.

Developments in

quasihydrodynamics

Developments in quasihydrodynamics

Theory and applications

Luca<EMAIL_ADDRESS>
A thesis presented for the degree of
Doctor of Philosophy in Physics and Nanoscience
Theoretical Physics curriculum

###### Contents

  1. Referees and defense details
  2. Summary
  3. Publications
  4. Aknowledgment
  5. 1 Introduction and context
    1. 1.1 Hydrodynamic regime and applications
    2. 1.2 Canonical approach to hydrodynamics
    3. 1.3 Other approaches
    4. 1.4 Hydrodynamics in condensed matter
      1. 1.4.1 Overview
      2. 1.4.2 Gurzhi effect
      3. 1.4.3 Nonlocal transport experiment
      4. 1.4.4 Wiedemann-Franz law violation
    5. 1.5 The content of this thesis
    6. 1.6 Notations
  6. 2 Hydrodynamics and Linear Response Theory
    1. 2.1 Hydrodynamics
      1. 2.1.1 Hydrodynamic variables
      2. 2.1.2 Constitutive relations
      3. 2.1.3 Order zero: ideal fluid
      4. 2.1.4 Frame choice
      5. 2.1.5 Order one: Navier-Stokes
      6. 2.1.6 Entropy current
      7. 2.1.7 Hydrostatic generating functional
      8. 2.1.8 Hydrostatic conditions
    2. 2.2 Linear Response Theory
      1. 2.2.1 Martin-Kadanoff method
      2. 2.2.2 Variational method
      3. 2.2.3 Discrete symmetries: Onsager relations
      4. 2.2.4 Thermoelectric transport
      5. 2.2.5 Modes and linear stability
  7. 3 Charge Density Waves hydrodynamics
    1. 3.1 Introduction
    2. 3.2 Breaking the translation symmetry and Ward identities
    3. 3.3 Spontaneous case
      1. 3.3.1 Constitutive relations
      2. 3.3.2 AC conductivities
    4. 3.4 Pseudo-spontaneous case
      1. 3.4.1 Constitutive relations
      2. 3.4.2 AC conductivities
    5. 3.5 Holographic model
      1. 3.5.1 Setup and background
      2. 3.5.2 Analytic DC conductivities
      3. 3.5.3 AC conductivities and matching hydrodynamics
      4. 3.5.4 On the spurious pole
    6. 3.6 Summary, discussion and outlook
  8. 4 Electrically-driven fluids
    1. 4.1 Introduction
    2. 4.2 Boost-agnostic fluids
      1. 4.2.1 Aristotelian geometry
      2. 4.2.2 Geometry, thermodynamics and hydrostatic flows
      3. 4.2.3 Constitutive relations and thermodynamics
    3. 4.3 Adding relaxations
      1. 4.3.1 Order zero
      2. 4.3.2 Order one
      3. 4.3.3 Entropy current
      4. 4.3.4 Linearised stability and DC conductivities
    4. 4.4 Dissipative corrections and optical conductivity
    5. 4.5 Summary, discussion and outlook
  9. 5 Onsager relations in relaxed hydrodynamics
    1. 5.1 Introduction
    2. 5.2 Martin-Kadanoff for quasihydrodynamics
    3. 5.3 Variational approach and Onsager relations
    4. 5.4 Summary, discussion and outlook
  10. 6 Anomalous hydrodynamics and Weyl semimetals
    1. 6.1 Introduction
    2. 6.2 Anomalous hydrodynamics and transport
      1. 6.2.1 Standard approach to anomalous hydrodynamics
      2. 6.2.2 Positivity of entropy production
      3. 6.2.3 Frame-dependent conductivities
      4. 6.2.4 Killing the ambiguity
    3. 6.3 The problem with DC conductivities
      1. 6.3.1 Weyl semimetals, hydrodynamics and transport
      2. 6.3.2 DC limits and relaxation terms
    4. 6.4 Kinetic theory
      1. 6.4.1 A primer on kinetic theory and hydrodynamics
      2. 6.4.2 Adding momentum relaxation
      3. 6.4.3 Adding charge relaxations
      4. 6.4.4 Generalized relaxations from RTA
      5. 6.4.5 Two-currents model
      6. 6.4.6 Modified RTA: charge conservation
    5. 6.5 Summary, discussion and outlook
  11. 7 Conclusions
    1. 7.1 Summary of main results
    2. 7.2 Outlook and future avenues
  12. A Integrals for kinetic theory
  13. B Aspects of Symmetry
  14. C Weyl semimetals

## Referees and defense details

Referees for this thesis:

  * •

Prof. Koenraad Schalm – Leiden University

  * •

Prof. Piotr Surówka – Wrocław University of Science and Technology

The defense of the doctoral degree in Physics was held at the University of
Genoa in the Department of Physics on the 1${}^{\text{st}}$ of March 2024.

Examination Committee for the defense:

  * •

Prof. Paolo Solinas – University of Genoa

  * •

Prof. Piotr Surówka – Wrocław University of Science and Technology

  * •

Prof. Daniele Passerone – ETH Zürich

  * •

Prof. Fernando Dominguez – Technische Universität Braunschweig

Supervisor: Prof. Andrea Amoretti

Candidate: Luca Martinoia
University ID: 3630563
Italian Ph.D. cycle: XXXVI

## Summary

At its core, hydrodynamics is a many-body low-energy effective theory for the
long-wavelength, long-timescale dynamics of conserved charges in systems close
to thermodynamic equilibrium. It has a wide range of applications, that span
from nuclear physics, astrophysics, cosmology, and more recently strongly-
interacting electronic phases of matter. In solid state systems, however,
symmetries are often only approximate, and softly broken by the presence of
the lattice, impurities and defects, or because the symmetry is accidental.
Therefore, the hydrodynamic regime must be expanded to include weak non-
conservation effects, which lead to a theory known as quasihydrodynamics.

In this thesis we make progress in understanding the theory of (quasi)
hydrodynamics, with a specific focus on applications to condensed matter
systems and their holographic description. First, we consider an electron
fluid in a strong magnetic field for which translations are broken by the
presence of Charge Density Waves. Therefore, the low-energy theory contains
Goldstone modes associated with the broken symmetries, which modify the
spectrum and transport properties. We focus on a new regime at non-zero
lattice pressure and compare with holographic models, finding perfect
agreement between the two descriptions.

Next we consider a simple system that mimics the weakly-coupled Drude model
from a hydrodynamic perspective. Specifically, a charged fluid in an external
electric field in the presence of impurities that relax momentum and energy.
We look for steady states, and we find that stationarity constraints should be
modified to include relaxations, which consequently give novel predictions for
the thermoelectric transport.

Finally, we study the effect of the axial anomaly on the transport properties
of Weyl semimetals in the hydrodynamic regime. We suggest a better approach to
deal with the derivative counting of the magnetic field, correcting mistakes
in the literature. Subsequently, we discuss the DC values of the
conductivities and look for models that obey fundamental and phenomenological
considerations. We find that generalized relaxations, which we study in depth
using variational methods and kinetic-theory approaches, are a necessary
ingredient to have finite DC conductivity, conserve electric charge, and have
the correlators obey Onsager relations. Moreover, our model provides
qualitatively new predictions for the thermoelectric transport, which could be
used to probe the hydrodynamic regime in such materials.

## Publications

This thesis is based on the following original publications:

  * •

Chapter 3 is based on [1], [1]

  * •

Chapter 4 is based on [2], [2]

  * •

Chapter 5 is based on [3], [3]

  * •

Chapter 6 is based on [4], [4]
and [5], [5]

During the Ph.D. I also worked on the following paper, which will not be
discussed here:

  * •

[6] [6].

## Aknowledgment

First, I express my sincere gratitude to my advisor, Andrea Amoretti, for his
thoughtful guidance throughout the Ph.D. I am especially thankful for his
support and trust in my research, but also for encouraging me to step out of
my comfort zone.

I would also like to acknowledge my close collaborators: Danny Brattan,
Ioannis Matthaiakakis, and more recently, Jonas Rongen. I had many interesting
and insightful discussions about physics with them over the years, which have
taught me a lot and helped me better understand many topics.

On a more personal note, I am extremely grateful to my wonderful wife,
Valeria, who has patiently supported me throughout the doctorate and always
manages to help in challenging times. I am a better person because of her.
These years together have been amazing, and I am truly excited to see what
awaits us in the future.

Finally, I would really thank my parents, Sergio and Laura, my family, Simone,
Anna, Giovanni, Anouck, Nhoack, Annina, and all my friends, for still keeping
me around after all these years.

##  Chapter 1 Introduction and context

> “I don’t see how he can ever finish if he doesn’t begin.”
>
> Lewis Carroll, _Alice in Wonderland_

###  1.1 Hydrodynamic regime and applications

Hydrodynamics and hydrostatics are very old subjects, which date back more
than two thousands years ago to Archimedes’ work, however it is only in the
18th century that physicists started a systematic study of hydrodynamics to
understand the flow of water and liquids in general.

