hydrodynamic excitations are given by the constitutive relations,

$\displaystyle\delta j_{t}$ $\displaystyle=-\chi_{QQ}\,\delta\mu\,,$
$\displaystyle\delta j_{x}$
$\displaystyle=-\sigma_{d}\,\partial_{x}\delta\mu\,,$ (3.30)

with $\delta\mu$ the local chemical potential and $\sigma_{d}$ the incoherent
conductivity. We therefore see that above the critical temperature we will
have a single diffusive mode determined by the diffusion constant
$D_{inc}=\chi_{QQ}^{-1}\,\sigma_{d}$. So, even though our sound modes tend to
become diffusive close to the transition, the attenuation constant does not
continuously connect to the diffusion constant of the normal phase incoherent
mode at the critical temperature.

###  3.3 Green’s functions

In this section we will introduces sources for the conserved $U(1)$ current in
our system which should be of order $\delta
s_{\alpha}\approx\mathcal{O}(\varepsilon)$. The construction of the
hydrodynamic perturbation in the bulk is identical to that of equation (3.2).
However, this time the frequency is fixed by the external sources and our
ultimate goal is to express the boundary current of the system in terms of the
sources and the phase of the condensed operator.

The above suggests that we will simply need to replace the boundary conditions
(3.5) by

$\displaystyle\delta\hat{v}_{t}=\delta\hat{s}_{t}-i\omega\,\delta\hat{c},\qquad\delta\hat{v}_{x}=\delta\hat{s}_{x}+i\varepsilon
k\,\delta\hat{c}\,,$ (3.31)

with $\delta\hat{s}_{\alpha}\propto\mathcal{O}(\varepsilon)$. Effectively, the
whole analysis of section 3.2 goes through after replacing
$-i\omega\,\delta\hat{c}\to\delta\hat{s}_{t}-i\omega\,\delta\hat{c}$ and
$i\varepsilon k\,\delta\hat{c}\to\delta\hat{s}_{x}+i\varepsilon
k\,\delta\hat{c}$. In terms of spacetime coordinates on the boundary, this is
equivalent to replacing the partial derivatives of the phase by the gauge
invariant combination

$\displaystyle\partial_{\alpha}\delta c\to\delta
s_{\alpha}+\partial_{\alpha}\delta c\,.$ (3.32)

One can of course check this explicitly but here we will just state the final
result for the expressions of the components of the boundary current

$\displaystyle q\,\delta j_{t}$ $\displaystyle=-\chi_{QQ}\,\left(\delta
s_{t}+\partial_{t}\delta c\right)+\Xi\,\partial_{t}\left(\delta
s_{t}+\partial_{t}\delta c\right)\,,$ $\displaystyle q\,\delta j_{x}$
$\displaystyle=-\chi_{JJ}\,\left(\delta s_{x}+\partial_{x}\delta
c\right)-\sigma_{d}\,\partial_{t}\left(\delta s_{x}+\partial_{x}\delta
c\right)\,.$ (3.33)

After having obtained the constitutive relations (3.3), we can impose the
continuity equation (3.6) to obtain an equation for the Goldstone mode. Doing
this in momentum space, gives us an expression for $\delta\hat{c}$ in terms of
the sources $\delta\hat{s}_{\alpha}$ which we can plug back in (3.3). From
that linear relation between the VEVs for the current and the sources we read
off the retarded Green’s functions

$\displaystyle G_{J^{t}J^{t}}(\omega,\varepsilon k)$
$\displaystyle=(\varepsilon k)^{2}\,\mathcal{G}(\omega,\varepsilon k),\qquad
G_{J^{x}J^{x}}(\omega,\varepsilon
k)=\omega^{2}\,\mathcal{G}(\omega,\varepsilon k)\,,$ $\displaystyle
G_{J^{t}J^{x}}(\omega,\varepsilon k)$
$\displaystyle=G_{J^{x}J^{t}}(\omega,\varepsilon k)=\varepsilon
k\,\omega\,\mathcal{G}(\omega,\varepsilon k)\,,$ (3.34)

where we have defined

$\displaystyle\mathcal{G}(\omega,\varepsilon
k)=\frac{\left(\chi_{QQ}+i\,\Xi\,\omega\right)\left(i\,\chi_{JJ}+\sigma_{d}\,\omega\right)}{\omega^{2}\left(-i\,\chi_{QQ}+\Xi\,\omega\right)+(\varepsilon
k)^{2}\left(i\,\chi_{JJ}+\sigma_{d}\,\omega\,\right)}\,.$ (3.35)

