Dissipative effects in finite density holographic superfluids

Aristomenis Donos and Polydoros Kailidis

Centre for Particle Theory and Department of Mathematical Sciences,
Durham University, Durham, DH1 3LE, U.K.

Abstract

> We derive the leading dissipative corrections of holographic superfluids at
> finite temperature and chemical potential by employing our recently
> developed techniques to study dissipative effects in the hydrodynamic limit
> of holographic theories. As part of our results, we express the incoherent
> conductivity, the shear and the three bulk viscosities in terms of
> thermodynamics and the black hole horizon data of the dual bulk geometries.
> We use our results to show that all three bulk viscosities exhibit singular
> behaviour close to the critical point.

##  1 Introduction

Holography provides large classes of examples of strongly coupled field
theories where exact computations can be carried out [1, 2]. In a certain
limit, non-trivial questions about field theory can be mapped to well defined
problems in Einstein’s classical theory of gravity in an asymptotically Anti
de-Sitter spacetime (AdS) of dimensionality larger by one. The fall-off
conditions of the classical gravitational fields close to the conformal
boundary of AdS set the sources of local operators on the field theory side.
In the classical limit, the holographic principle states that the partition
functions of the two sides are equal, providing a powerful tool to compute
expectation values of local operators.

In thermal equilibrium, the geometric dual of the thermal state is a black
hole geometry with the temperature set by the Hawking temperature of the
Killing horizon. Moreover, the holographic dictionary suggests that global
$U(1)$ symmetries on the field theory side are gauged in the bulk. The
asymptotic flux of the corresponding gauge fields set the electric charge
density of the field theory making holography an invaluable laboratory to
study large classes of strongly coupled systems at finite temperature and
number density [3].

An interesting application of holography concerns the study of systems
exhibiting spontaneous breaking of a global symmetry. In general, continuous
phase transitions are driven via perturbative instabilities of black holes
against fluctuations of classical bulk fields. The first examples of symmetry
breaking were due to bulk fields which are charged under continuous internal
symmetry groups, leading to superfluid phases of holographic matter [4, 5].
Spontaneous breaking of spacetime symmetries were realised later in [6, 7, 8,
9] making holography even more appealing for applications in condensed matter
systems.

Holographic systems reach local thermal equilibrium with their long wavelength
excitations obeying the laws of hydrodynamics. This has been a very active
area of research over the past years [10, 11, 12, 13, 14] from which new
lessons about low energy effective field theory have been learned [3, 15]. The
hydrodynamic limit of broken symmetry phases where the standard hydrodynamic
degrees of freedom of charged fluids combine with gapless Goldstone modes has
also been considered in holography [16, 17, 18]. More recently, the amplitude
mode which becomes gapless at continuous phase transitions has been realised
holographically in [19]. This is a first step towards constructing effective
theories which include fluctuations of the amplitude of the order parameter
apart from its phase.

The hydrodynamic limit of hologrpahic superfluids has been studied extensively
since their first discovery. As standard in hydrodynamics, the stress tensor
of the theory and the electric current admit a derivative expansion in terms
of the local temperature, the normal fluid velocity, the chemical potential
and the phase of the order parameter. The inequivalent terms that can appear
in the first few orders of the hydrodynamic series of relativistic superfluids
have been classified in [16, 17, 18]. This is an necessary step in order to
extract the number of the transport coefficients that parametrise the
different terms in the expansion series. These numbers are essentially the
invariants that one can have under different choices of a fluid frame. For an
isotropic relativistic fluid, the coefficients that need to be specified in
the small superfluid velocity limit are the incoherent electric conductivity
$\sigma$, the shear viscosity $\eta$ and three bulk viscosities $\zeta_{i}$.
Similarly to normal fluids, conformal symmetry constrains the form of the bulk
viscosities allowing only one of them to be non-zero [16, 17]. By introducing
scales through relevant scalar operators, we will retain as many independent
transport coefficients as in any relativistic superfluid.

A number of previous works have considered various aspects of the hydrodynamic
limit of holographic superfluids. However, most of them have either resorted
to numerical techniques [20, 21, 22, 23] or they have focused on specific
models where analytic solutions for the gravitational problem can be obtained
infinitesimally close to the transition [24, 16]. In this paper, we will
employ the techniques developed in [25, 19], based on the Crnkovic-Witten
symplectic current [26], to derive the first dissipative corrections in the
hydrodynamic limit of holographic superfluids. A significant advantage of this
approach is that an explicit solution of the gravitational fields is not
required. Instead, it focuses on the universal aspects of the black holes dual
to the thermal states.

As a by-product of our derivation, we will be able to fix the five non-trivial
dissipative transport coefficients that we expect to have. The expressions we
will obtain will be in terms of horizon data of the black holes dual to the
thermal states of our system. Our results reproduce a well known result for
the shear viscosity $\eta$ to entropy density $s$ ratio which is proportional
to $1/4\pi$ [10, 27, 16]. However, our results for the incoherent conductivity
and the bulk viscosities are new. In particular, our formula for the bulk
viscosity $\zeta_{1}$ generalises the results of [28] and [29] to include the
bulk massive vector on the horizon. Moreover, as we will see our expression
for $\zeta_{3}$ generalises the result of [25] which was obtained for
holographic superfluids at zero chemical potential.

Given our analytic expressions in terms of horizon data, we are able study the
behaviour of the first dissipative corrections close to the phase transition.
By following general arguments about the behaviour of our bulk fields near the
phase transition, we are able to show that the shear viscosity and the
electric conductivity are continuous functions across the transition. At the
same time, we show that all three bulk viscosities diverge at the critical
point.

As one might anticipate, the hydrodynamic modes of our system consist of the
two longitudinal sound modes and the transverse shear mode responsible for the
diffusion of momentum density. Taking the limit of the dispersion relations
close to the critical temperature we find that the speed of the first sound
mode remains finite while the speed for the second sound vanishes. These
results can be shown through general considerations of ideal superfluid
hydrodynamics. More interestingly, using the explicit expressions for the
dissipative coefficients we show the the attenuation of the first sound
diverges. This is in contrast to the second sound whose attenuation part
remains finite.

Our paper is organised in six sections which are further divided in
subsections. In section 2 we present the class of holographic models we wish
to study along with the thermodynamic properties of the geometries dual to the
field theory thermal states. In section 3 we discuss perturbations around our
black holes and present the static perturbations which are the infinite
wavelength and zero frequency limits of our hydrodynamic expansion. In section
4 we extract the leading dissipative corrections to the ideal superfluid based
on the techniques we developed in [25, 19]. We conclude our analytic results
in section 5 where we consider the limit of our hydrodynamic expansion close
to the critical point. We also take the limit of zero chemical potential and
compare our results with those in [25].

##  2 Setup

To model a holographic superfluid phase at finite density, we will consider a
bulk theory which contains a Maxwell field $A_{\mu}$, a neutral scalar $\phi$
and a complex scalar $\psi$ which is charged under the local $U(1)$ symmetry.
The neutral scalar $\phi$ is not a necessary ingredient but we will use it in
order to introduce additional scales into the system. This will allow all the
bulk viscosities we expect to find to be non-zero.

The system is described by the bulk action,

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

with the covariant derivative
$D_{\mu}\psi=\nabla_{\mu}\psi+iq_{e}A_{\mu}\,\psi$ and the field strength
$F=dA$. It is easy to see that the above action is invariant under the local
gauge transformations $A\to A+d\Lambda$ and $\psi\to
e^{-iq_{e}\Lambda}\,\psi$.

Our focus will be on the superfluid phase of our system corresponding to
backgrounds with a non-trivial profile for the complex bulk scalar $\psi$. In
this case, the field redefinition $\psi=\rho\,e^{iq_{e}\theta}$ is well
defined bringing the action to the form,

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

where we have set $B=A+\partial\theta$ and $F=dB$. The equations of motion
which extremise the bulk action are,

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}V-\frac{\tau}{2}\left(F_{\mu\rho}F_{\nu}{}^{\rho}-\frac{1}{4}g_{\mu\nu}\,F^{2}\right)\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle-\frac{1}{2}\partial_{\mu}\phi\,\partial_{\nu}\phi-\frac{1}{2}\partial_{\mu}\rho\,\partial_{\nu}\rho-\frac{1}{2}q_{e}^{2}\rho^{2}B_{\mu}\,B_{\nu}=$
$\displaystyle 0\,,$
$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi-\partial_{\phi}V-\frac{1}{4}\partial_{\phi}\tau\,F^{2}=$
$\displaystyle 0\,,$
$\displaystyle\nabla_{\mu}\nabla^{\mu}\rho-\partial_{\rho^{2}}V\,\rho-\frac{1}{4}\partial_{\rho^{2}}\tau\,\rho\,F^{2}-q_{e}^{2}\,\rho\,B^{2}=$
$\displaystyle 0\,,$ $\displaystyle\nabla_{\mu}\left(\rho^{2}B^{\mu}\right)=$
$\displaystyle 0\,,$
$\displaystyle\nabla_{\mu}(\tau\,F^{\mu\nu})-q_{e}^{2}\,\rho^{2}\,B^{\nu}=$
$\displaystyle 0\,.$ (2.3)

We will consider electrically charged black brane solutions which are dual to
thermal states of the deformed CFT by the relevant operator
$\mathcal{O}_{\phi}$ dual to the bulk field $\phi$. Moreover, we will assume
that below a critical temperature $T_{c}$, the system exhibits spontaneous
breaking of the field theory global $U(1)$.

The corresponding ansatz for the background fields is,

$\displaystyle ds^{2}$
$\displaystyle=-U(r)\,dt^{2}+\frac{dr^{2}}{U(r)}+e^{2g(r)}\,\left(dx^{2}+dy^{2}\right)\,,$
$\displaystyle B=$ $\displaystyle
a(r)\,dt,\qquad\phi=\phi(r),\qquad\rho=\rho(r)\,.$ (2.4)

The above choice of coordinates fixes the radial coordinate apart from a
global shift. We will use this freedom to always have the event horizon
sitting at $r=0$. Near the horizon, regularity implies the Taylor expansions

$\displaystyle U(r)\approx\,4\pi T\,r+\mathcal{O}(r^{2}),\qquad g(r)\approx
g^{(0)}+\mathcal{O}(r),\qquad a(r)\approx r\,a^{(0)}+\cdots\,,$
$\displaystyle\phi(r)\approx\phi^{(0)}+\mathcal{O}(r),\qquad\rho(r)\approx\rho^{(0)}+\mathcal{O}(r)\,,$
(2.5)

where $T$ is the Hawking temperature.

Close to the conformal boundary at $r\to\infty$ we impose the expansions,

$\displaystyle U(r)\approx(r+R)^{2}+\cdots+\frac{g_{(v)}}{r+R}+\cdots,\qquad
g(r)\approx\ln(r+R)+\mathcal{O}(r^{-1})\,,$ $\displaystyle
a(r)\approx\mu_{t}-\frac{\varrho}{r+R}+\cdots\,,$
$\displaystyle\phi(r)\approx\phi_{(s)}\,(r+R)^{\Delta_{\phi}-3}+\cdots+\phi_{(v)}\,(r+R)^{-\Delta_{\phi}}+\cdots\,,$
$\displaystyle\rho(r)\approx\rho_{(s)}\,(r+R)^{\Delta_{\psi}-3}+\cdots\rho_{(v)}\,(r+R)^{-\Delta_{\psi}}+\cdots\,,$
(2.6)

where we have defined the scalar operators sources $\phi_{(s)}$ and
$\rho_{(s)}$ for the neutral and complex operators correspondingly. Moreover,
we have defined the chemical potential $\mu_{t}$ and as we will see in the
next section, the field theory charge density is given by the constant of
integration $\varrho$. In this paper we will consider spontaneous breaking of
the global $U(1)$ and we will be setting the non-perturbative complex scalar
source $\rho_{(s)}$ equal to zero. Finally, the global shift in the radial
coordinate which fixes the horizon at $r=0$ is reflected by the constant of
integration $R$.

In general, when the complex scalar source $\rho_{(s)}$ is set to zero, a
perturbation of the bulk vector will admit the UV expansion [25],

$\displaystyle\delta
B_{\alpha}=\frac{\partial_{\alpha}\delta\theta_{(s)}}{(r+R)^{3-2\,\Delta_{\rho}}}+\cdots+\delta
m_{\alpha}+\cdots+\frac{\delta j_{\alpha}}{r+R}+\cdots\,,$ (2.7)

where $m_{\alpha}=\partial_{\alpha}\theta_{(v)}+\mu_{\alpha}$ is a gauge
invariant combination of the superfluid velocity
$\partial_{\alpha}\theta_{(v)}$ and the source $\mu_{\alpha}$ for the $U(1)$
current. As we will explain in the next section, the constant of integration
$\theta_{(s)}$ is essentially a source for the complex scalar operator and the
constants $\delta j_{\alpha}$ correspond to perturbations of the $U(1)$
electric current. As we will see these in the next section, these constants of
integration are subject to a scalar constraint which is equivalent to the Ward
identity satisfied by the electric current.

