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#  Revisiting the Landau criterion: a hydrodynamic and holographic approach to
superfluid instabilities

Filippo Sottovia  Filippo Sottovia

(04/01/2024)

## Acknowledgements

First of all I would like to express my gratitude to Prof. Karl Landsteiner
and Prof. Julian Sonner for accepting to read the manuscript and to be part of
the committee. For the same reason I also want to thank Prof. Francesco Nitti,
Prof. Laurent Sanchez-Palencia and Prof. Daniel Areán Fraga.
My supervisor Blaise Goutéraux deserves special credit for guiding me
throughout these three years. I am especially grateful to him for his patience
and all the opportunities he gave me, including the workshops and conferences
that I had the chance to attend. I also learnt a lot from Eric Mefford and
Daniel Areán Fraga. I would really like to thank them for bearing with me
through my numerous attempts at tackling long calculations: their help was
extremely valuable.
I would also like to thank Balt van Rees and, again, Laurent Sanchez-Palencia,
for being on my mini-defence committee.

This journey would not have been possible without those with whom I shared the
road. In the first place, the other PhD students and the postdoctoral
researchers. Yorgo, Erik, Gabriele, Matthieu, Adrien, Majdouline, David,
Vaios, Mikel, David, Aditya, Danny, Ashish: it has been a pleasure to share
ideas (and laughs) with you.

My days would have been much duller without my wonderful friends. First, those
who have always been there: gli amici del liceo, tra cui Luisa, Federico ed
Eva, Chiara (i pranzi domenicali non saranno dimenticati) e tutti gli amici
del Posta, und meine Schweizer Freunde, Stefania, Lidia, Nicolas, Robin und
die Spitzenköche. Thank you Dimitrios for our weekly dinners. Grazie Mendun
per le lunghe chiacchierate.
Merci aussi à mes colocataires, Bianca, Lucie, Manon et Lucas pour tous les
moments, quotidiens ou non, que nous avons partagés (mention spéciale à
Robin).
Un grand merci aux scouts de Fresnes, chefs et jeunes, de m’avoir accueilli
comme l’un des leurs. Vous êtes trop nombreux pour que j’écrive tous vos
prénoms, mais sachez que ce fut un plaisir et un grand honneur.

Last, but not least, I shall thank my family for their constant support. A
mamma Ilaria e papà Lucio, a zia Cristina e a tutti gli altri zii e zie, ai
miei nonni e cugini un grande grazie per l’affetto che mi avete dato in questi
anni.

## Abstract

Superfluidity is a state of matter whose main feature is the absence of
viscous flow and heat transport. It originates from the spontaneous breaking
of a $U(1)$ continuous global symmetry below a critical temperature $T_{c}$.
The gradient of the Goldstone field associated to the symmetry breaking
determines the superfluid velocity. At finite temperature a superfluid and a
normal fluid component coexist; both of them contribute to charge flow. Landau
predicted that the superfluid becomes unstable at large superfluid velocities.
He determined the critical velocity at low temperature by assuming boost
invariance and that the normal fluid is composed of free bosonic
quasiparticles with Bose-Einstein statistics.
Hydrodynamics, on the other hand, gives a valid description of interacting
systems at length- and timescales longer than the typical scales at which
local thermal equilibrium is reached. In this thesis we employ a
hydrodynamical approach in order to investigate the instabilities of
superfluids at finite superflow. In particular, we find that a hydrodynamic
collective mode crosses to the upper half complex frequency plane, which
signals a dynamical instability. At the same time, however, this instability
is also thermodynamic, as its onset is controlled by one of the second
derivatives of the free energy changing sign.
We study the onset of this instability in two main setups: the “probe limit”,
where the fluctuations of the temperature and the normal fluid’s velocity are
frozen, and a complete approach, which includes them.
In both cases we test our findings with the help of gauge-gravity duality
(holography). Up to our numerical precision, we find agreement between the
hydrodynamic modes of the boundary theory and the quasinormal modes of the
gravity theory. We are therewith able to verify our criterion in a holographic
setup.
Most importantly, our criterion, which is formulated in a model-independent
way, applies to interacting systems irrespective of the strength of
interactions, does not rely on boost invariance and does not assume any
specific quantum statistics. As a final check, we also show that it yields the
Landau critical velocity for Galilean superfluids with Bose-Einstein
quasiparticles.

## Résumé en français

La suprafluidité est un état de la matière caractérisé par l’absence de flux
visqueux et de transfert de chaleur. Elle survient lorsqu’une symétrie globale
continue de type $U(1)$ est brisée spontanément. Le gradient du champ de
Goldstone associé détermine la vélocité suprafluide. À temperature non-nulle,
le fluide possède une composante normale, qui contribue également au flot de
la charge. Une prédiction célèbre de Landau est que le système souffre d’une
instabilité lorsque la vitesse suprafluide dépasse une certaine valeur
critique. Landau détermina cette valeur dans un régime de basse température en
supposant le système invariant sous les transformations de Galilée et que le
fluide normal est constitué de particules libres avec une statistique de Bose-
Einstein. Cependant, afin d’évaluer la vitesse critique selon son critère, il
est nécessaire de connaître le spectre des excitations microscopiques.

L’approximation hydrodynamique permet de décrire les systèmes avec
interactions aux échelles d’espace et de temps longues comparées à celles où
l’équilibre thermodynamique est établi localement. Dans cette thèse nous
introduisons d’abord le formalisme hydrodynamique des systèmes suprafluides et
donnons une classification complète des contributions dissipatives. En suite,
nous appliquons cette approche hydrodynamique afin de determiner la vitesse
critique des suprafluides.

En particulier, aux grandes valeurs de la vitesse suprafluide, nous trouvons
qu’un mode collectif hydrodynamique passe dans le demi-plan supérieur des
fréquences complexes, ce qui signale une instabilité dynamique. Cette
instabilité est en fait d’origine thermodynamique, car elle est provoquée par
un changement de signe d’une dérivée seconde de l’énergie libre. Elle a lieu
lorsque:

$\xi+\partial_{\xi}n_{\text{s}}=0\,\text{,}$ (1)

où $\xi$ indique la partie spatiale du gradient du champ de Goldstone et
$n_{\text{s}}$ la densité de charge suprafluide.
Nous étudions l’occurrence de cette instabilité dans deux configurations
différentes : la “limite-test”, où la temperature et la vitesse du fluide
normal sont figées, et une approche complète, qui les inclut.

Dans les deux cas nous vérifions nos résultats à l’aide de la dualité entre
théorie de jauge et de gravité. Formulée en 1997 par Maldacena, cette
conjecture postule l’équivalence entre une théorie de gravité en $d+1$
dimensions dans un espace asymptotiquement anti-de Sitter (c’est à dire, à
courbature négative constante) et une théorie des champs invariante sous les
transformations conformes (dite “théorie de bord”). En particulier, on peut
montrer qu’une théorie des champs fortement couplée correspond à une théorie
de (supra)gravité classique, décrite par les équations d’Einstein. En outre,
on peut décrire une théorie de bord à temperature finie en étudiant une
solution de trou noir des équations d’Einstein. La temperature et l’entropie
du systéme sont données respectivement par la “temperature de Hawking” et
“l’entropie de Bekenstein-Hawking” du trou noir.

D’un point de vue pratique, nous suivons l’approche introduite en 2008 par
Hartnoll, Herzog et Horowitz et considérons une géometrie couplée à un champ
de jauge et un champ scalaire. En dessous d’une temperature critique le champ
scalaire forme un condensat en acquérant une solution non nulle. Il y a donc
brisure spontanée d’une symétrie : le gradient du champ de Goldstone qui en
provient correspond á la velocité suprafluide dans la théorie de bord. Dans la
“limite-test” nous negligéons les interactions entre le tenseur de la métrique
et les champs de matière.
Nous trouvons une très bonne concordance entre les fréquences caractéristiques
du système hydrodynamique et les fréquences quasinormales de la théorie de
gravitation. Notamment, nous pouvons constater qu’une fréquence quasinormale
va à zéro et acquiert une partie imaginaire positive quand la condition (1)
est satisfaite. Cela nous permet donc de vérifier notre critère dans un modèle
holographique.
Notamment, notre critére (formulé sans référence à un modèle en particulier)
est valide indépendamment du couplage et ne requiert pas d’invariance sous les
transformations de Galilée. En dernier lieu, nous montrons aussi qu’il
reproduit la limite de Landau une fois appliqué à des suprafluides galiléens
avec statistiques de Bose-Einstein.

###### Contents

  1. 1 Introduction
  2. 2 Hydrodynamics: an overview
    1. 2.1 Constitutive relations and frame choice
    2. 2.2 Susceptibilities and Green’s functions
      1. 2.2.1 Onsager relations
      2. 2.2.2 Computation of the Green’s functions
    3. 2.3 Instabilities: dynamical and thermodynamic
    4. 2.4 Superfluid hydrodynamics
      1. 2.4.1 The Landau criterion
      2. 2.4.2 Relativistic superfluid hydrodynamics
      3. 2.4.3 Modes in conformal limit with no background superfluid velocity
  3. 3 Gauge-gravity duality
    1. 3.1 Prerequisites
      1. 3.1.1 Conformal symmetry
      2. 3.1.2 AdS spaces
    2. 3.2 The GKPW prescription
    3. 3.3 Holographic thermodynamics
      1. 3.3.1 The Hawking temperature
      2. 3.3.2 Black hole thermodynamics
    4. 3.4 Definition of propagators and quasinormal modes
      1. 3.4.1 Asymptotic behaviour of the fields
      2. 3.4.2 Quasinormal modes and retarded functions
  4. 4 Holographic Superfluids
    1. 4.1 The holographic dictionary for holographic superfluids
    2. 4.2 The renormalized boundary action
    3. 4.3 One- and two-point functions
  5. 5 Probe limit
    1. 5.1 Hydrodynamics
    2. 5.2 Holographic approach
      1. 5.2.1 Background equations of motion and fields
      2. 5.2.2 Equations of motion for the fluctuations
    3. 5.3 Results
      1. 5.3.1 Numerical analysis
      2. 5.3.2 Analytical approach near $T_{c}$
  6. 6 The full picture
    1. 6.1 Hydrodynamics
      1. 6.1.1 Intermediate approach: breaking translations
      2. 6.1.2 “Full” superfluid hydrodynamics
    2. 6.2 Holographic check
      1. 6.2.1 Results
  7. 7 Conclusions and outlook
  8. Appendices
    1. Appendix A: Hydrodynamic frames
    2. Appendix B: recovering Landau’s criterion
    3. Appendix C: pseudospectral methods
      1. Computing the quasinormal modes with pseudospectral methods

##  Chapter 1 Introduction

Superfluidity is a state of matter characterized by the absence of dissipative
flow and of heat transport at zero temperature [1, 2, 3]. It arises from the
spontaneous breaking of a $U(1)$ global symmetry of a scalar field, caused by
an order parameter acquiring a vacuum expectation value. The associated
gapless Goldstone mode $\phi$ is responsible for the onset of an associated
charge current: the corresponding velocity, i.e. the superfluid velocity, is
proportional to the gradient of the Goldstone:
$\mathbf{v}_{\text{s}}\sim\nabla\phi$. In fact, under $U(1)$ transformations
the Goldstone transforms non-linearly $\phi\to\phi+\lambda$ and as a
consequence only the superfluid velocity can be measured, not the phase
itself.

The conventional phenomenological framework adopted in the description of such
systems, the so-called Landau-Tisza model, describes the coexistence of a
superfluid component with a normal fluid one. At a critical temperature
$T_{c}$ the superfluid component vanishes and the system transitions to a
normal fluid. For 4He, the best-known superfluid, $T_{c}\approx 2.2$K. In such
a composite system the superfluid component relies on a Bose-Einstein
condensate (BEC). The normal fluid component, on the other hand, consists of
the bath of thermal excitations above the ground state forming the condensate.
On top of the critical temperature superfluids also feature a critical
velocity $v_{c}$. Landau determined it in terms of the microscopic excitation
spectrum of the thermal theory [4, 5, 6, 7]:

$v_{c}=\min\frac{\epsilon_{p}}{p}\,.$ (1.1)

This criterion has a simple formulation; however, in order to apply it, one
needs to know the microscopic details of the system. It is therefore best
suited for systems with weakly interacting degrees of freedom, where the
calculation of the energy spectrum $\epsilon(p)$ can be successfully carried
out with field theory methods.

Strongly coupled theories, on the other hand, entail a different approach.
Rather than directly diagonalizing the Hamiltonian in perturbation theory, one
usually studies them by means of effective field theories (EFT). A classical
example of an effective field theory, and the one that we are going to employ
in this thesis, is hydrodynamics [4, 8, 9]. Assuming small perturbations out
of thermal equilibrium, it describes the collective excitations of thermal
systems in the low energy limit (or, alternatively, at times and length scales
much larger than the typical relaxation scales of the underlying microscopic
theory).
Another common strategy for tackling strongly coupled systems is the use of
dualities. A prime example thereof is the AdS-CFT correspondence [10, 11, 12,
13], or gauge-gravity duality, which states the equivalence between a
$d+1$-dimensional gravity theory in an asymptotically anti-de Sitter (AdS)
geometry and a conformal field theory (CFT) living on its $d$-dimensional
boundary. To wit, one can send the string coupling to zero, thus obtaining
classical supergravity on the gravity side, while also keeping a large
effective coupling in the boundary theory. This is the so-called ‘t Hooft
limit: it allows us to translate the AdS-CFT correspondence into a duality
between a classical gravity theory, governed by Einstein’s equations of
motion, and a hydrodynamic effective field theory. Hence the jargon “fluid-
gravity correspondence”.

In this thesis we make use of both these approaches in order to determine a
criterion for the critical velocity of superfluids that only depends on
collective quantities and not on microscopical degrees of freedom.
In short, we are going to show that the system undergoes an instability when:

$\partial_{\xi}\left(\xi n_{\text{s}}\right)=0\,,$ (1.2)

where $\xi$ is the norm of the spatial gradient of the Goldstone $\phi$ and
$n_{\text{s}}$ is the superfluid charge density. We verify this condition both
in hydrodynamics and using gauge-gravity duality. In the latter approach, we
use a Lagrangian involving (in its most complete form) fluctuations of the
background metric, of a gauge field and of a complex scalar field, which will
be dual to the fluctuations of the stress-energy tensor, current and
condensate respectively.

This criterion, being formulated in terms of collective variables, manifestly
holds for strongly coupled system described by effective theories. It is
nevertheless also valid for weakly coupled systems: we show that it predicts
the same critical velocity as the Landau criterion (2.39) in the case of 4He
with roton excitations.
Another relevant point concerns the nature of the instability signalled by
(1.2). We demonstrate that this is a dynamical and a thermodynamical
instability at the same time. In fact, when the criterion is fulfilled the
imaginary part of the eigenmodes of the collective excitations of the system
becomes positive, giving rise to perturbations which grow exponentially at
late times. Concurrently, however, the diagonal susceptibility associated to
the superfluid velocity diverges and changes sign, indicating the onset of a
thermodynamical instability.
This, together with the agreement with the Landau criterion (2.39) for 4He, is
a non-trivial check that highlights the power of hydrodynamics as an effective
field theory.

The “holographic check” (i.e. the verification by means of gauge-gravity
duality) of our criterion gives a further assessment of a correspondence that
has had an appreciable success since 2008, when Herzog, Hartnoll and Horowitz
published their pioneering papers [14, 15]. In particular, here we are adding
a non-zero background superfluid velocity to the picture, which translates
into a finite boundary value of the spatial part of the gauge field. In spite
of its rather technical character, then, such a test of the AdS-CFT
correspondence is significant in and of itself.

In this thesis, after introducing the necessary tools from hydrodynamics and
the gauge-gravity correspondence, we are going to show criterion (1.2) in a
handful of frameworks.
As a first example we are going to take a look at the so-called “probe limit”,
where we only consider perturbations to the Goldstone and the chemical
potential. We compute the eigenmodes of the system and derive our criterion
(1.2) by determining the value of $\xi$ for which the imaginary part of one
mode becomes positive. On the gravity side of the AdS-CFT correspondence this
amounts to neglecting the backreaction between the metric and the matter
fields. In the probe limit we also present an analytical solution to the
Einstein equations of motion, valid in the limit of small condensate and
background superfluid velocity. We are able to explicitly show the
correspondence of such a model with the hydrodynamical description of probe
superfluids, and to check the consistency of our criterion.
We then proceed with the hydrodynamical description of the full Landau-Tisza
model, including fluctuations of the temperature and the normal fluid
velocity. Correspondingly, on the gravity side we are going to include
backreaction. The eigenmodes of the system are much too involved to be written
down, but we are still able to check the validity of our criterion.

Throughout this thesis we are using natural units; in particular, we set
$c=\hbar=1$. Some of the results are given in [16], while a more detailed
analysis is presented in [17].

##  Chapter 2 Hydrodynamics: an overview

In the introduction to their book [4] Landau and Lifschitz define fluid
mechanics as “the theory of fluids and gases”. Although this definition works
for most practical applications, modern hydrodynamics deserves a more precise
framing.
In general, complex systems such as fluids and gases can behave quite
differently depending on the scales at which they are probed. In particular,
typical microscopic operators relax on length- and timescales $\ell_{th}$,
$\tau_{th}$. Thus fast variables, which do no overlap with conserved
operators, decay on short scales, becoming irrelevant for the effective
description of the state at long length- and timescales. This implies that the
system can be described by studying the operators $n_{a}$ overlapping with
quantities protected by conservation laws. These operators form the ensemble
of the state. 111Note that the choice of ensemble is not unequivocal,
different ensembles can be used to describe the same state.

###  2.1 Constitutive relations and frame choice

Assuming thermal equilibrium and the existence of a thermal density matrix
$\rho$ one can define the expectation values $\langle n_{a}\rangle$ of the
conserved operators $\hat{n}_{a}$:

$\langle
n_{a}\rangle\equiv\frac{1}{Z}\,\text{Tr}\,\left[\hat{n}_{a}\rho\right]\,\text{,}$
(2.1)

where $Z\equiv\text{Tr}\rho$ and the trace is taken over a complete set of
energy eigenstates of the system (for instance in the canonical ensemble
$\rho=e^{-\beta H}$ and in the grand-canonical ensemble $\rho=e^{-\beta
H+N\mu}$). Then one can write the conservation laws as:

$\partial_{t}\,\langle
n_{a}(t,\mathbf{x})\rangle+\nabla\cdot\langle\mathbf{j}_{a}(t,x)\rangle=0$
(2.2)

The currents $\mathbf{j}_{a}$ are fast variables; as such, their expectation
values in the thermal ensemble can be locally expanded in gradients of the
conserved densities:

$\langle\mathbf{j}_{a}\rangle=\alpha_{ab}\,\langle n_{b}\rangle-
D_{ab}\,\nabla\langle n_{b}\rangle+\mathcal{O}(\nabla^{2})$ (2.3)

Such relations are called constitutive relations. They are expressed up to
first order in gradients (or, in Fourier space, momenta). The zeroth order
$\alpha_{ab}$ terms define an ideal hydrodynamic system. The first-order terms
$D_{ab}$, on the other hand, are the so-called transport coefficients. They
can be hydrostatic (that is, originating from higher derivative corrections to
the hydrostatic pressure and signalling deviations out of homogeneous
equilibrium), or non-hydrostatic, describing slight deviations out of
equilibrium. Furthermore, they may or may not contribute to entropy production
(which has to stay non-negative, as dictated by the second law of
thermodynamics). Those which do are called dissipative terms. By imposing
parity and time-reversal invariance [18] one can show that relativistic (and
Galilean) superfluids do not admit any non-dissipative transport coefficients
at first order in gradients.
Note that the convergence of the hydrodynamic series with contributions at
arbitrarily high orders in gradients is not guaranteed a priori. For the
purpose of this thesis we will assume that a truncation of the series to
linear terms in gradients gives a valid low-energy description of the system
at hand.