To this day hydrodynamics is fundamentally well understood [7], but it remains
one of the topics at the forefront of research. This is due to the numerical
and theoretical challenges on one side, but also to its universality, which
makes it a great tool to describe the phenomenology of many different effects
in all branches of physics: astrophysics (like neutron stars [8], mergers [9],
accretion disks [10]), cosmology [11], condensed matter (graphene [12, 13],
high-$T_{c}$ superconductors [14, 15], Dirac and Weyl semimetals [16, 17], …),
active and soft matter [18], and high-energy physics (quark-gluon-plasma
produced at RHIC and LHC [19, 20, 21]), …

From a theoretical perspective, hydrodynamics has gained a lot of attentions
in recent years, thanks to a renewed interest following the discovery of the
fluid/gravity duality in holography [22, 23, 24]. This has opened the paths to
many new research avenues in theoretical hydrodynamics, of which some examples
are: a classification of dissipative superfluid terms [25], and of second-
[26, 27, 28] and third-order hydrodynamics [29, 30], parity-odd fluids [31,
32, 33], quantum anomalies [34, 35], spin hydrodynamics [36, 37, 38],
viscoelastic and viscoplastic fluids with broken translation symmetry (both
spontaneous and explicit) [39, 40, 41, 1, 15], developments in fluctuating
hydrodynamics as an EFT from an action principle [42, 43, 44, 45, 46], a full
classification of the possible classes of hydrodynamic transport [47, 48],
quasihydrodynamics [49, 50], Hydro+ to extend hydrodynamics to include slow
modes near critical points [51], fracton hydrodynamics [52, 53, 54],
hydrodynamics with higher-form symmetries [55, 56, 57].

Before dealing with some of these research paths, however, we should discuss
what hydrodynamics is first. It can be understood as a universal low-energy
theory for many-body thermal systems that describes the collective macroscopic
dynamics of conserved charges (or other low-energy modes) in the long-
wavelength and small-frequency regime $\omega,\boldsymbol{\mathbf{k}}\ll T$,
where $T$ is the temperature of the system. To understand the emergence of
hydrodynamics we will follow the heuristic argument given in [44].

Consider some quantum many-body system. At zero temperature, the low-energy
theory is characterized by massless quasiparticle excitations above the ground
state. Generically, these modes are long-lived and give rise to e.g. Fermi-
liquid behaviour. If the system is thermal, however, then there is a large
bath of gapless quasiparticles and any new excitation becomes quickly
incoherent in the bath, on a time (and length) scale fixed by the microscopic
dynamics $\tau$, $l$. This effectively means that the quasiparticles become
gapped due to the presence of the thermal bath, and decay on very short scale,
leading to a system that thermalizes quickly and can thus be assumed in local
thermodynamic equilibrium on macroscopic scales. This should be understood
from a coarse-grained perspective by dividing the system in small volume
elements, such that each element is small enough to be effectively point-like
in the effective-theory description, but microscopically large enough to have
a well-defined thermodynamic limit.

Figure 1.1: Figure taken from [44]. (a) Microscopic degrees of freedom
thermalize quickly to equilibrium. On the other hand, (b) conserved charges
cannot be destroyed locally and diffuse slowly. (c) The typical microscopic
length scale $l$ must be small compared to the length over which conserved
charges vary.

Thus, in this regime, the low-energy dynamics is governed by the conserved
charges, because they cannot be destroyed locally, but can only diffuse from
regions with an excess of charge to regions with a deficit of charge, and this
process is parametrically slower than the local microscopic thermalization,
see Figure 1.1. Therefore, the hydrodynamic limit is reached, and we end up
with a low-energy theory for long-wavelength/long-timescale dynamics that is
governed by the local conservation of charges (and eventually other low-energy
collective degrees of freedom which might survive, like in a superfluid
phase).

Clearly, the separation of scales between microscopic thermalization and
macroscopic diffusion needed for the emergence of hydrodynamics happens more
sharply in phases which are strongly coupled and clean of impurities. In these
systems the scattering time between quasiparticles becomes extremely short,
see Figure 1.2. In particular, we will be interested in describing the low-
energy dynamics of strongly-coupled condensed matter systems. While the
intention of this thesis is not to describe the detailed phenomenology of
specific condensed matter phases, the low-energy description of solid-state
physics is indeed one of the main motivation to expand on the theory of
hydrodynamics, specifically in its relaxed version, i.e. quasihydrodynamics.

###  1.2 Canonical approach to hydrodynamics

In the canonical approach, (relativistic) hydrodynamics is understood as a
perturbative expansion in a small parameter, the Knudsen number
$\text{Kn}=l/L\ll 1$, where $L$ is the system size or the length over which
thermodynamic quantities vary, while $l$ is the microscopic length scale.
Specifically, we can construct the derivative operator
$l\partial\sim\text{Kn}$ which act as a perturbative parameter expansion. From
this point of view, hydrodynamics is a theory based on the following
constraints:

  * •

a derivative expansion,

  * •

the symmetries (continuous, discrete and broken),

  * •

the local form of the second law of thermodynamics.

This leads to the formulation of hydrodynamics as a formal power series in
which the order zero (the ideal fluid) is non-dissipative, while the higher-
order corrections produce entropy, thus being dissipative. Like in any
effective theory, ideally adding higher-order terms should lead to a more
accurate description, at least within some convergence radius111There is
strong evidence that the hydrodynamic series does not converge [58, 59],
nonetheless it can be Borel resummed, leading to all-order hydrodynamics..
From the above discussion, hydrodynamics stops being a sensible effective
theory at length scales shorter than the microscopic mean free path $l$.

Each term in the derivative expansion at order one or higher comes with its
own _transport coefficient_ , which parametrizes the non-universal part of
fluid dynamics. These are functions of the thermodynamic quantities whose
values cannot be obtained directly from hydrodynamics alone, but depends on
the underlying microscopic theory, and can be computed from Kubo formulae
using various methods depending on the situation (lattice calculations,
kinetic theory, holography, QFT, …).

Although, as we claimed, hydrodynamics has been very successful in describing
the physics of many different systems, as a classical theory it suffers from
fundamental problems. The first one is related to the fact that the standard
equations of hydrodynamics are not causal, and contain solutions with super-
luminal velocity, which ruin the stability of the theory (super-luminal modes
in one frame correspond to modes that travel back in time in a boosted frame).
Specifically, the group velocity diverges
$v_{g}=\frac{\mathop{}\\!\mathrm{d}\omega}{\mathop{}\\!\mathrm{d}k}=2\gamma_{\eta}k$
for the shear mode dispersion relation, and becomes super-luminal, which
renders the theory acausal and unstable [60, 61, 62]. This prevents
numerically solving the equations of relativistic hydrodynamics from a set of
initial condition on a time-slice, ruining the predictability of the theory.

The classical solution to this problem comes from Maxwell and Cattaneo for
diffusion first [63, 64], and Müller, Israel, Stewart (MIS) for hydrodynamics
later [65, 66, 67]. Formally, the idea amounts to adding soft UV cut-offs
$\tau_{i}$ that tame the high-momentum behaviour of the hydrodynamic
fluctuations. Specifically, defining as $\pi^{\mu\nu}$ the dissipative shear
tensor, $\Pi$ the dissipative shear bulk, and $\nu^{\mu}$ the charge flux, the
MIS theory paradigm prescribes to modify the constitutive relations in
dynamical equations as

$\displaystyle\left(\tau_{\pi}u^{\mu}\partial_{\mu}+1\right)\pi^{\mu\nu}$
$\displaystyle=\dots$ (1.1a)
$\displaystyle\left(\tau_{\Pi}u^{\mu}\partial_{\mu}+1\right)\Pi$
$\displaystyle=\dots$ (1.1b)
$\displaystyle\left(\tau_{\nu}u^{\mu}\partial_{\mu}+1\right)\nu^{\mu}$
$\displaystyle=\dots$ (1.1c)

where the dots represent the standard terms appearing in the constitutive
relations. The corrections provided by this approach enter only at very large
frequency, however now the equations of hydrodynamics are strongly-hyperbolic,
can be solved numerically given a set of initial conditions, and have a well-
behaved thermal equilibrium state that is stable and causal. The new
parameters $\tau_{i}$ are associated with UV degrees of freedom: indeed in MIS
theory one finds a new non-hydrodynamic mode
$\omega(\boldsymbol{\mathbf{k}}=0)\sim-\frac{i}{\tau_{\pi}}$ which decays
quickly. To this day MIS theory, which can be formally obtained from kinetic-
theory arguments, is the most used formalisms in hydrodynamic simulations
[68].

Figure 1.2: If the system is strongly coupled and clean, scatterings with
impurities are rare, while electron-electron interactions happen on a
microscopic timescale. Then the system thermalizes locally very quickly,
developing hydrodynamic behaviour.

In recent years there has been a new proposal to solve the problem of
causality and stability of relativistic hydrodynamics [69, 70, 71, 72, 73].
This approach is based on taking advantage of an ambiguity that is present in
any hydrodynamic description. What happens is that the fluid variables are not
well-defined out of equilibrium, hence they are ambiguous up to derivative
corrections, the so-called frame redefinitions. Following this approach it is
possible to find a hydrodynamic frame (known as BDNK frame) in which first-
order relativistic hydrodynamics is strongly-hyperbolic, causal and stable, at
the cost of having more complicated constitutive relations.

The solution by Müller, Israel and Stewart requires working with second-order
hydrodynamics to be fully consistent [26]. This, however, leads to new
fundamental issues related to hydrodynamics. In particular, hydrodynamic modes
$\omega(\boldsymbol{\mathbf{k}})$ are long-lived solutions to the conservation
equations which can cause instabilities in the system, since they are against
the assumption of local thermal equilibrium [74].