From the above we see that the phase $\theta$ of the complex scalar $\psi$ has
become part of the massive vector $B$. It will useful for us to note that, in
the absence of a background source $\rho_{(s)}$ the asymptotic expansion for
the bulk phase close to the conformal boundary is given by,

$\displaystyle\delta\theta\approx(r+R)^{2\Delta_{\psi}-3}\,\delta\theta_{(s)}+\cdots+\delta\theta_{(v)}+\cdots\,,$
(2.8)

allowing us to write the expansion for the complex scalar perturbation,

$\displaystyle\psi$
$\displaystyle\approx(r+R)^{\Delta_{\psi}-3}\,e^{iq_{e}\,\theta_{(v)}}\,(iq_{e}\,\rho_{(v)}\,\delta\theta_{(s)}+\delta\rho_{(s)})$
$\displaystyle\qquad\qquad+\cdots+(r+R)^{-\Delta_{\psi}}\,e^{iq_{e}\,\theta_{(v)}}\,(\delta\rho_{(v)}+iq_{e}\,\rho_{(v)}\,\delta\theta_{(v)})+\cdots\,.$
(2.9)

From the above, we can read off the perturbative source for the complex
operator to be,

$\displaystyle\delta\lambda=e^{iq_{e}\,\theta_{(v)}}\,(iq_{e}\,\rho_{(v)}\,\delta\theta_{(s)}+\delta\rho_{(s)})\,.$
(2.10)

###  2.1 Thermodynamics and Ward Identities

In order to extract quantities which are relevant to the field theory living
on the boundary, our bulk action (2.1) needs to be supplemented with
appropriate counterterms that will render it finite on-shell [30]. Apart from
regularisation, the appropriate counterterms make the variational problem well
defined [31]. For the bulk action we are interested in, a set universal terms
are contained in,

$\displaystyle S_{bdr}=$ $\displaystyle-\int_{\partial
M}d^{3}x\,\sqrt{-\gamma}\,\left(-2K+4+R_{bdr}\right)$
$\displaystyle\quad-\frac{1}{2}\int_{\partial
M}d^{3}x\,\sqrt{-\gamma}\,[(3-\Delta_{\phi})\phi^{2}-\frac{1}{2\Delta_{\phi}-5}\,\partial_{a}\phi\partial^{a}\phi]$
$\displaystyle\quad-\frac{1}{2}\int_{\partial
M}d^{3}x\,\sqrt{-\gamma}\,[(3-\Delta_{\psi})|\psi|^{2}-\frac{1}{2\Delta_{\psi}-5}\,D_{a}\psi
D^{a}\psi^{\ast}]+\cdots\,,$ (2.11)

where $\gamma_{\alpha\beta}$ is the induced metric on the asymptotic
hypersusface $\partial M$ of constant radial coordinate $r$.

The grand canonical free energy density $w_{FE}$ for the thermal states
captured by the geometries of equation (2) coincides with the value of the
total action $S_{tot}=S+S_{bdr}$ after a Wick rotation to Euclidean time
$t=-i\,\tau$. This yields the total Euclidean action $I_{tot}$ such that
$w_{FE}=T\,I_{tot}$ and,

$\displaystyle w_{FE}=\epsilon-T\,s-\mu_{t}\,\varrho\,.$ (2.12)

In the expression above, $\epsilon$ is the conserved energy density, $\varrho$
is the electric charge density and,

$\displaystyle s=4\pi\,e^{2\,g^{(0)}}\,,$ (2.13)

is the Bekenstein-Hawking entropy density of the system. Another horizon
quantity we can define and which will become useful later is related to the
flux of the one-from $B_{\mu}$ through the black hole horizon. We will can
this the horizon charge density $\varrho_{h}$ and by using our near horizon
expansion (2) we can write,

$\displaystyle\varrho_{h}=e^{2g^{(0)}}\tau^{(0)}a^{(0)}\,.$ (2.14)

In our analysis, we are aiming to use the set of techniques that were
developed in [25, 19, 29] in order to extract holographic information for the
perturbations in real time. For this reason, we will perform an integration by
parts in the Einstein-Hilbert in order to obtain a Lagrangian density that
will only contain first order partial derivatives of the metric. As we argued
in [29], the boundary counterterms $\delta S^{\prime}_{bdr}$ for that action
will be given by,

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

After integrating by parts, the total action is given by,

$\displaystyle
S^{\prime}=\int\,d^{4}x\,\mathcal{L}\left(g_{\mu\nu},\partial_{\lambda}g_{\mu\nu},\varphi^{I},\partial_{\lambda}\varphi^{I}\right)+S_{bdr}^{\prime}\,.$
(2.16)

As we explained in [29], this action is suitable to extract real time boundary
quantities as we drop a term that can potentially arise on the black holes
horizon. Notice that dropping this term is standard within the holographic
dictionary. However, for the Euclidean solutions this term is responsible for
the entropy term in the expression (2.12) for the grand canonical free energy.

The VEV of the boundary stress tensor $T_{\mu\nu}$, the electric current
$J_{\mu}$ and the scalar operators $\mathcal{O}_{\phi}$ and
$\mathcal{O}_{\psi}$ are obtained via the variations,

$\displaystyle\langle T^{\mu\nu}\rangle$
$\displaystyle=\lim_{r\to\infty}\frac{2\,r^{5}}{\sqrt{-\gamma}}\left[\frac{\partial\mathcal{L}}{\partial(\partial_{r}g_{\mu\nu})}+\frac{\delta
S_{bdr}^{\prime}}{\delta\gamma_{\mu\nu}}\right]\,,$ $\displaystyle\langle
J^{\mu}\rangle$
$\displaystyle=\lim_{r\to\infty}\frac{r^{5}}{\sqrt{-\gamma}}\left[\frac{\partial\mathcal{L}}{\partial(\partial_{r}B_{\mu})}+\frac{\delta
S_{bdr}^{\prime}}{\delta B_{\mu}}\right]\,,$
$\displaystyle\langle\mathcal{O}_{\phi}\rangle$
$\displaystyle=\lim_{r\to\infty}\frac{r^{3}}{\sqrt{-\gamma}}\left[\frac{\partial\mathcal{L}}{\partial(\partial_{r}\phi)}+\frac{\delta
S_{bdr}^{\prime}}{\delta\phi}\right]\,,$
$\displaystyle\langle\mathcal{O}_{\psi}\rangle$
$\displaystyle=\lim_{r\to\infty}\frac{r^{3}}{\sqrt{-\gamma}}\left[\frac{\partial\mathcal{L}}{\partial(\partial_{r}\psi^{\ast})}+\frac{\delta
S_{bdr}^{\prime}}{\delta\psi^{\ast}}\right]\,,$ (2.17)

This form will be particularly useful to us as we will be able to read off
directly the dissipative parts of the VEVs for the stress tensor and the
electric current by using the techniques developed in [25, 19, 29].

In our analysis, the gravitational and vector field constraints in the bulk
will be the last set of equations to be imposed. From the field theory point
of view, when imposed on a hypersurface close to conformal boundary, they are
equivalent to the Ward identities of diffeomorphism and gauge invariance for
the sources,

$\displaystyle\nabla_{a}\langle T^{ab}\rangle$ $\displaystyle=F^{ba}\langle
J_{a}\rangle+\nabla^{b}\varphi_{(s)}^{I}\,\langle\mathcal{O}_{I}\rangle+\nabla^{b}\lambda\,\langle\mathcal{O}_{\psi^{\ast}}\rangle+\nabla^{b}\lambda^{\ast}\,\langle\mathcal{O}_{\psi}\rangle\,,$
$\displaystyle\nabla_{a}\langle J^{a}\rangle$
$\displaystyle=\frac{q_{e}}{2i}\,\left(\lambda\,\langle\mathcal{O}_{\psi^{\ast}}\rangle-\lambda^{\ast}\,\langle\mathcal{O}_{\psi}\rangle\right)\,.$
(2.18)

In the above we have define the field strength $F=dm=d\mu$ of the external
source of the current operator. Our main aim is to first obtain a set of
constitutive relations for the stress tensor and electric current in terms of
an appropriate set of hydrodynamic variables by solving the radial equations.
The Ward identities (2.1) will play the role of conservation laws in the
theory of hydrodynamics.

The background geometries of equation (2) yield the stress tensor,

$\displaystyle\langle T_{tt}\rangle=\epsilon\,,\quad\langle
T_{xx}\rangle=\langle T_{yy}\rangle=p\,,\quad\langle J^{t}\rangle=\varrho\,,$
(2.19)

where the energy density $\epsilon$ is given by,

$\displaystyle\epsilon=-2\,g_{(v)}-\varphi_{(s)}\,\langle\mathcal{O}_{\phi}\rangle\,,$
(2.20)

with $\langle\mathcal{O}_{\phi}\rangle=(2\Delta_{\phi}-3)\,\phi_{(v)}$.
Moreover, due to the translational invariance of the background thermal states
under consideration, the pressure $p$ is related to the free energy density
according to $p=-w_{FE}$.

Finally, the VEV of our complex scalar operators is given by the constants of
integration appearing in the asymptotic expansions (2.7) and (2.10) according
to,

$\displaystyle\langle\mathcal{O}_{\psi}\rangle=\frac{1}{2}(2\Delta_{\psi}-3)\,\rho_{(v)}e^{i\,q_{e}\,\theta_{(v)}}\,,$
(2.21)

which after we perturb yields,

$\displaystyle\delta\langle\mathcal{O}_{\psi}\rangle=\langle\mathcal{O}_{\psi}\rangle_{b}\,i\,q_{e}\,\delta\theta_{(v)}+e^{i\,\arg(\langle\mathcal{O}_{\psi}\rangle_{b})}\,\delta|\langle\mathcal{O}_{\psi}\rangle|\,.$
(2.22)

The thermal states we are considering in equation (2) are parametrised by
temperature $T$, the chemical potential $\mu_{t}$ and the neutral scalar
deformation parameter $\phi_{(s)}$. However, superfluids are also
characterised by a non-trivial electric current susceptibility $\chi_{JJ}$
allowing for thermal states which have a persistent electric with no heat
current flow. This leads to a form of the first law,

$\displaystyle
dw_{FE}=-s\,dT-j^{a}\,dm_{a}-\langle\mathcal{O}_{\phi}\rangle\,d\phi_{(s)}\,,$
(2.23)

where once again we have the gauge invariant combination
$m_{a}=\mu_{a}+\partial_{a}\theta_{(v)}$. Even though we are not considering
thermal background states with a persistent current, their perturbative form
will become important in the construction of our hydrodynamic modes, through
the supercurrent susceptibility. We will come back to this point in the next
subsection when we discuss our static, thermodynamic perturbations.

For later reference, we will also define the thermodynamic susceptibilities
through the variations,

$\displaystyle ds=T^{-1}\,c_{\mu}\,dT+\xi\,d\mu_{t}\,,\quad\quad
d\varrho=\xi\,dT+\chi_{QQ}\,d\mu_{t}\,.$ (2.24)

##  3 Perturbations

In this section we will discuss the various perturbations around the black
hole geometries (2) that will be relevant to our construction. In subsection
3.1 we discuss properties of general space-time dependent perturbations before
considering any hydrodynamic limit. In subsections 3.2 and 3.3 we discuss
static perturbations that we can obtain through variations of the
thermodynamic variables parametrising our background geometries as well as
perturbations that we can obtain through large coordinate transformations. As
we will see, these will play a dual role in obtaining the constitutive
relations. Finally, in subsection 3.4 we present our hydrodynamic expansion
along with the leading dissipative corrections we are after.

###  3.1 Real Time Perturbations

Before discussing the derivative expansion of hydrodynamics, it will be
beneficial to consider space-time dependent fluctuations from the boundary
theory point of view. Due to the translational symmetry in both space and
time, we find it convinient to perform a Fourier mode expansion of the general
form,

$\displaystyle\delta\mathcal{F}(t,x^{i};r)=e^{-iw\,(t+S(r))+ik_{i}x^{i}}\,\delta
f(r)\,,$ (3.1)

where $\delta\mathcal{F}$ represents represents perturbations of the scalars
as well as the metric field components. Moreover, by choosing the function
$S(r)$ to approach $S(r)\to\frac{\ln r}{4\pi T}+\cdots$ close to the black
hole horizon at $r=0$, we are guaranteed to the correct infalling boundary
conditions provided that $\delta f(r)$ admits a Taylor series expansion there.
Finally, in order to for the holographic dictionary to be solely dictated by
the asymptotics of $\delta f(r)$, we will choose $S(r)$ to behave as
$S(r)\to\mathcal{O}(1/r^{3})$ close to the conformal boundary.