Notice that the last equation in (3.3) is compatible with the Onsager relation
$G_{J^{t}J^{x}}(\omega,\varepsilon k)=-G_{J^{x}J^{t}}(\omega,-\varepsilon k)$
given that $\mathcal{G}$ is even in $k$. Moreover, the positions of poles of
the Green’s functions (3.3) are set by the roots of the denominator of the
function (3.35). As expected, these are located precisely on the curves
$\omega=\omega_{\pm}(\varepsilon k)$ set by the dispersion relations we found
in section 3.2 for the superfluid sound mode.

The final result we would like to present in this section is the Kubo formulae
for our transport coefficients $\sigma_{d}$ and $\Xi$. By using the Green’s
functions (3.3) we can write

$\displaystyle\sigma_{d}=\lim_{\omega\to 0}\lim_{k\to
0}\frac{\mathrm{Im}\,G_{J^{x}J^{x}}}{\omega},\qquad\Xi=\lim_{k\to
0}\lim_{\omega\to 0}\frac{\mathrm{Im}\,G_{J^{t}J^{t}}}{\omega}\,.$ (3.36)

##  4 Scalar Sources and Pinning

In this section we would like to explicitly deform our theory by a
perturbatively small pinning parameter $\delta\rho_{(s)}$ and study the
resulting pseudo-Goldstone mode. In other words, we would like to explicitly
break the global $U(1)$ in a controlled fashion. In order to do this, we will
have to deform the backgrounds we have considered so far by adding a small
perturbative source
$\rho_{(s)}=\delta\rho_{(s)}\propto\mathcal{O}(\varepsilon^{2})$ in the near
conformal boundary expansion of equation (2).

Before applying our logic to the class of holographic theories we are
considering, it is worth revisiting our expectations in field theory terms.
Suppose that we couple our theory to an external gauge field $A_{\alpha}$ and
that we also introduce a source $\lambda$ for our charged scalar operators
$\mathcal{O}_{\psi}^{\ast}$. If the resulting theory is invariant under the
infinitesimal gauge transformations

$\displaystyle\delta
A_{\alpha}=\partial_{\alpha}\delta\Lambda,\qquad\delta\lambda=-iq\lambda\,\delta\Lambda\,,$
(4.1)

then we can show that the corresponding Ward identity for the current gets
modified to

$\displaystyle\nabla_{\alpha}\langle
J^{\alpha}\rangle=iq\,\left(\langle\mathcal{O}_{\psi}\rangle\lambda^{\ast}-\langle\mathcal{O}_{\psi}^{\ast}\rangle\lambda\right)\,.$
(4.2)

This equation makes clear that the order parameter and the source need to
align in the complex plane in equilibrium. In other words we can no longer
freely rotate the order parameter in the complex plane after fixing a source
without generating a current.

In holography, the small background perturbation we want to consider changes
the interpretation of the angle $\theta_{(v)}$ that we introduced in equation
(2.9). To see this, we need to consider the asymptotic expansion of the
charged scalar close to the conformal boundary. To do this, we first need to
better understand the asymptotics of the vector field $B_{\mu}$ to include
sources $\theta_{(s)}$ for the Goldstone mode. In the absence of background
sources for the charged scalar field, we have

$\displaystyle
B_{\alpha}=\partial_{\alpha}\theta_{(s)}\,(r+R)^{2\Delta-3}+\cdots+v_{\alpha}+\cdots+q\,j_{\alpha}\,(r+R)^{-1}+\cdots\,,$
(4.3)

where $\theta_{(s)}$, $v_{\alpha}$ and $j_{\alpha}$ are constants of
integration, with the equations of motion implying the constraint