The constitutive relations are constrained by the symmetries of the system.

First of all, notice that there is usually some ambiguity in the constitutive
relations beyond ideal order: global thermodynamic variables such as
temperature, chemical potential, or fluid velocity, are not well-defined out
of equilibrium. Rather, they are to be understood as parameters occurring in
the definition of the conserved densities and the related currents, which do
have a microscopic definition. This leaves room for different, equivalent
definitions of first order hydrodynamics, which come under the name of frame
choices. In relativistic hydrodynamics one can for instance require [4] that
the expectation values of the current and the stress-energy tensor be
orthogonal to the fluid velocity. This frame choice is called the Landau frame
[19, 20].

In addition, according to the second law of thermodynamics entropy production
must be positive definite. This leads to constraints on the components of the
matrix $D_{ab}$.

Finally, time reversal invariance gives rise to the so-called Onsager
relations, which are essentially symmetry relations between the Green’s
functions (which we are going to introduce in the next sections). As such,
they translate into symmetry conditions for thermodynamic derivatives.

###  2.2 Susceptibilities and Green’s functions

Hydrodynamics describes deviations from thermodynamic equilibrium. Therefore
together with writing down the constitutive relations the first step in
describing a hydrodynamic system is the definition of the pressure, i.e. the
first law. It will be written as

$\text{d}p=\sum_{a}n_{a}\,\text{d}s_{a}$ (2.4)

$s_{a}$ is called the source of $n_{a}$. One can express the fluctuations of
the ensemble in terms of those of the sources. The matrix $\chi_{ab}$
responsible for the change of basis is the susceptibility matrix:

$\delta n_{a}=\chi_{ab}\,\delta s_{b}$ (2.5)

Owing to its definition and recalling that in the grand-canonical ensemble the
free energy density $f$ equals the pressure up to a sign, we see that the
susceptibility matrix is the Hessian matrix of $f$. As such, a necessary
condition for thermodynamic equilibrium is its positive definiteness. In
particular, all of its diagonal components have to be nonnegative.

We can now take a look back at the equations of motion (2.2) and introduce the
retarded Green’s functions $G^{R}_{ab}(t,\mathbf{x})\equiv
G^{R}_{n_{a},n_{b}}(t,\mathbf{x})$. Formally, the retarded functions are given
by the commutator of the two operators involved [21, 8, 22, 23]:

$G_{ab}^{R}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})\,\equiv
i\theta(t-t^{\prime})\,\langle[n_{a}(t,\mathbf{x}),n_{b}(t^{\prime},\mathbf{x}^{\prime})]\rangle$
(2.6)

On perturbing the sources at $t\geq 0$ by $\delta s_{i}(t,\mathbf{x})$ the
solution of the equations of motion can be written as:

$\delta\langle
n_{a}(t,\mathbf{x})\rangle=\int\text{d}t^{\prime}\text{d}^{d}\mathbf{x}^{\prime}\,G^{R}_{ab}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})\,\delta
s_{b}(t^{\prime},\mathbf{x}^{\prime})$ (2.7)

From their definition (2.6) one can see that $G^{R}(t,\mathbf{k})$ vanishes at
negative times. Hence, upon Fourier transforming $(t,x)\to(\omega,k)$, we see
that $G^{R}(\omega,\mathbf{k})$ is analytic in the upper half-plane of complex
$\omega$.

####  2.2.1 Onsager relations

Let $\Theta$ denote the time-reversal operator, under which any hermitian
operator $n_{a}$ behaves as

$\Theta
n_{a}(t,\mathbf{x})\Theta^{-1}=\eta_{a}n_{a}(-t,\mathbf{x})\quad\quad\quad\quad\text{with
the time-reversal eigenvalue }\eta_{a}=\pm 1$ (2.8)

The expectation value of an operator is taken over a complete set of states
(labeled as $|\phi\rangle$) with the density operator $\hat{\rho}$. Imposing
that the system be invariant under time-reversal one gets $[H,\Theta]=0$ and
so $\Theta H\Theta^{-1}=H$. Similarly,
$\Theta\hat{\rho}\Theta^{-1}=\hat{\rho}$.
We can now compute the correlation function $\langle
n_{a}(t,\mathbf{x})n_{b}(0,\mathbf{x})\rangle$:

$\displaystyle\begin{split}\langle
n_{a}(t,\mathbf{x})n_{b}(0,\mathbf{0})\rangle&\equiv\sum_{\phi}\langle\phi|\hat{\rho}\,n_{a}(t,\mathbf{x})n_{b}(0,\mathbf{0})|\phi\rangle\\\
&=\sum_{\phi}\langle\phi|\Theta^{-1}\Theta\,\hat{\rho}\,\Theta^{-1}\Theta
n_{a}(t,\mathbf{x})\Theta^{-1}\Theta
n_{b}(0,\mathbf{0})\Theta^{-1}\Theta|\phi\rangle\\\
&=\eta_{a}\eta_{b}\sum_{\phi}\langle\phi|\Theta^{-1}\,\hat{\rho}\,n_{a}(-t,\mathbf{x})n_{b}(0,\mathbf{0})\Theta|\phi\rangle\\\
&=\eta_{a}\eta_{b}\sum_{\phi}\langle\Theta\phi|\hat{\rho}\,n_{b}(0,\mathbf{0})n_{a}(-t,\mathbf{x})\,|\Theta\phi\rangle\\\
\end{split}$ (2.9)

where we exploited the invariance of $\hat{\rho}$ under time-reversal. The
time-reversed states $|\Theta\phi\rangle$ form a complete set, so we have

$\langle
n_{a}(t,\mathbf{x})n_{b}(0,\mathbf{0})\rangle=\eta_{a}\eta_{b}\,\langle
n_{b}(0,\mathbf{0})n_{a}(-t,\mathbf{x})\rangle$ (2.10)

We can rewrite $\langle n_{b}(0,\mathbf{0})n_{a}(t,\mathbf{x})\rangle$ in a
similar fashion. We then get:

$G^{R}_{ab}(t,\mathbf{x})=-i\theta(t)\langle[n_{a}|(t,\mathbf{x}),n_{b}(0,\mathbf{0})]\rangle=-i\theta(t)\,\eta_{a}\eta_{b}\,\langle[n_{b}(t,-\mathbf{x},n_{a}(0,\mathbf{0})]\rangle=\eta_{a}\eta_{b}\,G^{R}_{ba}(t,-\mathbf{x})$
(2.11)

If the constitutive relations depend on a quantity $\xi$ (such as a magnetic
field or, in our case, the background superfluid velocity) that flips under
time-reversal we have

$\Theta H(\xi)\Theta^{-1}=H(-\xi)$ (2.12)

giving

$G^{R}_{ab}(t,\mathbf{x},\xi)=\eta_{a}\eta_{b}\,G^{R}_{ba}(t,-\mathbf{x},-\xi)$
(2.13)

and in Fourier space:

$G^{R}_{ab}(\omega,\mathbf{k},\xi)=\eta_{a}\eta_{b}\,G^{R}_{ba}(\omega,-\mathbf{k},-\xi)$
(2.14)

####  2.2.2 Computation of the Green’s functions

To compute the retarded Green’s functions one first defines a matrix $M_{ab}$
so that the equations of motion can be written as: 222$M_{ab}$ should not
depend on time. In order to be able to write the equations of motion as in
(2.15), one should choose a frame where non-ideal contributions to the
constitutive relations do not carry time derivatives. In the rest of this
chapter we are going to assume that this is the case.

$\partial_{t}\,\langle n_{a}(t,\mathbf{x})\rangle+M_{ab}\,\delta\langle
n_{b}(t,\mathbf{x})\rangle=0$ (2.15)

In Fourier space:

$G^{R}_{ab}(\omega,\mathbf{k})=(\mathbf{1}+i\omega
K^{-1})_{ac}\,\chi_{cb}\,\text{,}$ (2.16)

with

$K\equiv-i\omega\,\mathbf{1}+M.$ (2.17)

Note that $\lim_{\omega,\mathbf{k}\to 0}G^{R}_{ab}=\chi_{ab}$: hence the
Onsager relations for the susceptibilities

$\chi_{ab}(\xi)=\eta_{a}\eta_{b}\,\chi_{ba}(-\xi)$ (2.18)

To show (2.16) we follow Kadanoff and Martin’s approach [24, 19]:

  * •

First, we Fourier transform (2.15). $\langle\delta n_{b}(t,\mathbf{x})\rangle$
is only defined for $t>0$, so we have to employ a Laplace transformation in
time. In frequency-momentum space we obtain

$\int_{0}^{\infty}\text{d}t\,e^{i\omega t}\partial_{t}\delta\langle
n_{a}(t,\mathbf{k})\rangle=-\delta\langle
n_{a}(t=0,\mathbf{k})\rangle-i\omega\,\delta\langle
n_{a}(\omega,\mathbf{k})\rangle$ (2.19)

And so (2.15) becomes:

$-i\omega\,\delta\langle n_{a}(\omega,\mathbf{k})\rangle+M_{ab}\,\delta\langle
n_{b}(\omega,\mathbf{k})\rangle=\delta\langle n_{a}(t=0,\mathbf{k})\rangle$
(2.20)

Recalling the definition of $K$ and $\chi$ we can invert this relation and
find:

$\delta\langle n_{a}(\omega,\mathbf{k})\rangle=(K^{-1})_{ab}\chi_{bc}\,\delta
s_{b}(t=0,\mathbf{k})$ (2.21)

  * •

We want to compare this expression with its equivalent formulation in terms of
the retarded Green’s functions $G^{R}_{ab}$ in frequency-momentum space.
Suppose that at $t=0$ we turn off the sources $\delta\langle
s_{b}(t,\mathbf{k})\rangle$.333We could as well turn them on at $t=0$, and get
the same final result. This particular choice comes from the fact that
$G^{R}(\omega,\mathbf{k})$ is analytic in the upper half-plane of complex
$\omega$; otherwise we would have to define an analytical continuation in the
lower-half plane.  Then, by definition of the retarded Green’s functions, we
obtain:

$\delta\langle
n_{a}(t,\mathbf{k})\rangle=-\int_{-\infty}^{0}\text{d}t^{\prime}\,e^{\epsilon
t^{\prime}}G^{R}_{ab}(t-t^{\prime},\mathbf{k})\,\delta
s_{b}(t^{\prime}=0,\mathbf{k})\quad\text{,}$ (2.22)

where we imposed that the time dependence of the source is given by an
adiabatic damping $e^{\epsilon t^{\prime}}$. This choice ensures convergence
of the Fourier integrals.

  * •

The Fourier transformation of the retarded function satisfies:

$G^{R}_{ab}(t-t^{\prime},\mathbf{k})=\frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}G^{R}_{ab}(\omega^{\prime},\mathbf{k})\,e^{-i\omega^{\prime}(t-t^{\prime})}$
(2.23)

Together with (2.22), after integrating over $\text{d}t^{\prime}$, this
yields:

$\delta\langle n_{a}(t,\mathbf{k})\rangle=-\frac{1}{2\pi}\langle
s_{b}(t^{\prime}=0,\mathbf{k})\rangle\int_{-\infty}^{\infty}\text{d}\omega^{\prime}G^{R}_{ab}(\omega^{\prime},\mathbf{k})\,\frac{e^{-i\omega^{\prime}t}}{i\omega^{\prime}+\epsilon}$
(2.24)

  * •

By taking another Laplace transformation we get (again adding a factor
$e^{-\tilde{\epsilon}t}$ to ensure convergence):

$\begin{split}\delta\langle
n_{a}(\omega,\mathbf{k})\rangle&=-\frac{1}{2\pi}\langle
s_{b}(t^{\prime}=0,\mathbf{k})\rangle\int_{-\infty}^{\infty}\text{d}\omega^{\prime}\int_{0}^{\infty}\text{d}t\,G^{R}_{ab}(\omega^{\prime},\mathbf{k})\,\frac{e^{it(\omega-\omega^{\prime}+i\tilde{\epsilon})}}{i\omega^{\prime}+\epsilon}\\\
&=-\frac{1}{2\pi}\langle
s_{b}(t^{\prime}=0,\mathbf{k})\rangle\int_{-\infty}^{\infty}\text{d}\omega^{\prime}G^{R}_{ab}(\omega^{\prime},\mathbf{k})\,\frac{1}{(\omega^{\prime}-\epsilon)(\omega^{\prime}-\omega-i\tilde{\epsilon})}\end{split}$
(2.25)

We evaluate this integral by means of Cauchy’s theorem in the upper half-
plane, where $G^{R}_{ab}(\omega,\mathbf{k})$ is analytic. In the
$\epsilon,\tilde{\epsilon}\to 0$ limit we finally find:

$\delta\langle
n_{a}(\omega,\mathbf{k})\rangle=-\frac{1}{i\omega}s_{b}(t^{\prime}=0,\mathbf{k})\left(G^{R}_{ab}(\omega=0,\mathbf{k})-G^{R}_{ab}(\omega,\mathbf{k})\right)$
(2.26)

  * •

By comparing (2.26) with (2.21) we finally obtain (2.16).

From (2.7) and (2.26) one can see that the poles of the Green’s functions
determine the eigenfrequencies of the system. In momentum space the poles
$\omega(\mathbf{k})$ of $G^{R}_{ab}(\omega,\mathbf{k})$ are called the modes
of the system. They are usually written in powers of $k\equiv|\mathbf{k}|$:

$\omega(\mathbf{k})=v\,k+\frac{i\Gamma}{2}\,k^{2}+\mathcal{O}(k^{3})$ (2.27)

where $v$ is called the speed of sound and $\Gamma$ the attenuation constant.
In ideal hydrodynamics $\omega(\mathbf{k})$ is real for real $\mathbf{k}$,
giving $\Gamma=0$.
Notably, the modes $\omega_{j}$ ($1\leq j\leq\text{dimension of the
ensemble}$) of the system determine the propagation of (small) initial
perturbations out of equilibrium, which is given by the wave functions
$e^{-i\omega_{j}t+ikx}$. One sees that whenever $\text{Im}(\omega)$ crosses
the real axis and becomes positive a dynamical instability is produced,
signalled by the exponential growth in time of the corresponding perturbation
[21].
Note: in complex Fourier space the collision of the hydrodynamic modes with
the first non-gapped non-hydrodynamic mode, as a function of complex
wavevector, determines the radius of convergence of the hydrodynamic series
[25, 26, 27]. In fact, one can write the propagators as a Laurent series
around the modes, which are their poles. It is a known property of Laurent
series that their radius of convergence is set by the next pole. If the latter
does not correspond to a hydrodynamic mode, then hydrodynamics will not be
able to describe the physics beyond it.

A final note: practically, one can compute the retarded Green’s functions by
means of the so-called variational method [19].
To do so one couples the currents to external fields $s_{\mathcal{O}}$, so
that the variation of the grand potential 444We work in the grandcanonical
ensemble, where the system is assumed to be in thermodynamic equilibrium with
a reservoir with which it can exchange heat and particles. The grand potential
$\Omega=-\ln Z$ satisfies $\Omega=U-TS-\mu N$, $U$ being the internal energy.
$\Omega\equiv-\ln Z$ ($Z$ being the static partition function) receives
contributions of the form:

$\Omega\to\Omega+\int\text{d}^{d}x\sqrt{|g|}\,\langle\mathcal{O}\rangle\phi_{\mathcal{O}}$
(2.28)

Let us take relativistic superfluid hydrodynamics as an example. As we are
going to see, it includes a charge current $j^{\mu}$ coupling to a gauge field
$A_{\mu}$, a stress-energy tensor $T^{\mu\nu}$ coupling to a perturbation to
the metric $g_{\mu\nu}$ and Goldstone field coupling to a scalar source
$s_{\phi}$.
The external fields will appear on the right-hand side of the equations of
motion. For instance, in our case:

$\partial_{\mu}j^{\mu}=s_{\phi}\qquad\qquad\nabla_{\mu}T^{\mu\nu}=j_{\rho}F^{\rho\sigma}+s_{\phi}\left(\partial^{\nu}\phi-A^{\nu}\right)$
(2.29)

with the antisymmetric field strength tensor
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. These relations
can be derived by imposing gauge and diffeomorphism invariance:

  * •

Under a gauge transformation the Goldstone field and the gauge field are
shifted by $\phi\to\phi+\lambda$ and $A_{\mu}\to
A_{\mu}+\partial_{\mu}\lambda$. Then the action varies by:

$S\to
S+\int\text{d}^{d}x\sqrt{-g}\,s_{\phi}\lambda+\int\text{d}^{d}x\sqrt{-g}\,j^{\mu}\partial_{\mu}\lambda\sim\int\text{d}^{d}x\sqrt{-g}\,\lambda(s_{\phi}-\partial_{\mu}j^{\mu})$
(2.30)

Imposing invariance we get $\partial_{\mu}j^{\mu}=s_{\phi}$.

  * •

Similarly, under a diffeomorphism parametrized by a vector $\theta_{\mu}$ the
fields are shifted by their Lie derivative with respect to $\theta_{\mu}$:
$\phi\to\phi+\theta^{\mu}\partial_{\mu}\phi$, $A_{\mu}\to
A_{\mu}+\theta^{\rho}\partial_{\rho}A_{\mu}+A_{\rho}\partial_{\mu}\theta^{\rho}$
and $g_{\mu\nu}\to
g_{\mu\nu}+\nabla_{\mu}\theta_{\nu}+\nabla_{\nu}\theta_{\mu}$. Then upon
integration per parts the action varies by:

$S\to
S+\int\text{d}^{d}x\sqrt{-g}\,\theta_{\mu}\left(s_{\phi}\partial^{\mu}\phi+j_{\nu}F^{\mu\nu}-\nabla_{\nu}T^{\mu\nu}\right)$
(2.31)

After restoring gauge invariance by shifting
$\partial^{\nu}\phi\to\partial^{\nu}\phi-A^{\nu}$ (the $s_{\phi}A^{\nu}$ term
is quadratic in the external fields, and thus does not contribute at linear
order in perturbation theory) we then obtain the second equation in (2.29).

One then solves the equations of motion and computes the expectation values of
the currents on shell, whose variation with respect to the external fields
yields the Green’s functions.

Starting from the expectation values in thermal equilibrium we define:

$\mathcal{J}^{\mu}\equiv\sqrt{|g|}\langle
j^{\mu}\rangle\quad\quad\quad\mathcal{T}^{\mu\nu}\equiv\sqrt{|g|}\langle
T^{\mu\nu}\rangle\quad\quad\quad\Phi\equiv\sqrt{|g|}\langle\phi\rangle$ (2.32)

The factor of $\sqrt{-g}$ arises due to the functional form of the action in
theories with non-trivial metric: $S=\int\text{d}^{d}x\,\sqrt{-g}\mathcal{L}$.
The retarded functions can then be computed as follows:

$G^{R}_{T^{\mu\nu}T^{\rho\sigma}}=2\,\frac{\partial\mathcal{T}^{\mu\nu}}{\partial
h_{\rho\sigma}}{\bigg{|}_{\begin{subarray}{c}h_{\alpha\beta\neq\rho\sigma}=\\\
A_{\alpha}=s_{\phi}=0\end{subarray}}}\quad\,\,G^{R}_{T^{\mu\nu}J^{\sigma}}=\frac{\partial\mathcal{T}^{\mu\nu}}{\partial{A_{\sigma}}}{\bigg{|}_{\begin{subarray}{c}h_{\alpha\beta}=A_{\alpha\neq\nu}\\\
=s_{\phi}=0\end{subarray}}}\quad\,\,G^{R}_{T^{\mu\nu}\phi}=\frac{\partial\mathcal{T}^{\mu\nu}}{\partial
s_{\phi}}{\bigg{|}_{h_{\alpha\beta}=A_{\alpha}=0}}$ (2.33)

The definitions of $G^{R}_{j^{\mu}\star}$ and $G^{R}_{\phi\star}$ are
analogous.
The retarded Green’s functions computed with this procedure are found to agree
with (2.16) up to so-called contact terms, which are independent of $\omega$.