Classical hydrodynamics is only concerned with the dissipative part of the
system, without systematically taking into account the effects of thermal
fluctuations. However, fluctuations should always appear on par with
dissipative effects, according to the fluctuation-dissipation theorem [75].
This can be amended via a bottom-up construction by including fluctuations as
noise in the equations, thus turning hydrodynamics into a proper EFT,
following the Martin-Siggia-Rose formalism [76, 77, 78, 42, 74]. More
recently, also a top-down view on dissipative fluctuating hydrodynamics has
been developed [44, 43, 79, 80], based on the closed time-path formalism of
Schwinger and Keldysh [81, 82], which has the advantage of working at the full
non-linear level, providing access to the $n$-point correlators.

Specifically, beyond linear order, modes can interact with themselves via
thermal fluctuations, and contribute to the renormalization of the transport
coefficients compared to their bare, linear-response, values. The
renormalization of transport coefficients depends on the UV scale
$\Lambda_{\text{UV}}$ at which hydrodynamics is expected to fail and is due to
interaction between modes close to the cut-off. These interactions can appear
even in equilibrium due to thermal fluctuations, and can cause the breakdown
of the hydrodynamic derivative expansion [74]. From a technical perspective,
this is because the interactions between modes modify the retarded correlators
by non-analytical terms which scale like $\mathcal{O}(\omega^{3/2})$ in
$3+1d$222In $2+1$-dimensions this is a logarithmic correction [83]. [84, 85],
hence these corrections are effectively larger than the first-order
hydrodynamics, which are linear in $\mathcal{O}(\omega)$, but smaller than
second order corrections, which scale as $\mathcal{O}(\omega^{2})$, and cannot
be captured by local derivative corrections. This non-analytic behaviour is
related to the well-known effect of long-time tails: namely, the non-analytic
$\omega^{3/2}$ dependence of e.g. $3+1d$ hydrodynamics corresponds to a real-
time evolution as $t^{3/2}$ of the response function [86], which is slower
than exponential and decays slowly.

Finally, as we explained above, hydrodynamics stops being a sensible EFT at
length scales shorter than the microscopic scale $l$. This assumption, while
being at the core of the canonical approach, is challenged by the observation
that hydrodynamics seems to work well as a model for systems in which the
separation of scales does not happen sharply. This has sparkled a lot of
research in this direction, trying to understand this _unreasonable
effectiveness of hydrodynamics_. Research in this direction focused on
resummation methods for all-order hydrodynamics [58, 68, 59, 87] and the
hydrodynamic attractor [88, 89], but also an action-like description of
hydrodynamics via the Schwinger-Keldysh formalism [44, 80, 43].

We remark that the problems associated with causality, stability and
fluctuations appear in interpreting the equations of hydrodynamics as a set of
classical PDE, when solving for the real-time evolution of the fluid, and are
due to high-frequency, high-wavevector modes, far from the hydrodynamic regime
(hydrodynamics is not expected to hold at microscopic length scale anyway). We
decided to include this section for completeness and to give a better context
on the state of hydrodynamics, however in this thesis we will be mostly
concerned with the linear response of fluids in the hydrodynamic regime,
focusing on the $\omega,\boldsymbol{\mathbf{k}}\rightarrow 0$ properties of
small fluctuations about some global thermodynamic equilibrium state. Hence,
we will simply assume that suitable corrections to standard hydrodynamics
exist, which render the theory sensible, and that can be expanded to include
the various extensions of hydrodynamics discussed in this thesis.

###  1.3 Other approaches

In the previous section we discussed the canonical approach, some of its
pitfalls and recent research avenues. There are however other possible points
of view to formulate hydrodynamics, that might give different insights into
what it is, how it relates to other theories, and how to expand upon the
framework of hydrodynamics.

##### Kinetic theory

One of the first methods, also from an historical perspective, comes from
kinetic theory [90, 91]. This might seem strange at first sight, since
hydrodynamics is mostly understood as describing strongly-coupled phases of
matter that thermalize quickly, while kinetic theory sits on the other side of
the spectrum and is usually understood as describing weakly-coupled theories.
This apparent contradiction disappears once we appreciate that hydrodynamics
is extremely universal: in the presence of a single species of quasiparticles
(as it is often the case in kinetic theory) and no impurities there is nothing
that can stops the onset of hydrodynamics, which appears naturally at long
wavelength even at small couplings.

MIS hydrodynamics comes quite naturally from kinetic theory [66, 91, 92],
providing a justification for the timescales $\tau_{i}$ which were introduced
phenomenologically in the previous section. More recently, the kinetic theory
approach to hydrodynamics has been generalized to naturally include different
hydrodynamic frames [93, 94], in particular the already-cited BDNK frame.
Furthermore, since kinetic theory is a microscopic description, this approach
also gives direct access to expressions for the transport coefficients in
terms of microscopic and thermodynamic parameters.

##### Schwinger-Keldysh formalism

Hydrodynamics is a very powerful tool, but from a microscopic point of view
the arguments used in the first Section 1.1 to explain the emergence of
hydrodynamics are purely heuristic and not satisfactory. First, the canonical
formulation does not systematically deal with thermal and quantum
fluctuations, which must be included separately in bottom-up approaches, even
if they are important in many situations such as turbulence, chaotic systems,
and the renormalization of transport coefficients (long-time tails). Another
aspect is related to the regime of validity of the theory, which in classical
hydrodynamics is subject to the scale separation $l\ll L$, but experiments in
heavy-ion collisions suggest this bound is too strict. Indeed, recent action-
principle formulations of fluid dynamics and resummed all-order hydrodynamics
manage to go beyond this paradigm. Finally, the symmetries and the
thermodynamic constraints, such as first and second law of thermodynamics, are
imposed phenomenologically, while in the effective-action approach these come
directly from fundamental properties of the dissipative effective action,
which provides the theory with a strong theoretical ground.

From a theoretical perspective, there has always been a strong interest in
trying to develop field theory action-like methods (based e.g. on path
integrals) to describe dissipative systems like hydrodynamics [95, 96, 97].
While perfect fluids and certain other non-dissipative aspects of
hydrodynamics are already captured by action-like principles [98, 99], a full
discussion of the dissipative corrections from path integral methods is still
a topic of research [100, 101, 102, 49].

In the past decade however the Schwinger-Keldysh formalism has been
successfully applied to formalize hydrodynamics (and more generally thermal
dissipative systems) from an action-like principle [46, 103, 44, 43]. This
framework is based on realizing that in order to describe thermal dissipative
systems it is necessary to recast the standard problem of statistical field
theory on a closed time path, which leads to a doubling of the hydrodynamic
degrees of freedom (advanced and retarded fields depending on which branch of
the time contour they live on). After integrating out the UV degrees of
freedom one finds a Wilsonian effective action that must obey certain
constraints, such as unitarity and the novel $\mathbb{Z}_{2}$ dynamical KMS
symmetry. This construction provides an effective action that encodes all the
information related to the constitutive relations and the conservation
equations of hydrodynamics. Furthermore, contrary to the canonical approach,
this method automatically incorporates the effects of thermal (and eventually
quantum) fluctuations at the full non-linear level, in agreement with the
fluctuation-dissipation theorem.

##### Fluid/gravity duality

This approach is based on the gauge/gravity duality, grounded in string theory
[104, 105, 106]. It comes in various different form, but the simplest and most
practical one, which does not make reference to quantum gravity and strings,
is that it is a duality between a strongly-coupled (large ’t Hooft coupling)
large-$N$ quantum field theory in $d$ dimensions living on the boundary, and a
theory of classical gravity in $d+1$ dimensions that lives in bulk Anti de
Sitter space. In particular, a black hole in the interior of AdS space gives
rise to thermodynamics in the dual boundary theory, whose temperature and
entropy are fixed by the black-hole Gibbons-Hawking temperature and
Bekenstein–Hawking entropy.

Even before the discovery of holography, after the realization that black
holes are thermal systems, it was quickly argued that the analogy is not
limited to thermal equilibrium, and works for generic hydrodynamic
perturbations of the event horizon, implying that black holes are dissipative
systems, which is summarized by the membrane paradigm [107]. This
correspondence is clearer in AdS space [108, 109, 110], in which it acts as a
basis for the fluid/gravity duality.

What holography managed to show is that the membrane paradigm analogy actually
goes beyond linear perturbations, therefore the fluid/gravity correspondence
works at the full non-linear level [27], allowing one to obtain not only the
full set of hydrodynamic correlators, but also the equations of hydrodynamics
from General Relativity, see [24, 111, 23, 22, 112] for some books and
reviews.

Holography is a powerful tool, which can be used in conjunction with
hydrodynamics to study strongly-coupled systems from two different approaches,
and as a cross-check on the correctness of the hydrodynamic effective theory.
Historically, many interesting features of hydrodynamics (such as the effect
of anomalies on fluids, to name one) were first discovered using holographic
methods, and only subsequently obtained in hydrodynamics. Furthermore, like
kinetic theory, holographic models also provide us with the non-universal part
of the fluid description, namely the equation of state and the transport
coefficients, which can then be used to set bounds on transport coefficients,
e.g. the famous $\eta/s\geq\frac{1}{4\pi}$ KSS bound [113].