$\displaystyle\partial_{\alpha}j^{\alpha}=2q\,\rho_{(v)}^{2}\,(2\Delta-3)\,\theta_{(s)}+\mathcal{O}(\partial^{2}\theta_{(s)})\,,$
(4.4)

where we dropped terms of higher order in derivatives of $\theta_{(s)}$. The
reason for doing this is that we will be interested in satisfying this
equation up to order $\mathcal{O}(\varepsilon^{3})$ and such terms would be
higher order provided that $\theta_{(s)}\propto\mathcal{O}(\varepsilon^{2})$.
We will show this later when we consider the proper boundary conditions,
compatible with the absence of time dependent sources for the charged scalar
operator $\mathcal{O}_{\psi}$.

After including the perturbative source terms, the angle $\theta$ and the
background source term $\delta\rho_{(s)}$ enter the asymptotic expansion of
the charged scalar perturbations according to,

$\displaystyle\delta\psi=i(\rho_{(v)}\,\theta_{(s)}+\delta\rho_{(s)}\theta_{(v)})\,(r+R)^{\Delta-3}+\cdots+i\rho_{(v)}\,\theta_{(v)}\,(r+R)^{-\Delta}+\cdots\,.$
(4.5)

This shows that equation (4.4) is nothing but the Ward identity (4.2) after
identifying

$\displaystyle\langle J^{\alpha}\rangle$
$\displaystyle=j^{\alpha}+\mathcal{O}(\partial\theta_{(s)}),$
$\displaystyle\lambda$
$\displaystyle=\delta\rho_{(s)}+i(\rho_{(v)}\,\theta_{(s)}+\delta\rho_{(s)}\theta_{(v)})+\mathcal{O}(\delta\rho_{(s)}\,\theta_{(s)}),$
$\displaystyle\langle\mathcal{O}_{\psi}\rangle$
$\displaystyle=\langle\mathcal{O}_{\psi}\rangle_{b}\,(1+i\theta_{(v)})+\mathcal{O}(\delta\rho_{(s)}\,\theta_{(s)})\,.$
(4.6)

From the above we see that in order to correctly identify the time dependent
perturbative source $\delta s_{\psi}$ for the scalar field, we need to impose

$\displaystyle\theta_{(s)}=\frac{1}{\rho_{(v)}}\left(-\delta\rho_{(s)}\,\theta_{(v)}+\delta
s_{\psi}\right)\,.$ (4.7)

Note that in this notation, the time dependent source of the complex scalar
introduces only a source $\delta s_{Y}=\delta s_{\psi}$ for the operator
$\mathcal{O}_{Y}$ that we introduced in section 2. The above equation shows
that the source term $\theta_{(s)}$ for the perturbation of $B_{\mu}$ and the
background perturbation source $\delta\rho_{(s)}$ need to be of the same order
in the $\varepsilon$ expansion. After this observation, the constraint
equation (4.4) becomes

$\displaystyle\partial_{\alpha}j^{\alpha}=-2\,q\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,\delta\rho_{(s)}\,\theta_{(v)}+2\,q\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,\delta
s_{Y}\,.$ (4.8)

From the point of view of hydrodynamics, the goal is to have the correct
constitutive relations for the currents $j_{\alpha}$ up to second order in
$\varepsilon$. That would allow us to satisfy equation (4.8) up to third order
in $\varepsilon$ provided that we take the source for the charged scalar to be
of order $\mathcal{O}(\varepsilon^{2})$. This is justified by the fact that
the terms we dropped in equation (4.4) are of order
$\mathcal{O}(\varepsilon^{4})$. This argument shows that the constitutive
relations of equations (3.3) and (3.27) are sufficient for this task. Finally,
one might worry that we should take in account the fact that the background
quantities which enter equation (2.12) will bring their own $\varepsilon$
corrections to the perturbation of $B_{\mu}$. However, as we argued above, the
correction of the bulk scalar due to the perturbative source
$\delta\rho_{(s)}$ will be of order $\mathcal{O}(\varepsilon^{2})$. That would
induce corrections to our perturbation $\delta B_{\mu}$ at order
$\mathcal{O}(\varepsilon^{3})$ which is certainly beyond our scope.