###  2.3 Instabilities: dynamical and thermodynamic

As we anticipated above, a necessary condition for the system to be in a
stable state is that the imaginary part of the modes be negative. This can be
seen by considering the evolution of the vacuum expectation value of a
quantity $n_{a}$ in the ensemble, after turning on a source $s_{b}$ at $t=0$.

Recall equation (2.7):

$\begin{split}\delta\langle
n_{a}(t,\mathbf{x})\rangle&=\int\text{d}t^{\prime}\text{d}^{d}\mathbf{x}^{\prime}\,G^{R}_{ab}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})\,\delta
s_{b}(t^{\prime},\mathbf{x}^{\prime})\\\
&=-i\int\text{d}t^{\prime}\text{d}^{d}\mathbf{x}^{\prime}\,\theta(t-t^{\prime})\rho_{ab}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})\,\delta
s_{b}(t^{\prime},\mathbf{x}^{\prime})\end{split}$ (2.34)

where in the second line we have employed the spectral function $\rho_{ab}$
[19]:

$G_{ab}^{R}(t-t^{\prime},\mathbf{x}-\mathbf{x^{\prime}})\equiv-i\theta(t-t^{\prime})\rho_{ab}(t-t^{\prime},\mathbf{x}-\mathbf{x^{\prime}})$

The above integral is the convolution of $G_{ab}^{R}$ with $\delta s_{b}$, so
upon Fourier transforming this becomes:

$\delta\langle
n_{a}(t,\mathbf{x})\rangle=-i\int\text{d}t^{\prime}\text{d}^{d}\mathbf{x}^{\prime}\,\int\text{d}\omega\text{d}^{d}\mathbf{k}\,e^{-i\omega(t-t^{\prime})+i\mathbf{k}(\mathbf{x}-\mathbf{x^{\prime}})}\theta(t-t^{\prime})\rho_{ab}(\omega,\mathbf{k})\,\delta
s_{b}(\omega,\mathbf{k})$ (2.35)

We can evaluate the integral over $\text{d}\omega$ by means of Cauchy’s
integral theorem. Let us choose a contour in the upper half plane consisting
of the real axis, together with the half circle $\omega=Re^{i\theta}$, with
$0\leq\theta\leq\pi$ and $R\to\infty$. Assuming
$\rho_{ab}(\omega,\mathbf{k})\,\delta s_{b}(\omega,\mathbf{k})$ to fall off
sufficiently quickly at $|\omega|\to\infty$, the integral over the whole
contour reduces to the integral over the real axis, which is the one that we
want to compute. With Cauchy’s theorem we find then:

$\delta\langle
n_{a}(t,\mathbf{x})\rangle=-\frac{1}{2\pi}\int\text{d}t^{\prime}\text{d}^{d}\mathbf{x}^{\prime}\,\int\text{d}^{d}\mathbf{k}\,e^{i\mathbf{k}(\mathbf{x}-\mathbf{x^{\prime}})}\theta(t-t^{\prime})\sum_{\begin{subarray}{c}\omega_{i}\text{
pole of }\rho_{ab}\\\
\text{Im}(\omega_{i})>0\end{subarray}}e^{-i\omega_{i}(t-t^{\prime})}\delta
s_{b}(\omega_{i})\text{Res}\,\rho_{ab}(\omega_{i})$ (2.36)

The Heaviside function $\theta(t-t^{\prime})$ implies that we have to check
whether $\delta\langle n_{a}(t,\mathbf{x})\rangle$ is well-behaved for
$t\to+\infty$. But in this limit whenever $\text{Im}(\omega)>0$ we get an
exponential growth in time. Therefore if we assume the perturbation
$\delta\langle n_{a}(t,\mathbf{x})\rangle$ to be well-behaved at large times
we see that the retarded propagator $G_{ab}^{R}$ shall not have any poles in
the upper half plane.

Whenever the imaginary part of a hydrodynamic mode crosses over to the upper
half plane one obtains a dynamic instability.

In thermodynamics, on the other hand, instabilities are signalled by the free
energy not being at a minimum anymore.
This condition can be detected by looking at the susceptibility matrix (2.5).
Since it is the Hessian matrix of the free energy, it has to be positive
definite for the system to sit in a (local) minimum of the free energy. This
also implies that all its diagonal entries must be positive.

In this thesis we are going to show that at a critical superfluid velocity
$\xi_{c}$ one encounters a dynamical instability signalled by the imaginary
part of a hydrodynamic mode crossing over to the upper half-plane. Together
with the imaginary part, also the real part of the mode is found to undergo a
sign change. Furthermore, we demonstrate that this dynamic instability is also
a thermodynamic instability by showing that the susceptibility associated to
the superfluid velocity also changes sign.
Finally we show that our criterion reduces to the well-known Landau criterion
for superfluids with roton excitations.

###  2.4 Superfluid hydrodynamics

Superfluidity is a phase of matter whose main feature is frictionless charge
transport (vanishing shear viscosity). It was first observed in 4He at
temperatures below $T_{c}\approx 2.2$K [4, 28, 29] and more recently in
ultracold atomic gases [30]. It is also expected to take place inside neutron
stars [31], in quark plasma above $10^{11}$K and in metals at low temperature
[1].

A relativistic superfluid is a charged fluid with a spontaneously broken
global symmetry (in its simplest form a $U(1)$ symmetry) and a related
Goldstone boson. The superfluid velocity is proportional to the gradient of
the Goldstone:

${v_{\text{s}}}_{\nu}\equiv\frac{\partial_{\nu}\phi}{\mu}$ (2.37)

This makes its spacelike component irrotational:
$\nabla\times\mu\mathbf{v}_{\text{s}}=0$, hence the absence of unbound
vortices in the system 555The formation of vortices after applying external
angular momentum is a quantum effect, stemming from the fact that the
condensate vanishes in the core of the vortex. This effectively change the
topology of the fluid - see [32], section 2.3.1 for more details. The vortices
counterbalance the externally applied angular momentum. .

The conventional phenomenological framework adopted in the description of such
systems, the so-called Landau-Tisza model [33, 34], considers the coexistence
of a superfluid component with a normal fluid one. At a critical temperature
$T_{c}$ the superfluid component vanishes and the system transitions to a
normal fluid. For 4He, the best-known superfluid, $T_{c}\approx 2.2$K. On the
other hand, at $T=0$ the charge fraction carried by the normal component
vanishes and one is left with a pure superfluid.

In such a composite system the charge density associated with the superfluid
component is given by a Bose-Einstein condensate (BEC). The normal fluid, on
the other hand, consists of the bath of thermal excitations above the ground
state forming the condensate.

Graphical representation of the $U(1)$ spontaneous symmetry breaking
mechanism.

####  2.4.1 The Landau criterion

Superfluids become unstable when the relative velocity between the normal
fluid and the superfluid component exceeds a critical velocity $v_{c}$. A
first criterion to compute this bound was first derived by Landau and is
called the Landau criterion [3, 32] .

Let us assume that the superfluid moves through a capillary with velocity
$v_{\text{s}}$; $\epsilon_{p}$ is its excitation spectrum in its rest frame.
Imposing Galilean boost invariance, its spectrum in the capillary’s reference
frame is given by
$\epsilon_{p}^{\prime}=\epsilon_{p}+\mathbf{p}\cdot\mathbf{v}_{\text{s}}$. At
zero temperature, in order for it not to be energetically convenient to
thermally excite the condensate, the excitation spectrum in the capillary’s
rest frame must be nonnegative. Its minimum is attained when $\mathbf{p}$ and
$\mathbf{v}_{s}$ are antiparallel [3]:

$\min\epsilon_{p}^{\prime}=\epsilon_{p}-p\,v_{\text{s}}$ (2.38)

This quantity has to be nonnegative:

$v_{\text{s}}\leq v_{c}\equiv\min\frac{\epsilon_{p}}{p}$ (2.39)

This criterion, albeit generally effective in foreseeing the critical velocity
of common superfluids such as Helium 4, has some limitations. First of all, it
assumes Galilei invariance, which is necessary to compute the excitation
spectrum in the capillary’s rest frame. Also, the derivation given above is
only valid at low temperatures, and does not take dissipation into account.
From a practical point of view, moreover, in order to actually compute the
bound one needs to know the microscopic theory responsible for the excitation
spectrum.

Hence the main purpose of this thesis: to formulate a criterion for the
critical velocity of superfluids which does not rely on the microscopic
details of the system or invariance under boosts.

####  2.4.2 Relativistic superfluid hydrodynamics

The hydrodynamics of a relativistic superfluid is built upon two background
scalar quantities, the chemical potential $\mu$ and the temperature $T$,
together with the Goldstone boson $\phi$ and $u^{\mu}$, the normal fluid
velocity. It is normed: $u^{\mu}u_{\mu}=-1$.

The superfluid velocity is proportional to the gradient of the Goldstone,
denoted as $\xi^{\mu}$:

${v_{\text{s}}}_{\nu}=\frac{\xi_{\nu}}{\mu}\equiv\frac{\partial_{\nu}\phi}{\mu}$
(2.40)

The Josephson relation links the gradient of the Goldstone boson and the
chemical potential: 666Following [35], one can express the Josephson relation
as the constitutive relation for a higher-form current, putting all
hydrodynamic equations on equal footing. In this thesis, however, we are going
to stick to the conventional formulation of superfluid hydrodynamics.

$u^{\mu}\xi_{\mu}+\mu=0$ (2.41)

Slightly moving out of equilibrium one can modify this relation by adding
dissipative contributions packaged in $\mu_{\text{diss}}$:

$u^{\mu}\xi_{\mu}+\mu+\mu_{\text{diss}}=0$ (2.42)

We define the projector onto the subspace perpendicular to the normal fluid
velocity $u^{\mu}$:

$\Delta^{\mu\nu}\equiv\eta^{\mu\nu}+u^{\mu}u^{\nu}$ (2.43)

This allows us to write down the component of the superfluid velocity
orthogonal to the normal fluid velocity:

$\zeta^{\mu}\equiv\Delta^{\mu}_{\nu}\xi^{\nu}$ (2.44)

We can now write down the constitutive relations up to linear order in
gradients (labeled by the ${}_{\text{diss}}$ subscript). We will justify the
form of the constitutive relations at ideal order in the box at the end of
this section.
The constitutive relation for the current can then be written as:

$j^{\mu}=nu^{\mu}+\frac{n_{\text{s}}}{\mu}\zeta^{\mu}+j_{\text{diss}}^{\mu}$
(2.45)

Note that at ideal order, given the Josephson relation (2.41), this is
equivalent to writing

$j^{\mu}=n_{n}u^{\mu}+n_{\text{s}}\frac{\xi^{\mu}}{\mu}+\mathcal{O(\partial)}$
(2.46)

with the normal charge density $n_{n}\equiv n-n_{\text{s}}$.

The stress-energy tensor takes the following form:

$T^{\mu\nu}=(\epsilon+p)\,u^{\mu}u^{\nu}+p\,\eta^{\mu\nu}+n_{\text{s}}\left(\zeta^{\mu}u^{\nu}+\zeta^{\nu}u^{\mu}+\frac{1}{\mu}\zeta^{\mu}\zeta^{\nu}\right)+\pi_{\text{diss}}^{\mu\nu}$
(2.47)

For example, with zero background superfluid velocity (that is, when the
spatial part of $\xi^{\mu}$ appears at first order in fluctuations) we
consider the following dissipative terms:

$j_{\text{diss}}^{\mu}=-\frac{\kappa}{T}\,\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}\qquad\quad\mu_{\text{diss}}=-\zeta_{3}\,\partial_{\mu}\left(n_{\text{s}}\zeta^{\mu}\right)\qquad\quad\pi_{\text{diss}}^{\mu\nu}=-\eta\,\sigma^{\mu\nu}\,,$
(2.48)

where
$\sigma_{\mu\nu}\equiv\Delta^{\alpha}_{\mu}\Delta^{\beta}_{\nu}\left(\partial_{(\alpha}u_{\beta)}-\eta_{\alpha\beta}\frac{\partial_{\rho}u^{\rho}}{d}\right)$is
the traceless symmetric part of $\partial_{\mu}u_{\nu}$, projected
perpendicular to $u_{\mu}$ [36] 777We define
$A_{(\mu}B_{\nu)}\equiv(A_{\mu}B_{\nu}+A_{\nu}B_{\mu})/2$ . $\eta$ is the
shear viscosity, $\kappa$ the thermal diffusivity and $\zeta_{3}$ the
superfluid diffusivity [37].

The Smarr relation relates energy density, pressure, charge density and
entropy density:

$\epsilon+p=n\mu+sT$ (2.49)

Following [38], it can be proven by recalling that the entropy $S$, the
internal energy $E$, the volume $V$ and the charge $N$ are extensive
quantities, meaning that $E(\lambda z)=\lambda E(z)$ (and similarly for $S,V$,
and $N$) for a scaling parameter $\lambda$ which rescales the “size” $z$ of
the system. But then the energy is given by $E(z)=\frac{\partial
E(z,\lambda)}{\partial\lambda}\big{|}_{\lambda=1}$, and therefore
$E=\frac{\partial E}{\partial\lambda}{\big{|}_{\lambda=1}}=(\frac{\partial
E}{\partial S}{\big{|}_{V,N}}S+\frac{\partial E}{\partial
V}{\big{|}_{S,N}}V+\frac{\partial E}{\partial
N}{\big{|}_{V,S}}N)\big{|}_{\lambda=1}$ (since just as we do for $E$, we have
$S(z)=\frac{\partial S(z,\lambda)}{\partial\lambda}\big{|}_{\lambda=1}$ etc.).
The first law of thermodynamics states $\text{d}E\supset
T\text{d}S+\mu\text{d}N-p\text{d}V$: hence we find $E=TS-pV+\mu N$. Dividing
both sides of this equation by $V$ we finally get (2.49) since the densities
are defined as $\epsilon\equiv E/V$, $s\equiv S/V$ and $\rho\equiv N/V$.

We impose conservation of the current and the stress-energy tensor:

$\displaystyle\begin{split}\partial_{\mu}j^{\mu}&=0\\\
\partial_{\mu}T^{\mu\nu}&=0\end{split}$ (2.50) Ideal superfluid hydrodynamics:
fixing the constitutive relations and the Josephson equation
In this box we motivate the constitutive relation (2.47) and derive the
Josephson equation (2.41) by imposing that entropy production vanish at ideal
level (following [39]).

First, let us give the most general parametrization for the stress-energy
tensor. For a normal fluid it is just given by
$T^{\mu\nu}=(\epsilon+p)\,u^{\mu}u^{\nu}+p\,\eta^{\mu\nu}$. Adding the
superfluid component to the picture we can build two symmetric tensors,
$\zeta^{\mu}u^{\nu}+\zeta^{\nu}u^{\mu}$ and $\zeta^{\mu}\zeta^{\nu}$ and one
more scalar, $\partial_{\mu}\phi\partial^{\mu}\phi$ or, equivalently,
$\zeta^{2}$ (from the definition of $\zeta^{\mu}$ it follows that
$\zeta^{\mu}u_{\mu}=0$). We can then write:
$T_{\mu\nu}=(\epsilon+p)\,u^{\mu}u^{\nu}+p\,\eta^{\mu\nu}+\alpha_{1}\zeta^{2}\,\Delta_{\mu\nu}+2\alpha_{2}\,\zeta_{(\mu}u_{\nu})+\alpha_{3}\,\zeta_{\mu}\zeta_{\nu}$
(2.51) for some coefficients $\alpha_{i}(\mu,T,\zeta^{2})$.
We define [39]: $\mu_{\text{s}}\equiv-u^{\mu}\xi_{\mu}$ (2.52) Then (2.44)
becomes $\zeta^{\mu}=\xi^{\mu}-\mu_{\text{s}}u^{\mu}$.
The constitutive relation for the current is simpler - we assume the normal
and superfluid components to carry an amount of charge given by their charge
densities:
$j^{\mu}=n_{n}u^{\mu}+n_{\text{s}}\frac{\partial^{\mu}\phi}{\mu_{s}}=nu^{\mu}+n_{\text{s}}\zeta^{\mu}\,$
(2.53) with the total charge $n\equiv n_{n}+n_{\text{s}}$. Finally, we
parametrize the first law as:
$\text{d}p=s\text{d}T+n\text{d}\mu+\frac{X}{2}\text{d}(\partial_{\mu}\phi\partial^{\mu}\phi+\mu^{2})\,$
(2.54) where $X=X(\mu,T,\zeta^{2})$.