###  1.4 Hydrodynamics in condensed matter

####  1.4.1 Overview

In conventional metals the behaviour of electrons at low temperature is
observed to be diffusive [114, 115]. They are assumed to be (almost) non-
interacting, and relax only via electron-impurity scatterings on some
characteristic time $\tau_{\text{ei}}$, so that a Fermi-liquid picture in
terms of long-lived quasiparticles applies. Indeed, electrons in metals are
$\sim\qty{2}{}$ apart, however their mean free path at room temperature is of
order $\qty{e4}{}$, implying a weakly-coupled phase. Furthermore, contrary to
molecules in standard fluids, electrons move in a lattice background and
interact with impurities, defects and phonons, thus losing momentum. Hence, if
the sample is large $L\gg l=v_{F}\tau_{\text{ei}}$ with $v_{F}$ the Fermi
velocity, and the temperature is low $T\tau_{\text{ei}}\ll 1$, the system
reaches a diffusive regime [116]. In terms of collective excitations,
diffusion is characterized by a single decaying mode, while fluids also have
propagating sound modes with linear dispersion relation associated with energy
and momentum conservation333The modern point of view on hydrodynamics is that
it also includes diffusion, which is based on the same fundamental principles,
only with a different set of conserved charges. In the context of condensed
matter, however, we keep the distinction.. One exception, that emerges in very
clean samples for which the system is similar to the mean free path in size,
is the _ballistic_ motion of electrons, see Figure 1.2.

Scattering mechanisms in solids are characterized by mean free paths that
depend on the temperature. In conventional metals the low-temperature regime
is dominated by electron-impurity elastic scatterings, leading to the residual
resistance. On the other hand, at high temperatures usually electron-phonon
scatterings dominate, leading to a characteristic $T$-dependence behaviour of
the conductivity [114]. For most materials, at any temperature either
electron-impurity or electron-phonon scatterings are dominant compared to
electron-electron interactions $l_{\text{ee}}\gg l_{\text{ei}},l_{\text{ep}}$,
leaving no room for hydrodynamic electronic transport.

Despite the above discussion, historically, hydrodynamics has been
successfully applied in condensed matter to study the dynamics of phonons
[117] (see [118] for experiments and references on the topic), spin waves
(magnons) [119], and more recently, based on the work by Gurzhi [117], also to
electronic systems [120, 12, 16, 121, 122, 123, 124, 125], see also the
reviews [126, 127].

From an experimental perspective, only recently very clean materials in which
$l_{\text{ee}}\ll l_{\text{ei}},l_{\text{ep}}$ have been realized, paving the
road to electron hydrodynamics [128, 129, 130, 130, 131, 132, 133, 134, 135,
136, 137]. In particular, in $2d$ materials $l_{\text{ee}}\sim T^{-2}$, while
acoustic phonons interactions scale as $l_{\text{ep}}\sim T^{-1}$ and in
ultra-pure samples the impurity scattering are rare $l_{\text{ei}}\gg
l_{\text{ee}}$, allowing for a parameter range with hydrodynamic behaviour. In
graphene [128, 129, 138, 139, 140, 141], for example, the mean free path with
impurities and phonons remains very large
$l_{\text{ei}},l_{\text{ep}}\geq\qty{1}{\micro}$ up to room temperature, while
at $T\geq\qty{150}{}$ the electron-electron scattering has a characteristic
length of $l_{\text{ee}}\sim\qty{0.1}{\micro}$, which opens a window to
hydrodynamic transport.

More recently, nonlocal resistance [128] and violation of the Wiedemann-Franz
law [129] have been used as signatures of hydrodynamic behaviour [139, 140,
141]. In particular, hydrodynamic flow in electronic solid state systems
should exhibit an enhanced conductivity, higher than the ballistic limit [138,
142]. This can happen because the Sharvin limit is obtained by assuming no
electron-electron interactions in the ballistic regime, however collective
hydrodynamic motion driven by electron-electron scatterings can overcome the
ballistic bound. Recently, experiments [131, 135] employed various imaging
methods to directly observe the viscous flow of electrons in graphene.

A different class of materials which also show hydrodynamic behaviour are
ultra-pure heterostructure semiconductors [131, 133, 137], since the first
experimental observation of the Gurzhi effect [143]. Furthermore, other
materials such as Weyl semimetals [16, 17] might exhibit fluid dynamic
behaviour [132, 144]. Finally, hydrodynamic transport is often associated with
the unusual linear-in-$T$ dependence of the resistivity in the strange metal
phase of high-temperature superconductors [145, 146, 147].

####  1.4.2 Gurzhi effect

The Gurzhi effect [117] is often considered a defining signature of
hydrodynamic transport in solid state systems. Consider an electric current
flowing in a thin, clean wire of metal. In this situation electrons can
undergo two different scattering processes: scatterings with the walls of the
wire (the boundaries), and electron-electron scatterings (eventually mediated
by phonons). Assume that at very low temperature the electron-electron
scattering length is larger than the width of the wire $l_{\text{ee}}\gg W$.
In this regime, scatterings off the walls of the wire dominate, leading to a
characteristic temperature-independence of the resistivity
$\rho_{\text{lowT}}\sim 1/W$.

Figure 1.3: Left: Figure taken from [117]. Theoretical prediction of the
Gurzhi effect, with its characteristic
$\mathop{}\\!\mathrm{d}\rho/\mathop{}\\!\mathrm{d}T\leq 0$. Right: Figure
taken from [133]. In red and blue samples with an obstacle, in black without
it, and in purple a macroscopic sample. The blue triangles are a theoretical
prediction and the solid thin line represent the resistivity due to
scatterings with acoustic phonons. Notice the negative slope for
$T\leq\qty{40}{}$.

On the other hand, the electron-electron interaction is instead temperature
dependent, e.g. $l_{\text{ee}}\sim T^{-2}$ for direct scatterings, and
$l_{\text{ee}}\sim T^{-5}$ for phonon-mediated ones. Hence, as $T$ increases,
the electron-electron scattering length decreases, until $l_{\text{ee}}\ll W$
and the resistivity becomes dominated by electron-electron interactions,
leading to a dependence $\rho_{\text{highT}}\sim
l_{\text{ee}}/W^{2}<\rho_{\text{lowT}}$ which is smaller than the low-
temperature resistivity444In terms of standard fluids, the Gurzhi effect can
be understood as the electronic realization of the crossover between Knudsen
flow (particle based) and Poiseuille (viscous) flow.. This result clearly
assumes that the effective mean free path $W^{2}/l_{\text{ee}}$ in the wire,
is much smaller than bulk momentum-relaxing processes, such as electron-
impurities and electron-phonon scatterings, implying that we are in the
hydrodynamic regime. As the temperature increases $W^{2}/l_{\text{ee}}\gg
l_{\text{ei}}$ and a crossover between hydrodynamics and diffusive regime
happens, leading to the usual $\rho(T)$ dependence. Thus, the Gurzhi effect
identifies a minimum in the resistivity, which decreases $\rho\sim T^{-2}$ as
a function of $T$, and is understood as a signature of hydrodynamic transport.

From an experimental perspective, several factors hinder the observation of
the Gurzhi effect in real metals, such as impurities and Umklapp scatterings,
or the non-spherical shape of the Fermi surface, which contribute to
corrections to the $T$ dependence of the resistivity, eventually washing away
its characteristic signature. Nonetheless, it was observed in [143] in
heterostructures and more recently in [133].

####  1.4.3 Nonlocal transport experiment

Some experiments manage to measure negative vicinity resistance [128] and
Wiedemann-Franz law violation [129] in graphene, both of which are usually
interpreted as smoking guns of hydrodynamic electronic transport in [130].

Standard transport measurements at two or four terminals are concerned with
extracting the current-voltage characteristic, thus they focus only on the
total current flowing inside the sample to obtain the resistance $R$. On the
contrary, nonlocal transport experiments consist in measuring the spatial
dependence of the current density in the system, by studying the voltage drops
measured from multiple gates, see Figure 1.4.

Figure 1.4: Figure taken from [148]. Conventional four-terminals measurements
consist in having a current flow between Source $S$ and Drain $D$, while
recording the voltage drop between 1 and 3 (or 2 and 4). The resistance is
then $R_{SD,13}=V_{13}/I_{SD}$ and is related to the longitudinal resistivity
$\rho_{xx}=R_{SD,13}W/L$, with $W$ and $L$ the width and the length of the
sample, as in Figure. In nonlocal measurements, instead, a current flows
between 1 and 2, while the voltage is measured from 3 to 4. In the case of
diffusive transport the voltage drop $V_{34}$ should be exponentially
suppressed $R_{12,35}\sim\rho_{xx}e^{-L/W}$, but it is not in hydrodynamic
transport.

Nonlocal measurements were first intended to identify ballistic transport in
mesoscopic systems, however they have been recently argued to be possible
indicators of hydrodynamic transport [120, 12, 128, 139, 140]. In particular,
hydrodynamic flows can give rise to vortices in confined geometries, which
lead to a characteristic negative nonlocal resistance at the opposite side of
the vortex called _vicinity resistance_. Although this effect can be a sign of
hydrodynamic behaviour [128, 149], it can also appear in ballistic transport
[150], thus to differentiate between the two regimes one needs either to check
the temperature dependence [139] or to employ spatial imaging techniques to
directly observe the electron flow [151, 152, 153].