Close to the conformal the radial functions that parametrise our space-time
dependent perturbations will have to behave according to,

$\displaystyle\delta g_{ab}(r)$ $\displaystyle=(r+R)^{2}\,\left(\delta
s_{ab}+\cdots+\frac{\delta t_{ab}}{(r+R)^{3}}+\cdots\right)\,,$
$\displaystyle\delta g_{ra}(r)$
$\displaystyle=\mathcal{O}\left(\frac{1}{r^{3}}\right),\quad\delta
g_{rr}(r)=\mathcal{O}\left(\frac{1}{r^{4}}\right)\,,$ $\displaystyle\delta
B_{a}(r)$ $\displaystyle=\delta m_{a}+\frac{\delta
j_{a}}{r+R}+\cdots,\quad\delta
B_{r}(r)=\mathcal{O}\left(\frac{1}{r^{3}}\right)\,,$
$\displaystyle\delta\phi(r)$
$\displaystyle=\frac{\delta\phi_{(v)}}{(r+R)^{\Delta_{\phi}}}+\cdots,\quad\delta\rho(r)=\frac{\delta\rho_{(v)}}{(r+R)^{\Delta_{\psi}}}+\cdots\,.$
(3.2)

Notice that we do not choose to work in a particular coordinate system, we
will only our coordinates asymptotically through the decays of the metric
components $\delta g_{r\mu}$. Moreover, the above stated decay for the one-
form field components $\delta B_{r}$ is correct provided that we impose the
Ward identity for the electric current in equation (2.1).

Close to the black hole horizon at $r=0$, we need to impose ingoing boundary
conditions which we can achieve through the asymptotics,

$\displaystyle\delta g_{tt}(r)$ $\displaystyle=4\pi T\,r\,\delta
g_{tt}^{(0)}+\cdots\,,\quad\delta g_{rr}(r)=\frac{\delta g_{rr}^{(0)}}{4\pi
T\,r}+\cdots\,\,,$ $\displaystyle\delta g_{ti}(r)$ $\displaystyle=\delta
g_{ti}^{(0)}+r\,\delta g_{ti}^{(1)}+\cdots\,,\quad\delta
g_{ri}(r)=\frac{\delta g_{ri}^{(0)}}{4\pi T\,r}+\delta g_{ri}^{(1)}+\cdots\,,$
$\displaystyle\delta g_{ij}(r)$ $\displaystyle=\delta
g_{ij}^{(0)}+\cdots\,,\quad\quad\delta g_{tr}(r)=\delta
g_{tr}^{(0)}+\cdots\,,$ $\displaystyle\delta B_{t}(r)$ $\displaystyle=\delta
b_{t}^{(0)}+\delta b_{t}^{(1)}\,r+\cdots,\quad\quad\delta
B_{r}(r)=\frac{\delta b_{r}^{(0)}}{4\pi T\,r}+\delta b_{r}^{(1)}+\cdots\,,$
$\displaystyle\delta B_{i}(r)$ $\displaystyle=\delta b_{i}^{(0)}+\cdots\,,$
$\displaystyle\delta\phi(r)$
$\displaystyle=\delta\phi^{(0)}+\cdots,\quad\quad\delta\rho(r)=\delta\rho^{(0)}+\cdots\,.$
(3.3)

In order to achieve regular infalling boundary conditions, the above need to
be supplemented by additional conditions,

$\displaystyle-2\pi T(\delta g_{tt}^{(0)}+\delta g_{rr}^{(0)})=-4\pi T\,\delta
g_{rt}^{(0)}$ $\displaystyle\equiv\delta T_{h}\,,$ $\displaystyle\delta
g_{ti}^{(0)}=\delta g_{ri}^{(0)}$ $\displaystyle\equiv-\delta u_{i}\,,$
$\displaystyle\delta b_{r}^{(0)}=\delta b_{t}^{(0)}$
$\displaystyle\equiv\delta\mu_{h}\,.$ (3.4)

In the above equations we have defined some horizon quantities which can be
thought of as local temperature, fluid velocity and chemical potential and
which can be defined on the black hole horizon. These will not be directly
relevant to us since we will study the fluid from the boundary theory point of
view.

###  3.2 Thermodynamics Perturbations

In this subsection we will consider all static perturbations which can be
realised as small variations of thermodynamic variables and the Goldstone
mode. As such, these will be the starting point of our hydrodynamic expansion
which we will have to correct by using the techniques introduced in [25, 19,
29].

The first perturbation we will consider is generated by a variation in the
temperature $T$ of the background black holes of equation (2). However, in
order to achieve the regular infalling boundary conditions of equations (3.1)
and (3.1), we need to accompany the temperature variation with an
infinitesimal coordinate transformation,

$\displaystyle t\to t-\partial_{T}S\,\delta T\,,$ (3.5)

yielding the fluctuation,

$\displaystyle\delta_{T}g_{tt}$
$\displaystyle=-\partial_{T}U\,,\quad\delta_{T}g_{rr}=-\frac{\partial_{T}U}{U^{2}}\,,\quad\delta_{T}g_{tr}=U\,\partial_{T}S^{\prime}\,,$
$\displaystyle\delta_{T}g_{ij}$
$\displaystyle=2\,\delta_{ij}\,e^{2g}\,\partial_{T}g\,,\quad\delta_{T}B_{t}=\partial_{T}a\,,$
$\displaystyle\delta_{T}\phi$
$\displaystyle=\partial_{T}\phi\,,\quad\delta_{T}\rho=\partial_{T}\rho\,.$
(3.6)

Another thermodynamic variation which we wish to consider is with respect to
the external field source $\delta\mu_{a}$ and superfluid velocity
$\partial_{a}\delta\theta$. These two variables are packaged in the gauge
invariant quantity $m_{a}=\mu_{a}+\partial_{a}\theta$. In order to generate
the variations with respect to $m_{t}$, we can simply consider the derivative
of our backgrounds with respect to the chemical potential to obtain,

$\displaystyle\delta_{m_{t}}g_{tt}$
$\displaystyle=-\partial_{\mu_{t}}U\,,\quad\delta_{m_{t}}g_{rr}=-\frac{\partial_{\mu_{t}}U}{U^{2}}\,,\quad\delta_{m_{t}}g_{ij}=2\,\delta_{ij}\,e^{2g}\,\partial_{\mu_{t}}g$
$\displaystyle\delta_{m_{t}}B_{t}$
$\displaystyle=\partial_{\mu_{t}}a\,,\quad\delta_{m_{t}}\phi=\partial_{\mu_{t}}\phi\,,\quad\delta_{m_{t}}\rho=\partial_{\mu_{t}}\rho\,.$
(3.7)

In contrast to the case of the temperature variation, we don’t need to shift
the time coordinate here. The derivative of $U$ with respect to the chemical
potential results in a perturbation such that $\delta g_{tt}$ and $\delta
g_{rr}$ exhibit a regular behaviour near the horizon, as can be seen from the
asymptotics (2). In the notation of equations (3.1) and (3.1) it leads to
$\delta T_{h}=0$.

The perturbation with respect to the spatial components $m_{i}$ will result in
a static solution representing a static, isotropic current flow captured by,

$\displaystyle\delta_{m_{j}}g_{ti}$ $\displaystyle=\delta^{j}_{i}\,\delta
f_{g}\,,\quad\delta_{m_{j}}B_{i}=\delta^{j}_{i}\,\delta f_{b}\,.$ (3.8)

Since this solution is not directly generated through a derivative of our
backgrounds (2), it is worth discussing it in more detail. An important point
is that in order to define the current-current susceptibility $\chi_{JJ}$
through this fluctuation, we have to fix a frame where there is no heat
current flow. This is necessary in order to obtain a unique solution since in
the opposite case, we would be able to perform a Lorentz boost an still have a
solution with a different current and heat flow. From a physics point of view,
this is simply the fact that we want to consider a frame where only the
superfluid component carries electric current. In this situation, there cannot
be a heat current as the superfluid cannot transport it.

The absence of a heat current flow on the boundary imposes that the constant
term in $\delta f_{g}(r)$ will vanish in the near horizon expansion,

$\displaystyle\delta f_{g}(r)=\delta f_{g}^{(1)}\,r+\cdots\,,\quad\delta
f_{b}(r)=\delta f_{b}^{(0)}+\cdots\,.$ (3.9)

This can be justified through either earlier works relating the horizon data
of static perturbations to boundary quantities [32, 33] or by following the
techniques used in the present paper. By using our techniques, we can show
this statement by e.g. considering the symplectic current constructed from the
perturbation (3.8) and $\delta_{s_{ti}}$ discussed in the next subsection.
This is also the reason that we don’t need to include a perturbation for the
metric component $\delta g_{ri}$ since this produces a perturbation which has
$\delta u_{i}=0$, in the notation of the boundary conditions (3.1) and (3.1).
Asymptotically we must have,

$\displaystyle\delta
f_{g}(r)=\frac{1}{3}\frac{\mu_{t}\,\chi_{JJ}}{r+R}+\cdots\,,\quad\delta
f_{b}=1-\frac{\chi_{JJ}}{r+R}+\cdots\,,$ (3.10)

which imposes the zero heat current condition.

The final static solution we would like to consider is generated by boundary
by infinitesimal Lorentz boosts with parameter $\delta v_{i}$. As one can
easily see, at finite chemical potential this can be generated by a
combination of a metric and a constant current source,

$\displaystyle t\to t-\delta v_{i}\,x^{i}\,,\quad x^{i}\to x^{i}-\delta
v^{i}\,t\,,\quad\delta m_{i}=\mu_{t}\,\delta v_{i}\,,$ (3.11)

which will guarantee the absence of a net current source in the boundary
theory. In order to obtain a regular solution in the bulk, we need to see the
above as the asymptotics of a large coordinate transformation which is
otherwise regular everywhere in the bulk. This is achieved by the coordinate
transformation,

$\displaystyle t\to t-\delta v_{i}\,x^{i}\,,\quad x^{i}\to x^{i}-\delta
v^{i}\,(t+S(r))\,,$ (3.12)

combined with the static solution (3.8) with the appropriate sources to cancel
the external gauge field source. This leads to the bulk perturbation,

$\displaystyle\delta_{v_{j}}g_{tt}$
$\displaystyle=\delta_{v_{j}}g_{rr}=\delta_{v_{j}}g_{tr}=0\,,\quad\delta_{v_{j}}\varphi=0\,,$
$\displaystyle\delta_{v_{j}}g_{ti}$
$\displaystyle=\delta^{j}_{i}\,\left(U-e^{2g}+\mu_{t}\,\delta
f_{g}\right)\,,\quad\delta_{v_{j}}g_{ri}=-\delta^{j}_{i}\,\left(e^{2g}\,S^{\prime}+\mu_{t}\,\delta
f_{c}\right)\,,$ $\displaystyle\delta_{v_{j}}B_{i}$
$\displaystyle=\delta^{j}_{i}\,\left(\mu_{t}\,\delta
f_{b}-a\right)\,,\quad\delta_{v_{j}}B_{t}=0\,,\quad\delta_{v_{j}}B_{r}=0\,.$
(3.13)

The perturbations discussed in this section will be the building blocks for
the zero frequency, infinite wavelength limit of the hydrodynamic
perturbations we will start constructing in section 3.4. From the boundary
point of view, the perturbations we discussed here give rise to the stress
tensor and electric current fluctuations of the ideal superfluid,

$\displaystyle\delta\langle T^{tt}\rangle$
$\displaystyle=\partial_{T}\epsilon\,\delta
T+\partial_{\mu_{t}}\epsilon\,\delta
m_{t}=\left(c_{\mu}+\mu_{t}\,\xi\right)\,\delta
T+\left(T\,\xi+\mu_{t}\,\chi_{QQ}\right)\,\delta m_{t}\,,$
$\displaystyle\delta\langle T^{ij}\rangle$
$\displaystyle=\delta^{ij}\,\left(\partial_{T}p\,\delta
T+\partial_{\mu_{t}}p\,\delta m_{t}\right)=\delta^{ij}\,\left(s\,\delta
T+\varrho\,\delta m_{t}\right)\,,$ $\displaystyle\delta\langle T^{ti}\rangle$
$\displaystyle=\left(\epsilon+p-\mu_{t}^{2}\,\chi_{JJ}\right)\,\delta
v^{i}-\mu_{t}\,\chi_{JJ}\,\delta m^{i}\,,$ $\displaystyle\delta\langle
J^{t}\rangle$ $\displaystyle=\partial_{T}\varrho\,\delta
T+\partial_{\mu_{t}}\varrho\,\delta m_{t}=\xi\,\delta T+\chi_{QQ}\,\delta
m_{t}\,,$ $\displaystyle\delta\langle J^{i}\rangle$
$\displaystyle=\left(\varrho-\mu_{t}\,\chi_{JJ}\right)\,\delta
v^{i}-\chi_{JJ}\,\delta m^{i}\,.$ (3.14)

It is also useful to check the heat current,

$\displaystyle\delta\langle Q^{i}\rangle=\delta\langle
T^{ti}\rangle-\mu_{t}\,\delta\langle
J^{i}\rangle=\left(\epsilon+p-\mu_{t}\,\varrho\right)\,\delta v^{i}=Ts\,\delta
v^{i}\,,$ (3.15)

is indeed transferred by the normal fluid component which contributes to the
entropy density of the system. Moreover, it will be useful to define the
normal and superfluid charge densities,

$\displaystyle\varrho_{n}$ $\displaystyle=\varrho-\mu_{t}\,\chi_{JJ}\,,$
$\displaystyle\varrho_{s}$ $\displaystyle=\mu_{t}\,\chi_{JJ}\,,$ (3.16)

respectively.