We now make a crucial assumption: entropy is only carried by the normal
component. In fact, the charge carriers composing the superfluid sit in the
Bose-Einstein condensate, thereby only occupying the ground state: at
sufficiently small temperature there will be no disorder in the superfluid
component (the total number of possible microstates being 1) and thus zero
entropy. With the entropy density $s$ the entropy current will then be
proportional to the normal fluid’s four-velocity $u^{\mu}$:
$j_{s}^{\mu}=su^{\mu}$ (2.55) We make use of the equations of motion in order
to determine the coefficients $\alpha_{i}$ and $X$. Consider the conservation
of the current, multiplied by $\mu$: $\mu\partial_{\mu}j^{\mu}=0$ (2.56) With
the Smarr relation $\epsilon+p=n\mu+sT$ it becomes:
$\displaystyle\begin{split}\mu\partial_{\mu}j^{\mu}&=\mu\partial_{\mu}\big{(}n_{\text{s}}\zeta^{\mu}\big{)}+\partial_{\mu}\big{(}(\epsilon+p)u^{\mu}\big{)}-T\partial_{\mu}(su^{\mu})-s\partial_{\mu}(Tu^{\mu})-nu^{\mu}\partial_{\mu}\mu\\\
&=\mu_{\text{s}}\mu\partial_{\nu}\frac{n_{\text{s}}}{\mu_{\text{s}}}+\mu\frac{n_{\text{s}}}{\mu_{\text{s}}}\partial_{\mu}\big{(}\mu_{\text{s}}\zeta^{\mu}\big{)}+\partial_{\mu}\big{(}(\epsilon+p)u^{\mu}\big{)}-T\partial_{\mu}(su^{\mu})-s\partial_{\mu}(Tu^{\mu})-nu^{\mu}\partial_{\mu}\mu\end{split}$
(2.57) Now, we assume the entropy current to be conserved at ideal order, so
$\partial_{\mu}(su^{\mu})=0$. Extracting $T\partial_{\mu}(su^{\mu})$ from
(2.57) this allows us to write:
$T\partial_{\mu}(su^{\mu})=\mu_{\text{s}}u^{\nu}\partial_{\nu}\frac{n_{\text{s}}}{\mu_{\text{s}}}+\mu\frac{n_{\text{s}}}{\mu_{\text{s}}}\partial_{\mu}\big{(}\mu_{\text{s}}\zeta^{\mu}\big{)}+\partial_{\mu}\big{(}(\epsilon+p)u^{\mu}\big{)}-s\partial_{\mu}(Tu^{\mu})-nu^{\mu}\partial_{\mu}\mu=0$
(2.58) Note that this equation is not automatically equivalent with the
equation of motion $\mu\partial_{\mu}J^{\mu}=0$ because of the additional
assumption that $T\partial_{\mu}(su^{\mu})=0$. Still, it could be a trivial
equation if the other equation of motion, $\partial_{\nu}T^{\mu\nu}=0$,
implied it. To investigate this, we contract the derivative of the stress-
energy tensor with $u_{\mu}$. We find:
$\displaystyle\begin{split}u_{\mu}\partial_{\nu}T^{\mu\nu}=&u^{\nu}\partial_{\nu}p-\partial_{\mu}\big{(}(\epsilon+p)u^{\mu}\big{)}\\\
&-\alpha_{1}\zeta^{2}\partial_{\mu}u^{\mu}-\frac{\alpha_{2}}{\mu_{\text{s}}}\partial_{\nu}(\mu_{\text{s}}n^{\nu})-\mu_{\text{s}}n^{\nu}\partial_{\nu}\frac{\alpha_{2}}{\mu_{\text{s}}}\\\
&+\frac{\alpha_{3}}{\mu_{\text{s}}^{2}}\left(\frac{1}{2}u_{\nu}\partial^{\nu}(\mu_{\text{s}}^{2}\zeta^{2})-\mu_{\text{s}}u_{\mu}u_{\nu}\partial^{\nu}\big{(}\mu_{\text{s}}\zeta^{\mu}\big{)}+\mu_{\text{s}}\zeta^{\mu}\partial_{\nu}\mu_{\text{s}}\right)\end{split}$
(2.59) where we used that $u^{\mu}u_{\mu}=-1$ and so
$u^{\mu}\partial_{\nu}u_{\mu}=0$.
Now, imposing $u_{\mu}\partial_{\nu}T^{\mu\nu}=0$ we can substitute for
$\partial_{\mu}\left((\epsilon+p)u^{\mu}\right)$ in (2.58). Furthermore, from
the first law (2.54) we can compute the derivatives of $p$. We finally get:
$\displaystyle\begin{split}T\partial_{\mu}(su^{\mu})\,=0=\,&\frac{\mu_{\text{s}}^{2}X+\alpha_{3}}{2\mu_{\text{s}}^{2}}\,u_{\nu}\partial^{\nu}(\mu_{\text{s}}^{2}n^{2})\\\
&+\frac{\alpha_{2}-\alpha_{3}}{2\mu_{\text{s}}}\,u_{\mu}u_{\nu}\partial^{\nu}(\mu_{\text{s}}n^{\mu})\\\
&+\mu_{\text{s}}n^{\nu}\bigg{(}-\partial_{\nu}\frac{\alpha_{2}}{\mu_{\text{s}}}-\alpha_{3}\,\partial_{\nu}\frac{1}{\mu_{\text{s}}}+\mu\partial_{\nu}\frac{\rho_{\text{s}}}{\mu_{\text{s}}}\bigg{)}\\\
&+\frac{\mu
n_{\text{s}}-\alpha_{2}}{\mu_{\text{s}}}\partial_{\nu}(\mu_{\text{s}}n^{\nu})\\\
&-\alpha_{1}\,\zeta^{2}\partial_{\rho}u^{\rho}\\\
&+\frac{X}{2}u_{\nu}\partial^{\nu}(\mu^{2}-\mu_{\text{s}}^{2})\end{split}$
(2.60) The important point to stress here is that in order to obtain this
equation we already employed the equations of motion: therefore this equation,
which describes the conservation of the entropy and is an additional
requirement that we impose, constrains $\alpha_{i}$ and $X$.
We see that in order to let the third and the last line vanish we need
$\mu=\mu_{\text{s}}$. This is the Josephson relation (2.41).
We also find $\alpha_{1}=0$ and $\alpha_{3}=\alpha_{2}=\mu n_{\text{s}}$ and
so $X=-\frac{\alpha_{3}}{\mu}=-\frac{n_{\text{s}}}{\mu}$. We thus recover the
constitutive relation (2.47) for the stress-energy tensor.

##### Modified phase frame and entropy production

We work in the so-called modified phase frame. Assuming a dissipative
contribution to the chemical potential as in (2.42) we define:

$\displaystyle\begin{split}\xi_{0}^{\mu}&\equiv\xi^{\mu}-\mu_{\text{diss}}u^{\mu}\\\
{\xi_{0}}&\equiv\sqrt{-{\xi_{0}}_{\mu}{\xi_{0}}^{\mu}}\end{split}$ (2.61)

The Josephson relation gives $u^{\mu}\xi_{0,\mu}=-\mu$. Furthermore, using
${\xi_{0}}_{\mu}$ instead of $\xi_{\mu}$ the definition of $\zeta^{\mu}$ is
unchanged:
$\zeta^{\mu}\equiv\Delta^{\mu\nu}\xi_{\nu}=\Delta^{\mu\nu}{\xi_{0}}_{\nu}$.
Notice that in the modified phase frame we can rewrite the constitutive
relations (2.45) and (2.47) as (with $f\equiv n_{\text{s}}\mu$):

$\displaystyle\begin{split}j^{\mu}&=n_{\text{n}}u^{\mu}+f{\xi_{0}}^{\mu}+j_{\text{diss}}^{\mu}\\\
T^{\mu\nu}&=(\rho_{\text{n}}+p)u^{\mu}u^{\nu}+p\,\eta^{\mu\nu}+f{\xi_{0}}^{\mu}{\xi_{0}}^{\nu}+\pi^{\mu\nu}_{\text{diss}}\end{split}$
(2.62)

Note that, due to the change of frame, the dissipative corrections are in
general going to be different from those defined previously.
In order to write down dissipative coefficients that ensure positivity of
entropy production one first has to compute the entropy current. Starting from
the first law (2.89) one can show that in the modified phase frame its
divergence is given by:

$\partial_{\mu}j_{s}^{\mu}=-\bigg{(}\partial_{\mu}\frac{\mu}{T}\bigg{)}j_{\text{diss}}^{\mu}-\mu_{\text{diss}}\,\Delta^{\mu\nu}\partial_{\nu}\bigg{(}\frac{f}{T}\zeta_{\mu}\bigg{)}-\bigg{(}\partial_{\mu}\frac{u_{\nu}}{T}\bigg{)}\pi_{\text{diss}}^{\mu\nu}\,\text{,}$
(2.63)

where $f\equiv n_{\text{s}}/\mu$. We are going to show this formula in a box
at the end of this section.

One can easily check that the formula for entropy production (2.88) does not
carry any time derivatives. This is the main practical advantage of the
modified phase frame. In fact, as we are going to see, it is a natural choice
to define $j_{\text{diss}}^{\mu}$, $\pi_{\text{diss}}^{\mu\nu}$ and
$\mu_{\text{s}}$ as linear combinations of the factors they come with in
$\partial_{\mu}j_{s}^{\mu}$, since in this way it is immediate to impose
positivity of entropy production. Thus we see that working in the modified
phase frame we can cast the equations of motion as in (2.15).
Moreover, from a formal point of view the ideal level constitutive relations
are expressed in terms of the chemical potential $\mu$, not of
$u^{\mu}\xi_{\mu}$. Hence the use of ${\xi_{0}}_{\mu}$, which satisfies
$u^{\mu}{\xi_{0}}_{\mu}=-\mu$, provides for a cleaner formulation of the
constitutive relations.

Entropy production in the modified phase frame: derivation
In this box we follow [36] (with a different sign convention for the Josephson
relation).
As a first step we want to show that:
${\xi_{0}}_{\mu}(u^{\rho}\partial_{\rho}){\xi_{0}}^{\mu}+{\xi_{0}}u^{\mu}\partial_{\mu}{\xi_{0}}=-\mu_{\text{diss}}\,\xi_{\mu}u^{\rho}\partial_{\rho}(u^{\mu})-\zeta^{\rho}\partial_{\rho}\mu_{\text{diss}}$
(2.64) We start by computing (up to linear order in $\mu_{\text{diss}}$):
$\displaystyle\begin{split}{\xi_{0}}_{\mu}(u^{\rho}\partial_{\rho}){\xi_{0}}^{\mu}&=\,\xi_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-\mu_{\text{diss}}\,u_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-\xi_{\mu}u^{\rho}\partial_{\rho}(\mu_{\text{diss}}u^{\mu})\\\
&\stackrel{{\scriptstyle
u^{\mu}\xi_{\mu}=-\mu-\mu_{\text{diss}}}}{{=}}\,\xi_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-\mu_{\text{diss}}\,\big{(}u_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}+\xi_{\mu}u^{\rho}\partial_{\rho}u^{\mu}\big{)}-\mu\,u^{\rho}\partial_{\rho}\mu_{\text{diss}}\end{split}$
(2.65) Now consider:
$\displaystyle\begin{split}u_{\mu}{\xi_{0}}^{\rho}\partial_{\rho}{\xi_{0}}^{\mu}=&\,u_{\mu}\xi^{\rho}\partial_{\rho}\xi^{\mu}-\mu_{\text{diss}}\,u_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-u_{\mu}\xi^{\rho}\partial_{\rho}(\mu_{\text{diss}}u^{\mu})\\\
&\stackrel{{\scriptstyle
u_{\mu}\partial_{\rho}u^{\mu}=0}}{{=}}u_{\mu}\xi^{\rho}\partial_{\rho}\xi^{\mu}-\mu_{\text{diss}}\,u_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}+\xi^{\rho}\partial_{\rho}\mu_{\text{diss}}\end{split}$
(2.66) Now subtract the two terms:
$\displaystyle\begin{split}{\xi_{0}}_{\mu}u^{\rho}\partial_{\rho}{\xi_{0}}^{\mu}-u_{\mu}{\xi_{0}}^{\rho}\partial_{\rho}{\xi_{0}}^{\mu}=&\,\xi_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-u_{\mu}\xi^{\rho}\partial_{\rho}\xi^{\mu}\\\
&-\mu_{\text{diss}}\,\xi_{\mu}u^{\rho}\partial_{\rho}u^{\mu}+\big{(}\mu
u^{\rho}-\xi^{\rho}\big{)}\partial_{\rho}\mu_{\text{diss}}\end{split}$ (2.67)
$\xi^{\mu}$ is irrotational, so
$\xi_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-u_{\mu}\xi^{\rho}\partial_{\rho}\xi^{\mu}=\xi_{\mu}u^{\rho}\partial_{\rho}\xi^{\mu}-\xi_{\mu}u^{\rho}\partial_{\mu}\xi^{\rho}=0$.
Then:
${\xi_{0}}_{\mu}u^{\rho}\partial_{\rho}{\xi_{0}}^{\mu}-u_{\mu}{\xi_{0}}^{\rho}\partial_{\rho}{\xi_{0}}^{\mu}=-\mu_{\text{diss}}\,\xi_{\mu}u^{\rho}\partial_{\rho}u^{\mu}+\big{(}\mu
u^{\rho}-\xi^{\rho}\big{)}\partial_{\rho}\mu_{\text{diss}}$ (2.68) So we get:
${\xi_{0}}_{\mu}(u^{\rho}\partial_{\rho}){\xi_{0}}^{\mu}-u_{\mu}{\xi_{0}}^{\rho}\partial_{\rho}{\xi_{0}}^{\mu}=-\mu_{\text{diss}}\,\xi_{\mu}u^{\rho}\partial_{\rho}u^{\mu}-\zeta^{\rho}\partial_{\rho}\mu_{\text{diss}}$
(2.69) But:
${\xi_{0}}u^{\mu}\partial_{\mu}{\xi_{0}}={\xi_{0}}u^{\mu}\cdot\frac{1}{2{\xi_{0}}}\big{(}-2{\xi_{0}}^{\nu}\partial_{\mu}{\xi_{0}}_{\nu}\big{)}=-u^{\mu}{\xi_{0}}^{\nu}\partial_{\mu}{\xi_{0}}_{\nu}$
(2.70) And so we finally get (2.64).
Our goal is to compute the divergence of the entropy current:
$\partial_{\mu}(su^{\mu})=s\,\partial_{\mu}u^{\mu}+u^{\mu}\,\partial_{\mu}s$
(2.71) To this end we rewrite the Smarr relation as:
$sT+\mu\,n_{\text{n}}=\epsilon-\mu n_{\text{s}}+p=\rho_{\text{n}}+p\,,$ (2.72)
where $\rho_{\text{n}}\equiv\epsilon-\mu n_{\text{s}}$. We can then recast the
first law as
$\text{d}\rho_{\text{n}}=\mu\,\text{d}n_{\text{n}}+T\,\text{d}s+f{\xi_{0}}\,\text{d}{\xi_{0}}$.
Then:
$u^{\mu}\partial_{\mu}s=\frac{1}{T}\big{(}u^{\mu}\partial_{\mu}\rho_{\text{n}}-\mu\,u^{\mu}\partial_{\mu}n_{\text{n}}-fu^{\mu}{\xi_{0}}\partial_{\mu}{\xi_{0}}\big{)}$
(2.73) And thus:
$\partial_{\mu}\big{(}su^{\mu}\big{)}=\frac{1}{T}\big{(}u^{\mu}\partial_{\mu}\rho_{\text{n}}+(\rho_{\text{n}}+p)\partial_{\mu}u^{\mu}\big{)}-\frac{\mu}{T}\big{(}u^{\mu}\partial_{\mu}n_{\text{n}}+n_{\text{n}}\partial_{\mu}u^{\mu}\big{)}-\frac{f{\xi_{0}}}{T}u^{\mu}\partial_{\mu}{\xi_{0}}$
(2.74) Now we can make use of the equations of motion. From the conservation
of the stress-energy tensor (2.62) we obtain (using the Josephson relation
$u_{\mu}{\xi_{0}}^{\mu}=-\mu$)
$\begin{split}0=u_{\nu}\partial_{\mu}T^{\mu\nu}&=u_{\nu}u^{\mu}u^{\nu}\partial_{\mu}(\rho_{\text{n}}+p)+\eta^{\mu\nu}u_{\nu}\partial_{\mu}p+(\rho_{\text{n}}+p)u_{\nu}u^{\mu}\partial_{\mu}u^{\nu}+u_{\nu}\partial_{\mu}\big{(}f{\xi_{0}}^{\mu}{\xi_{0}}^{\nu}\big{)}\\\
&\quad\quad+u_{\nu}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}\\\
&=-u^{\mu}\partial_{\mu}\rho_{\text{n}}-(\rho_{\text{n}}+p)\,\partial_{\mu}u^{\mu}-\mu\,\partial_{\mu}\big{(}f{\xi_{0}}^{\mu}\big{)}+fu_{\nu}\,{\xi_{0}}^{\mu}\partial_{\mu}{\xi_{0}}^{\nu}+u_{\nu}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}\end{split}$
(2.75) So we can write:
$u^{\mu}\partial_{\mu}\rho_{\text{n}}+\big{(}\rho_{\text{n}}+p\big{)}\,\partial_{\mu}u^{\mu}=-\mu\,\partial_{\mu}\big{(}f{\xi_{0}}^{\mu}\big{)}+fu_{\nu}\,{\xi_{0}}^{\mu}\partial_{\mu}{\xi_{0}}^{\nu}+u_{\nu}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}$
(2.76) Furthermore current conservation $\partial_{\mu}j^{\mu}=0$ implies:
$\partial_{\mu}j^{\mu}=0=n_{\text{n}}\partial_{\mu}u^{\mu}+u^{\mu}\partial_{\mu}n_{\text{n}}+\partial_{\mu}\left(f{\xi_{0}}^{\mu}\right)+\partial_{\mu}j_{\text{diss}}^{\mu}$
(2.77) This yields:
$u^{\mu}\partial_{\mu}n_{\text{n}}+n_{\text{n}}\partial_{\mu}u^{\mu}=-\partial_{\mu}\big{(}f{\xi_{0}}^{\mu}\big{)}-\partial_{\mu}j_{\text{diss}}^{\mu}$
(2.78) Plugging everything together we find:
$\begin{split}\partial_{\mu}\big{(}su^{\mu}\big{)}&=\frac{1}{T}\big{(}-\mu\partial_{\mu}\big{(}f{\xi_{0}}^{\mu}\big{)}+fu_{\nu}{\xi_{0}}^{\mu}\partial_{\mu}{\xi_{0}}^{\nu}+u_{\nu}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}\big{)}\\\
&\quad\qquad+\frac{\mu}{T}\big{(}\partial_{\mu}\big{(}f{\xi_{0}}^{\mu}\big{)}+\partial_{\mu}j_{\text{diss}}^{\mu}\big{)}-\frac{{\xi_{0}}f}{T}u^{\mu}\partial_{\mu}{\xi_{0}}\\\
&=\frac{1}{T}\big{(}fu_{\nu}{\xi_{0}}^{\mu}\partial_{\mu}{\xi_{0}}^{\nu}+u_{\nu}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}\big{)}+\frac{\mu}{T}\partial_{\mu}j_{\text{diss}}^{\mu}-\frac{{\xi_{0}}f}{T}u^{\mu}\partial_{\mu}{\xi_{0}}\end{split}$
(2.79) Using that
$\partial_{\nu}{\xi_{0}}^{\mu}-\partial_{\nu}{\xi_{0}}^{\mu}=\mathcal{O}(\partial^{2})$
this becomes:
$\partial_{\mu}\big{(}su^{\mu}\big{)}=-\frac{f}{T}\big{(}u^{\mu}{\xi_{0}}^{\nu}\partial_{\mu}{\xi_{0}}^{\nu}+{\xi_{0}}u^{\mu}\partial_{\mu}{\xi_{0}}\big{)}+\frac{1}{T}u_{\nu}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}+\frac{\mu}{T}\partial_{\mu}j_{\text{diss}}^{\mu}$
(2.80) Now we make use of (2.64) and obtain:
$\partial_{\mu}\big{(}su^{\mu}\big{)}=\frac{f}{T}\big{(}\mu_{\text{diss}}\,\xi_{\mu}u^{\rho}\partial_{\rho}(u^{\mu})+\zeta^{\rho}\partial_{\rho}\mu_{\text{diss}}\big{)}+\frac{u_{\nu}}{T}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}+\frac{\mu}{T}\partial_{\mu}j_{\text{diss}}^{\mu}$
(2.81) Notice that, as expected, this expression vanishes at linear order in
fluctuations.