####  1.4.4 Wiedemann-Franz law violation

Unconventional charge and heat transport in hydrodynamic electronic systems
leads to a violation of the Wiedemann-Franz law [114, 115]. If both charge and
heat are due to the transport of the same quasiparticles, and are influenced
by the same scattering processes (as is the case in most microscopic models of
non-interacting electrons), then the ratio between charge and heat
conductivity must be a constant

$\frac{\kappa}{\sigma}=\mathcal{L}T\qquad\qquad\mathcal{L}=\mathcal{L}_{0}=\frac{\pi^{2}}{3e^{2}}$
(1.2)

where $\sigma$ and $\kappa$ are the electric and thermal conductivities, while
$\mathcal{L}$ is called Lorenz ratio and $\mathcal{L}_{0}$ is valid for free
carriers. In most conventional metals electrons are indeed weakly interacting
and departures from the WF law are usually weak. On the contrary, a strong
violation of the WF law should be regarded as due to new effects, like
strongly-coupled hydrodynamic transport [16, 12, 14].

Figure 1.5: Left: Figure taken from [144]. Strong WL law violation in a micro-
ribbon of $\mathrm{WP_{2}}$. Center: Figure taken from [132]. Same violation
in bulk crystals of $\mathrm{WP_{2}}$. Right: Figure taken from [154]. WF law
violation is observed also in $\mathrm{MoP}$.

Theory predicts a very large value of the Lorenz ratio in graphene
$\mathcal{L}\propto\tau_{\text{ei}}/\tau_{\text{ee}}\gg 1$ in the hydrodynamic
regime [12], which was indeed observed in the graphene Dirac fluid in [129].

Signatures of electron hydrodynamics also appear in topological materials
(Dirac and Weyl semimetals) [132, 144, 154], from the observation of strong WL
law violations. In these compounds the WL law violation happens in the
opposite direction compared to graphene: now the Lorenz ratio is found to be
very small $\mathcal{L}\ll\mathcal{L}_{0}$, again attributed to the existence
of hydrodynamic electronic transport, which onsets at lower temperature
compared to graphene.

###  1.5 The content of this thesis

Hydrodynamics, as we discussed, is concerned with the dynamics of conserved
charges in closed systems. Although this can be a very good approximation,
real systems are rarely exactly closed and void of impurities. As a
consequence, we can expect the conservation laws of hydrodynamics to be only
approximate and, in general, to be broken by some effective relaxation term
characterized by a timescale parameter $\Gamma\sim 1/\tau_{1}$ which dictates
the damping of fluctuations towards equilibrium. These kinds of theories are
called relaxed hydrodynamics or quasihydrodynamics.

Clearly, these corrections to the conservation equations must be small in
amplitude, for the hydrodynamic description to still be approximately valid
[50]. Physically, this corresponds to saying that whatever is ruining the
exact conservation acts on scales which are still much larger than the
microscopic mean-free-path and thermalization scale (the non-hydrodynamic
modes), so that a separation of scale is still possible, see Figure 1.6.

Figure 1.6: Figure taken from [50]. Left: If the relaxation times for the non-
hydrodynamic modes are all comparable, then hydrodynamics dominates in the IR
and there is no quasihydrodynamics regime, since $\tau_{1}\sim\tau_{2}$.
Right: On the other hand, if there is a mode with a much larger decay time
$\tau_{1}\gg\tau_{2}$, then hydrodynamics has a reduced regime of validity,
close to the origin. However, quasihydrodynamics describes the IR physics on a
wider regime, by including the effect of non-hydrodynamic mode with
parametrically small decay time $\tau_{1}$.

From a phenomenological perspective, we can expect the quark-gluon-plasma
created in heavy-ion collisions in a vacuum to be an almost perfect fluid
(_the_ most perfect fluid [155]), with virtually no impurities, thus
quasihydrodynamics might have limited applicability in the context of nuclear
physics or astrophysics555Although there are no impurities, close to thermal
critical points fluctuations become parametrically large and the correlation
length diverges, requiring a modification of hydrodynamics to include the
presence of these slow modes [51].. On the other hand, condensed matter
systems are in comparison much more complicated structure: the electron fluid
always moves in the presence of a background lattice and interacts with
phonons, which can degrade energy and momentum; furthermore, even clean
samples have some level of impurities and Umklapp scatterings which can
disrupt the electron flow and take away momentum [121]. The presence of
boundaries and layers can also contribute to losses of would-be conserved
charges, since the system is never perfectly closed. Finally, in many
condensed matter systems symmetries are only approximate and accidental, thus
they are not protected by any fundamental law. Consequently, the associated
Noether charges are not exactly conserved, and the conservation equations are
explicitly weakly broken. This is the case e.g. of the emergent Lorentz
symmetry in graphene, or the chiral symmetry in Weyl semimetals, which is not
exact and broken at high quasiparticle momenta, far from the Fermi surface
(the band structure is not linear up to arbitrary high momenta).

In [51, 39, 156], but also in Chapter 3, the extra mode is assumed to be very
light and dynamical, so that it thermalizes quickly and the regime of validity
of hydrodynamics is naturally extended to include it. On the other hand, for
most of this thesis, like in [50, 49, 14, 157, 158, 159, 160], we will assume
that the associated microscopic non-hydrodynamical mode is integrated out and
not dynamical, and its effect is that of modifying the conservation law of
some quantity with a relaxation term. Weak effective breaking of symmetries,
and momentum relaxations in particular, are also of great interest from a
holographic perspective, see e.g. [161, 162, 163, 164] and many others.

Alternatively, another point of view is that of [50]: hydrodynamics works on
scales much larger than the mean free path $l$ and its associated microscopic
time $\tau$. Normally, non-hydrodynamic operators decay quickly as $\langle
O(t)O(0)\rangle\sim e^{-t/\tau_{2}}$, and what survives the long-
wavelength/long-timescale regime is the dynamics of conserved operators.
However, it can happen that there is some operator whose decay is
parametrically slower $\langle P(t)P(0)\rangle\sim e^{-t/\tau_{1}}$ than other
$\tau_{2},\tau_{3},\dots\ll\tau_{1}$. Strictly speaking, hydrodynamics applies
only on scales $\tau\gtrsim\tau_{1}$ and breaks down when
$\tau_{1}\partial_{t}\sim 1$. Nonetheless, because $\tau_{1}$ is so much
larger than the other decay rates, a separation of scales is still possible,
and we can include $\langle P\rangle$ as a quasi-conserved operator, with an
effective decay time $\tau_{1}$, which is a valid description as long as
$\tau_{2}\partial_{t}\ll 1$.

This is the main context of the thesis. In most of this work we will expand on
these ideas, trying to learn general lessons from quasihydrodynamics that can
be useful to characterize condensed matter systems in which the standard
hydrodynamic charges are not exactly conserved, but instead decay to some
fixed equilibrium value. The thesis is structured as follows.

In Chapter 2 we review the canonical approach to hydrodynamics, we discuss how
to obtain the retarded responses of fluids to external linear perturbations,
and we introduce the tool of the hydrostatic generating functional.

Subsequently, in Chapter 3 we expand on previous works and discuss the
hydrodynamic theory of electronic Charge Density Waves in terms of
viscoelastic fluids which contains Goldstone bosons for broken spatial
translations (spontaneous and explicit breaking), while including the effects
of a strong external order-zero magnetic field and a new transport
coefficient, the lattice pressure. We study the transport properties of such
fluids both analytically from hydrodynamics, and numerically from holography,
and find a very good match in the hydrodynamic regime between the two
approaches. This work sits in the broader line of research that tries to
better understand the strange-metal strongly-coupled phase of cuprates.

Chapter 4 is devoted to constructing a theory of hydrostatic fluids in strong
external electric fields for systems in which energy and momentum are not
exactly conserved, but rather slowly relax. In particular, we find that
momentum relaxation modifies the stationary state, hence it should be included
in the hydrostatic constraints of the fluid, leading to new predictions for
the thermoelectric transport.

In Chapter 5 we introduce the concept of generalized relaxations in linearized
hydrodynamics, which are decay terms that mix the charges (e.g. a charge
fluctuation which induces energy relaxation and vice versa). We obtain a set
of very general constraints that these relaxations must obey, based on
microscopic time-reversal invariance, positivity of entropy production and
linear stability. Then, we show how to make them consistent with curved
spacetime and background gauge fields by adding new source terms proportional
to relaxations and imposing Onasger relations on the system. This procedure
thus allows us to obtain the full spectrum of retarded correlators for the
relaxed theory.

Finally, in Chapter 6 first we correct previous results in the literature
regarding the longitudinal magnetotransport of anomalous fluids, emphasizing
the importance of working at the correct derivative order for the anomaly to
appear in the conductivities. Subsequently, we use this new result to study
the transport properties of Weyl semimetals in the hydrodynamic regime. We
find that generalized relaxations are a necessary ingredient if we want our
model to satisfy basic fundamental principles (namely Onsager relations,
finite DC conductivities and electric charge conservation). However, the set
of relaxations we find consistent with the constraints above do not obey
positivity of entropy production, implying the system is open. Furthermore,
our model predicts qualitatively different DC values for the longitudinal
thermoelectric magnetoresistance compared to older works, suggesting a way to
test the hydrodynamic regime in Weyl semimetals. To conclude, we also
introduce a mechanism to reproduce generalized relaxations from kinetic
theory, by using appropriate collision integrals with energy-dependent
Relaxation Time Approximation.