###  3.3 Static Perutrbations from Diffeomorphisms

Another class of static perturbations we will find useful are simply generated
by large diffeomorphisms. Similarly with the thermodynamic perturbations of
the previous subsection, their role in our construction will be twofold.
Firstly, they will later be used in order to introduce sources in our
hydrodynamic expansion as they will be part of the zero frequency, infinite
wavelength limit. Secondly, they will be used in the construction of the
symplectic current. As described in [29], this will help us read off the
leading dissipative corrections to the stess tensor.

Similarly to [29], we will consider the combination of large coordinate
transformations along with constant sources for the electric current sources,

$\displaystyle x^{a}$ $\displaystyle\to x^{a}+\delta
s^{a}{}_{b}\,(x^{b}+\delta^{b}_{t}\,S(r))\,,$ $\displaystyle\delta\mu_{a}$
$\displaystyle=-\mu_{t}\,\delta s^{t}{}_{a}\,,$ (3.17)

where the term involving $S(r)$ takes care of regular, infalling boundary
conditions on the black hole horizon at $r=0$. It is easy to see that from the
boundary theory point of view, this coordinate transformation induces a change
in the metric,

$\displaystyle\delta g_{ab}$ $\displaystyle=\eta_{ac}\,\delta
s^{c}{}_{b}+\eta_{bc}\,\delta s^{c}{}_{a}=2\,\delta s_{\left(ab\right)}\,,$
(3.18)

while the explicit current sources of equation (3.3) combine with the
transformation of the background source to give zero.

From the bulk point of view the bulk perturbation corresponding to the source
$\delta s_{tt}$ is realised by,

$\displaystyle\delta_{s_{tt}}g_{tt}$
$\displaystyle=2\,U-\mu_{t}\,\partial_{\mu_{t}}U\,,\quad\delta_{s_{tt}}g_{tr}=U\,(S^{\prime}+\mu_{t}\,\partial_{\mu_{t}}S^{\prime})\,,$
$\displaystyle\delta_{s_{tt}}g_{rr}$
$\displaystyle=-\mu_{t}\,\frac{\partial_{\mu_{t}}U}{U^{2}}\,,\quad\delta_{s_{tt}}g_{ij}=2\,\delta_{ij}\,\mu_{t}\,e^{2g}\,\partial_{\mu_{t}}g\,,$
$\displaystyle\delta_{s_{tt}}B_{t}$
$\displaystyle=-a+\mu_{t}\,\partial_{\mu_{t}}a\,,\quad\delta_{s_{tt}}B_{r}=-a\,S^{\prime}\,,$
$\displaystyle\delta s_{tt}\phi$
$\displaystyle=\mu_{t}\,\partial_{\mu_{t}}\phi\,,\quad\delta
s_{tt}\rho=\mu_{t}\,\partial_{\mu_{t}}\rho\,.$ (3.19)

For the rest of the perturbations generated by (3.3) we have,

$\displaystyle\delta_{s_{tj}}g_{ti}$
$\displaystyle=\delta^{j}_{i}\,\left(U+\mu_{t}\,\delta
f_{g}\right)\,,\quad\delta_{s_{tj}}B_{i}=\delta^{j}_{i}\,\left(-a+\mu_{t}\,\delta
f_{b}\right)\,,$ $\displaystyle\delta_{s_{jt}}g_{ti}$
$\displaystyle=\delta^{j}_{i}\,e^{2g}\,,\quad\delta_{s_{jt}}g_{ri}=\delta^{j}_{i}\,e^{2g}\,S^{\prime}\,,$
$\displaystyle\delta_{s_{ij}}g_{kl}$
$\displaystyle=2\,\delta^{\left(i\right.}_{k}\delta^{\left.j\right)}_{l}\,e^{2g}\,.$
(3.20)

It is a simple matter to combine variations and transformations the VEVs of
the background stress tensor and electric current to obtain the perturbations,

$\displaystyle\delta\langle T^{tt}\rangle$
$\displaystyle=\left(2\,\epsilon+\mu_{t}\,\left(T\,\xi+\mu_{t}\,\chi_{QQ}\right)\right)\,\delta
s^{tt}\,,$ $\displaystyle\delta\langle T^{ij}\rangle$
$\displaystyle=\delta^{ij}\mu_{t}\,\varrho\,\delta s^{tt}-2\,p\,\delta
s^{\left(ij\right)}\,,$ $\displaystyle\delta\langle T^{ti}\rangle$
$\displaystyle=\epsilon\,\delta
s^{it}-\left(p-\mu_{t}^{2}\,\chi_{JJ}\right)\,\delta s^{ti}\,,$
$\displaystyle\delta\langle J^{t}\rangle$
$\displaystyle=\left(\varrho+\mu_{t}\,\chi_{QQ}\right)\,\delta s^{tt}\,,$
$\displaystyle\delta\langle J^{i}\rangle$ $\displaystyle=\varrho\,\delta
s^{it}+\mu_{t}\,\chi_{JJ}\,\delta s^{ti}\,.$ (3.21)

The above variations with respect to the sources, along with the ones coming
from thermodynamics in equation (3.2), will become the constitutive relations
for the ideal superfluid.

###  3.4 Hydrodynamic Perturbations

In this subsection we will construct the hydrodynamic modes in the bulk. In
order to do this, we think of them as finite frequency and long wavelength
deformations of the static modes we discussed in subsections 3.2 and 3.3. To
make things technically more tractable, we will consider the Fourier
decomposition (3.1) with $k_{i}=\varepsilon\,q_{i}$ and
$w=\varepsilon\,\omega$.

Our hydrodynamic can be expanded in $\varepsilon$ according to,

$\displaystyle\delta_{H}f$ $\displaystyle=\varepsilon\,\delta
f^{(1)}+\varepsilon^{2}\,\delta f^{(2)}+\cdots$
$\displaystyle=\delta_{T}f\,\delta T+\delta_{v_{i}}f\,\delta
v_{i}+\delta_{m_{a}}f\,\delta m_{a}+\delta_{s_{ab}}f\,\delta
s_{ab}+\varepsilon^{2}\,\delta f^{(2)}+\cdots\,,$ (3.22)

where $\delta f^{(1)}$ is a linear combination of the static modes that we
discussed in subsections 3.2 and 3.3. After dressing this linear combination
with the exponential of the Fourier modes, we expect that the radial function
will admit an $\varepsilon$ expansion which after writing back in position
space, we can think of as derivative corrections. In order to have this
interpretation, we will need to show that the VEV corrections induced by the
leading correction $\delta f^{(2)}$ can be indeed expressed in terms of
derivatives of the local temperature $\delta T$, fluid velocity $\delta v_{i}$
and superfluid velocity $\partial_{a}\delta\theta$.

To organise the expansion of the boundary stress tensor and electric current,
we can write,

$\displaystyle\delta\langle T^{tt}\rangle$
$\displaystyle=\left(c_{\mu}+\mu_{t}\,\xi\right)\,\delta
T+\left(T\,\xi+\mu_{t}\,\chi_{QQ}\right)\,\delta\mu+2\,\epsilon\,\delta
s^{tt}+\varepsilon^{2}\,\delta\langle T^{tt}\rangle_{(2)}+\cdots\,,$
$\displaystyle\delta\langle T^{it}\rangle$ $\displaystyle=\delta\langle
T^{ti}\rangle=\left(Ts+\mu_{t}\,\varrho_{n}\,\right)\,\delta
v^{i}-\varrho_{s}\,\delta m^{i}+\epsilon\,\delta
s^{it}+\varepsilon^{2}\,\delta\langle T^{ti}\rangle_{(2)}+\cdots\,,$
$\displaystyle\delta\langle T^{ij}\rangle$
$\displaystyle=\delta^{ij}\,\left(s\,\delta
T+\varrho\,\delta\mu\right)-2\,p\,\delta
s^{\left(ij\right)}+\varepsilon^{2}\,\delta\langle
T^{ij}\rangle_{(2)}+\cdots\,,$ $\displaystyle\delta\langle J^{t}\rangle$
$\displaystyle=\xi\,\delta T+\chi_{QQ}\,\delta\mu+\varrho\,\delta
s^{tt}+\varepsilon^{2}\,\delta\langle J^{t}\rangle_{(2)}\,,$
$\displaystyle\delta\langle J^{i}\rangle$ $\displaystyle=\varrho_{n}\,\delta
v^{i}-\chi_{JJ}\,\delta m^{i}+\varrho\,\delta
s^{it}+\varepsilon^{2}\,\delta\langle J^{i}\rangle_{(2)}\,,$ (3.23)

where we have used our definitions (3.2) for the normal and superfluid charge
densities. Moreover, we have defined the variation of the new chemical
potential,

$\displaystyle\delta\mu=\delta m_{t}+\mu_{t}\,\delta s^{tt}\,,$ (3.24)

which as we will later see in section 4.2, it will receive dissipative
corrections. In the above, the corrections $\delta\langle T^{ab}\rangle_{(2)}$
and $\delta\langle J^{a}\rangle_{(2)}$ are precisely the corrections due to
$\delta f^{(2)}$ in the expansion (3.4). Already at leading order, the
$\mathcal{O}(\varepsilon)$ terms in the constitutive relations (3.4) are
enough to are enough to yield the equations of motion of the ideal superfluid,
when combined with the Ward identities (2.1),

$\displaystyle\left(Ts+\mu_{t}\varrho_{n}\right)\,\partial_{t}\delta
v_{i}+\varrho_{n}\,\partial_{t}\delta m_{i}+s\,\partial_{i}\delta
T-Ts\,\partial_{i}\delta s^{tt}=0\,,$ $\displaystyle\xi\,\partial_{t}\delta
T+\chi_{QQ}\,\partial_{t}\delta\mu+\varrho_{n}\,\partial_{i}\delta
v^{i}-\chi_{JJ}\,\partial_{i}\delta m^{i}+\varrho\left(\partial_{i}\delta
s^{it}+\delta_{ij}\partial_{t}\delta s^{ij}\right)=0\,,$
$\displaystyle\left(c_{\mu}+\mu_{t}\,\xi\right)\,\partial_{t}\delta
T+\left(T\xi+\mu_{t}\chi_{QQ}\right)\,\partial_{t}\delta\mu+\left(Ts+\mu_{t}\varrho_{n}\right)\,\partial_{i}\delta
v^{i}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\varrho_{s}\,\partial_{i}\delta
m^{i}+\left(\epsilon+p\right)\left(\partial_{i}\delta
s^{it}+\delta_{ij}\partial_{t}\,\delta s^{ij}\right)=0\,.$ (3.25)

The task of the next section will be to employ the techniques of [29] and
express these corrections in terms of $\delta T$, $\delta v_{i}$ and
$\partial_{a}\delta\theta$. As a byproduct, we will obtain specific
expressions for the shear and the three bulk viscosities, along with the two
independent dissipative coefficients that enter the currents or the Josephson
relation, depending on the frame one would like to use. For completeness we
will express our constitutive relations in the transverse frame [16, 17].

##  4 Constitutive Relations

In this section we will derive the constitutive relations for the stress
tensor and electric current relevant to the hydrodynamic fluctuations of our
system. We will achieve this in two steps which we discuss in detail in the
following subsections. The first step is to extract the constitutive relations
in terms of our hydrodynamic variables. As we will see, we will land in an
unusual fluid frame which will include bulk integrals as part of its
artefacts. In the second step we will change our description to the so called
transverse frame from which we will be able to read off the shear viscosity
$\eta$, the three bulk viscosities $\zeta_{i}$ and the incoherent conductivity
$\sigma$.

###  4.1 Constitutive Relations in Terms of Horizon Data

The aim of this section is to write the dissipative corrections of the
constitutive relations (3.4) in terms of derivatives of the hydrodynamic
variables $\delta T$, $\delta v^{i}$ and $\delta\theta$. In order to achieve
this, we will follow closely the logic developed in [29].

The main tool in our construction will be the Crnkovic-Witten symplectic
current defined for any classical theory of a collection of fields $\phi^{I}$
whose equations of motion can be obtained from a first order Lagrangian
density $\mathcal{L}(\phi^{I},\partial\phi^{I})$. For any two perturbations
$\delta_{1}\phi^{I}$ and $\delta_{1}\phi^{I}$ around a background
$\phi_{b}^{I}$ which solve the Euler-Lagrange equations of motion, the vector
density,

$\displaystyle
P^{\mu}_{\delta_{1},\delta_{2}}=\delta_{1}\phi^{I}\,\delta_{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi^{I}}\right)-\delta_{2}\phi^{I}\,\delta_{1}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi^{I}}\right)\,,$
(4.1)

is divergence free,

$\displaystyle\partial_{\mu}P^{\mu}_{\delta_{1},\delta_{2}}=0\,.$ (4.2)

For completeness, we write list the contributing terms to (4.1),

$\displaystyle\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}$
$\displaystyle=\sqrt{-g}\,\Gamma^{\mu}_{\gamma\delta}\left(\,g^{\gamma\alpha}\,g^{\delta\beta}-\frac{1}{2}\,g^{\gamma\delta}g^{\alpha\beta}\right)-\sqrt{-g}\,\Gamma^{\kappa}_{\kappa\lambda}\,\left(g^{\mu\left(\alpha\right.}g^{\left.\beta\right)\lambda}-\frac{1}{2}\,g^{\mu\lambda}g^{\alpha\beta}\right)\,,$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\partial_{\mu}B_{\nu}}$
$\displaystyle=-\sqrt{-g}\,\tau\,F^{\mu\nu}\,,\quad\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi}=-\sqrt{-g}\,\partial^{\mu}\phi\,,\quad\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\rho}=-\sqrt{-g}\,\partial^{\mu}\rho\,,$
(4.3)

through the derivatives of the first order bulk action (2.16).