We can now determine the divergence of the entropy current.
First we compute
$\frac{f}{\mu}\zeta^{\nu}\partial_{\nu}\mu_{\text{diss}}=\partial_{\nu}\bigg{(}\frac{f}{T}\zeta^{\nu}\mu_{\text{diss}}\bigg{)}-\mu_{\text{diss}}\,\partial_{\nu}\bigg{(}\frac{f}{T}\zeta^{\nu}\bigg{)}$
(2.82) and (using that $\zeta^{\mu}={\xi_{0}}^{\mu}-\mu u^{\mu}$ and
$u^{\mu}\zeta_{\mu}=0$ so that
$u^{\mu}\partial_{\nu}\zeta_{\mu}=-\zeta^{\mu}\partial_{\nu}u_{\mu}$):
$\begin{split}\frac{f}{T}\mu_{\text{diss}}{\xi_{0}}^{\mu}u^{\nu}\partial_{\nu}u_{\mu}=-\frac{f}{T}\mu_{\text{diss}}u^{\mu}u^{\nu}\partial_{\nu}\zeta_{\mu}\end{split}$
(2.83) And so:
$\frac{f}{T}\big{(}\mu_{\text{diss}}\,{\xi_{0}}^{\mu}u^{\nu}\partial_{\nu}u_{\mu}+\zeta^{\nu}\partial_{\nu}\mu_{\text{diss}}\big{)}=\partial_{\nu}\bigg{(}\frac{f}{T}\zeta^{\nu}\mu_{\text{diss}}\bigg{)}-\mu_{\text{diss}}\,\left(\partial_{\nu}\left(\frac{f}{T}\zeta^{\nu}\right)+\frac{f}{T}u^{\mu}u^{\nu}\partial_{\nu}\zeta_{\mu}\right)$
(2.84) The last term can be rewritten as:
$\begin{split}\partial_{\nu}\bigg{(}\frac{f}{T}\zeta^{\nu}\bigg{)}+\frac{f}{T}u^{\mu}u^{\nu}\partial_{\nu}\zeta_{\mu}=\Delta^{\mu\nu}\partial_{\nu}\bigg{(}\frac{f}{T}\zeta_{\mu}\bigg{)}\end{split}$
(2.85) So that (2.81) finally becomes:
$\partial_{\mu}\big{(}su^{\mu}\big{)}=-\partial_{\mu}\bigg{(}\frac{f}{T}\zeta^{\mu}\mu_{\text{diss}}\bigg{)}-\mu_{\text{diss}}\,\Delta^{\mu\nu}\partial_{\nu}\bigg{(}\frac{f}{T}\zeta_{\mu}\bigg{)}+\frac{u_{\nu}}{T}\partial_{\mu}\pi^{\mu\nu}_{\text{diss}}+\frac{\mu}{T}\partial_{\mu}j_{\text{diss}}^{\mu}$
(2.86) Now we define the entropy current as: $j_{s}^{\mu}\equiv
su^{\mu}-\frac{\mu}{T}j_{\text{diss}}^{\mu}-\frac{u_{\nu}}{T}\pi^{\mu\nu}-\frac{f}{T}\zeta^{\mu}\,\mu_{\text{diss}}\,,$
(2.87) with which we finally recover (2.88):
$\partial_{\mu}j_{s}^{\mu}=-\bigg{(}\partial_{\mu}\frac{\mu}{T}\bigg{)}j_{\text{diss}}^{\mu}-\mu_{\text{diss}}\,\Delta^{\mu\nu}\partial_{\nu}\bigg{(}\frac{f}{T}\zeta_{\mu}\bigg{)}-\bigg{(}\partial_{\mu}\frac{u_{\nu}}{T}\bigg{)}\pi_{\text{diss}}^{\mu\nu}\,.$
(2.88)

##### Ensemble and sources

A commonly used ensemble is the entropy ensemble, consisting of the timelike
component of the charge current (which, with vanishing normal fluid velocity,
corresponds to the charge density), entropy density, momentum density and the
spatial gradient of the Goldstone boson $\boldsymbol{\xi}\equiv\nabla\phi$:

$(j^{t},s,\boldsymbol{\pi},\boldsymbol{\xi})$

We saw at the beginning of this section that the first law takes the form:
[39]

$\displaystyle\begin{split}\text{d}p&=s\,\text{d}T+n\,\text{d}\mu-\frac{n_{\text{s}}}{2\mu}\,\text{d}(\xi^{2}+\mu^{2})\\\
&=s\,\text{d}T+\left(n+\frac{n_{\text{s}}}{\mu}\,\boldsymbol{\xi}\cdot\mathbf{u}\right)\,\text{d}\mu-\frac{n_{\text{s}}}{\mu}\boldsymbol{\xi}\cdot(\text{d}\boldsymbol{\xi}-\mu\,\text{d}\mathbf{u})\end{split}$
(2.89)

Take the constitutive relation for the current (2.45). At ideal order
$j^{t}=n+(n_{\text{s}}/\mu)\zeta^{t}$. But the Josephson relation (2.41)
yields $\xi^{t}=\boldsymbol{\xi}\cdot\mathbf{u}+\mu$ and so
$\zeta^{t}=\xi^{t}-\mu$. Thus we can write:

$j^{t}=n+\frac{n_{\text{s}}}{\mu}\,\boldsymbol{\xi}\cdot\mathbf{u}$ (2.90)

We can then take a look back at the first law (2.89) together with definition
(2.4) and read off the sources (recall that the spacelike part of the normal
fluid velocity $u^{\mu}$ comes at first order in fluctuations):

$s_{j^{t}}=\mu\qquad\qquad s_{s}=T$ (2.91)
$s_{\xi}^{i}\equiv\mathbf{h}_{\xi}^{i}=\frac{n_{\text{s}}\xi^{i}}{\mu}$ (2.92)

On top of this, the source of the momentum density is, as, usual, the normal
fluid velocity:

$s_{\boldsymbol{\pi}}=\mathbf{u}$ (2.93)

We can now write down the susceptibility matrix $\chi_{ab}$, so that:

$\begin{pmatrix}\delta j^{t}\\\ \delta s\\\ \delta\boldsymbol{\pi}\\\
\delta\boldsymbol{\xi}\end{pmatrix}=\chi\cdot\begin{pmatrix}\delta\mu\\\
\delta T\\\ \delta\mathbf{u}\\\ \delta\mathbf{h}_{\xi}\end{pmatrix}$ (2.94)

Onsager reciprocity fixes some of the susceptibilities. In particular, in the
collinear limit $\boldsymbol{\zeta}//\mathbf{u}//\mathbf{k}//\hat{e}_{x}$ one
finds ($\xi\equiv\boldsymbol{\xi}^{x}$):

$\partial_{\mu}n_{\text{s}}|_{\xi,T,u_{x}}=\frac{n_{\text{s}}}{\mu}-\,\frac{\mu}{\xi}\chi_{nh_{\xi}}$
(2.95)
$\partial_{T}n_{\text{s}}|_{\xi,\mu,u_{x}}=-\frac{\mu}{\xi}\chi_{sh_{\xi}}|_{\mu,T,u_{x}}$
(2.96)
$\partial_{u_{x}}n_{\text{s}}|_{\xi,\mu,T}=-\mu\,\partial_{\xi}n_{\text{s}}|_{\mu,T,u_{x}}$
(2.97) $\chi_{n\pi}=-\mu\,\chi_{nh_{\xi}}$ (2.98)
$\chi_{s\pi}=-\mu\,\chi_{sh_{\xi}}$ (2.99)
$\chi_{ph_{\xi}}=\partial_{\xi}n_{\text{s}}|_{\mu,T,u_{x}}+\frac{n_{\text{s}}}{\xi}$
(2.100)

These relations yield the following susceptibility matrix:

$\begin{pmatrix}\delta j^{t}\\\ \delta s\\\ \delta\pi^{x}\\\
\delta\xi\end{pmatrix}=\begin{pmatrix}\chi_{nn}+\chi_{nh_{\xi}}^{2}\chi_{\xi\xi}&\chi_{ns}+\chi_{nh_{\xi}}\chi_{sh_{\xi}}\chi_{\xi\xi}&\frac{\xi
n_{\text{s}}}{\mu}&-\chi_{nh_{\xi}}\chi_{\xi\xi}\\\
\chi_{ns}+\chi_{nh_{\xi}}\chi_{sh_{\xi}}\chi_{\xi\xi}&\chi_{ss}+\chi_{sh_{\xi}}^{2}\chi_{\xi\xi}&0&\chi_{sh_{\xi}}\\\
\frac{\xi
n_{\text{s}}}{\mu}&0&\epsilon+p+\frac{\xi}{n_{\text{s}}}\xi^{2}&\mu\\\
\chi_{nh_{\xi}}\chi_{\xi\xi}&\chi_{sh_{\xi}}\chi_{\xi\xi}&\mu&\chi_{\xi\xi}\end{pmatrix}\begin{pmatrix}\delta\mu\\\
\delta T\\\ \delta u_{x}\\\ \delta h_{\xi}\end{pmatrix}$ (2.101)

Where

$\chi_{\xi\xi}=\frac{\mu}{n_{\text{s}}+\xi\,\partial_{\xi}n_{\text{s}}|_{T,\mu,u_{x}}}$
(2.102)

Note that here we are expressing the static susceptibilities at fixed
$h_{\xi}$ in terms of those computed at fixed $\xi$; for instance
$\chi_{nn}^{h_{\xi}}=\chi_{nn}+\chi_{nh_{\xi}}^{2}\chi_{\xi\xi}$.

##### Defining the dissipative coefficients

The contribution of the stress-energy tensor to entropy production (2.88) can
be rewritten as [36]

$\begin{split}-\frac{1}{T}\,\pi_{diss}^{\mu\nu}\partial_{\mu}u_{\nu}&=-\frac{1}{T}\,\pi_{diss}^{\mu\nu}\sigma_{\mu\nu}-\frac{1}{d\,T}\pi_{diss,\,\mu}^{\mu}\,\partial_{\rho}u^{\rho}\end{split}$
(2.103)

Note that $\sigma_{\mu\nu}$, by construction, only couples to the traceless
part of $\pi_{diss}^{\mu\nu}$, which we call $\pi_{diss,\,TL}^{\mu\nu}$. This
implies that the contribution of $\pi_{diss}^{\mu\nu}$ to entropy production
consists of two independent terms (corresponding to two irreducible
representations of $SO(1,d)$):

$\partial_{\mu}\tilde{J}_{s}^{\mu}=-\bigg{(}\partial_{\mu}\frac{\mu}{T}\bigg{)}\tilde{J}_{diss}^{\mu}+\mu_{\text{diss}}\,\Delta^{\mu\nu}\partial_{\nu}\bigg{(}\frac{f}{T}\zeta_{\mu}\bigg{)}-\frac{1}{T}\pi_{diss,\,TL}^{\mu\nu}\sigma_{\mu\nu}-\frac{1}{d\,T}\partial_{\rho}u^{\rho}{\pi_{diss}}^{\mu}_{\mu}$
(2.104)

This motivates the following definition of the dissipative contributions:

$\begin{pmatrix}J_{diss}^{\mu}\\\ \mu_{diss}\\\ \pi_{diss,TL}^{\mu\nu}\\\
\pi^{\rho}_{diss,\rho}\end{pmatrix}=-\begin{pmatrix}A^{\mu\rho}&B^{\mu}&C^{\mu\rho\sigma}&D^{\mu}\\\
E^{\rho}&F&G^{\rho\sigma}&H\\\
I^{\mu\nu\rho}&K^{\mu\nu}&L^{\mu\nu\rho\sigma}&M^{\mu\nu}\\\
N^{\rho}&Q&R^{\rho\sigma}&S\end{pmatrix}\begin{pmatrix}\Delta_{\rho}^{\alpha}\partial_{\alpha}\frac{\mu}{T}\\\
\Delta^{\mu\nu}\partial_{\nu}\big{(}\frac{n_{\text{s}}}{\mu
T}\zeta_{\mu}\big{)}\\\ \frac{1}{T}\,\sigma_{\rho\sigma}\\\
\frac{1}{d\,T}\,\partial_{\alpha}u^{\alpha}\end{pmatrix}$ (2.105)

This matrix will have to be positive (semi)definite in order to achieve
positivity of entropy production. In particular, its diagonal elements must be
nonnegative.
We work in the transverse frame:
$u_{\mu}j_{diss}^{\mu}=u_{\mu}\pi_{diss}^{\mu\nu}=0$, so that its elements can
be parametrized as follows:

$\displaystyle A^{\mu\rho}$
$\displaystyle=\alpha_{A}\,\Delta^{\mu\rho}+\beta_{A}\,\tilde{\zeta}^{\mu}\tilde{\zeta}^{\rho}$
(2.106) $\displaystyle B^{\mu}$ $\displaystyle=\alpha_{B}\zeta^{\mu}$ (2.107)
$\displaystyle C^{\mu\rho\sigma}$
$\displaystyle=\alpha_{C}\,\big{(}\tilde{P}^{\mu\sigma}\tilde{\zeta}^{\rho}+\tilde{P}^{\mu\rho}\tilde{\zeta}^{\sigma}\big{)}+\beta_{C}\,\zeta^{\mu}P^{\rho\sigma}$
(2.108) $\displaystyle D^{\mu}$ $\displaystyle=\alpha_{D}\,\zeta^{\mu}$
(2.109) $\displaystyle E^{\sigma}$ $\displaystyle=\alpha_{E}\,\zeta^{\sigma}$
(2.110) $\displaystyle G^{\rho\sigma}$
$\displaystyle=\alpha_{G}\,P^{\rho\sigma}$ (2.111) $\displaystyle
I^{\mu\nu\rho}$
$\displaystyle=\alpha_{I}\,\big{(}\tilde{P}^{\mu\rho}\tilde{\zeta}^{\nu}+\tilde{P}^{\nu\rho}\tilde{\zeta}^{\mu}\big{)}+\beta_{I}\,\zeta^{\rho}P^{\mu\nu}$
(2.112) $\displaystyle K^{\mu\nu}$ $\displaystyle=\alpha_{K}\,P^{\mu\nu}$
(2.113) $\displaystyle L^{\mu\nu\rho\sigma}$
$\displaystyle=\alpha_{L}\,P^{\mu\nu}P^{\rho\sigma}+\beta_{L}\,\Delta^{\mu\rho}\Delta^{\nu\sigma}+\gamma_{L}\,\big{(}\zeta^{\mu}\tilde{\zeta}^{\sigma}\tilde{P}^{\nu\rho}+\zeta^{\nu}\tilde{\zeta}^{\sigma}\tilde{P}^{\mu\rho}\big{)}$
(2.114) $\displaystyle M^{\mu\nu}$ $\displaystyle=\alpha_{M}\,P^{\mu\nu}$
(2.115) $\displaystyle N^{\rho}$ $\displaystyle=\alpha_{N}\,\zeta^{\rho}$
(2.116) $\displaystyle R^{\rho\sigma}$
$\displaystyle=\alpha_{R}\,P^{\rho\sigma}$ (2.117) $\displaystyle F$
$\displaystyle=\alpha_{F}\qquad H=\alpha_{H}\qquad Q=\alpha_{Q}\qquad
S=\alpha_{S}\,\text{,}$ (2.118)

where $\tilde{\zeta}^{\mu}\equiv\zeta^{\mu}/\sqrt{\zeta^{\nu}\zeta_{\nu}}$,
$P^{\mu\nu}\equiv\tilde{\zeta}^{\mu}\tilde{\zeta}^{\nu}-\Delta^{\mu\nu}/d$ is
a traceless, transverse tensor and
$\tilde{P}^{\mu\nu}\equiv\Delta^{\mu\nu}-\tilde{\zeta}^{\mu}\tilde{\zeta}^{\nu}$
is the projector onto the subspace orthogonal to both $u^{\mu}$ and
$\zeta^{\mu}$.
Onsager reciprocity requires the matrix appearing in (2.105) to be symmetric.
We get:

$\alpha_{B}=\alpha_{E}\quad\alpha_{C}=\alpha_{I}\quad\beta_{C}=\beta_{I}\quad\alpha_{D}=\alpha_{N}\quad\alpha_{G}=\alpha_{K}\quad\alpha_{H}=\alpha_{Q}\quad\alpha_{M}=\alpha_{R}$
(2.119)

We thus obtain a total of 14 dissipative parameters, as stated in [40].

Note that in the limit of vanishing background superfluid velocity
$\zeta^{\mu}\to 0$ the normed vector $\tilde{\zeta}^{\mu}$ is not well-defined
[36], so that one must set $\alpha_{G}=\alpha_{L}=0$. After imposing Onsager
invariance this means that the only non-vanishing dissipative contributions to
the constitutive relations will be parametrized by a total of five
coefficients: $\alpha_{A}$, $\beta_{L}$, $\alpha_{F}$, $\alpha_{H}$ and
$\alpha_{S}$.

Imposing Weyl invariance amounts to requiring that the stress-energy tensor be
traceless; in this case this condition translates to
$\alpha_{N}=\alpha_{Q}=\alpha_{R}=\alpha_{S}=0$, leaving us with a total of 10
dissipative coefficients after imposing Onsager reciprocity, as in [40]. In
the limit of zero background superfluid velocity this set further reduced to
three coefficients: $\alpha_{A}$, $\beta_{L}$ and $\alpha_{F}$.

In the collinear limit, where the spatial parts of $u^{\mu}$ and $\zeta^{\mu}$
are both parallel to the wavevector, the linearized equations of motion can be
written as follows:

$-i\omega\,\begin{pmatrix}\delta J^{t}\\\ \delta s\\\ \delta\pi^{x}\\\
\delta\xi\end{pmatrix}+\tilde{M}\cdot\begin{pmatrix}\delta\mu\\\ \delta T\\\
\delta u_{x}\\\ \delta h_{\xi}\end{pmatrix}=0$ (2.120)

with

$\tilde{M}=\begin{pmatrix}\frac{\alpha_{A}+\beta_{A}}{T}\,k^{2}&\frac{-\mu^{2}(\alpha_{A}+\beta_{A})+\xi^{2}n_{\text{s}}\,\alpha_{B}}{T^{2}\mu}\,k^{2}&ikn+\frac{(\alpha_{D}+2\beta_{C})\xi}{3T}\,k^{2}&ik+\frac{k^{2}\alpha_{B}\xi}{T}\\\
\frac{-\mu^{2}(\alpha_{A}+\beta_{A})+\xi^{2}n_{\text{s}}\,\alpha_{B}}{T^{2}\mu}\,k^{2}&\frac{\mu^{4}(\alpha_{A}+\beta_{A})+n_{\text{s}}(n_{\text{s}}\alpha_{F}-2\mu^{2}\alpha_{B})}{T^{2}\mu^{2}}\,k^{2}&iks-\frac{\left(n_{\text{s}}(2\alpha_{G}+\alpha_{H})+(\alpha_{D}+2\beta_{C})\mu^{2}\right)\xi}{3T^{2}\mu}\,k^{2}&-\frac{k^{2}\xi\left(n_{\text{s}}\alpha_{F}+\mu^{2}\alpha_{B}\right)}{T^{2}\mu}\\\
ikn+\frac{(\alpha_{D}+2\beta_{C})\xi}{3T}\,k^{2}&iks-\frac{\left(n_{\text{s}}(2\alpha_{G}+\alpha_{H})+(\alpha_{D}+2\beta_{C})\mu^{2}\right)\xi}{3T^{2}\mu}\,k^{2}&2n_{\text{s}}\xi\,ik+\frac{6\beta_{L}+4(\alpha_{L}+\alpha_{M})+\alpha_{S}}{9T}\,k^{2}&ik\xi+\frac{k^{2}(2\alpha_{G}+\alpha_{H})}{3T}\\\
ik+\frac{k^{2}\alpha_{B}\xi}{T}&\frac{k^{2}\xi\left(n_{\text{s}}\alpha_{F}+\mu^{2}\alpha_{B}\right)}{T^{2}\mu}&ik\xi+\frac{k^{2}(2\alpha_{G}+\alpha_{H})}{3T}&\frac{\alpha_{F}}{T}k^{2}\end{pmatrix}$
(2.121)

Note that this matrix is Onsager symmetric, as expected.
We see that in the collinear limit the derivative coefficients appear in the
equations of motion in only six independent linear combinations. We can thus
set:

$\sigma_{0}=\frac{\alpha_{A}+\beta_{A}}{T}\,,\quad\zeta_{1}=\frac{\alpha_{G}+\alpha_{H}}{2T}\,,\quad\zeta_{2}=\frac{\alpha_{B}}{T}\,,\quad\zeta_{3}=\frac{\alpha_{F}}{T}\,,\quad\zeta_{6}=\frac{(\alpha_{D}+\beta_{C})}{2T}\,,$
(2.122)
$\eta=\frac{\beta_{L}+\xi\gamma_{L}}{2T}\,,\quad\zeta=\frac{\alpha_{S}+\alpha_{L}+\alpha_{M}-2\xi\gamma_{L}}{4T}$
(2.123)

Notice that since, as we saw above, $\alpha_{G}=0$ for vanishing background
superfluid velocity, for a Weyl invariant theory, featuring $\alpha_{H}=0$,
also $\zeta_{1}$ will vanish at $\zeta^{\mu}=0$.
Interestingly, at nonzero superfluid velocity, the bulk viscosity is non-
vanishing when conformal invariance is imposed.

As we mentioned earlier, positivity of entropy production sets bounds on the
dissipative coefficients. In particular, the diagonal components of (2.105)
must be nonnegative: here we can immediately conclude that
$\zeta_{3},\sigma_{0}\geq 0$. Furthermore, all principal minors of (2.105)
must be positive definite. For example, this yields
$\sigma_{0}\zeta_{3}\geq\xi^{2}\zeta_{2}^{2}$.