###  1.6 Notations

We work in natural units $\hbar=c=k_{B}=1$. Greek indexes $\mu,\nu,\dots$
identify spacetime directions, while Latin indexes $i,j,\dots$ only the
spatial ones. Latin capital indexes $I,J,\dots$, mostly in Chapter 3,
represent a subset of the spatial directions. The spacetime metric is taken
with signature $g_{\mu\nu}=\text{diag}\left(-1,1,\dots,1\right)$. Vectors are
indicated both in bold $\boldsymbol{\mathbf{v}}$ or with the arrow $\vec{v}$.
In this thesis I will use both $G^{R}_{ab}$ and
$\langle\phi_{a}\phi_{b}\rangle$ to indicate retarded Green functions
interchangeably.

Name |  |  Acronym
---|---|---
Weyl semimetal |  |  WSM
Relaxation Time Approximation |  |  RTA
Direct Current |  |  DC
Alternating Current |  |  AC
Charge Density Wave |  |  CDW
Negative MagnetoResistance |  |  NMR
Partial Differential Equation |  |  PDE
Local Thermodynamic Equilibrium |  |  LTE
Anti de Sitter |  |  AdS
Wiedemann-Franz |  |  WF
Right-Hand Side |  |  RHS
Left-Hand Side |  |  LHS

##  Chapter 2 Hydrodynamics and Linear Response Theory

> “Begin at the beginning”, the King said gravely, “and go on till you come to
> the end: then stop.”
>
> Lewis Carroll, _Alice in Wonderland_

###  2.1 Hydrodynamics

In the previous chapter we discussed hydrodynamics in broad terms. Now we will
focus on the relativistic normal fluid (without superfluid component) in $d+1$
spacetime dimensions, which will be the pedagogical example in this thesis.
The approach presented here, however, applies to fluids with any symmetry
content (e.g. Galilean, Carrolian, boost-agnostic, …). In this review we will
mostly follow [74].

####  2.1.1 Hydrodynamic variables

As we have already discussed, hydrodynamics is concerned with the long
wavelength dynamics of conserved charges. Following Noether’s theorem, we know
that conservation laws are a consequence of the continuous symmetries of the
fundamental theory, and this in turn implies the existence of conserved
currents. In particular here we will focus on a charged relativistic fluid,
which means we not only assume Poincaré symmetry (boosts, rotations and
translations), but also an internal $\mathrm{U(1)}$ symmetry, such that we
also have a conserved vector current (e.g. baryon number, electric charge, …).

Symmetry under spacetime translations gives us a conserved symmetric stress-
energy tensor $T^{\mu\nu}$ that couples to the external spacetime metric
$g_{\mu\nu}$ and can be defined as the variation of the action with respect to
the metric [11, 165]

$T^{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}}$ (2.1)

Although the stress-energy tensor obtained from Noether’s theorem is not
always symmetric, the one obtained from the above definition automatically is.
Anyway, following the Belinfante’s construction, it is always possible to add
extra antisymmetric terms to $T^{\mu\nu}$ which are identically conserved and
that lead to a fully symmetric stress-energy tensor, thus we will only
consider symmetric $T^{\mu\nu}$ in this work. We remark that if spin-angular
momentum is a conserved quantity relevant in hydrodynamics, one indeed needs
to consider antisymmetric corrections to the stress-energy tensor [37],
however spin hydrodynamics will not be covered in this thesis.

The conservation law for spacetime translations (or diffeomorphism invariance)
is simply

$\partial_{\mu}T^{\mu\nu}=0$ (2.2a) We can also compute the conserved currents
that correspond to the other spacetime symmetries, namely boosts and
rotations, and we obtain
$\mathcal{M}^{\mu\nu\alpha}=x^{\mu}T^{\nu\alpha}-x^{\nu}T^{\mu\alpha}$, which
is automatically conserved thanks to (2.2a) and the relativistic Ward Identity
$T^{\mu\nu}=T^{\nu\mu}$, hence there are no other conserved quantities related
to the spacetime symmetries.

We still have to discuss the internal symmetries, but for a global
$\mathrm{U(1)}$ symmetry this simply implies that there exists a conserved
current $J^{\mu}$

$\partial_{\mu}J^{\mu}=0$ (2.2b)

It is also possible to introduce in hydrodynamics higher-form symmetries,
which lead to the conservation of totally antisymmetric currents with more
than one index, see for example [55, 57].

It is important to stress that (2.2b) and (2.2a) are to be understood as non-
trivial equations of motion for the conserved charges, which are the dynamical
fields in hydrodynamics. While these equations are solved identically on shell
for a given microscopic action $S$, in hydrodynamics they provide the
dynamical content of the theory.

It is then immediately obvious that, without further assumptions, it is not
possible to solve the conservation equations. In particular, by simply
counting the degrees of freedom, we see that a symmetric stress-energy tensor
has $(d+1)(d+2)/2$ components and the $\mathrm{U(1)}$ current has $d+1$,
however we only have $d+1$ equations from (2.2a) and one equation from (2.2b),
not enough to determine uniquely the evolution of the system.

The simplifying assumption of hydrodynamics is that we can express
$T^{\mu\nu}$ and $J^{\mu}$ as functions of $d+2$ local fields, which are
called _hydrodynamics fields_ or variables. In the context of relativistic
fluid dynamics, the independent fields are usually taken to be: a local
temperature $T(x)$, a local chemical potential $\mu(x)$ and a local fluid
velocity, described by the four-vector $u^{\mu}(x)$.

To better understand this choice of variable we can consider a thermal system
in the grand canonical ensemble whose equilibrium is characterized by a
density operator $\hat{\rho}$ which is proportional to the exponential of the
conserved charges [166]. In our present case these are described by the four-
momentum
$\hat{P}^{\mu}=\int\mathop{}\\!\mathrm{d}\Sigma_{\nu}\hat{T}^{\mu\nu}$ and
particle number operator
$\hat{N}=\int\mathop{}\\!\mathrm{d}\Sigma_{\mu}\hat{J}^{\mu}$, where
$\Sigma_{\mu}$ is some spacelike hypersurface. Then, we can express the
density operator as

$\hat{\rho}=\frac{1}{Z}e^{\beta_{\mu}\hat{P}^{\mu}+\Lambda_{\beta}\hat{N}}$
(2.3)

where $Z=\text{Tr}\ e^{\beta_{\mu}\hat{P}^{\mu}+\Lambda_{\beta}\hat{N}}$ is
the partition function. We introduced two Lagrange multipliers: a timelike
vector $\beta^{\mu}$, called the thermal vector, and a scalar
$\Lambda_{\beta}$, the thermal twist [47]. It is customary in relativistic
hydrodynamics to rewrite these two parameters as $\beta^{\mu}=\beta u^{\mu}$
and $\Lambda_{\beta}=\beta\mu$ in terms of a four-velocity $u^{\mu}$
(normalized such that $u^{2}=-1$), a chemical potential $\mu$ and an inverse
temperature $\beta=T^{-1}$.

In hydrodynamics, we move away from the uniform equilibrium state
characterized by (2.3), and we consider temperature, chemical potential and
velocity as slowly varying functions of spacetime. The idea being that, as the
operators $\hat{T}^{\mu\nu}$ and $\hat{J}^{\mu}$ are always well-defined (for
a given theory), their expectation values
$T^{\mu\nu}=\langle\hat{T}^{\mu\nu}\rangle$ and
$J^{\mu}=\langle\hat{J}^{\mu}\rangle$ can be expressed in terms of equilibrium
quantities when these vary slowly in spacetime.

We comment here that while a specific equilibrium state breaks the boost
symmetry, by picking a specific timelike vector $\beta^{\mu}$, there are many
physically equivalent equilibrium states with different spatial velocities. It
is thus only the state the breaks the symmetry, while the theory is still
perfectly Lorentz covariant. Indeed, it can be shown explicitly that the
Goldstone boson associated with the spontaneous breaking of the boost symmetry
is not an extra dynamical field, as it must be identified with the fluid
velocity [167].

####  2.1.2 Constitutive relations

Having discussed the equations of motion, i.e. the conservation laws, and the
hydrodynamic variables, it is now time to turn to the constitutive relations.
This means that we need to write the stress-energy tensor and the
$\mathrm{U(1)}$ current as the most general expressions in terms of the
hydrodynamic fields compatible with the symmetries of the theory.

In the context of relativistic hydrodynamics it is useful to decompose the
stress-energy tensor and the current with respect to the timelike vector
$u^{\mu}$, as this allows us to work with quantities that are easily
recognized as Lorentz covariant objects. We work in flat Minkowski space
$\eta_{\mu\nu}=\text{diag}(-1,1,\dots,1)$ and we define the explicitly
symmetric projector orthogonal to the four-velocity, namely
$\Delta^{\mu\nu}=\eta^{\mu\nu}+u^{\mu}u^{\nu}$.111$\Delta^{\mu\nu}$ is a
projector orthogonal to $u^{\mu}$ in the sense that $\Delta^{\mu\nu}u_{\nu}=0$
and $\Delta^{\mu\lambda}\Delta_{\lambda}^{\nu}=\Delta^{\mu\nu}$. The
decomposition takes the form [168, 74]

$\displaystyle T^{\mu\nu}$
$\displaystyle=\mathcal{E}u^{\mu}u^{\nu}+\mathcal{P}\Delta^{\mu\nu}+2\mathcal{Q}^{(\mu}u^{\nu)}+\mathcal{T}^{\mu\nu}$
(2.4a) $\displaystyle J^{\mu}$
$\displaystyle=\mathcal{N}u^{\mu}+\mathcal{J}^{\mu}$ (2.4b)

where we used the standard notation in which the brackets mean symmetrized
indices. The coefficients $\mathcal{E}$, $\mathcal{P}$ and $\mathcal{N}$ are
scalars, $\mathcal{J}^{\mu}$ and $\mathcal{Q^{\mu}}$ are vectors transverse
with respect to $u^{\mu}$ (such that
$\mathcal{J}^{\mu}u_{\mu}=\mathcal{Q}^{\mu}u_{\mu}=0$), while
$\mathcal{T}^{\mu\nu}$ is a traceless and transverse symmetric tensor.