Following very similar arguments with [29], one can show that the asymptotic
behaviour of the radial components of the symplectic current is,

$\displaystyle P^{r}_{\delta_{1},\delta_{2}}=$
$\displaystyle\frac{1}{r^{3}}\,\left(\delta_{1}\phi_{(s)}\,\delta_{2}\left(\sqrt{-\gamma}\,\langle\mathcal{O}_{\phi}\rangle\right)-\delta_{2}\varphi_{(s)}\,\delta_{1}\left(\sqrt{-\gamma}\,\langle\mathcal{O}_{\phi}\rangle\right)\right)$
$\displaystyle+\frac{1}{r^{3}}\,\left(\delta_{1}\rho_{(s)}\,\delta_{2}\left(\sqrt{-\gamma}\,\langle\mathcal{O}_{\rho}\rangle\right)-\delta_{2}\rho_{(s)}\,\delta_{1}\left(\sqrt{-\gamma}\,\langle\mathcal{O}_{\rho}\rangle\right)\right)$
$\displaystyle+\frac{1}{r^{3}}\,\left(\delta_{1}m_{a}\,\delta_{2}\left(\sqrt{-\gamma}\,\langle
J^{a}\rangle\right)-\delta_{2}m_{a}\,\delta_{1}\left(\sqrt{-\gamma}\,\langle
J^{a}\rangle\right)\right)$
$\displaystyle+\frac{1}{r^{3}}\frac{1}{2}\left(\delta_{1}\gamma_{ab}\,\delta_{2}\left(\sqrt{-\gamma}\,\langle
T^{ab}\rangle\right)-\delta_{2}\gamma_{ab}\,\delta_{1}\left(\sqrt{-\gamma}\,\langle
T^{ab}\rangle\right)\right)+\cdots\,,$ (4.4)

with the VEVs being correct up to second derivatives of the sources. This is
useful to us as we are interested in obtaining the VEVs of the stress tensor
and the electric current up to order $\mathcal{O}(\varepsilon^{2})$. The above
expression shows that by using appropriate combinations of our hydrodynamic
mode (3.4) and the static solutions of subsections 3.2 and 3.3, we can read
off the VEVs of our system.

By introducing the Fourier decomposition (3.1) of the symplectic current
itself and by integrating the condition (4.2) we can express,

$\displaystyle\left.P^{r}_{\delta_{1},\delta_{2}}\right|_{r=\infty}=\left.P^{r}_{\delta_{1},\delta_{2}}\right|_{r=0}+B_{\delta_{1},\delta_{2}}\,,$
(4.5)

where we have set,

$\displaystyle
B_{\delta_{1},\delta_{2}}=i\,\int_{0}^{\infty}dr\,\left(-w\,\left(S^{\prime}\,P^{r}_{\delta_{1},\delta_{2}}+P^{t}_{\delta_{1},\delta_{2}}\right)+k_{i}\,P^{i}_{\delta_{1},\delta_{2}}\right)\,.$
(4.6)

This shows that when e.g. one of the two perturbations $\delta_{2}$ is chosen
to be hydrodynamic mode $\delta_{H}$ we can expand in $\varepsilon$ to obtain,

$\displaystyle P^{\mu}_{\delta_{1},\delta H}$
$\displaystyle=\varepsilon\,P^{\mu(1)}_{\delta_{1},\delta
H}+\varepsilon^{2}\,P^{\mu(2)}_{\delta_{1},\delta H}+\cdots\,.$ (4.7)

The leading term $P^{r(1)}_{\delta_{1},\delta H}$ is the symplectic current
formed by the $\varepsilon$ terms in the hydrodynamic expansion (3.4). As
such, it will not offer us additional information apart from facts about the
static perturbations since $\delta_{1}$ always refers to static perturbations
of the background. At second order in the $\varepsilon$ expansion, we have,

$\displaystyle\left.P^{r(2)}_{\delta_{1},\delta_{2}}\right|_{r=\infty}=\left.P^{r(2)}_{\delta_{1},\delta_{2}}\right|_{r=0}+B^{r(2)}_{\delta_{1},\delta_{2}}\,,$
(4.8)

with,

$\displaystyle
B^{(2)}_{\delta_{1},\delta_{2}}=i\,\int_{0}^{\infty}dr\,\left(-\omega\,\left(S^{\prime}\,P^{r(1)}_{\delta_{1},\delta_{2}}+P^{t(1)}_{\delta_{1},\delta_{2}}\right)+q_{i}\,P^{i(1)}_{\delta_{1},\delta_{2}}\right)\,.$
(4.9)

We therefore see that bulk integrals will in principle be present in our final
expressions. However, after moving to a fluid frame in which the transport
coefficients are of physical significance and using properties of the static
perturbations, the bulk integral will disappear from the final constitutive
relations.

Before we embark on our journey to fix the dissipative parts of the
constitutive relations (3.4), we would like to point at some piece of
information that will be useful and that we can obtain from the symplectic
current formed entirely from our static solutions. By considering the
symplectic current $P^{\mu}_{\delta v^{i},\delta m^{j}}$ and applying equation
(4.5) we obtain,

$\displaystyle\delta f_{g}^{(1)}=-\frac{4\pi\varrho_{h}}{s}\,\delta
f_{b}^{(0)}+\frac{4\pi\varrho_{n}}{s}\,.$ (4.10)

In Appendix A we list all the constraints we can obtain by forming the
symplectic current of all possible combinations of the static perturbations in
subsections 3.2 and 3.3. In contrast to the case of normal fluids that were
studied in [29], these bulk constraints will be used in order to show all the
transport coefficients are fixed by the event horizon of our thermal states
(2).

The static perturbations of subsection 3.3 which contain sources for the
asymptotic metric, will let us express the boundary stress tensor in terms of
horizon data and bulk integrals of the perturbation. Similarly, the
perturbation $\delta m_{a}$ discussed in subsection 3.2 will let us do the
same for the electric current. As we will see, this will leave us with
constants of integration on the horizon which still need to be expressed in
terms of our hydrodyanmic variables. Similarly to [29], we will achieve this
by considering the symplectic current formed by using the hydrodynamic mode
(3.4) along with the temperature variations $\delta_{T}$ and boosts $\delta
v_{i}$ of subsection 3.2.

In order to read off the corrections $\delta\langle T^{ab}\rangle_{(2)}$, we
will consider the symplectic current $P^{\mu}_{\delta s_{ab},\delta_{H}}$ to
obtain,

$\displaystyle\varepsilon\,\delta\langle T^{tt}\rangle_{(2)}=$
$\displaystyle\frac{i}{4\pi}\left(-q_{i}s\,\left(\delta s^{it}+\delta
v^{i}\right)+\omega\left(s-\mu_{t}\,\xi\right)\,\delta_{ij}\delta
s^{ij}+\omega\frac{c_{\mu}}{sT}\left(s-\mu_{t}\,\xi\right)\delta T\right.$
$\displaystyle+\omega\,\frac{\xi}{s}\,\left(s-\mu_{t}\,\xi\right)\,\delta\mu+\omega\mu_{t}\,\xi\delta
s^{tt}+\varepsilon\,4\pi i\,\partial_{\mu}\varrho_{h}\,\mu_{t}\,\delta
b_{t}^{(2)(0)}-\varepsilon\,8i\,\pi^{2}\,T\,\delta^{ij}\delta g_{ij}^{(2)(0)}$
$\displaystyle+\varepsilon\,2\pi i\,T\,\mu_{t}\,\xi\,\left(\delta
g_{rr}^{(2)(0)}+\delta
g_{tt}^{(2)(0)}\right)+s\mu_{t}\,\omega\left(\partial_{T}\rho^{(0)}\partial_{\mu}\rho^{(0)}+\partial_{T}\phi^{(0)}\partial_{\mu}\phi^{(0)}\right)\,\delta
T$
$\displaystyle\left.+s\mu_{t}\,\omega\,\left(\left(\partial_{\mu}\rho^{(0)}\right)^{2}+\left(\partial_{\mu}\phi^{(0)}\right)^{2}\right)\,\delta\mu\right)+\varepsilon\,B^{(2)}_{\delta_{s_{tt}},\delta_{H}}\,,$
$\displaystyle\varepsilon\,\delta\langle T^{ti}\rangle_{(2)}=$
$\displaystyle-\varepsilon\,\frac{4\pi}{s}\left(\mu_{t}\varrho_{n}+Ts\right)\delta
g_{ti}^{(2)(0)}+i\omega\mu_{t}(\delta f_{b}^{(0)})^{2}\left(\delta
m^{i}+\mu_{t}\delta
v^{i}\right)+\varepsilon\,B^{(2)}_{\delta_{s_{ti}},\delta_{H}}\,,$
$\displaystyle\varepsilon\,\delta\langle T^{ij}\rangle_{(2)}=$
$\displaystyle-\frac{is}{4\pi}\left(2\left(q^{(i}\delta s^{j)t}+q^{(i}\delta
v^{j)}\right)-\delta^{ij}q_{k}\left(\delta s^{kt}+\delta v^{k}\right)\right)$
$\displaystyle-\frac{i\omega}{4\pi}\left(\delta^{ij}\left(2\,s\,\delta_{kl}\delta
s^{kl}-s\,\delta s^{tt}+\frac{c_{\mu}}{T}\delta
T+\xi\,\delta\mu\right)-2s\,\delta s^{(ij)}\right)$
$\displaystyle-\varepsilon\,\delta^{ij}\frac{s\,T}{2}\left(\delta
g_{rr}^{(2)(0)}+\delta
g_{tt}^{(2)(0)}\right)-\varepsilon\,\varrho_{h}\delta^{ij}\,\delta
b_{t}^{(2)(0)}+\varepsilon\,B^{(2)}_{\delta_{s_{ij}},\delta_{H}}\,.$ (4.11)

By considering the symplectic current $P^{\mu}_{\delta s_{it},\delta_{H}}$, we
can obtain the form,

$\displaystyle\varepsilon\,\delta\langle T^{ti}\rangle_{(2)}=$
$\displaystyle-\frac{i}{4\pi T}\left(q^{i}\,\left(sT\delta^{kl}\delta
s_{kl}+c_{\mu}\delta T+T\xi\delta\mu\right)+\omega\,\delta
v^{i}\,\left(sT+\mu\varrho_{n}\right)\right)$ $\displaystyle-\frac{i}{4\pi
T}\left(\omega\,\varrho_{n}\,\delta m^{i}-\delta
f_{b}^{(0)}\varrho_{h}\left(\delta m^{i}+\mu\delta
v^{i}\right)\omega\right)+\varepsilon\,\frac{s}{4\pi}\delta^{ij}\left(\delta
g_{tj}^{(2)(1)}-2\delta g_{tj}^{(2)(0)}g_{(1)}\right)$
$\displaystyle+\varepsilon\,\frac{is}{4\pi}\omega\left(\delta m^{i}+\mu\delta
v^{i}\right)\delta f_{c}^{(1)}+\varepsilon\,\varrho_{h}\delta^{ij}\delta
b^{(2)(0)}_{j}+\varepsilon\,B^{(2)}_{\delta_{s_{it}},\delta_{H}}\,,$ (4.12)

This form for $\delta\langle T^{ti}\rangle_{(2)}$ is equivalent to the one we
listed in (4.1) because $P^{\mu}_{\delta s_{it},\delta_{H}}$ and
$P^{\mu}_{\delta s_{ti},\delta_{H}}$ linearly combine to $P^{\mu}_{\delta
v_{i},\delta_{H}}$ which is source free.

The above expressions express the first dissipative correction to the ideal
superfluid as functions of the our hydrodynamic variables as well as the
horizon constants of integration $\delta^{ij}\delta g_{ij}^{(2)(0)}$ and
$\delta b_{t}^{(2)(0)}$ for the correction $\delta f^{(2)}$ in our
hydrodynamic expansion (3.4). Moreover, we have the appearance of integrals
over the bulk which, as we will later see, are artifacts of the fluid frame.
Moreover, the appearance of the constants of integration $\delta
g_{tt}^{(2)(0)}+\delta g_{rr}^{(2)(0)}$ and $\delta g_{ti}^{(2)(0)}$ should
not bother us as they can be considered as corrections to the local
temperature $\delta T$ and fluid velocity $\delta v_{i}$ respectively. In
other words, we will be above to absorb them in the definition of $\delta T$
and $\delta v_{i}$ via a change of fluid frame.