In order to compute the retarded Green’s functions of the system of our
operators, we need to solve the Ward identity (4.8) after identifying
$\theta_{(v)}=\delta c$. By doing so we obtain the explicit expressions

$\displaystyle G_{J^{t}J^{t}}(\omega,\varepsilon k)$
$\displaystyle=\frac{f\,(w+(\varepsilon\,k)^{2}\,g)}{h}\,,\quad
G_{J^{x}J^{x}}(\omega,\varepsilon k)=\frac{(-w+\omega^{2}f)\,g}{h}$
$\displaystyle G_{J^{x}J^{t}}(\omega,\varepsilon k)$
$\displaystyle=G_{J^{t}J^{x}}(\omega,\varepsilon
k)=\varepsilon\,k\,\omega\frac{f\,g}{h}\,,$ $\displaystyle
G_{J^{t}\mathcal{O}_{Y}}(\omega,\varepsilon k)$
$\displaystyle=-i\frac{q\omega\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,f}{h}\,,\quad
G_{J^{x}\mathcal{O}_{Y}}(\omega,\varepsilon
k)=-i\frac{qk\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,g}{h}$ $\displaystyle
G_{\mathcal{O}_{Y}J^{t}}(\omega,\varepsilon k)$
$\displaystyle=i\frac{q\omega\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,f}{h}\,,\quad
G_{\mathcal{O}_{Y}J^{x}}(\omega,\varepsilon
k)=i\frac{qk\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,g}{h}$ $\displaystyle
G_{\mathcal{O}_{Y}\mathcal{O}_{Y}}(\omega,\varepsilon k)$
$\displaystyle=\frac{q^{2}\,|\langle\mathcal{O}_{\psi}\rangle_{b}|^{2}}{h}$
(4.9)

where we have defined

$\displaystyle f$ $\displaystyle=\chi_{QQ}+i\,\Xi\,\omega,\qquad
g=\chi_{JJ}-i\,\sigma_{d}\,\omega\,,$ $\displaystyle h$
$\displaystyle=w-\omega^{2}\,f+\varepsilon^{2}\,k^{2}\,g,\qquad
w=2q^{2}\,|\langle\mathcal{O}_{\psi}\rangle_{b}|\,\delta\rho_{(s)}\,.$ (4.10)

The poles of the Green’s functions in equation (4) reveal that the second
sound mode has acquired a resonance frequency as well as a gap. More
specifically, we find the dispersion relation

$\displaystyle\omega=\pm\sqrt{\frac{w+k^{2}\,\chi_{JJ}}{\chi_{QQ}}}-\frac{i}{2\chi_{QQ}^{2}}\left(w\,\Xi+k^{2}\,(\Xi\,\chi_{JJ}+\sigma_{d}\,\chi_{QQ})\right)\,,$
(4.11)

from which we can read off the resonance frequency
$\omega_{r}=\sqrt{w/\chi_{QQ}}$ and the gap666Notice that in the language of
[14], we have $\omega_{gap}\sim\chi_{YY}^{-1}$ with $\chi_{YY}$ the
susceptibility of the operator $\mathcal{O}_{Y}$. Moreover, in the language of
[12] we have $\omega_{gap}\sim\omega_{r}^{2}$.
$\omega_{gap}=w\,\Xi/(2\,\chi_{QQ}^{2})$.

It is interesting to notice that, when $w<0$, the theory develops an
instability. This can be easily seen from the square root in (4.11) which will
become imaginary in this case. This instability is simply a mode of the system
which wants to align the VEV and explicit deformation of the system in the
complex plane.