These quantities are explicitly defined by projecting the stress-energy tensor
and current. Specifically they can be written as

$\displaystyle\mathcal{E}$
$\displaystyle=u_{\mu}u_{\nu}T^{\mu\nu}\qquad\qquad\mathcal{P}=\frac{1}{d}\Delta_{\mu\nu}T^{\mu\nu}\qquad\mathcal{N}=-u_{\mu}J^{\mu}$
(2.5a) $\displaystyle\mathcal{Q}_{\mu}$
$\displaystyle=-\Delta_{\mu\lambda}u_{\sigma}T^{\lambda\sigma}\qquad\mathcal{J}_{\mu}=\Delta_{\mu\nu}J^{\nu}$
(2.5b) $\displaystyle\mathcal{T}_{\mu\nu}$
$\displaystyle=\frac{1}{2}\left(\Delta_{\mu\lambda}\Delta_{\nu\sigma}+\Delta_{\nu\lambda}\Delta_{\mu\sigma}-\frac{2}{d}\Delta_{\mu\nu}\Delta_{\lambda\sigma}\right)T^{\lambda\sigma}$
(2.5c)

This decomposition is an identity and holds for all symmetric tensors and
vectors. The hydrodynamic assumption enters when we express the coefficients
of (2.4) in terms of the hydrodynamic variables $\mu,T$ and $u^{\mu}$. In
particular, we will follow the logic of a derivative expansion, common in EFT,
by assuming that the hydrodynamic fields differ from equilibrium only on
scales much larger than the microscopic mean free path, that plays the role of
a UV cut-off. This means that first-order corrections are larger than the
second-order ones, second-order terms are larger than third, and so on.
Formally we can write

$\displaystyle T^{\mu\nu}$
$\displaystyle=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{(1)}+T^{\mu\nu}_{(2)}+\dots+T^{\mu\nu}_{(n)}+\mathcal{O}(\partial^{(n+1)})$
(2.6a) $\displaystyle J^{\mu}$
$\displaystyle=J^{\mu}_{(0)}+J^{\mu}_{(1)}+J^{\mu}_{(2)}+\dots+J^{\mu}_{(n)}+\mathcal{O}(\partial^{(n+1)})$
(2.6b)

The ideal or perfect fluid contains zero derivatives, while Navier-Stokes has
one-derivative corrections.

####  2.1.3 Order zero: ideal fluid

Since all the transverse vectors and transverse traceless tensor we can
construct with $\mu$, $T$, $u^{\mu}$ and their derivatives are at least order
one in gradients, we can already recognize that the ideal fluid will have
$\mathcal{Q}^{\mu}=\mathcal{J}^{\mu}=\mathcal{T}^{\mu\nu}=0$. Thus, we need to
write $\mathcal{E},\mathcal{P}$ and $\mathcal{N}$ in terms of the order-zero
scalars, namely our hydrodynamic variables $\mu$ and $T$. To make progress we
notice that, in global thermodynamic equilibrium (in the fluid rest frame),
the stress-energy tensor and the current take a universal form, namely
$T^{\mu\nu}_{\text{RF}}=\text{diag}(\varepsilon,P,\dots,P)$ and
$J^{\mu}_{\text{RF}}=(n,\boldsymbol{\mathbf{0}})$. These can be understood as
the definitions of the equilibrium energy density $\varepsilon$, pressure $P$
and charge density $n$. These expressions are valid in the rest frame of the
fluid, thus to obtain the covariant expressions for a generic fluid moving
with four-velocity $u^{\mu}$ we simply need to perform a Lorentz boost. The
final expressions are [11, 7]

$\displaystyle T^{\mu\nu}$ $\displaystyle=\varepsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}=(\varepsilon+P)u^{\mu}u^{\nu}+P\eta^{\mu\nu}$
(2.7a) $\displaystyle J^{\mu}$ $\displaystyle=nu^{\mu}$ (2.7b)

By construction these equations hold in global thermodynamic equilibrium,
however ideal hydrodynamics simply corresponds to assuming that the same form
holds when the thermodynamic variables are promoted to slowly varying fields
in local thermal equilibrium, thus we find that for an ideal fluid
$\mathcal{E}(x)=\varepsilon(x)$, $\mathcal{P}(x)=P(x)$ and
$\mathcal{N}(x)=n(x)$.

It is clear from this discussion that one also needs an equilibrium equation
of state $P(T,\mu)$ in order to know the explicit constitutive relations in
terms of the fundamental hydrodynamic fields. Thus, we assume that a sensible
equation of state exists, and using the thermodynamic relations [169, 75]

$\displaystyle\mathop{}\\!\mathrm{d}P$ $\displaystyle=\frac{\partial
P}{\partial T}\biggr{\rvert}_{\mu}\mathop{}\\!\mathrm{d}T+\frac{\partial
P}{\partial\mu}\biggr{\rvert}_{T}\mathop{}\\!\mathrm{d}\mu=s\mathop{}\\!\mathrm{d}T+n\mathop{}\\!\mathrm{d}\mu$
(2.8a) $\displaystyle\varepsilon$ $\displaystyle=-P+Ts+\mu n$ (2.8b)

we first define the entropy density $s$ and the charge density from the
Gibbs–Duhem equation and subsequently the energy density via the Euler
relation.

It is possible to combine the longitudinal component of (2.2a), namely
$u_{\mu}\partial_{\mu}T^{\mu\nu}=0$, together with the equation of motion for
the current (2.2b), expressed for the constitutive relations of an ideal fluid
(2.7) to obtain

$\partial_{\mu}\left(su^{\mu}\right)=0$ (2.9)

The term in the bracket is interpreted as the ideal fluid entropy current and
this equation tells us that, for a perfect fluid, entropy is locally conserved
and therefore ideal hydrodynamics is non-dissipative.

####  2.1.4 Frame choice

Before discussing first-order corrections to the perfect fluid, leading to the
so-called relativistic Navier-Stokes equations, we must first deal with a
problem that appears in all EFT constructions, related to redefinitions of the
fundamental fields used to perform the gradient expansion.

Specifically, at order one in derivatives we can redefine the hydrodynamic
variables in terms of order-one quantities that vanish in equilibrium, so that
different definitions match in global thermodynamic equilibrium, when the
gradients are zero. A specific definition of the fields $T$, $\mu$ and
$u^{\mu}$ is usually called a _frame choice_. Physically this is due to the
fact that there is no microscopic definition of the hydrodynamic fields when
out of equilibrium, e.g. there is no temperature operator whose expectation
value would give us $T(x)$ for out-of-equilibrium states. The correct way to
think about the constitutive relations is to consider $T(x)$, $\mu(x)$ and
$u^{\mu}(x)$ as auxiliary parameters used to parametrize $T^{\mu\nu}$ and
$J^{\mu}$, which do have a microscopic out-of-equilibrium definition. A change
of frame may redefine the hydrodynamic fields, but will not change the
expectation value of stress-energy tensor and the current (only how they
appear in terms of the local variables) [170, 25].

From an EFT perspective, it is natural to have the freedom to redefine fields
that differ by derivative quantities, and this is indeed the case that appears
when discussing EFT approach for dissipative systems [44, 171]. Since frame
choice are just a redundancy of the theory, they can be interpreted as gauge-
like transformation [172]. Furthermore, exactly like gauge theories, they can
be recast in a gauge invariant manner, meaning that it is possible to develop
a theory of hydrodynamics without fixing a frame, only in terms of frame
invariant scalars and vectors [25].

Formally, this means that the coefficients $\mathcal{E},\mathcal{P}$ and
$\mathcal{N}$ take the generic form

$\displaystyle\mathcal{E}$
$\displaystyle=\varepsilon(T,\mu)+f_{\mathcal{E}}(\partial
T,\partial\mu,\partial u)$ (2.10a) $\displaystyle\mathcal{P}$
$\displaystyle=p(T,\mu)+f_{\mathcal{P}}(\partial,\partial\mu,\partial u)$
(2.10b) $\displaystyle\mathcal{N}$
$\displaystyle=n(T,\mu)+f_{\mathcal{N}}(\partial,\partial\mu,\partial u)$
(2.10c)

where the first terms are obtained by the equilibrium equation of state, as
discussed in Section 2.1.3, while the $f$s depends on the gradients of the
hydrodynamic variables and are specified by our choice of the out-of-
equilibrium definitions of local temperature, chemical potential and fluid
velocity.