In order to obtain similar expressions for the leading dissipative corrections
$\delta\langle J^{a}\rangle$ of the electric current, we will apply equation
(4.8) for the symplectic current $P^{\mu}_{\delta_{m_{a}},\delta_{H}}$. This
allows us to write,

$\displaystyle\varepsilon\,\delta\langle J^{t}\rangle_{(2)}=$
$\displaystyle-\frac{i\omega}{4\pi}\xi\left(\delta^{kl}\delta s_{kl}-\delta
s^{tt}\right)-\frac{i\omega}{4\pi s}\xi\left(\frac{c_{\mu}}{T}\delta
T+\xi\delta\mu\right)$ $\displaystyle+\frac{i\omega
s}{4\pi}\left(\left(\partial_{T}\rho^{(0)}\partial_{\mu}\rho^{(0)}+\partial_{T}\phi^{(0)}\partial_{\mu}\phi^{(0)}\right)\delta
T+\left(\left(\partial_{\mu}\rho^{(0)}\right)^{2}+\left(\partial_{\mu}\phi^{(0)}\right)^{2}\right)\delta\mu\right)$
$\displaystyle-\varepsilon\,\partial_{\mu}\varrho_{h}\,\delta
b^{(2)(0)}_{t}-\varepsilon\,\frac{T\xi}{2}\left(\delta g_{rr}^{(2)(0)}+\delta
g_{tt}^{(2)(0)}\right)+\varepsilon\,B^{(2)}_{\delta_{m_{t}},\delta_{H}}\,,$
$\displaystyle\varepsilon\,\delta\langle J^{i}\rangle_{(2)}=$ $\displaystyle
i\omega\left(\delta f_{b}^{(0)}\right)^{2}\,\left(\delta m^{i}+\mu_{t}\delta
v^{i}\right)-\varepsilon\,\frac{4\pi}{s}\varrho_{n}\delta^{ij}\delta
g_{tj}^{(0)}+\varepsilon\,B^{(2)}_{\delta_{m_{i}},\delta_{H}}\,.$ (4.13)

Once again, the above expressions contain constants of integration for the
perturbation which need to be fixed in terms of our hydrodynamic variables.

To achieve this, we will consider equation (4.8) for the symplectic currents
$P^{\mu}_{\delta v^{i},\delta_{H}}$ and $P^{\mu}_{\delta T,\delta_{H}}$ which
don’t contain any sources. Moreover, we will examine the radial component of
the vector field equation of motion in (2) close to the black hole horizon.
This step was also necessary in [25] and in [29] in order to eliminate the
horizon degree of freedom.

By considering the symplectic current $P^{\mu}_{\delta v^{i},\delta H}$, we
obtain the equation,

$\displaystyle i\omega\left(\delta f_{b}^{(0)}\right)^{2}\mu_{t}\left(\delta
m_{i}+\mu_{t}\delta v_{i}\right)-\varepsilon\,\varrho_{h}\delta
b^{(2)(0)}_{i}-\varepsilon\,\frac{4\pi}{s}\left(s\,T+\mu_{t}\varrho_{n}\right)\delta
g_{ti}^{(2)(0)}$ $\displaystyle+\frac{i}{4\pi
T}\left(q_{i}\left(s\,T\,\delta^{kl}\delta s_{kl}+c_{\mu}\delta
T+T\xi\,\delta\mu\right)+\omega\left(sT+\mu_{t}\varrho_{n}\right)\delta
v_{i}\right)$ $\displaystyle+\frac{i}{4\pi T}\left(\omega\varrho_{n}\,\delta
m_{i}-\delta f_{b}^{(0)}\varrho_{h}\,\omega\left(\delta m_{i}+\mu_{t}\delta
v_{i}\right)\right)$ $\displaystyle-\varepsilon\,\frac{s}{4\pi}\left(\delta
g_{ti}^{(2)(1)}-2\delta
g_{ti}^{(2)(0)}g_{(1)}\right)-\frac{is}{16\pi^{2}T}\omega\,\delta
f_{c}^{(1)}\left(\delta m_{i}+\mu_{t}\delta
v_{i}\right)=\varepsilon\,B^{(2)}_{\delta_{v^{i}},\delta H}\,.$ (4.14)

The above can used to show the equivalence between the expressions (4.1) and
the one given in (4.1) after noting that,

$\displaystyle B^{(2)}_{\delta_{v_{i}},\delta
H}=B^{(2)}_{\delta_{s_{ti}},\delta H}-B^{(2)}_{\delta_{s_{it}},\delta H}\,.$
(4.15)

Our next step is to consider equation (4.8) for the symplectic current
$P^{\mu}_{\delta T,\delta H}$ yielding,

$\displaystyle\frac{c_{\mu}^{2}}{sT^{2}}\,\omega\,\delta
T+\frac{c_{\mu}\xi}{sT}\,\omega\,\delta\mu+\frac{c_{\mu}}{T}\omega\left(\delta^{ij}\delta
s_{ij}-\delta s^{tt}\right)-\varepsilon\,2\pi i\,c_{\mu}\left(\delta
g_{rr}^{(2)(0)}+\delta g_{tt}^{(2)(0)}\right)$
$\displaystyle-\varepsilon\,4\pi i\,\partial_{T}\varrho_{h}\,\delta
b^{(2)(0)}_{t}-\varepsilon\,8\,i\,\pi^{2}\,\delta^{ij}\delta
g_{ij}^{(2)(0)}-s\omega\left(\left(\partial_{T}\rho^{(0)}\right)^{2}+\left(\partial_{T}\phi^{(0)}\right)^{2}\right)\delta
T$
$\displaystyle-s\,\omega\left(\partial_{T}\rho^{(0)}\partial_{\mu}\rho^{(0)}+\partial_{T}\phi^{(0)}\partial_{\mu}\phi^{(0)}\right)\delta\mu=\varepsilon\,\pi
i\,B^{(2)}_{\delta_{T},\delta H}\,.$ (4.16)

The above equation in combination with the radial component of the vector
equation in (2) evaluated on the horizon,

$\displaystyle\varepsilon\,\delta b^{(2)(0)}_{t}=\frac{4\pi
i}{s\,q_{e}^{2}\left(\rho^{(0)}\right)^{2}}\left(\varrho_{h}q_{i}\left(\delta
s^{it}+\delta v^{i}\right)-\omega\left(\varrho_{h}\delta^{ij}\delta
s_{ij}+\partial_{T}\varrho_{h}\delta
T+\partial_{\mu}\varrho_{h}\delta\mu\right)\right)\,,$ (4.17)

can be used to eliminate the constants $\delta^{ij}\delta g_{ij}^{(2)(0)}$ and
$\delta b_{t}^{(2)(0)}$ from the constitutive relations (4.1) and (4.1). By
eliminating those we can express all our constitutive relations in terms of
the hydrodynamic variables $\delta T$, $\delta v_{i}$ and
$\delta\theta_{(v)}$. However, there is still a set of bulk integrals which
are not equal to zero. By performing the change of frame given by,

$\displaystyle\delta T$ $\displaystyle\to\delta
T-\frac{\varepsilon}{c_{\mu}+\mu_{t}\,\xi}\left(B^{(2)}_{\delta_{s_{tt}},\delta
H}-T\,B^{(2)}_{\delta_{T},\delta H}\right)\,,$ $\displaystyle\delta v^{i}$
$\displaystyle\to\delta
v^{i}-\frac{\varepsilon}{s\,T+\mu_{t}\,\varrho_{n}}B^{(2)}_{\delta_{s_{ti}},\delta
H}\,,$ (4.18)

we can eliminate the bulk integrals from the dissipative corrections
$\delta\langle T^{ti}\rangle_{(2)}$ and $\delta\langle T^{tt}\rangle_{(2)}$.
However, the change of fluid frame will make them appear in the dissipative
corrections $\delta\langle T^{ij}\rangle_{(2)}$, $\delta\langle
J^{t}\rangle_{(2)}$ and $\delta\langle J^{i}\rangle_{(2)}$. The important
observation is that the combinations,

$\displaystyle-\frac{s}{c_{\mu}+\mu_{t}\,\xi}\left(B^{(2)}_{\delta_{s_{tt}},\delta
H}-T\,B^{(2)}_{\delta_{T},\delta
H}\right)\delta^{ij}+B^{(2)}_{\delta_{s_{ij}},\delta H}$ $\displaystyle=0\,,$
$\displaystyle-\frac{\xi}{c_{\mu}+\mu_{t}\,\xi}\left(B^{(2)}_{\delta_{s_{tt}},\delta
H}-T\,B^{(2)}_{\delta_{T},\delta H}\right)+B^{(2)}_{\delta_{m_{t}},\delta H}$
$\displaystyle=0\,,$
$\displaystyle-\frac{\varrho_{n}}{s\,T+\mu_{t}\,\varrho_{n}}B^{(2)}_{\delta_{s_{ti}},\delta
H}+B^{(2)}_{\delta_{m_{i}},\delta H}$ $\displaystyle=0\,,$ (4.19)

that would appear, are actually equal to zero. This is a non-trivial result
which in order to be shown, requires the use of the constraints that we list
in Appendix A along with equations of motion of the ideal fluid (3.4).

After cancelling out the bulk integrals, we land in a non-standard fluid frame
which we need to change in order to bring our constitutive relations to a more
conventional form. In the next subsection we will rewrite our theory of
hydrodynamics in the so called transverse frame. This will help us read off
the transport coefficients without resorting to the relevant Kubo formulae.

###  4.2 The Transverse Fluid Frame

In this subsection we wish to bring our constitutive relations to a form
compatible with a well established frame in the literature. In the transverse
frame that we wish to consider, the constitutive relations of our system can
be decomposed to ideal and dissipative pieces according,

$\displaystyle\langle T^{ab}\rangle$ $\displaystyle=\langle
T^{ab}\rangle_{ideal}+\langle T^{ab}\rangle_{diss}\,,$ $\displaystyle\langle
J^{a}\rangle$ $\displaystyle=\langle J^{a}\rangle_{ideal}+\langle
J^{a}\rangle_{diss}\,,$ (4.20)

with the perturbations of the ideal parts being given by the
$\mathcal{O}(\varepsilon)$ terms of equations (3.4). The constraints that fix
the transverse frame that we wish to consider are,

$\displaystyle u_{a}\,\langle T^{ab}\rangle_{diss}=0\,,\quad u_{a}\,\langle
J^{a}\rangle_{diss}=0\,.$ (4.21)

In a spacetime of $d+1$ spacetime dimensions, the above equations constitute
$d+2$ constraints. These can be achieved by transforming the local temperature
$\delta T$, chemical potential $\delta\mu$ and fluid velocity $\delta v^{a}$.
Given the fact that the normal fluid velocity, $u_{a}$ satisfies the
normalisation condition $u^{2}=-1$, we have exactly $d+2$ variables to achieve
the conditions (4.21).

It can be shown that the most general form for the constitutive relations
after imposing these constraints takes the form,

$\displaystyle\langle T_{ab}\rangle_{diss}=$
$\displaystyle-\eta\,\sigma^{n}_{ab}-\zeta_{1}\,P_{ab}\,\nabla_{c}u^{c}-\zeta_{2}\,P_{ab}\,\nabla_{c}\left(\varrho_{s}n^{c}\right)\,,$
$\displaystyle\langle J_{a}\rangle_{diss}=$
$\displaystyle-\sigma\,P_{ab}\,\left(\nabla^{b}\left(\frac{\mu}{T}\right)-\frac{1}{T}\,F^{bc}\,u_{c}\right)\,,$
(4.22)

where we have defined,

$\displaystyle\sigma^{n}_{ab}=$ $\displaystyle
P_{a}{}^{c}P_{b}{}^{d}\left(\nabla_{c}u_{d}+\nabla_{d}u_{c}\right)-P_{ab}\,P^{cd}\,\nabla_{c}u_{d}\,,$
$\displaystyle n_{a}=$
$\displaystyle-\frac{1}{\mu_{s}}\,P_{a}{}^{b}\,m_{b}\,,$
$\displaystyle\mu_{s}=$ $\displaystyle u^{a}\,m_{a}\,,$ (4.23)

and the projection operator is as usual $P_{ab}=g_{ab}+u_{a}\,u_{b}$. In order
to transform to this frame, we need to perform a redefinition of the local
chemical potential to obtain a Josephson relation,

$\displaystyle\mu=u^{a}\,m_{a}+\mu_{diss}\,,$ (4.24)

with dissipative corrections given by,

$\displaystyle\mu_{diss}=\zeta_{3}\,\nabla_{c}\left(\varrho_{s}\,n^{c}\right)+\zeta_{2}\,\nabla_{c}u^{c}\,.$
(4.25)

Notice that even though the components $\langle T_{tt}\rangle_{diss}$ and
$\langle J_{t}\rangle_{diss}$ are trivial, the fact that the chemical
potential (4.24) contains dissipative terms it is enough for the ideal parts
to lead to dissipation.