Finally, we would like to flesh out some of the global effects of the explicit
breaking on our observables. In order to do this, we will consider the Green’s
functions in equation (4) at $k=0$. In that limit, from the Green’s function
$G_{\mathcal{O}_{Y}J^{x}}$ we see that the transport current $J^{x}$ and the
operator $\mathcal{O}_{Y}$ completely decouple. Similarly, we will see that
$J^{x}$ also decouples from the charge density $J^{t}$. It is therefore
natural to expect that the small explicit breaking will have no effect on the
low frequency electric conductivity,

$\displaystyle\sigma_{AC}(\omega)=\frac{G_{J^{x}J^{x}}(\omega,k=0)}{i\,\omega}=\frac{i\,\chi_{JJ}}{\omega}+\sigma_{d}\,.$
(4.12)

However, we see that at $k=0$ the charge density and the scalar operator
$\mathcal{O}_{Y}$ remain coupled turning the electric charge to an almost
conserved quantity in the deformed theory.

##  5 Numerical checks

In this section we carry out a series of numerical checks of the results
derived in the previous sections and in particular, we numerically confirm the
sound mode dispersion relation (3.28), the Green’s functions (3.3)-(3.35) and
the formula for the gap (4.11).

We consider the action (2.2) with the following potential and gauge coupling

$\displaystyle
V=-6+m_{\rho}^{2}\rho^{2}+\frac{c^{2}}{2}\phi^{4}+\frac{m_{\phi}^{2}}{2}\phi^{2}+\lambda\,\rho^{2}\phi^{2}\,,\quad
m_{\rho}^{2}=-2\,,\quad m^{2}_{\phi}=-2\,,$ $\displaystyle\tau=\cosh{\phi},$
(5.1)

where we note that, for $\rho=0$, $\partial_{\phi}V=0$ both at $\phi=0$ and at
$\phi=1/c$.

Given the choices above, the corresponding equations of motion admit a unit-
radius $AdS_{4}$ vacuum solution with $\rho=\phi=A=\theta=0$, which is dual to
a $d=3$ CFT. Placing the CFT at finite temperature corresponds to considering
the Schwarzschild black hole, which typically serves as the configuration dual
to the normal phase of holographic systems. However, here we choose to deform
our boundary theory by a relevant operator, $O_{\phi}$, with scaling dimension
$\Delta_{\phi}=2$. Then, the corresponding backreacted solution dual to the
normal phase of our system will be given by black brane with a non-trivial
profile for the scalar field $\phi$. As the temperature goes to zero, $T\to
0$, these configurations will approach a flow between the unit-radius
$AdS_{4}$ in the UV, with $\phi=0$, and an IR $AdS_{4}$ with radius
$L_{IR}^{2}=12c^{2}/(1+12c^{2})$ supported by $\phi=1/c$.

To construct these solutions explicitly we consider the ansatz (2) with
$\rho(r)=0$ and the IR and UV boundary conditions, (2) and (2) respectively,
with $\rho^{(0)}=$$\rho_{(s)}=\rho_{(v)}=0$. This boundary condition problem
is then solved using a double-sided shooting technique. In figure 1, we plot
the logarithmic derivative of the entropy of the system with respect to the
temperature, $TS^{\prime}(T)/S(T)$ for $\phi_{(s)}=1$ and $c=1$. We clearly
see that both at very high and very low temperatures the entropy scales like
$T^{2}$ which is compatible with having $AdS_{4}$ on both sides of the RG
flow.

Figure 1: Plot of the logarithmic derivative of the entropy
$TS^{\prime}(T)/S(T)$ as a function of the temperature indicating that our
solutions interpolate between two $AdS_{4}$ geometries. Here $\phi_{(s)}=1$
and $c=1$.

On top of these thermal states, we will consider instabilities associated to
the scalar field $\rho$. To ensure that such instabilities exist in our model
we need to make sure that the scalar field $\rho$ violates the BF bound
associated with the $AdS_{4}$ in the IR, i.e.