Finally, in the collinear case the matrix $\tilde{M}$ describing the equations
of motion (2.120) becomes:

$\tilde{M}=\begin{pmatrix}\sigma_{0}\,k^{2}&\frac{-\mu^{2}\,\sigma_{0}-n_{\text{s}}\xi^{2}\zeta_{2}}{T\mu}\,k^{2}&ikn+\xi\zeta_{6}\,k^{2}&ik+k^{2}\zeta_{2}\xi\\\
\frac{-\mu^{2}\,\sigma_{0}-n_{\text{s}}\xi^{2}\zeta_{2}}{T\mu}\,k^{2}&\frac{n_{\text{s}}\xi^{2}(n_{\text{s}}\zeta_{3}+2\mu^{2}\zeta_{2})+\mu^{4}\sigma}{T^{2}\mu^{2}}\,k^{2}&iks+\frac{\xi\left(n_{\text{s}}\zeta_{1}+\mu^{2}\zeta_{6}\right)}{T\mu}\,k^{2}&-\frac{k^{2}\xi\left(n_{\text{s}}\zeta_{3}+\mu^{2}\zeta_{2}\right)}{T\mu}\\\
ikn+\xi\zeta_{6}\,k^{2}&iks+\frac{\xi\left(n_{\text{s}}\zeta_{1}+\mu^{2}\zeta_{6}\right)}{T\mu}\,k^{2}&2n_{\text{s}}\xi\,ik+(\eta+\zeta_{b})\,k^{2}&ik\xi
k^{2}\,\zeta_{1}\\\ \
ik+k^{2}\zeta_{2}\xi&-\frac{k^{2}\xi\left(n_{\text{s}}\zeta_{3}+\mu^{2}\zeta_{2}\right)}{T\mu}&ik\xi
k^{2}\,\zeta_{1}&\zeta_{3}\,k^{2}\end{pmatrix}$ (2.124)

##### Kubo relations

The dissipative coefficients can be extracted from the $\omega\to 0$, $k\to 0$
behaviour of the retarded Green’s functions. As we are going to see these
relations are especially useful in the context of the applications of gauge-
gravity duality. Simply put, they allow us to extract the dissipative
coefficients appearing in the gradient expansion of the vacuum expectation
values of the boundary theory.

In the collinear limit we have:

$\begin{split}\sigma_{0}&=-\lim_{\omega\to 0,k\to
0}\,\frac{1}{\omega}\,\text{Im}\,G^{R}_{J^{x}J^{x}}(\omega,k)\\\
\zeta_{1}&=-\lim_{\omega\to 0,k\to
0}\,\frac{1}{k\omega}\,\text{Im}\,G^{R}_{T^{xx}\xi^{x}}(\omega,k)\\\
\xi\,\zeta_{2}&=-\lim_{\omega\to 0,k\to
0}\,\frac{1}{k}\,\text{Im}\,G^{R}_{J^{x}\xi^{x}}(\omega,k)\\\
\zeta_{3}&=-\lim_{\omega\to 0,k\to
0}\,\frac{\omega}{k^{2}}\,\text{Im}\,G^{R}_{\xi^{x}\xi^{x}}(\omega,k)\\\
\xi\,\zeta_{6}&=-\lim_{\omega\to 0,k\to
0}\,\frac{1}{\omega}\,\text{Im}\,G^{R}_{J^{x}T^{xx}}(\omega,k)\\\
\eta+\zeta_{b}&=-\lim_{\omega\to 0,k\to
0}\,\frac{1}{\omega}\,\text{Im}\,G^{R}_{T^{xx}T^{xx}}(\omega,k)\\\
\eta&=-\lim_{\omega\to 0,k\to
0}\,\frac{1}{\omega}\,\text{Im}\,G^{R}_{T^{xy}T^{xy}}(\omega,k)\end{split}$
(2.125)

We can check that in the $\xi\to 0$ limit $\zeta_{1}$ vanishes. In fact, we
can rewrite the above Kubo relation as $\zeta_{1}=\lim_{\omega\to
0}\,\text{Re}\,G^{R}_{T^{xx}\phi}(\omega,k=0)$. Now, $T^{xx}$ is even under
time-reversal, while $\phi$ is odd (since the superfluid velocity is odd and
is given by its spatial gradient). Thus their correlator must be odd. At
vanishing wavevector the only background quantity that is odd under time
reversal is $\xi$ itself; hence $\text{Re}G^{R}_{T^{xx}\phi}(\omega,k=0)$ must
have an expansion in odd powers of $\xi$, implying that it vanishes at $\xi\to
0$. This confirms what we saw above, i.e. that in the limit of vanishing
background superfluid velocity only three dissipative coefficients enter the
equations of motion.

##### Back to the start: the conductivity

Having defined the propagators, given the constitutive relations and written
down the dissipative coefficients of superfluid hydrodynamics it is now a good
time to take a step back and reflect on the physical implications of the
superfluid component.
In particular, we should take a look at the conductivity. It is obtained from
the current-current propagator in the vanishing momentum limit. In superfluid
hydrodynamics with zero background superfluid velocity $\xi$ and a momentum
relaxation rate $\Gamma$ (which appears in the equation of motion for the
momentum density $T^{tx}$ as
$(\partial_{t}+\Gamma)T^{tx}+\partial_{x}T^{xx}=0$ ) one can show that it is
given by 888Non-dissipative, non-hydrostatic terms contribute to $\sigma$ at
the same order at $\sigma_{0}$, see [41] for the case of hydrodynamic metals.
Here we are assuming the absence of such terms.: [24, 42]

$\sigma(\omega)\equiv\frac{i}{\omega}G_{J^{x}J^{x}}(\mathbf{k}=0,\omega)=\frac{1}{\Gamma-i\omega}\frac{n_{\text{n}}^{2}}{\mu
n_{\text{n}}+sT}+\frac{i}{\omega}\frac{n_{\text{s}}}{\mu}+\sigma_{0}+\mathcal{O}(\omega)$
(2.126)

We can see that, in the presence of a finite $\Gamma$, the conductivity
$\sigma(\omega)$ has a pole at $\omega=0$ if $n_{\text{s}}\neq 0$. This is
indeed a token of superconductivity.
On the other hand, if $\Gamma=0$, $\sigma(\omega)$ does have a pole at
$\omega=0$ even in the absence of a superfluid component. This feature,
however, arises from translation invariance, as an electric field applied onto
a translationally invariant system produces uniform acceleration [15].

##### Computing the modes

The hydrodynamic modes $\omega(k)$ (2.27) can be computed by solving for:

$\text{det}(-i\omega\mathbb{1}+\tilde{M}\chi^{-1})=0$ (2.127)

For small values of the wavevector $k$ the modes are sound modes.
Their real part, linear in $k$, depends on thermodynamic quantities such as
the charge and densities, the temperature and the susceptibilities. The
imaginary part, one the other hand, is quadratic in $k$ and depends on the
dissipative coefficients, too.
One commonly identifies two “first sound” and two “second sound” modes, as we
are going to see in the coming section.

####  2.4.3 Modes in conformal limit with no background superfluid velocity

At this point it would come natural to write down the modes of superfluid
hydrodynamics with the full set of dissipative coefficients described above.
Such modes can be computed; their analytic form, however, is too long and
complicated to write down.
Nevertheless, the modes do simplify in the limit where the background
superfluid velocity $\xi$ vanishes. In this section we are going to list them
in this particular case, for a conformal theory: from a qualitative point of
view, we would encounter the same types of mode (two pairs of sound modes and
a diffusive mode) with $\xi\neq 0$, too. Moreover, this is a demonstration of
the power of conformal symmetry, which allows us to derive relations between
the thermodynamic derivatives.
We take (2.48) as the definition of the three dissipative terms which are
relevant in this limit.

As a first step, we recall [39] that in the absence of a background superfluid
velocity conformal symmetry implies that the $\mu/T$ ratio is the only
available scale invariant background scalar quantity. Then any scalar can be
expressed as a function of $\mu/T$, multiplied with a power of $T$ (or,
equivalently, $\mu$), determining its dimensionality. So, in particular,

$p=T^{d+1}f\Big{(}\frac{\mu}{T}\Big{)}$ (2.128)

But from the first law (2.89) one gets:

$s=\frac{\partial p}{\partial T}{\bigg{|}_{\mu}}\quad\quad\quad\quad
n=\frac{\partial p}{\partial\mu}{\bigg{|}_{T}}$ (2.129)

By further differentiating with respect to $\mu$ and $T$ one can verify that:
999These relations can also be derived by imposing that the trace of the
stress-energy tensor vanish: $\langle T_{\mu}^{\mu}\rangle=0$. In fact, under
a conformal transformation the (infinitesimal) variation of the metric is
proportional to the metric itself: $\delta g^{\mu\nu}=\lambda g^{\mu\nu}$. We
can write $\langle
T_{\mu\nu}\rangle=-\frac{2}{\sqrt{-g}}\frac{\delta\Omega}{\delta g^{\mu\nu}}$.
But then under a variation of the metric one has that
$\delta\Omega=-\frac{\sqrt{-g}}{2}\delta g^{\mu\nu}\langle
T_{\mu\nu}\rangle\propto\lambda g^{\mu\nu}\langle
T_{\mu\nu}\rangle\propto\langle T_{\mu}^{\mu}\rangle$. So invariance under
conformal transformations implies that the stress-energy tensor is traceless.

$\frac{\partial s}{\partial
T}{\bigg{|}_{\mu}}=d\,\frac{s}{T}-\frac{\mu}{T}\frac{\partial
s}{\partial\mu}{\bigg{|}_{T}}\qquad\qquad\qquad\frac{\partial
n}{\partial\mu}{\bigg{|}_{T}}=d\,\frac{n}{\mu}-\frac{T}{\mu}\frac{\partial
s}{\partial\mu}{\bigg{|}_{T}}$ (2.130)

Furthermore, one can define the reduced entropy $\sigma$ as [4]:

$\sigma\equiv\frac{s}{n}$ (2.131)

and express the susceptibilities in terms of its thermodynamic derivatives.
For instance:

$\frac{\partial s}{\partial\mu}{\bigg{|}_{T}}=\frac{\partial n}{\partial
T}{\bigg{|}_{\mu}}=\frac{n\left(d\,s-nT\,\frac{\partial\sigma}{\partial
T}{\big{|}_{\mu}}\right)}{sT+n\mu}$ (2.132)

Writing $\omega=ck-\frac{i}{2}\Gamma k^{2}+\mathcal{O}(k^{3})$ as in (2.27),
one obtains the following modes [43, 8]:

  * •

One diffusion mode:

$v=0\quad\quad\quad\quad\Gamma=\frac{2\eta}{sT+\mu n_{n}}$ (2.133)

  * •

Two first sound modes:

$v=\pm\frac{1}{\sqrt{d}}\quad\quad\quad\quad\Gamma=\frac{2(d-1)\eta}{d(sT+n\mu)}$
(2.134)

  * •

Two second sound modes [44, 39, 42, 37, 45] :

$\displaystyle\begin{split}v&=\pm\frac{\sigma^{2}n_{\text{s}}}{(sT+\mu
n_{n})\frac{\partial\sigma}{\partial T}{\big{|}_{\mu}}}\\\\[4.30554pt]
\Gamma&=\frac{\mu(sT+\mu n)}{T^{2}\,n^{2}\,\frac{\partial\sigma}{\partial
T}{\big{|}_{\mu}}}\,\sigma_{0}+\frac{2(d-1)\,\mu n_{\text{s}}}{d(sT+\mu
n)(sT+\mu n_{n})}\,\eta+\frac{n_{\text{s}}(sT+\mu n)}{\mu(sT+\mu
n_{n})}\,\zeta_{3}\end{split}$ (2.135)

Notice that the imaginary parts of the modes are nonpositive, provided the
dissipative coefficients are nonnegative. This, in turn, directly follows from
positivity of entropy production (2.88).

##  Chapter 3 Gauge-gravity duality

The AdS-CFT correspondence was first proposed by Maldacena in 1997 [46]. It
posits the dynamical equivalence between a $d+1$-dimensional gravity theory in
an asymptotically anti-de-Sitter (AdS) space and a $d$-dimensional conformal
field theory (CFT). AdS spaces are maximally symmetric spaces with constant
curvature; a theory is called conformal if it is invariant under conformal
maps, i.e. transformations that locally preserve angles.

The original correspondence involves type IIB string theory on
AdS${}_{5}\times S^{5}$ and $\mathcal{N}=4$ Super Yang-Mills theory with gauge
group $SU(N)$. Yang-Mills theories are gauge theories with a non-abelian
symmetry group; super Yang-Mills theories make use of supersymmetry to
associate fermionic partners to the bosonic fields. In this case the gauge
bosons can be mapped onto $\mathcal{N}=4$ different fermionic partners.
Owing to the nature of the theories it links together, the AdS-CFT
correspondence is thus also called “gauge-gravity duality”.

On top of $N$ the main parameters governing the involved theories are the
string coupling $g_{s}$, the field theory coupling $g_{\text{YM}}$ and the AdS
curvature radius $L$.
Writing down the full Yang-Mills Lagrangian is beyond the scope of this
introduction; it will be sufficient to know that the Yang-Mills coupling
$g_{\text{YM}}$ appears in front of the kinetic term for the gauge field:
$\mathcal{L}\supset-\text{Tr}F_{\mu\nu}F^{\mu\nu}/(2g_{\text{YM}}^{2})$. On
the other hand, the string coupling $g_{s}$ is given by the expectation value
of the dilaton field in the gravity theory emerging from the string theory. It
controls the genus expansion of string theory scattering (a small $g_{s}$
suppresses the contributions of topologies with large genus to the scattering
amplitude) and is related to Newton’s gravitational constant in $d$ dimensions
$G_{d}$ by $G_{d}\sim l_{s}^{d-2}g_{s}^{2}$, with $l_{s}$ being the string
length.

The coupling of the field theory is related to the string coupling by:

$g_{\text{YM}}^{2}=2\pi g_{s}$ (3.1)

Furthermore, the string length $l_{s}$ satisfies the relation

$g_{\text{YM}}^{2}\sim\frac{L^{4}}{l_{s}g_{s}N}$ (3.2)

In particular, one can consider a matrix theory with fields in the adjoint
representation of $SU(N)$. Owing to the two-index structure of the matrix
theory the Feynman diagrams are going to be surfaces with an intrinsic
topology. One defines:

$\lambda\equiv g_{\text{YM}}^{2}N=\text{const}$ (3.3)

One can show that the Feynman diagrams go like $N^{2-2G}\lambda^{E-V}$, with
$G$ being the genus of the surface [47, 12]. In this case one can take the ‘t
Hooft limit: $N\to\infty$ with $g_{YM}^{2}\to 0$, so that $\lambda$ stays
finite. In this case planar diagrams with $G=0$ will dominate and $N$-point
correlation functions factorize [47]:

$<\mathcal{O}\cdots\mathcal{O}>\,=\,<\mathcal{O}>\cdots<\mathcal{O}>+\,\mathcal{O}(1/N^{2})$
(3.4)

We thus obtain a large-$N$ theory with nontrivial interactions, although we
see that correlators behave classically up to subleading corrections in $1/N$.
Indeed, from (3.1) and (3.3) we see that the $1/N$ expansion of the field
theory corresponds to the genus expansion of the string theory: in the
$N\to\infty$ limit one recovers classical string theory.
One can also assume $\lambda$ to be large. (3.2) and (3.3) imply that in this
limit one sends the string length to zero, yielding classical supergravity on
the gravity side of the correspondence. This “strong/weak coupling duality”
[13, 48] comes in handy because it allows us to study strongly coupled
conformal field theories in $d$ dimensions by solving the classical Einstein’s
equations of motion in $d+1$ dimensions.

In this thesis we are going to make use of gauge-gravity duality to study the
hydrodynamics of superfluids. Now, a hydrodynamic theory is the effective
description at long distances and late times of the universal behaviour at
finite temperature of some underlying interacting theory. So far, however, we
have not yet mentioned how one can obtain a thermal theory from a
gravitational one. As we are going to see, the main idea will be to consider a
black hole solution of the Einstein equations of motion. One can build a
thermodynamic theory of black holes; most notably, their entropy is
proportional to their area, and one can associate to them a temperature
depending on the horizon behaviour of the metric. This will allow us to
construct a finite temperature theory on the asymptotically AdS boundary of
such a space.
We will do so by establishing a dictionary relating the fields and the
propagators on both sides of the correspondence. In particular, in this thesis
we will consider a gravitational theory coupled to a gauge field and a complex
scalar field undergoing spontaneous symmetry breaking. The metric fluctuations
will be dual to those of the stress-energy tensor and the gauge field to the
current.

###  3.1 Prerequisites

####  3.1.1 Conformal symmetry

Note: this section closely follows some parts of the introduction to conformal
symmetry in Erdmenger and Ammon’s book [12].

Conformal field theories (CFT’s) are invariant under conformal symmetry. The
conformal symmetry group contains the transformations that locally preserve
angles: dilatations and special conformal transformations. Such
transformations leave the metric invariant up to an arbitrary (positive) scale
factor:

$g_{\mu\nu}(x)\mapsto\Omega(x)^{-2}g_{\mu\nu}(x)\equiv
e^{2\sigma(x)}g_{\mu\nu}(x)$ (3.5)

Infinitesimal metric transformations are generated by a vector $\xi^{\mu}$:

$g_{\mu\nu}\to g_{\mu\nu}+\nabla_{(\mu}\xi_{\nu)}$ (3.6)

A Killing vector leaves the metric unaltered. One can show [12] that in
$d$-dimensional flat spacetime $g_{\mu\nu}=\eta_{\mu\nu}$ any conformal vector
$\xi_{\mu}$ satisfies the conformal Killing equation:

$\left(\eta_{\mu\nu}\partial_{\rho}\partial^{\rho}+(d-2)\,\partial_{\mu}\partial_{\nu}\right)\partial_{\rho}\xi^{\rho}=0$
(3.7)

In $d>2$ dimensions (in this thesis we will mainly work with conformal field
theories in 3 spacetime dimensions) the most general solution to (3.7) may be
parametrized as:

$\xi^{\mu}(x)=a^{\mu}+\omega^{\mu}_{\nu}x^{\nu}+\lambda
x^{\mu}+b^{\mu}x^{2}-2(b\cdot x)x^{\mu}$ (3.8)

From this expression we can extract the generators of the conformal algebra in
$d>2$ dimensions and read off the generators:

  * •

$a^{\mu}\Rightarrow$ translation: $\sigma(x)=0$. Generator:
$P_{\mu}=-i\partial_{\mu}$

  * •

$\omega^{\mu}_{\nu}x^{\nu}\Rightarrow$ Lorentz transformation: $\sigma(x)=0$.
Note that $\omega_{\mu\nu}=-\omega_{\nu\mu}$.
Generator: $J_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$

  * •

$\lambda x^{\mu}\Rightarrow$ dilatation:. $\sigma(x)=\lambda$. Generator:
$D=-ix^{\mu}\partial_{\mu}$

  * •

$b^{\mu}x^{2}-2(b\cdot x)x^{\mu}\Rightarrow$ special conformal transformation:
$\sigma(x)=-2(b\cdot x)$.
Generator: $K_{\mu}=i(x^{2}\partial_{\mu}-2x_{\mu}x^{\nu}\partial_{\nu})$

$a^{\mu}$ and $b^{\mu}$ have $d$ independent components, while
$\omega_{\mu\nu}$, being antisymmetric, has $d(d-1)/2$. So, together with
$\lambda$, we are left with $(d+1)(d+2)/2$ components. Notice that this is the
dimension of $SO(2,d)$, which gives a hint at the AdS-CFT correspondence.