By linearising the expression we have in fluid and source perturbations, we
obtain the constitutive relations,

$\displaystyle\delta\langle T^{ij}\rangle_{diss}=$
$\displaystyle\eta\left(-2\left(\partial^{(i}\delta
s^{j)t}+\partial^{(i}\delta v^{j)}\right)+\delta^{ij}\partial_{k}\left(\delta
s^{kt}+\delta v^{k}\right)-2\,\partial_{t}\,\delta
s^{(ij)}+\delta^{ij}\,\delta^{kl}\partial_{t}\delta s_{kl}\right)$
$\displaystyle-\zeta_{1}\delta^{ij}\left(\partial_{k}\left(\delta
s^{kt}+\delta v^{k}\right)+\partial_{t}\delta
s_{kl}\delta^{kl}\right)+\delta^{ij}\zeta_{2}\,\chi_{JJ}\,\partial_{k}\,\left(\delta
m^{k}+\mu_{t}\,\delta v^{k}\right)\,,$ $\displaystyle\delta\langle
J^{i}\rangle_{diss}=$
$\displaystyle\frac{\sigma}{T}\left(\partial^{i}\left(\delta
m_{t}-\delta\mu\right)+\frac{\mu_{t}}{T}\,\partial^{i}\delta
T-\partial_{t}\delta m^{i}\right)\,,$ (4.26)

with the linear piece of the dissipative part of the chemical potential
reading,

$\displaystyle\delta\mu_{diss}=\zeta_{2}\,\left(\partial_{i}\left(\delta
s^{it}+\delta v^{i}\right)+\partial_{t}\delta^{ij}\delta
s_{ij}\right)-\,\zeta_{3}\,\chi_{JJ}\,\partial_{i}\left(\delta
m^{i}+\mu_{t}\,\delta v^{i}\right)\,.$ (4.27)

In order to compare to the above form for the constitutive relations, we can
use the relations (4.1) and (4.17) to express the constitutive relations (4.1)
and (4.1) entirely in terms of our hydrodynamic variables. The relations (4.1)
guarantee that the bulk integrals are a frame artifact and that they can be
removed from our constitutive relations. After transforming to the transverse
frame, our constitutive relations take the form (4.2) with dissipative parts
(4.2) and (4.27) fixed by the incoherent conductivity and shear viscosity,

$\displaystyle\sigma$
$\displaystyle=\frac{s^{2}T^{3}}{(sT+\mu_{t}\varrho_{n})^{2}}\,(\delta
f_{b}^{(0)})^{2}\,\tau^{(0)}\,,$ $\displaystyle\eta$
$\displaystyle=\frac{s}{4\pi}\,.$ (4.28)

In order to express the three bulk viscosities $\zeta_{i}$ in terms of horizon
data and thermodynamic quantities, it is convenient to make the change of
background thermodynamic variables $(T,\mu_{t})\rightarrow(s,\varrho)$. For
any quantity $F$, using the chain rule, the thermodynamic derivatives are
connected through,

$\displaystyle\partial_{\mu_{t}}F$
$\displaystyle=\chi_{QQ}\,\partial_{\varrho}F+\xi\,\partial_{s}F\,,$ (4.29)
$\displaystyle\partial_{T}F$
$\displaystyle=\xi\,\partial_{\varrho}F+\frac{c_{\mu}}{T}\,\partial_{s}F\,.$
(4.30)

After performing this change of variables, the three bulk viscosities take the
form,

$\displaystyle\zeta_{1}=$
$\displaystyle\frac{s}{4\pi}\left(\left(s\,\partial_{s}\phi^{(0)}+\varrho\,\partial_{\varrho}\phi^{(0)}\right)^{2}+\left(s\,\partial_{s}\rho^{(0)}+\varrho\,\partial_{\varrho}\rho^{(0)}\right)^{2}\right)$
$\displaystyle\qquad+\frac{4\pi}{s\,q_{e}^{2}\left(\rho^{(0)}\right)^{2}}\left(\varrho_{h}-\varrho\,\partial_{\varrho}\varrho_{h}-s\,\partial_{s}\varrho_{h}\right)^{2}\,,$
$\displaystyle\zeta_{2}=$
$\displaystyle\frac{s}{4\pi}\left(\partial_{\varrho}\phi^{(0)}\left(s\,\partial_{s}\phi^{(0)}+\varrho\,\partial_{\varrho}\phi^{(0)}\right)+\partial_{\varrho}\rho^{(0)}\left(s\,\partial_{s}\rho^{(0)}+\varrho\,\partial_{\varrho}\rho^{(0)}\right)\right)$
$\displaystyle\qquad-\frac{4\pi}{s\,q_{e}^{2}\left(\rho^{(0)}\right)^{2}}\partial_{\varrho}\varrho_{h}\left(\varrho_{h}-\varrho\,\partial_{\varrho}\varrho_{h}-s\,\partial_{s}\varrho_{h}\right)\,,$
$\displaystyle\zeta_{3}=$
$\displaystyle\frac{s}{4\pi}\left(\left(\partial_{\varrho}\phi^{(0)}\right)^{2}+\left(\partial_{\varrho}\rho^{(0)}\right)^{2}\right)+\frac{4\pi}{s\,q_{e}^{2}\left(\rho^{(0)}\right)^{2}}\left(\partial_{\varrho}\varrho_{h}\right)^{2}\,,$
(4.31)

where $\varrho_{h}$ is the horizon charge density we defined in equation
(2.14). From these expressions obviously $\zeta_{1}>0$. Also, using the
Schwarz inequality we can show that111This corrects a typo in the relation
previously given in [16] which was based on the positivity of entropy
productioon. $\zeta_{1}\zeta_{3}\geq\zeta_{2}^{2}$.

##  5 Limits

In this following subsections we will consider the limit of our hydrodynamics
close to the phase transition as well as at zero chemical potential. This is
possible because of our explicit expressions for the five transport
coefficients in terms of horizon data.

###  5.1 Near Critical Point

Here, we wish to consider the limit of the hydrodynamic modes close to the
phase transition as we approach it from the broken phase. This is certainly
possible since we have the holographic expressions (4.2) and (4.2) for the
transport coefficients in terms of thermodynamics and horizon data.

In order to take the limit close to the critical point, we can vary either the
background chemical potential $\mu_{t}$, the temperature $T$ or the scalar
deformation parameter $\phi_{(s)}$. For simplicity, we will only consider
variations of the temperature away from its critical value $T_{c}$ for fixed
chemical potential and scalar deformation. Setting $\delta\lambda=T_{c}-T$, we
can write the expansion,

$\displaystyle\rho=\delta\lambda\,\rho_{0}+\delta\lambda^{3}\,\rho_{1}+\cdots,$
(5.1)

for the bulk scalar dual to the amplitude of the complex order parameter. A
simple analysis of the gauge field equation of motion in(2) reveals that close
to the transition we can write,

$\displaystyle\chi_{JJ}=\frac{c_{JJ}}{\delta\lambda^{2}}+\cdots\,,\quad\varrho_{h}=\varrho+\delta\lambda^{2}\,\varrho_{h1}+\cdots\,,\qquad\varrho_{n}=\varrho+\delta\lambda^{2}\,\varrho_{1}+\cdots\,.$
(5.2)

The above relations yield the singular limit for the bulk viscosities,

$\displaystyle\zeta_{i}=\frac{c_{i}}{\delta\lambda^{2}}+\cdots\,,$ (5.3)

with $c_{i}$ remaining finite at the transition.

The first task is to find the hydrodynamic modes of our superfluid away from
the critical point and then take the limit. More concretely, we would like to
turn of the sources $\delta s_{ab}$ for the metric and the gauge field
$\delta\mu_{a}$ and solve the Ward identities (2.1) given the constitutive
relations in the transverse frame that we discussed in section 4.2. Without
loss of generality, we can take the wavevector of the fluctuation to lie
entirely on the $x^{1}$ axis with components $k_{1}=\varepsilon\,q$ and
$k_{2}=0$ with $\varepsilon$ the hydrodynamic expansion parameter. The goal is
to fix the dispersion relations of the relevant quasi-normal modes whose
frequency we can expand according to,

$\displaystyle\omega=\omega_{[1]}\,\varepsilon+\omega_{[2]}\,\varepsilon^{2}+\cdots\,.$
(5.4)

In order to find the modes we are after, we Fourier expand our hydrodynamic
variables according to,

$\displaystyle\delta T$ $\displaystyle=e^{-i\omega\,t+i\varepsilon
qx^{1}}\,\varepsilon\left(\delta T_{0}+\varepsilon\,\delta
T_{1}+\cdots\right)\,,$ $\displaystyle\delta\theta_{(v)}$
$\displaystyle=e^{-i\omega\,t+i\varepsilon
qx^{1}}\,\left(\delta\theta_{0}+\varepsilon\,\delta\theta_{1}+\cdots\right)\,,$
$\displaystyle\delta v^{i}$ $\displaystyle=e^{-i\omega\,t+i\varepsilon
qx^{1}}\,\varepsilon\left(\delta v^{i}_{0}+\varepsilon\,\delta
v^{i}_{1}+\cdots\right)\,.$ (5.5)

The leading part of the modes along with $\omega_{[1]}$ are determined by the
ideal superfluid equations of motion (3.4). This leads to a linear algebraic
system for the constants $\delta T_{0}$, $\delta\theta_{0}$ and $\delta
v_{0}^{i}$ which is trivial unless the frequency satisfies the zero
determinant condition,

$\displaystyle\omega_{[1]}\left(\left(s\,T+\mu_{t}\,\varrho_{n}\right)\left(T\,\xi^{2}-c_{\mu}\,\chi_{QQ}\right)\omega_{[1]}^{4}-s^{2}\,T\,\chi_{JJ}\,q^{4}\right.$
$\displaystyle\qquad\left.+\left(c_{\mu}\,\varrho\,\varrho_{n}+s\,T\left(s\,\chi_{QQ}+\left(c_{\mu}+\mu_{t}\,\xi\right)\chi_{JJ}-\left(\varrho+\varrho_{n}\right)\xi\right)\right)\omega_{[1]}^{2}\,q^{2}\right)=0\,.$
(5.6)

By solving the above equation for $\omega_{[1]}$, we fix the leading piece of
the dispersion relations we are after. The first solution trivially yields,

$\displaystyle\omega^{shear}_{[1]}=0,$ (5.7)

which is a transverse mode with $\delta v_{0}^{1}=0$. This is essentially the
shear mode describing the diffusion of momentum along a transverse direction.

The remaining four modes come in two pairs corresponding to the first and the
second sounds modes of the superfluid. Here, we will only be interested in the
limit of the dispersion relations near the critical point. For the first sound
we have the asymptotic behaviour,

$\displaystyle(\omega^{f.s.}_{[1]})^{2}=\frac{Z}{\left(s\,T+\mu_{t}\,\varrho\right)\left(c_{\mu}\,\chi_{QQ}-T\,\xi^{2}\right)}\,q^{2}+\cdots\,,$
(5.8)

where for convenience we have defined the quantity,

$\displaystyle
Z=s^{2}\,T\,\chi_{QQ}+c_{\mu}\,\varrho^{2}-2s\,T\,\xi\,\varrho\,.$ (5.9)

leading to a finite speed of sound in the small $\delta\lambda$ limit. In
contrast, for the second sound we have the leading piece of the dispersion
relation,

$\displaystyle(\omega^{s.s.}_{[1]})^{2}=\frac{\chi_{JJ}\,s^{2}\,T}{Z}\,q^{2}+\cdots$
(5.10)

with a speed of sound approaching zero close to the critical point.

After imposing the vanishing of the determinant of the leading order linear
system of equations, the allowed modes are determined by the kernel of the
linear operator multiplying the vector with components $\delta_{0}$,
$\delta\theta_{0}$ and $\delta v_{0}^{i}$. By expanding the Ward identities up
to order $\varepsilon^{3}$, we can determine the higher order constants in the
hydrodynamic expansion (5.1). However, in order for this to be possible we
find that the $\omega_{[2]}$ part of the frequency expansion needs to satisfy
an algebraic condition which completely fixes it in terms of $\omega_{[1]}$.

More specifically, for the shear mode we have the correction,

$\displaystyle\omega^{shear}_{[2]}=-i\frac{\eta}{T\,s+\mu_{t}\,\varrho}q^{2}\,,$
(5.11)

leading to a purely diffusive mode, as expected. For the leading correction to
the first sound mode we obtain,

$\displaystyle\omega^{f.s.}_{[2]}=-\frac{i}{2}\frac{\zeta_{1}}{s\,T+\mu_{t}\,\varrho}q^{2}+\cdots\,,$
(5.12)

showing the the attenuation blows up close to the critical point for fixed
wavenumber. This is due the fact that the bulk viscosity of relativistic
fluids blows up close to the critical point when approached from the broken
phase. This has been discussed before in the context of simpler systems [29]
with scalar order parameters.