$L^{2}_{IR}\,{m^{IR}_{\rho}}^{2}=\frac{12}{1+12c^{2}}(c^{2}\,m_{\rho}^{2}+\lambda)<-\frac{9}{4}\,,$
(5.2)

but is nevertheless stable in the UV: $m_{\rho}^{2}\geq-\frac{9}{4}\,$. For
example, this is the case for $c=1,\lambda=-3/2$ and so we expect an
instability to occur for this choice of parameters. Thus, for temperatures
below a critical one, we expect a new branch of black holes to emerge
characterised by a non-trivial condensate for $\rho$. To determine the
critical temperature at which these instabilities set in we need to study the
associated zero mode. In particular, we consider a linearised perturbation
around the background constructed above of the form

$\displaystyle\rho$ $\displaystyle=0+\delta\rho\,.$ (5.3)

Plugging the above perturbation in the equations of motion, we obtain one
second order linear ODE, which we solve by imposing the following boundary
conditions at the black hole horizon

$\displaystyle\delta\rho$ $\displaystyle=\delta\rho_{h}+\cdots\,,$ (5.4)

and asymptotically

$\displaystyle\delta\rho$
$\displaystyle=0+\frac{\delta\rho_{v}}{r^{2}}+\cdots\,,$ (5.5)

where we have already set the source for $\delta\rho$ to $0$, so that the
emergence of the new phase is spontaneous. Overall, the boundary conditions
are determined by 2 constants $\delta\rho_{h}\,,\delta\rho_{v}$, one of which
can be set to $1$ because of the linearity of the equation. Consequently,
performing the numerical integration will fix the highest value of the
temperature, $T_{c}=0.022$, for which one can find non-trivial solutions for
$\delta\rho$.

The next step is to construct the backreacted solutions corresponding to the
broken phase. To achieve this we use the ansatz (2) and the IR and UV boundary
conditions, (2), (2). When plugging this ansatz in the equations of motion we
obtain a set of three second order ODEs and one first order. Thus, a solution
is specified in terms of 7 constants of integration. Looking at the expansion
(2), we see that it is specified in terms of 3 constants, in addition to the
temperature $T$. Similarly, the asymptotic expansion (2) is parametrised by 4
constants in addition to $\phi_{(s)}$ —note that we keep $\rho_{(s)}=0$ so
that the condensation is spontaneous. Overall, in the IR and UV expansions we
have a total of 7 constants as well as $T$ and $\phi_{(s)}$, which matches the
7 constants of integration. We proceed to solve this boundary condition
problem numerically using double-sided shooting. In figure 2 we show the
condensate as well as the free energy as functions of the temperature, $T$, in
support of the phase transition being second order. Here $\phi_{(s)}=1$ and
$c=1,\lambda=-3/2$.

Figure 2: (left) Plot of the condensate as a function of the temperature.
(right) Plot of the free energy of the system in the normal phase (red line)
and in the broken phase (blue line) as a function of the temperature. Both
plots demonstrate that there is a second order phase transition at $T=T_{c}$.

###  5.1 Static perturbations

Having constructed the backreacted black holes corresponding to the broken
phase, we now turn our attention to studying perturbations around them. In
particular, to check numerically the validity of the analytic expressions for
the dispersion relation of the sound mode and the spatially resolved two-point
functions, we need to construct the static perturbations as in section 2.1 and
extract from them $\chi_{JJ}$, $\chi_{QQ}$.

Specifically, considering the perturbations $\delta B_{t}^{(t)}(r)$ and
$\delta B_{x}^{(x)}(r)$ gives rise to two decoupled linear second order
equations, which we solve subject to boundary conditions (2.1),(2.1). Then,
given the numerical solutions, the susceptibilities are simply obtained by
dividing the corresponding expectation values by the associated sources as
extracted by the asymptotic expansion

$\chi_{QQ}=-\delta j_{t}/\delta\mu_{t}\,,\qquad\chi_{JJ}=-\delta
j_{x}/\delta\mu_{x}\,.$ (5.6)

On the other hand, $\Xi$, $\sigma_{d}$ are calculated using horizon data. For
$\\{T,c,\lambda,\phi_{(s)}\\}=\\{0.015,1,-3/2,1\\}$, we find
$\chi_{JJ}=0.12238$, $\chi_{QQ}=0.138$, $\Xi=0.408$, $\sigma_{d}=0.318$.