The only non-vanishing commutation relations between the generators are the
following:

$[J_{\mu\nu},K_{\rho}]=i\left(\eta_{\mu\rho}K_{\nu}-\eta_{\nu\rho}K_{\mu}\right)\qquad[P_{\nu},K_{\mu}]=2i\left(\eta_{\mu\nu}D-J_{\mu\nu}\right)$
(3.9) $[D,P_{\mu}]=iP_{\mu}\qquad[D,K_{\mu}]=-iK_{\mu}$ (3.10)

With these commutators one can construct the representations of the conformal
algebra. A field $\phi(x)$ has scaling dimension $\Delta$ if:

$[D,\phi(0)]=-i\Delta\phi(0)\,,$ (3.11)

so that under a rescaling $x\mapsto\lambda x$:

$\phi(x)\mapsto\phi^{\prime}(x^{\prime})=\lambda^{-\Delta}\phi(x)$ (3.12)

But then, using the commutation relations, it follows that:

$[D,P_{\mu}\,\phi(0)]=-i(\Delta-1)\phi(0)\qquad\text{and}\qquad[D,K_{\mu}\,\phi(0)]=-i(\Delta+1)\phi(0)$
(3.13)

Thus one can define primary operators, i.e. operators that are annihilated by
$K_{\mu}$, and construct a tower of operators of higher dimension (called
conformal descendants) by using $P_{\mu}$ as a creation operator.
The conformal dimension $\Delta$ is subject to unitarity bounds. In
particular, for a scalar operator one can show that $\Delta\geq(d-2)/2$ [12].

Conformal field theories have many interesting properties which we are not
going to examine here. For instance, two- and three-point correlation
functions of primary operators are determined by the scaling dimensions of the
fields. In particular, for a primary operator of dimension $\Delta$, one has:

$\langle\mathcal{O}(x_{1})\mathcal{O}(x_{2})\rangle=(x_{1}-x_{2})^{-2\Delta}$
(3.14)

Last but not least, an important feature of conformally invariant theories is
that the trace of the stress-energy tensor vanishes. This can be seen as
follows: the stress-energy tensor is given by the variation of the action with
respect to the metric. But then (3.5) gives:

$\delta S=-\frac{\sqrt{-g}}{2}\delta
g^{\mu\nu}T_{\mu\nu}\propto\Omega^{-2}g^{\mu\nu}T_{\mu\nu}\propto
T_{\mu}^{\mu}$ (3.15)

So in order for the theory to be conformally invariant we need
$T_{\mu}^{\mu}=0$.

####  3.1.2 AdS spaces

In this section we are going to follow the standard introduction to AdS spaces
given in textbooks such as [13, 12, 11].

AdS spaces are maximally symmetric and feature a negative curvature. In
particular, in $d+1$ dimensions they possess $(d+1)(d+2)/2$ Killing vectors
(as many as the number of independent components of the metric). Notice that
the the scalar curvature has to be constant owing to the space being maximally
symmetric.

A $(d+1)$-dimensional AdS space can be embedded into $(d+2)$-dimensional
Minkowski space with metric $\tilde{\eta}=\text{diag}(-,+,\dots,+,-)$ by
imposing the hyperboloid equation ($L$ is called the AdS radius):

$-X_{0}^{2}+\sum_{i=1}^{d}X_{i}^{2}-X_{d+1}^{2}=-L^{2}$ (3.16)

Note that the hyperboloid defined by this equations possesses the isometry
group $SO(d,2)$.
The solution to (3.16) can be parametrized as [12]:

$(X^{0},X^{i},X^{d+1})=L\,(\cosh\rho\cos\tau,\,\Omega_{i}\,\sin\rho\cos\tau,\,\cosh\rho\sin\tau)\qquad
i=1,\dots,d\\\ $ (3.17)

The $\Omega_{i}$ coordinates parametrize the $S^{d-1}$ sphere:
$\sum_{i=1}^{d}\Omega_{i}^{2}=1$. Then the metric takes the following form:

$\text{d}s^{2}=\frac{L^{2}}{\cos^{2}\theta}\left(-\text{d}\tau^{2}+\text{d}\theta^{2}+\sin^{2}\theta\text{d}\Omega_{d-1}\right)$
(3.18)

where we defined a new coordinate $\theta$ by $\tan\theta=\sinh\rho$. Note
that $\theta\in[0,\pi/2]$, so in this way we only cover half of the space. In
particular, the AdS boundary is located at $\theta=\pi/2$.
Another possible parametrization reads ($t\in\mathbb{R}$,
$r\in\mathbb{R}_{+}$, $\mathbf{x}\in\mathbb{R}^{d-1}$):

$X^{0}=\frac{L^{2}}{2r}+\frac{r}{2L^{2}}(\mathbf{x}^{2}-t^{2}+L^{2})\,\quad
X^{i}=\frac{rx^{i}}{L}\,\quad
X^{d}=\frac{L^{2}}{2r}+\frac{r}{2L^{2}}(\mathbf{x}^{2}-t^{2}-L^{2})\,\quad
X^{d+1}=\frac{rt}{L}$ (3.19)

These coordinates are called the Poincaré patch. Notice that, since $r$ is
positive, they also only cover half of the space. In this case the metric
becomes ($\eta_{\mu\nu}$ is the Minkowski metric):

$\text{d}s^{2}=\frac{L^{2}}{r^{2}}\text{d}r^{2}+\frac{r^{2}}{L^{2}}\eta_{\mu\nu}\,\text{d}x^{\mu}\text{d}x^{\nu}$
(3.20)

The AdS boundary is located at $r\to\infty$: in this limit one can rewrite the
defining equation (3.16) as
$-X_{0}^{2}+\sum_{i=1}^{d}X_{i}^{2}-X_{d+1}^{2}=0$.
As we are going to do in the rest of this thesis, with a simple coordinate
transformation $u\equiv 1/r$ we obtain:

$\text{d}s^{2}=\frac{L^{2}}{u^{2}}\left(\text{d}u^{2}+\eta_{\mu\nu}\,\text{d}x^{\mu}\text{d}x^{\nu}\right)$
(3.21)

In this case the AdS boundary is obviously located at $u\to 0$.

Notice [13] that this metric is invariant under rescalings
$(u,x^{\mu})\to\lambda(u,x^{\mu})$, and under the transformation
$(u,x^{\mu})\to(u,x^{\mu})(u^{2}+\eta_{\mu\nu}x^{\mu}x^{\nu})$. In the $u\to
0$ limit this last transformation becomes inversion symmetry.
Now recall that the generating algebra of the conformal group consists of the
generator of dilatations $D$, those of translations $P_{i}$ and those of
special conformal transformations $K_{i}$. Notably, one can write
$K_{i}=IP_{i}I$, where $I$ is an inversion: $I:x^{\mu}\to x^{\mu}/x^{2}$.
Therefore we see that the isometry group $SO(d,2)$ acts on the boundary as the
conformal group in $d$ dimensions.
Furthermore, note that the continuation of the metric (3.21) to the boundary
$u=0$ is realized by multiplying it with a defining function which has a
second order zero at $u=0$ [12]. The choice of the defining function is not
univocal: different defining functions describe boundary metrics related to
each other by conformal transformations. The equivalence class of such
boundary metrics is called conformal structure.

The AdSd+1 scalar curvature amounts to

$R=-\frac{d(d+1)}{L^{2}}$ (3.22)

It is then easy to show that the AdS metric satisfies the Einstein equations
of motion with the following cosmological constant:

$\Lambda=-\frac{d(d-1)}{2L^{2}}$ (3.23)

A final note: all theories we consider in this thesis feature gauge
symmetries: they are invariant under diffeomorphisms and gauge transformations
(when adding scalar and gauge fields to the picture). In particular, we are
going to use $U(1)$ gauge transformations, parametrized by a scalar field, and
diffeomorphisms whose infinitesimal version is obtained with a Lie derivative
with respect to a vector field.

###  3.2 The GKPW prescription

The AdS/CFT conjecture can be expressed as an equality between the generating
functional of the conformal field theory on one side and that of the dual
string theory on the other side. This induces a mapping, the so-called field-
operator correspondence, between the fields in the gravity theory and the
operators of the CFT.

Let us first consider the simple example of a massless scalar field $\phi$ in
the bulk. Let $\phi_{0}$ its restriction on the AdS boundary and $\mathcal{O}$
the conformal field to which it couples. The Gubser-Klepanov-Polyakov-Witten
(GKPW) prescription links the partition function of the CFT to the Euclidean
gravitational action of the bulk theory: [49, 47, 12, 50]

$\left\langle
e^{-\int_{\text{Bdy}}\phi_{0}\mathcal{O}}\right\rangle_{\text{CFT}_{d}}=\mathcal{Z}_{\text{strings
in AdS}_{d+1}}[\phi_{0}]$ (3.24)

Note that for the right-hand side to be well-defined the boundary field
$\phi_{0}$ has to uniquely determine the bulk field $\phi$. This is a known
property of AdSd+1: any square-integrable solution to the scalar equation of
motions is automatically the trivial solution, so that the boundary condition
at infinity $\phi_{0}$ fixes $\phi$ in the bulk [49]. In other words, the
imposition of regularity in the bulk is the second boundary condition that
fully determines the solution [23]. (This is strictly the case in Euclidean
signature; in Lorentzian signature there will be two independent solutions, as
we are going to see at the end of this chapter).

The generalization for different types of fields is straightforward. For
instance a gauge field $A^{M}$ with boundary restriction $A_{0}^{\mu}$ on the
boundary coupling to a current operator will produce an action
$\mathcal{S}=\int A_{0}^{\mu}J_{\mu}$, so the equivalent prescription to
(3.24) will be [49]:

$\left\langle
e^{-\int_{\text{Bdy}}A_{0}^{\mu}J_{\mu}}\right\rangle_{\text{CFT}_{d}}=\mathcal{Z}_{\text{strings
in AdS}_{d+1}}[A_{0}]$ (3.25)

We see that changing the boundary conditions on the gravity side amounts to
modifying the Lagrangian of the dual conformal field theory. Thus the choice
of the boundary conditions has direct physical implications: tuning the
leading behaviour of the bulk fields is equivalent to adjusting the sources of
the boundary theory. Moreover, imposing the correct boundary conditions on the
gravity side guarantees the realization of symmetries in the conformal field
theory - for instance gauge invariant boundary conditions ensure that the CFT
propagators respect the Ward identities.

In the classical limit one can take a saddle point approximation to the
gravitational path integral, yielding

$e^{-W_{\text{CFT}_{d}}}=e^{-S_{\text{bulk}}}$ (3.26)

In practice this makes it possible to compute $n$-point Euclidean correlation
functions [48] by computing the functional derivative of the action with
respect to the sources of the fields. For example the $n$-point function of a
scalar field is given by:

$\left\langle\mathcal{O}(x_{1})\cdots\mathcal{O}(x_{n})\right\rangle=(-1)^{n+1}\frac{\delta\mathcal{S}}{\delta\phi_{0}(x_{1})\cdots\delta\phi_{0}(x_{n})}\bigg{|}_{\phi_{0}=0}$
(3.27)

###  3.3 Holographic thermodynamics

Having presented the main ideas behind the AdS-CFT correspondence we can now
introduce some notions on holographic thermodynamics, which are necessary in
order to understand the emergence of a thermal theory on the boundary. Such
fundamental concepts are the Hawking temperature, the entropy and the free
energy of the black hole. We are going to take a look at the Schwarzschild and
Reissner-Nordström black holes as specific examples.

We denote our coordinates as $x^{M}=(u,x^{\mu})$, where $\mu$ runs over $d$
indices.

####  3.3.1 The Hawking temperature

The temperature of the black hole is identified with the Hawking temperature,
which is computed by imposing periodicity of the Wick-rotated metric at the
horizon. Here we are following [13, 12, 51]
The metric of a static black hole with spherical symmetry takes the following
form:

$\text{d}s^{2}=-g_{tt}(r)\,\text{d}t^{2}+\frac{1}{g^{rr}(r)}\,\text{d}r^{2}+r^{2}\text{d}\Omega^{2}_{d-1}$
(3.28)

A Wick rotation to Euclidean space yields:

$\text{d}s_{E}^{2}=g_{tt}(r)\,\text{d}\tau^{2}+\frac{1}{g^{rr}(r)}\,\text{d}r^{2}+r^{2}\text{d}\Omega^{2}_{d-1}$
(3.29)

At the horizon radius $r_{h}$ both and $g_{tt}$ and $g_{tt}$ vanish, so that
for $r\approx r_{h}$ one can Taylor expand:

$\text{d}s_{E}^{2}\approx
g_{tt}^{\prime}(r)(r-r_{h})\,\text{d}\tau^{2}+\frac{1}{{g^{rr}}^{\prime}(r_{h})(r-r_{h})}\,\text{d}r^{2}+r^{2}\text{d}\Omega^{2}_{d-1}$
(3.30)

We define the “polar coordinates” [12]

$\rho^{2}\equiv\frac{4(r-r_{h})}{{g^{rr}}^{\prime}(r_{h})}\qquad\qquad\phi\equiv\frac{1}{2}\sqrt{{g^{rr}}^{\prime}(r_{h}){g_{tt}}^{\prime}(r_{h})}\,\tau$
(3.31)

Then

$\text{d}\rho^{2}=\frac{\text{d}r^{2}}{{g^{rr}}^{\prime}(r_{h})(r-r_{h})}\qquad\qquad\text{d}\phi^{2}=\frac{1}{4}{g^{rr}}^{\prime}(r_{h}){g_{tt}}^{\prime}(r_{h})\,\text{d}\tau^{2}$
(3.32)

And in the near-horizon region we can write the metric as

$\text{d}s_{E}^{2}\approx\rho^{2}\,\text{d}\phi^{2}+\text{d}\rho^{2}+\cdots$
(3.33)

We see that we can interpret $\phi$ as an angular coordinate and $\rho$ as a
radial coordinate. In particular, in order to avoid conical singularities,
$\phi$ has to be $2\pi$-periodic:

$\phi\sim\phi+2\pi$ (3.34)

Converting back to the original coordinates this gives:

$\tau\sim\tau+\frac{4\pi}{\sqrt{{g^{rr}}^{\prime}(r_{h}){g_{tt}}^{\prime}(r_{h})}}$
(3.35)

Knowing that the periodicity of imaginary time corresponds to the inverse
temperature, we finally identify the temperature as:

$T_{H}=\frac{\sqrt{{g^{rr}}^{\prime}(r_{h}){g_{tt}}^{\prime}(r_{h})}}{4\pi}$
(3.36)

Note that the Hawking temperature of a (static) black hole was historically
identified with its surface gravity [52, 53].

Let us consider an asymptotically AdS Schwarzschild black hole in $d+1$
dimensions with a planar horizon as an example. In this case the metric is
given by (3.28) with ($L$ being the AdS radius)

$\text{d}s^{2}=-\frac{r^{2}}{L^{2}}f(r)\,\text{d}t^{2}+\frac{L^{2}}{r^{2}f(r)}\,\text{d}r^{2}+\text{d}\Omega^{2}$
(3.37)

with

$f(r)=1-\frac{m}{r^{d}}$ (3.38)

and $\text{d}\Omega^{2}$ is the angular part of the metric. The horizon radius
$r_{h}$ satisfies $f(r_{h})=0$ and therefore:

$m=r_{h}^{d}\,\text{,}$ (3.39)

so that one finds:

$T=\frac{r_{h}^{2}}{4\pi
L^{2}}\left(\frac{d\,m}{r_{h}^{d+1}}\right)=\frac{d\,r_{h}}{4\pi L^{2}}$
(3.40)

####  3.3.2 Black hole thermodynamics

One of the motivations of the AdS-CFT correspondence is the realization that
the Bekenstein-Hawking entropy of black holes is proportional to the area of
their event horizon:

$S=\frac{A}{4G}$ (3.41)

As we are going to see this identity motivates the intuition that the thermal
state of the field theory is dual to a black hole in the bulk. It is
instructive to see how this formula comes about, starting from thermodynamic
identities [12, 13, 11].

Let us start with the way thermodynamic entropy is computed, starting from the
free energy:

$S=-\frac{\partial F}{\partial T}=\beta^{2}\,\frac{\partial
F}{\partial\beta}=\left(\beta\frac{\partial}{\partial\beta}-1\right)\beta F$
(3.42)

We thus need an expression for the free energy. It is given in terms of the
thermodynamic partition function $\mathcal{Z}$:

$F=-\frac{1}{\beta}\ln\mathcal{Z}(\beta)$ (3.43)

Recall the GKPW formula (3.24), linking the gravity partition function to that
of the field theory. In general, the canonical partition function for gravity
at temperature $T=1/\beta$ reads

$\mathcal{Z}(\beta)=\int\mathcal{D}g\,e^{-\mathcal{S}[g]}\simeq
e^{-\mathcal{S}^{*}}\,,$ (3.44)

where in the classical limit in Euclidean signature we substituted the path
integral with its saddle point approximation ($\mathcal{S}^{*}$ being the on-
shell gravitational action). Then one can compute the free energy as:

$F\simeq\frac{\mathcal{S}^{*}}{\beta}$ (3.45)

Formally, one integrates over all metric with appropriate falloff; however,
the partition function can be approximately computed with a saddle point
approximation: $\mathcal{S}^{*}=\mathcal{S}(g^{*})$, where $g^{*}$ is the
metric solves the Einstein equations of motion.

For the purpose of this chapter the action $\mathcal{S}[g]$ will be the
Einstein-Hilbert (EH) action for classical gravity in Euclidean signature:

$\mathcal{S}_{\text{EH}}=-\frac{1}{16\pi
G}\int_{\mathcal{M}}\text{d}^{d+1}x\,\sqrt{|g|}\,(R-2\Lambda)$ (3.46)

where $R$ is the scalar Ricci curvature, $\Lambda$ the cosmological constant
and $\mathcal{M}$ the manifold over which the theory is defined. This action,
however, does not yield a well-defined variational problem for the
fluctuations of the bulk metric. In fact, after integration by parts it still
carries terms proportional to the derivatives of the metric, which are
incompatible with Dirichlet boundary conditions. These terms can by canceled
by supplementing the Einstein-Hilbert action with the so-called Gibbons-
Hawking-York boundary term [54]:

$\mathcal{S}=\mathcal{S}_{\text{EH}}+\mathcal{S}_{\text{GBY}}=-\frac{1}{16\pi
G}\int_{\mathcal{M}}\text{d}^{d+1}x\,\sqrt{|g|}\,(R-2\Lambda)-\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}\text{d}^{d}x\,\sqrt{-\gamma}\,K$ (3.47)

where $\gamma$ is the induced metric at the boundary and
$K=-\gamma^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}n_{\alpha}$ is the trace of the
extrinsic curvature ($n^{\mu}$ being the normal vector to
$\partial\mathcal{M}$).
Even after adding $\mathcal{S}_{\text{GHY}}$ the action still diverges if the
boundary is located at infinity, as is the case for (asymptotically) AdS
spaces. As we are going to see, one adds counterterms absorbing the
divergences of the action, or subtract the action of a reference background.