However, for the second sound we find the limiting behaviour,

$\displaystyle\omega^{s.s.}_{[2]}=$
$\displaystyle-\frac{i}{2}\left(\zeta_{3}\,\chi_{JJ}+\frac{\zeta_{1}\,\chi_{JJ}\left(s\,T\,\xi-
c_{\mu}\,\varrho\right)^{2}}{Z^{2}}\right.$
$\displaystyle\qquad\qquad\left.+\frac{2\,T^{2}\,\zeta_{2}\,\chi_{JJ}\left(s\,T\,\xi-
c_{\mu}\,\varrho\right)+\left(s\,T+\mu_{t}\,\varrho\right)^{2}\,\sigma}{Z\,T^{2}}\right)q^{2}+\cdots$
(5.13)

for the leading attenuation part showing that it remains finite close to the
phase transition eventhough the bulk viscosities seem to blow up close to the
transition according to (5.3). This is in parallel to the observation first
made in [25] for holographic superfluids at zero chemical potential. In fact,
by taking the zero chemical potential limit, we can match the limiting
behavious of the attenuation (5.1) to the expression of [25].

###  5.2 Zero Chemical Potential

It is interesting to consider the zero chemical potential limit of our
hydrodynamics and compare with our results in [25]. For the classes of
holographic theories we consider, it is easy to see that the background scalar
fields $\phi$ and $\rho$ are even under changing the sign of either the
chemical potential or the charge density $\varrho$. This suggests that at zero
chamical potential and therefore zero charge density, we must have vanishing
$\partial_{\varrho}\phi^{(0)}$ and $\partial_{\varrho}\rho^{(0)}$. Moreover,
the horizon charge density $\varrho_{h}$ and the thermodynamic susceptibity
$\xi$ are identically zero for any value of the entropy of the system. This
shows that $\zeta_{2}=0$ while $\zeta_{1}$ becomes the bulk viscosity of
relativistic holographic fluids [28, 29]. The final expressions for the bulk
viscosities in the zero chemical potential limit are,

$\displaystyle\zeta=\zeta_{1}$
$\displaystyle=\frac{s^{3}}{4\pi}\left(\left(\partial_{s}\phi^{(0)}\right)^{2}+\left(\partial_{s}\rho^{(0)}\right)^{2}\right)=\frac{s}{4\pi}\left(\frac{Ts}{c_{\mu}}\right)^{2}\left(\left(\partial_{T}\phi^{(0)}\right)^{2}+\left(\partial_{T}\rho^{(0)}\right)^{2}\right)\,,$
$\displaystyle\zeta_{2}$ $\displaystyle=0\,,$ $\displaystyle\zeta_{3}$
$\displaystyle=\frac{2\pi}{s\,q_{e}^{2}\,\left(\rho^{(0)}\right)^{2}}\left(\partial_{\varrho}\varrho_{h}\right)^{2}=\frac{2\pi}{s\,q_{e}^{2}\left(\rho^{(0)}\right)^{2}\chi_{QQ}^{2}}\left(\partial_{\mu}\varrho_{h}\right)^{2}\,.$
(5.14)

In order to compare with the constitutive relations of [25] for the electric
current, we will express our constitutive relations in terms of the phase
$\delta\theta_{(v)}$. Combining our expressions (4.2) along with (4.2), (3.4)
and (4.27) we obtain,

$\displaystyle\delta\langle J^{t}\rangle$
$\displaystyle=\chi_{QQ}\,\delta\mu=\chi_{QQ}\,\partial_{t}\delta\theta_{(v)}-\chi_{QQ}\,\chi_{JJ}\,\zeta_{3}\,\partial_{i}\partial^{i}\,\delta\theta_{(v)}=\chi_{QQ}\,\partial_{t}\delta\theta_{(v)}-\chi_{QQ}^{2}\,\zeta_{3}\,\partial_{t}^{2}\,\delta\theta_{(v)}\,,$
$\displaystyle\delta\langle J^{i}\rangle$
$\displaystyle=-\chi_{JJ}\,\partial^{i}\theta_{(v)}-\frac{\sigma}{T}\,\partial^{i}\partial_{t}\theta_{(v)}\,$
(5.15)

where in the first line we used the ideal superfluid equations of motion
(3.4). We see that the above agrees with [25] after matching
$\Xi=\chi_{QQ}^{2}\,\zeta_{3}$ and $\sigma_{d}=\sigma/T$, noting that $\delta
c=q_{e}\,\delta\theta_{(v)}$. To make the comparison more precise we note that
for small chemical potential
$\left.\delta\varrho_{h}\right|_{here}=\left.e^{2g^{(0)}}\tau^{(0)}a_{t}^{(0)}\,\delta\mu_{t}\right|_{there}$.
For the thermodynamic supercurrent perturbation we have $\left.\delta
f_{b}^{(0)}\right|_{here}=\left.a_{x}^{(0)}\right|_{there}$. Moreover, due to
different normalisation of the bulk scalar we have
$\left.\rho^{(0)}\right|_{here}=\sqrt{2}\,\left.\rho^{(0)}\right|_{there}$.

##  6 Discussion

In this paper we have used the techniques that were recently developed in [25,
19, 29] to study the hydrodynamic limit of fluctuations in a holographic
superfluid at finite chemical potential. Based on general arguments, we
expected organise the long wavelength limit of the stress tensor and the
electric current in terms of a derivative expansion of appropriate
hydrodynamic variables which have a clear interpretation in the infinite
wavelength, thermodynamic limit. After fixing a specific fluid frame, we would
then expect the leading dissipative corrections to be parametrised by a set of
independent transport coefficients. For a relativistic fluid which preserves
homogeneity and isotropy, we expect five independent coefficients, the
incoherent conductivity $\sigma$, the shear viscosity $\eta$ and the three
bulk viscosities $\zeta_{i}$.

An important by-product of our derivation was the explicit expressions for the
dissipative transport coefficients in equations (4.2) and (4.2) in terms of
thermodynamics and the black hole horizon data. Our results confirm [10, 27,
16] that in the leading gravitational limit and while preserving translations
and isotropy, the shear viscosity $\eta$ is fixed by the entropy density of
the theory according to (4.2). The results for the incoherent conductivity and
the bulk viscosities are new as far as we know.

As an application of our results, in section 5.1 we studied the limit of the
hydrodynamic fluctuations close to the critical temperature. As we saw, the
bulk viscosities $\zeta_{i}$ blow up close to the critical, signalling the
breakdown of the hydrodynamic expansion. The two longitudinal sound modes
behave differently close to the critical temperature. The first sound behaves
in a way similar to the behaviour that was discusses earlier in [29]. The
speed of sound remains finite while the attenuation blows up dues to the
leading behaviour of the bulk viscosity $\zeta_{1}$. The second sound, which
is due to the superfluid component, has a vanishing speed of sound close to
the transition. However, the attenuation remains finite, similarly to what
happens in superfluids at zero chemical potential [25].

Given the fact that our expressions for the transport coefficients are
determined by data on the black hole horizon, it is possible to consider their
low temperature limit for a given ground state geometry. Large classes of
holographic ground states have been considered over the years in different
works [34, 35, 36] in models which are sub cases of the general model in
equation (2.1). It would be interesting to use our results to study the
effects of dissipation in holographic superfluids at low temperatures by using
these geometries.

The fact that the three bulk viscosities blow up close to the phase transition
is due to the amplitude mode of the superfluid becoming exactly gapless. Below
the phase transition this mode becomes gapped, joining the rest of the UV
modes and we can safely integrate it out. An interesting direction to pursuit
further in the future is to include this mode in the hydrodynamic description
and obtain an effective theory which is valid up to energy scales which are
equal to the gap of this universal mode and even beyond that. This is
certainly possible given the recent progress that was made in [19].

##  Appendix A Constraints for static perturbations

In this Appendix we will list the constraints resulting from the symplectic
current when constructed from the static pairs of perturbations discussed in
subsections 3.2 and 3.3. In particular, given that for those pairs of
perturbations only the radial component is non-trivial, the divergence free
condition (4.2) yields the radial constraint,

$\displaystyle
P^{r}_{\delta_{1},\delta_{1}}=\left.P^{r}_{\delta_{1},\delta_{1}}\right|_{r=0}=\left.P^{r}_{\delta_{1},\delta_{1}}\right|_{r=\infty}\,.$
(A.1)

The inequivalent constraints we can obtain read,

$\displaystyle-e^{2g}a^{\prime}\delta
f_{b}+(aa^{\prime}-U^{\prime}+2e^{2g}g^{\prime})\delta f_{g}+aU\delta
f^{\prime}_{b}+(U-e^{2g})\delta f^{\prime}_{g}=-\varrho+\mu_{t}\,\chi_{jj}\,,$
$\displaystyle e^{2g}a^{\prime}\delta f_{b}-2e^{2g}g^{\prime}\delta
f_{g}+e^{2g}\delta f^{\prime}_{g}=\varrho-\mu_{t}\,\chi_{jj}\,,$
$\displaystyle-a^{\prime}\partial_{\mu_{t}}a+U\phi^{\prime}\partial_{\mu_{t}}\phi+U\rho^{\prime}\partial_{\mu_{t}}\rho+2U\partial_{\mu_{t}}g^{\prime}+\partial_{\mu_{t}}U^{\prime}=0\,,$
$\displaystyle
e^{2g}\Big{(}2(U^{\prime}-2Ug^{\prime}-aa^{\prime})\partial_{\mu_{t}}g-2g^{\prime}\partial_{\mu_{t}}U-U\phi^{\prime}\partial_{\mu_{t}}\phi-U\rho^{\prime}\partial_{\mu_{t}}\rho-4U\partial_{\mu_{t}}g^{\prime}-a\partial_{\mu_{t}}a^{\prime}\Big{)}=\xi\,T\,,$
$\displaystyle
e^{2g}\Big{(}2(U^{\prime}-2Ug^{\prime}-aa^{\prime})\partial_{T}g-2g^{\prime}\partial_{T}U-U\phi^{\prime}\partial_{T}\phi-U\rho^{\prime}\partial_{T}\rho-4U\partial_{T}g^{\prime}-a\partial_{T}a^{\prime}\Big{)}=c_{\mu}\,,$
$\displaystyle
e^{2g}\Big{(}-a^{\prime}\partial_{T}a+U\phi^{\prime}\partial_{T}\phi+U\rho^{\prime}\partial_{T}\rho+2U\partial_{T}g^{\prime}+\partial_{T}U^{\prime}\Big{)}=s\,,$
$\displaystyle e^{2g}\left(U^{\prime}-2Ug^{\prime}-aa^{\prime}\right)=s\,T\,.$
(A.2)

The above equations where obtained by considering the symplectic currents in
the order $P^{\mu}_{\delta_{v_{i}},\delta_{m_{i}}}$,
$P^{\mu}_{\delta_{s_{it}},\delta_{m_{i}}}$,
$P^{\mu}_{\delta_{s_{ii}},\delta_{m_{t}}}$,
$P^{\mu}_{\delta_{s_{tt}},\delta_{m_{t}}}$,
$P^{\mu}_{\delta_{T},2\delta_{s_{ii}}-\mu_{t}\,\delta_{m_{t}}}$,
$P^{\mu}_{\delta_{s_{ii}},\delta T}$ and
$P^{\mu}_{\delta_{s_{it}},\delta_{v_{i}}-\mu_{t}\,\delta_{m_{i}}}$. Finally,
from the symplectic current $P^{\mu}_{\delta_{T},\delta_{m_{t}}}$ we obtain
the bulkier constraint,

$\displaystyle-2e^{2g}\left(-a^{\prime}\partial_{T}a+U\phi^{\prime}\partial_{T}\phi+U\rho^{\prime}\partial_{T}\rho+2U\partial_{T}g^{\prime}+\partial_{T}U^{\prime}\right)\partial_{\mu_{t}}g$
$\displaystyle+2e^{2g}\left(-a^{\prime}\partial_{\mu_{t}}a+U\phi^{\prime}\partial_{\mu_{t}}\phi+U\rho^{\prime}\partial_{\mu_{t}}\rho+2U\partial_{\mu_{t}}g^{\prime}+\partial_{\mu_{t}}U^{\prime}\right)\partial_{T}g$
$\displaystyle-e^{2g}\left(2\partial_{T}g^{\prime}+\rho^{\prime}\partial_{T}\rho+\phi^{\prime}\partial_{T}\phi\right)\partial_{\mu_{t}}U+e^{2g}\left(2\partial_{\mu_{t}}g^{\prime}+\rho^{\prime}\partial_{\mu_{t}}\rho+\phi^{\prime}\partial_{\mu_{t}}\phi\right)\partial_{T}U$
$\displaystyle+e^{2g}U\left(\partial_{T}\rho^{\prime}\partial_{\mu_{t}}\rho-\partial_{T}\rho\partial_{\mu_{t}}\rho^{\prime}\right)+e^{2g}U\left(\partial_{T}\phi^{\prime}\partial_{\mu_{t}}\phi-\partial_{T}\phi\partial_{\mu_{t}}\phi^{\prime}\right)$
$\displaystyle+e^{2g}\left(\partial_{\mu_{t}}a^{\prime}\partial_{T}a-\partial_{\mu_{t}}a\partial_{T}a^{\prime}\right)=-\xi\,.$
(A.3)

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