###  5.2 Second sound and spatially resolved two-point functions

In order to compute the second sound and the two-point functions we need to go
beyond static perturbations. In particular, we consider the linearised
perturbation (3.3), with

$S(r)=\int_{\infty}^{r}\frac{dy}{U(y)}\,.$ (5.7)

Plugging this ansatz in the equations of motion, we obtain two second order
ODEs, along with an algebraic equation for $\delta B_{r}$. We now turn to the
boundary conditions for these functions. In the IR we impose in-falling
boundary conditions at the horizon which is located at $r=0$

$\displaystyle\delta B_{t}=b_{t}+\dots$ $\displaystyle\delta
B_{x_{1}}=b_{x_{1}}+\dots\,.$

Thus, we see that the expansion is fixed in terms of 2 constant
$b_{t},b_{x_{1}}$, in addition to $\omega,k$. The UV expansion takes the form

$\displaystyle\delta B_{t}=\delta s_{t}-i\omega\delta c+\frac{\delta
j_{t}}{r+R}+\dots$ $\displaystyle\delta B_{x_{1}}=\delta s_{x_{1}}+ik\delta
c+\frac{\delta j_{x_{1}}}{r+R}+\dots,\,$ (5.8)

where $\delta j_{t}$ is fixed in terms of the other parameters. Overall, this
expansion is determined in terms of the sources $\delta s_{t},\delta s_{x}$
and the 2 parameters $\delta c,\delta j_{x_{1}}$, in addition to $\omega,k$.

For constructing the second sound, we solve the above equations around the
numerical background of the previous section, imposing that the sources vanish
$\delta s_{t}=\delta s_{x_{1}}=0$. In addition, due to the linearity of the
equations we also impose $b_{x_{1}}=1$. Thus, for fixed $k$, the shooting
method determines $b_{t},\delta j_{x_{1}},\delta c,\omega$. In figure 3, we
compare our numerical results with the analytics of the previous chapter by
plotting certain derivatives of the dispersion relation. For small values of
$k$, where our analytic arguments are valid, we see a good quantitative
agreement.

For constructing the two point functions we again employ a double-sided
shooting technique, but now we set either $\delta s_{t}$ or $\delta s_{x_{1}}$
to zero and scale the remaining source to one using the linearity of the
equations. Thus, for fixed $\omega,k$ and say $\delta s_{t}=0,\delta s_{x}=1$,
the boundary condition system determines $b_{t},b_{x_{1}},\delta c,\delta
j_{x}$, allowing one to compute
$G_{xx}=\omega^{2}\,\mathcal{G}(k,\omega),G_{tx}=G_{xt}=k\omega\,\mathcal{G}(k,\omega),G_{tt}=k^{2}\,\mathcal{G}(k,\omega)$
where

$\mathcal{G}(k,\omega)=\frac{\delta j_{x}+i\omega-k\omega\delta
c}{\omega^{2}}\,.$ (5.9)

In figure 4 we plot $\mathcal{G}$ (scaled appropriately) as function of the
frequency, either for zero and finite $k$. In the small frequency limit, these
quantities approach the constants $\chi_{QQ}$ (top, left), $-\chi_{JJ}$ (top,
right), $\sigma_{d}$ (bottom, left) and $\Xi$ (bottom, right), in agreement
with the analytic expressions (3.3)-(3.35).

Figure 3: Plots of $\tfrac{\omega^{2}Re[\mathcal{G}]}{\partial k}$ and
$\frac{1}{2}\frac{\partial^{2}Im[\omega]}{\partial k^{2}}$ as functions of $k$
for the second sound dispersion relation. The dashed black line corresponds to
the analytic prediction. Here
$\\{T,c,\lambda,\phi_{(s)}\\}=\\{0.015,1,-3/2,1\\}$.

Figure 4: Plots of $\mathcal{G}$ as a function of $\omega$ for $k=0$ and $k=$
finite. The dashed black lines corresponds to $\chi_{QQ}$ (top, left),
$-\chi_{JJ}$ (top, right), $\sigma_{d}$ (bottom, left) and $\Xi$ (bottom,
right). Here $\\{T,c,\lambda,\phi_{(s)}\\}=\\{0.015,1,-3/2,1\\}$.