##### Example: Schwarzschild black hole

Let us see how this works concretely for a Schwarzschild AdSd+1 geometry [13].
At the boundary $r\to\infty$ the geometry becomes AdS and so the scalar
curvature is given by (3.22):

$\lim_{r\to\infty}R=-\frac{d(d+1)}{L^{2}}$ (3.48)

While the cosmological constant appearing in the EH action (3.46) is

$\Lambda=-\frac{d(d-1)}{2L^{2}}$ (3.49)

Wick rotating the metric ($\tau=it$) we obtain

$\text{d}s_{E}^{2}=\frac{r^{2}}{L^{2}}\left(f(r)\text{d}\tau^{2}+\text{d}\mathbf{x}^{2}\right)+\frac{L^{2}}{r^{2}f(r)}\text{d}r^{2}$
(3.50)

With $\lim_{r\to\infty}f(r)=1$. The determinant of the metric reads:

$\sqrt{g_{E}}=\left(\frac{r}{L}\right)^{d-1}$ (3.51)

We now have to determine the normal vector at the boundary, defined by sending
$r\to\infty$. It points in radial direction; since it has to be normed, it is
given by

$n^{\mu}=\frac{r\sqrt{f(r)}}{L}\,\delta^{\mu}_{r}$ (3.52)

The induced metric $\gamma_{\alpha\beta}$ is the simply the original metric,
minus the radial component, so that its determinant amounts to:

$\sqrt{|\gamma_{E}|}=\left(\frac{r}{L}\right)^{d}\sqrt{f(r)}\,\xrightarrow[r\to\infty]{\,}\left(\frac{r}{L}\right)^{d}$
(3.53)

One can now compute the trace of the extrinsic curvature
$K=\gamma_{E}^{\alpha\beta}\nabla_{\alpha}n_{\beta}=-\gamma_{E}^{\alpha\beta}\,\Gamma^{\mu}_{\alpha\beta}\,n_{\mu}$.
Since $n^{\mu}$ points in radial direction and $\gamma_{\alpha\beta}$ has no
radial component we only need $\Gamma^{r}_{\mu\nu}$ with $\mu,\nu\neq r$. The
only non-vanishing Christoffel symbols of interest are then (denoting with
$x^{i}$ all coordinates different from $r$ and $\tau$):

$\displaystyle\begin{split}\Gamma^{r}_{x^{i}x^{i}}&=-\frac{r^{3}f(r)}{L^{4}}\qquad\qquad\Gamma^{r}_{\tau\tau}=-\frac{r^{3}f(r)\big{(}2f(r)+rf^{\prime}(r)\big{)}}{2L^{4}}\\\
\end{split}$ (3.54)

Therefore one obtains:

$K=\frac{\sqrt{f(r)}}{L}\left(d+\frac{rf^{\prime}(r)}{2f(r)}\right)\,\xrightarrow[r\to\infty]{\,}\frac{d}{L}\
$ (3.55)

We can now compute the counterterm, at leading order in $1/r$. For this we
only need the asymptotic value of the Ricci scalar curvature (3.48).
Integrating the EH action over $\text{d}r$ we find (to leading order in $r$):

$\mathcal{S}_{\text{bdy}}=-\frac{d-1}{8\pi
G}\int_{\partial\mathcal{M}}\text{d}\tau\text{d}\mathbf{x}\,\frac{r^{d}}{L^{d+1}}=(\text{finite
terms})-\lim_{\epsilon\to 0}\,\frac{d-1}{8\pi
G}\int^{\beta}_{0}\text{d}\tau\int_{\mathbb{R}^{d-1}}\text{d}\mathbf{x}\,\frac{\epsilon^{-d}}{L^{d+1}}\quad\text{,}$
(3.56)

where we have recalled that the limits of integration for Euclidean time are
$0$ and $T=1/\beta$ and we have parametrized the UV boundary by setting
$r=\epsilon^{-1}$, with $\epsilon\to 0$. We can absorb this divergence by
adding a corresponding counterterm to the boundary action111Note that for
$d>2$ this is not the only needed correction. Further corrections come in
powers of the curvature of the boundary metric [55, 56]. For the purpose of
illustrating the computation of the entropy, though, in this chapter we are
restricting ourselves to the leading term.

$\mathcal{S}_{\text{ct}}\equiv\frac{d-1}{8\pi GL}\int_{r\to
0}\text{d}\tau\text{d}\mathbf{x}\,\sqrt{|\gamma_{E}|}$ (3.57)

With this counterterm we can finally perform the integration; recall that the
lower integration limit on $r$ is the horizon radius $r_{h}$. We get [13]:

$\mathcal{S}^{*}=-\frac{1}{4G}\left(\frac{r_{h}}{L}\right)^{d-1}\text{Vol}_{d-1}=-\frac{1}{4G}\frac{(4\pi)^{d-1}L^{d-1}T^{d-1}}{d^{d}}\text{Vol}_{d-1}\quad\text{,}$
(3.58)

where $\text{Vol}_{d-1}$ is the volume enclosed by the $d-1$ spatial
coordinates $\mathbf{x}$.
So, with (3.45) and (3.36), the saddle point approximation for the free energy
is:

$F\simeq T\mathcal{S}^{*}=-\frac{T^{d}}{4Gd}\left(\frac{4\pi
L}{d}\right)^{d-1}\text{Vol}_{d-1}$ (3.59)

We can then compute the entropy:

$S=-\frac{\partial F}{\partial T}=\frac{1}{4G}\text{Vol}_{d-1}\left(\frac{4\pi
LT}{d}\right)^{d-1}=\frac{1}{4G}\left(\frac{r_{h}}{L}\right)^{d-1}\text{Vol}_{d-1}$
(3.60)

With (3.50) we recognize that we can rewrite this last expression in terms of
the horizon area:

$S=\frac{A_{\text{horizon}}}{4G}$ (3.61)

This is the famous result predicted by Bekenstein and Hawking [57].

##### Example: Reissner-Nordström black hole

Note: from now on we are going to set $16\pi G=L=1$ for simplicity’s sake.

By coupling a gauge field to the metric one obtains the so-called Reissner-
Nordström (RN) black hole. In an asymptotically AdSd+1 geometry its bulk
action is given by the Einstein-Maxwell action, which (in Lorentzian
signature) reads:

$\mathcal{S}_{\text{EM}}=\int\text{d}^{d+1}x\sqrt{-g}\,\left[\big{(}R+d(d-1)\big{)}-\frac{1}{4e^{2}}F_{MN}F^{MN}\right]$
(3.62)

where $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$ is the field strength
tensor associated to the gauge field.
Assuming the gauge field to be timelike (that is, only $A_{t}\neq 0$), the
equations of motion obtained by extremizing this action are solved by [13]:

$\text{d}s^{2}=\frac{r^{2}}{L^{2}}\left(-f(r)\text{d}t^{2}+\text{d}\mathbf{x}^{2}\right)+\frac{L^{2}}{r^{2}f(r)}\text{d}r^{2}\qquad\text{with}\qquad
f(r)=1+\frac{Q^{2}}{r^{2d-2}}-\frac{M}{r^{d}}$ (3.63)
$A_{t}=\mu\left(1-\frac{r_{h}^{d-2}}{r^{d-2}}\right)$ (3.64)

The chemical potential $\mu$ and $Q$ are related by:

$Q=\sqrt{\frac{(d-2)}{2(d-1)e^{2}}}r_{h}^{d-2}\mu$ (3.65)

Note that in the limit $e\to\infty$ we recover the Schwarzschild black hole
solution. This is the so-called “probe limit”. $r_{h}$ denotes the larger root
of $f(r)$ and is called outer horizon radius. It is related to the charge $Q$
and the the mass $M$ of the black hole by:

$M=r_{h}^{d}+Q^{2}\,r_{h}^{2-d}$ (3.66)

The temperature reads:

$T=\frac{d\,r_{h}}{4\pi}-\frac{(d-2)^{2}\,\mu^{2}}{8\pi(d-1)\,e^{2}\,r_{h}}$
(3.67)

With a similar procedure to the Schwarzschild case one can compute the free
energy:

$F=-T\ln\mathcal{Z}\approx T\mathcal{S}^{*}=-\frac{V_{d-1}}{d}\left(4\pi
r_{h}^{d-1}T+\frac{(d-2)\mu^{2}}{e^{2}}r_{h}^{d-2}\right)$ (3.68)

We can then compute the entropy density and verify the Bekenstein-Hawking
conjecture:

$s\equiv-\frac{\partial F}{\partial T}\bigg{|}_{\mu}=4\pi r_{h}^{d-1}$ (3.69)

Note that this solution displays finite entropy at zero temperature.
Similarly, the charge density associated with the conserved current of the
boundary field theory is given by:

$n\equiv-\frac{\partial
F}{\partial\mu}\bigg{|}_{T}=\frac{d-2}{e^{2}}r_{h}^{d-2}\mu$ (3.70)

Note that the charge density obtained in this way is proportional to
$\partial_{r}A_{t}$ on the boundary, as prescribed by the holographic
dictionary that we are going to introduce in the upcoming sections.

Having obtained analytical expressions for both $s$ and $\rho$, together with
their sources $T$ and $\mu$, we can now compute the related susceptibilities.
Recall the thermodynamical definition of susceptibilities: given an ensemble
$\phi_{a}$, $a=1,\cdots,n$ and its sources $s_{a}$ the susceptibility
$\chi_{ab}$ is the variation of the vacuum expectation value $\phi_{a}$ with
respect to the source of $\phi_{b}$, keeping all other sources constant:

$\chi_{ab}=\frac{\partial\phi_{a}}{\partial s_{b}}\bigg{|}_{s_{c\neq b}}$
(3.71)

Here we find:

$\chi_{nn}=\frac{r_{h}^{d-2}}{e^{2}}\frac{\left(2d(d-2)(d-1)\,e^{2}r_{h}^{2}+(d-2)^{3}(2d-3)\,\mu^{2}\right)}{2d(d-1)\,e^{2}r_{h}^{2}+(d-2)^{2}\,\mu^{2}}$
(3.72) $\chi_{ss}=\frac{32(d-1)^{2}\,\pi
e^{2}r_{h}^{d}}{2d(d-1)\,e^{2}r_{h}^{2}+(d-2)^{2}\,\mu^{2}}$ (3.73)
$\chi_{ns}=\chi_{sn}=\frac{8(d-2)^{2}(d-1)\,\pi\mu
r_{h}^{d-1}}{2d(d-1)\,e^{2}r_{h}^{2}+(d-2)^{2}\,\mu^{2}}$ (3.74)

One can see that the susceptibility matrix is symmetric:
$\chi_{sn}=\chi_{ns}$, and that the diagonal susceptibilities $\chi_{nn}$ and
$\chi_{ss}$ are positive, as required by thermodynamic equilibrium.

The energy density and the pressure can be extracted from the boundary
behaviour of the metric, which is dual to the stress-energy tensor of the
field theory. In Fefferman-Graham coordinates, where
$g_{rm}=r^{-2}\delta_{rm}$, the subleading behaviour of the metric at the
boundary gives the expectation value of the stress-energy tensor in the dual
field theory [55].
In particular, as we are going to see in the upcoming chapter, the boundary
behaviour of the metric in $d+1$ dimensions is222If $d$ is odd logarithmic
terms appear in the expansion [58]:

$\lim_{r\to\infty}=r^{2}\left[g_{(0)\mu\nu}\left(1+\mathcal{O}(r^{-2})\right)+r^{-d}g_{(d)\mu\nu}\left(1+\mathcal{O}(r^{-2})\right)\right]$
(3.75)

For example for a Reissner-Nordström metric we find[13]:

$\displaystyle\epsilon$ $\displaystyle=d\langle
T^{00}\rangle=d\,g_{(d)}^{tt}=dM$ (3.76) $\displaystyle p$
$\displaystyle=d\langle T^{xx}\rangle=M$ (3.77)

Notice that $\langle T^{\mu}_{\mu}\rangle=0$, as required for a conformal
field theory.
Finally, with equations (3.66, 3.67, 3.68, 3.70) one can then check the Smarr
law:

$\epsilon+p=sT+n\mu$ (3.78)

###  3.4 Definition of propagators and quasinormal modes

From now on we are going to set $L=16\pi G=1$. Moreover, for the sake of
consistency with the upcoming chapters, we are going to use the coordinate
$u\equiv 1/r$, so that the AdS boundary is located at $u\to 0$. Finally, we
assume to be working in AdSd+1 with $d$ odd, as for even $d$ logarithmic terms
arise in the boundary expansion of the fields [58].

####  3.4.1 Asymptotic behaviour of the fields

In this section we examine the asymptotic behaviour of scalar and gauge field,
assuming that the metric is asymptotically AdS (3.21):

$\lim_{u\to
0}\text{d}s^{2}=\frac{1}{u^{2}}\left(\text{d}u^{2}+\eta_{\mu\nu}\,\text{d}x^{\mu}\text{d}x^{\nu}\right)$
(3.79)

So that the square root of the metric asymptotes to:

$\lim_{u\to 0}\sqrt{-g}=u^{-(d+1)}$ (3.80)

Let us start with the simple example of a scalar field $\phi$ whose bulk
action reads [47]:

$\mathcal{S}_{\phi}=\int\text{d}^{d+1}x\,\sqrt{-g}\left(m^{2}\phi^{2}+\partial_{M}\phi\partial^{M}\phi\right)$
(3.81)

The resulting equation of motion is [59]:

$\frac{1}{\sqrt{-g}}\partial_{M}\left(\sqrt{-g}\partial^{M}\right)\phi=m^{2}\phi$
(3.82)

We assume radial symmetry (that is, the fields only depend on $u$). On the
boundary equation (3.82) then becomes:

$\left(u^{d+1}\partial_{u}u^{1-d}\partial_{u}-m^{2}\right)\phi=u^{2}\partial_{u}^{2}\phi+(1-d)u\,\partial_{u}\phi-m^{2}\phi=0$
(3.83)

Assuming an asymptotic behaviour $\phi\sim u^{\Delta}$ we obtain an equation
for the falloff exponent $\Delta$:

$\Delta(\Delta-d)=m^{2}$ (3.84)

which is solved by

$\Delta=\Delta_{\pm}\equiv\frac{1}{2}\left(d\pm\sqrt{d^{2}+4m^{2}}\right)\,\text{,}$
(3.85)

so that we can write:

$\phi=\phi_{+}u^{\Delta_{+}}+\phi_{-}u^{\Delta_{-}}$ (3.86)

for two coefficients $\phi_{+}$ and $\phi_{-}$, which in general are functions
of the non-radial coordinates $x^{\mu}$.
Notice that under dilatations $u\mapsto\lambda u$ the scalar field is
invariant. Hence the coefficients transform as:

$\phi_{\pm}\mapsto\lambda^{-\Delta_{\pm}}\phi_{\pm}$ (3.87)

In particular, as we are going to see in the coming sections, $\phi_{+}$ gives
the vacuum expectation value of the dual field theory operator, which has
scaling dimension $\Delta_{+}$ [47, 13], determined by $m^{2}$.

One can see that a negative mass does not lead to an instability (that is,
$\phi$ gets a complex scaling dimension) as long as:

$m^{2}>m_{\text{BF}}^{2}\equiv-\frac{d^{2}}{4}$ (3.88)

This is the so-called Breitenlohner-Freedman (BF) bound [60, 61].

Let us now add a gauge field to the picture. Considering a complex scalar
together with a gauge field as in [15] the simplest form the bulk part of the
action is given by [59, 36]:

$\mathcal{S}_{A_{\mu},\psi}=\int\text{d}^{d+1}x\,\sqrt{-g}\left(\frac{1}{4e^{2}}F_{MN}F^{MN}+m^{2}|\psi|^{2}+|\partial\psi-
iA\psi|^{2}|\right)$ (3.89)

From this action one can derive the equations of motion for the scalar and the
gauge field [59]:

$\frac{1}{\sqrt{-g}}(\partial_{M}-iA_{M})\sqrt{-g}(\partial^{M}\psi-
iA^{M}\psi)=m^{2}\psi^{*}$ (3.90)
$\partial_{M}\left(\sqrt{-g}F^{MN}\right)=e^{2}g^{NK}J_{K}$ (3.91)

With the field strength tensor $F_{MN}=\partial_{M}F_{N}-\partial_{N}F_{M}$
and the current

$J_{M}=i\left(\psi^{*}(\partial_{M}-iA_{M})\psi-\psi(\partial_{M}+iA_{M})\psi^{*}\right)=2A_{M}|\psi|^{2}+i\left(\psi^{*}\partial_{M}\psi-\psi\partial_{M}\psi^{*}\right)$
(3.92)

We work in the so-called radial gauge, where $A^{u}=0$. Then the above
equations of motion (3.90,3.91) become:

$u^{2}\partial_{u}^{2}\psi-(d+1)u\,\partial_{u}\psi-
A_{\mu}A^{\mu}\psi-m^{2}\psi^{*}=0$ (3.93)
$\partial_{u}(u^{3-d}\,\partial_{u}A_{\mu})-2u^{2}e^{2}|\psi|^{2}A_{\mu}=0$
(3.94)

Assuming the same asymptotic behaviour for $\psi$ and $\psi^{*}$:
$\psi\sim\psi^{*}\sim u^{\Delta}$ with $\Delta$ as in (3.85) we see that
(3.93) is satisfied if the leading behaviour for the gauge field at infinity
is just a constant. If we then consider the second equation (3.94) without the
mixing term we see that it is indeed solved by

$A_{\mu}\sim\alpha+\beta u^{d-2}$ (3.95)

We can now plug this expansion back into (3.94) and see that the mixing term
is subleading in $u$ if $m^{2}<d+1$:

$u^{2}|\psi|^{2}A_{\mu}\sim u^{2+d-\sqrt{d^{2}+4m^{2}}}$ (3.96)

So for $m^{2}<0$, as will be the our case in the rest of this thesis, we can
retain the asymptotic behaviours given by (3.85) and (5.22).
Also, note that a positive squared mass gives the scalar field a diverging
boundary behaviour. Its dual will then be an irrelevant operator, which, if
sourced (i.e. if it gets a non-vanishing leading coefficient), alters the UV
behaviour of the theory, driving the theory to a different UV fixed point and
endangering the asymptotic “AdS-ness” of the metric [47].

As we are going to soon see in better detail, the subleading component of the
fields at the boundary give the vacuum expectation values of the operators in
the dual conformal field theory, while the leading components are their
sources. For instance the timelike component $A_{t}$ of the gauge field
delivers the conserved charge and its associated chemical potential in the
field theory:

$A_{t}\sim\mu-\rho\,u^{d-2}$ (3.97)

Notice that the Reissner-Nordström solution (3.64) for $A_{t}$ shows this
behaviour, with $\rho=r_{h}^{d-2}\mu$.
Similarly, having a finite background superfluid velocity $\sim\xi$ means
[59]:

$A_{x}=\xi-\frac{e^{2}}{2}\big{<}J_{x}\big{>}\,u^{d-2}+\mathcal{O}(r^{3})$
(3.98)

####  3.4.2 Quasinormal modes and retarded functions

From a practical point of view, in order to make use of gauge/gravity duality
one has to compute the quasinormal modes (QNMs) and the propagators of the
operators on the gravity side.

In finite temperature field theory there are two ways of computing response
functions. The first option is to perform a Wick rotation to Euclidean time,
compute the response functions for complex frequencies, and then analytically
continue them to real ones. This, however, is a rarely viable strategy: one
needs to know the correlators at extremely high precision (basically,
analytically) in order for the analytical continuation to deliver acceptable
results [13].
Alternatively, one can directly work in Lorentzian signature. It turns out
that, given a bulk theory with a black hole, one can holographically reproduce
the Schwinger-Keldysh structure of the retarded and advanced propagators of
the dual thermal field theory. This is done by selecting infalling and
outgoing solutions to the equations of motion at the black hole horizon. Here
we are not going to prove this construction; instead, we will limit ourselves
to outlining the practical procedure than one has to follow in order to
compute the correlators.

The first step consists in deriving the equations of motion for the background
by minimizing the bulk action. Once one has solved for the background (usually
numerically, although an analytical approach is possible around $T_{c}$ [39,
36] or, more generally, in the long wavelength limit [62, 63, 64, 65]) one can
compute the QNMs by allowing for small fluctuations of the fields around their
background values. The standard procedure is the following (note that for
simplicity we are setting the horizon radius $r_{h}=1$) [66]:

  1. 1.

One uses a plane wave ansatz for the perturbations to the background fields
$\Lambda_{i}$. :

$\delta\Lambda_{i}=\lambda_{i}(u)\,e^{-i\omega t+ikx}$ (3.99)

Note that one is allowed to use this ansatz if one considers flat spatial
sections for the metric; this will be the case in this thesis.