###  5.3 Pseudo-gapless modes

In this subsection we outline the numerical computation of the pseudo-gapless
modes in the presence of pinning. We perform a calculation similar to the one
for second sound, but we now consider linearised fluctuations (3.3), (5.7)
around a background configuration that has a small but finite source,
$\rho_{(s)}$, for the scalar $\rho$. The only difference to the previous
subsection is the expansion of the perturbations in the UV part of the
geometry. In this case we find that

$\displaystyle\delta B_{t}=0+\frac{\delta j_{t}}{r+R}+\dots$
$\displaystyle\delta B_{x_{1}}=0+\frac{\delta j_{x_{1}}}{r+R}+\dots\,.$ (5.10)

Note that this expansion differs from (5.2), not only because there are no
sources for the perturbations in this case, but also because having
$\rho_{(s)}\neq 0$ pushes the VEV of the goldstone mode to appear at order
$1/(r+R)$ leading to $\delta j_{t},\delta j_{x_{1}}$ being independent
constants. The parameter counting follows just like above, suggesting that for
fixed $k$ we expect to find a discrete set of solutions. In figure 5, we plot
our numerical results for the real and imaginary part of the gap and we
overlay them with the analytics of the previous chapter. We see a good
quantitative agreements for small values of $\rho_{(s)}$, as expected.

Figure 5: Plots of the real and imaginary part of the gap as function of the
source $\rho_{(s)}$. The dashed black lines corresponds to the analytic
prediction. Here $\\{T,c,\lambda,\phi_{(s)}\\}=\\{0.015,1,-3/2,1\\}$ and
$k=10^{-4}$.

##  6 Discussion

In this paper we studied first order dissipative effects in the hydrodynamic
regime of holographic superfluid phases of matter. At zero chemical potential
and charge density, the normal and the superfluid collective degrees of
freedom remain decoupled. This fact simplified our analysis in extracting the
transport coefficients relevant to the superfluid dissipation.

For a relativistic superfluid, in principle we would only have to determine
four invariant quantities that would fully fix the constitutive relation of
the conserved current in terms of the phase [6]. In our case, only two of them
were non-trivial and they were both fixed in terms of susceptibilities and
black hole horizon data. For a specific class of models the coefficient
$\sigma_{d}$ had been computed earlier in the literature [20, 22]. Here, we
presented a technique which is applicable also in inhomogeneous black hole
backgrounds relevant to phases of matter in which translations are broken
either spontaneously or explicitly. Interesting extensions of our work include
the reduced hydrodynamics of superfluid phases without momentum conservation
[23].

One interesting aspect of our work concerns the coefficient $\Xi$ which
appears in the time component of constitutive relations (3.3) for the current.
We have shown that this grows like $(T_{c}-T)^{-1}$ close to the transition
signalling the breakdown of the derivative expansion. More specifically,
judging from the next to leading term in the hydrodynamic expansion, we have
made explicit that this will converge for wavenumbers with $k\ll T_{c}-T$. It
would be interesting to explore the precise way that the hydrodynanic
expansion breaks down by following the logic of e.g. [24, 25, 26]. Finally, we
have studied the hydrodynamics of the system upon introducing sources which
break the global symmetry in a controlled manner.

An interesting extension we will report on in the future is the inclusion of a
background finite magnetic field [23]. That would imply the existence of
background vortices which have been previously studied in the framework of
holography in [27]. One step further would include the presence of disorder
and the resulting flux pinning which relaxes the supercurrent leading to
finite DC electric conductivity at finite temperature.

Finally, holography provides access to a plethora of superfluid phase ground
states [28, 29]. An interesting direction would be to study the low
temperature behaviour of the transport coefficients we computed in this paper.
This is possible by extracting the behaviour of low temperature black hole
horizons by following the techniques of e.g. [30].

## Acknowledgements

AD is supported by STFC grant ST/T000708/1. CP is supported by the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement HoloLif No 838644.

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