Consider now a generic frame transformation

$\displaystyle T(x)$ $\displaystyle\longrightarrow\quad
T^{\prime}(x)=T(x)+\delta T(x)$ (2.11a) $\displaystyle\mu(x)$
$\displaystyle\longrightarrow\quad\mu^{\prime}(x)=\mu(x)+\delta\mu(x)$ (2.11b)
$\displaystyle u^{\mu}(x)$ $\displaystyle\longrightarrow\quad
u^{\prime\mu}(x)=u^{\mu}(x)+\delta u^{\mu}(x)$ (2.11c)

where the $\delta$ represent first-order correction222Notice that $\delta
u^{\mu}$ is a transverse vector, such that $u_{\mu}\delta u^{\mu}=0$. This is
needed in order to preserve the normalization of the four-velocity at the
given order in derivatives $u^{2}=-1+\mathcal{O}(\partial^{2})$.. One can use
the new definition of $u^{\mu}$ to decompose the stress-energy tensor and the
current as in (2.5), remembering that $T^{\mu\nu}$ and $J^{\mu}$ are invariant
under frame choice and that $\mathcal{Q}^{\mu}$ and $\mathcal{J}^{\mu}$ are
themselves order one in derivatives, to determine how the coefficients of the
decomposition vary under a frame transformation. The result is

$\displaystyle\delta\mathcal{E}$
$\displaystyle=0\qquad\delta\mathcal{P}=0\qquad\delta\mathcal{N}=0$ (2.12a)
$\displaystyle\delta\mathcal{Q}^{\mu}$
$\displaystyle=-(\mathcal{E}+\mathcal{P})\delta
u^{\mu}\qquad\delta\mathcal{J}^{\mu}=-\mathcal{N}\delta u^{\mu}$ (2.12b)
$\displaystyle\delta\mathcal{T}^{\mu\nu}$ $\displaystyle=0$ (2.12c)

From these expressions it is immediately clear that one can always pick
$\delta u^{\mu}$ such that $\mathcal{J}^{\mu}=0$ or $\mathcal{Q}^{\mu}=0$. The
first case is often called Eckart frame [168, 11], while the second choice is
called Landau frame [7].

While frames are not physical in the hydrodynamic regime, they are simply
understood as redefinitions of the fields we use to parametrize the observable
quantities $T^{\mu\nu}$ and $J^{\mu}$, they can still have a physical
interpretation. In particular the Eckart frame is defined so that there is no
charge flow in the local rest frame of the fluid, while the Landau frame is
such that there is no energy current in the local rest frame of the fluid.
There are in principle infinite frames, however in practice only a handful of
these are really used in the literature. In particular two very useful ones
are the thermodynamic frame [98], which is actually a class of frame that we
will discuss later on, and the recently introduced BDNK frame [71, 69].

The first equation in (2.12) implies that
$\varepsilon(T,\mu)+f_{\mathcal{E}}(\partial T,\partial\mu,\partial
u)=\varepsilon(T^{\prime},\mu^{\prime})+f^{\prime}_{\mathcal{E}}(\partial
T^{\prime},\partial\mu^{\prime},\partial u^{\prime})$ and similar expressions
hold also for $\mathcal{N}$ and $\mathcal{P}$. This means that we can write

$\displaystyle f^{\prime}_{\mathcal{E}}$
$\displaystyle=f_{\mathcal{E}}-\left(\frac{\partial\varepsilon}{\partial
T}\right)_{\mu}\delta
T-\left(\frac{\partial\varepsilon}{\partial\mu}\right)_{T}\delta\mu$ (2.13a)
$\displaystyle f^{\prime}_{\mathcal{P}}$
$\displaystyle=f_{\mathcal{P}}-\left(\frac{\partial p}{\partial
T}\right)_{\mu}\delta T-\left(\frac{\partial
p}{\partial\mu}\right)_{T}\delta\mu$ (2.13b) $\displaystyle
f^{\prime}_{\mathcal{N}}$ $\displaystyle=f_{\mathcal{N}}-\left(\frac{\partial
n}{\partial T}\right)_{\mu}\delta T-\left(\frac{\partial
n}{\partial\mu}\right)_{T}\delta\mu$ (2.13c)

to relate the primed functions in terms of the old ones plus the frame choice
derivative corrections. In particular, we can always pick the out-of-
equilibrium definition of $\mu$ and $T$ such that two of the three functions
$f$ are zero. It is standard to set to zero $f^{\prime}_{\mathcal{E}}$ and
$f^{\prime}_{\mathcal{N}}$, such that $\mathcal{E}=\varepsilon$ and
$\mathcal{N}=n$ at all orders in derivatives.

####  2.1.5 Order one: Navier-Stokes

The ideal fluid we have discussed so far, however, predicts unphysical
solutions: if we only consider the constitutive relations truncated at order
zero in derivatives, then we find solutions to the conservation equations of
hydrodynamics that describe impossible flows. For example, an equilibrated
fluid flowing in the $x$ direction with a velocity gradient along $y$ is a
valid solution to the ideal fluid hydrodynamic equations, but it is unphysical
in actual systems, since fluid particles will transfer momentum between the
fluid layers and render the flow homogeneous, see Figure 2.1. Hence, it is
imperative to study the first-order dissipative corrections, for which such
flows are not equilibrium solutions.

Figure 2.1: Figure taken from [74].

We will work in the Landau frame, i.e. we will choose the out-of-equilibrium
definition of $u^{\mu}$ to be such that $\mathcal{Q}^{\mu}=0$ and furthermore
we also fix the definitions of $\mu$ and $T$ in such a way that
$\mathcal{E}=\varepsilon$ and $\mathcal{N}=n$. The Landau frame matching
conditions are usually written as

$u_{\mu}T^{\mu\nu}=-\varepsilon u^{\nu}\qquad\qquad u_{\mu}J^{\mu}=-n$ (2.14)

so that the fluid velocity is an eigenvector of the stress-energy tensor with
the energy as eigenvalue.

Following the hydrodynamic EFT prescription, we write the free coefficients
$\mathcal{P}$, $\mathcal{J}^{\mu}$, $\mathcal{T}^{\mu\nu}$ (those that are not
fixed by the choice of the frame) as the most general expressions in terms of
the derivatives of the fundamental fields. There are only three one-derivative
scalars at this order, namely the matter derivative of temperature and
chemical potential $u^{\alpha}\partial_{\alpha}T$,
$u^{\alpha}\partial_{\alpha}\mu$ and the expansion
$\theta=\partial_{\alpha}u^{\alpha}$. There are furthermore three transverse
vectors $\Delta^{\mu\nu}\partial_{\nu}T$, $\Delta^{\mu\nu}\partial_{\nu}\mu$
and $\Delta^{\mu\nu}a_{\nu}$ where
$a_{\mu}=u^{\alpha}\partial_{\alpha}u_{\mu}$ is the acceleration. To conclude,
there is only one traceless symmetric order-one tensor we can build, the shear
tensor

$\sigma^{\mu\nu}=\Delta^{\mu\alpha}\Delta^{\nu\beta}\left(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}-\frac{2}{d}\eta_{\alpha\beta}\partial_{\lambda}u^{\lambda}\right)$
(2.15)

Let us focus on the scalar part first. We already fixed $\mathcal{E}$ and
$\mathcal{N}$ from the Landau matching conditions, so we only have one scalar
left, $\mathcal{P}$, which should be written as the most generic expression in
terms of the order-zero and order-one scalars listed above.

$\mathcal{P}=P+c_{1}u^{\alpha}\partial_{\alpha}T+c_{2}u^{\alpha}\partial_{\alpha}\mu+c_{3}\theta+\mathcal{O}(\partial^{2})$
(2.16)

However, we remember that hydrodynamics should be understood as a formal
series, and by using the ideal fluid constitutive relations (2.7) in the two
scalars equations $\partial_{\mu}J^{\mu}=0$ and
$u_{\mu}\partial_{\nu}T^{\mu\nu}=0$ we find two equations that relate the
three order-one scalars. This means that, e.g. $u^{\alpha}\partial_{\alpha}T$
is proportional to $u^{\alpha}\partial_{\alpha}\mu$, up to corrections that
are higher order in derivatives $\mathcal{O}(\partial^{2})$, and the same
holds for the expansion. Thus, at order one in derivatives there is only one
independent scalar, and it is a convention to keep $\partial_{\mu}u^{\mu}$ as
the independent one. We can finally write down $\mathcal{P}$ as the order-zero
term plus the first-order correction

$\mathcal{P}=P-\zeta\partial_{\mu}u^{\mu}+\mathcal{O}(\partial^{2})$ (2.17)

where $\zeta=\zeta(T,\mu)$, the bulk viscosity, is the first transport
coefficient we encounter.

We now turn to the only transverse vector left unspecified, namely
$\mathcal{J}^{\mu}$. As before, there are three transverse vectors at order
one in derivatives, however we also have one transverse ideal-fluid equation
of motion, $\Delta_{\mu\nu}\partial_{\alpha}T^{\alpha\nu}=0$. This relates the
three order-one vectors up to higher-order corrections, which again means we
only have two independent terms. Thus, we write

$\mathcal{J}^{\mu}=-\sigma
T\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}+\chi_{T}\Delta^{\mu\nu}\partial_{\nu}T+\mathcal{O}(\partial^{2})$
(2.18)

where we reorganized the vectors in a way that will be useful later on. For
each term we introduced a new transport coefficient: the charge conductivity
$\sigma$ and $\chi_{T}$.

To end our discussion, we observe that since the shear tensor (2.15) is the
only transverse traceless symmetric tensor, it must be proportional to
$\mathcal{T}^{\mu\nu}$, hence we write

$\mathcal{T}^{\mu\nu}=-\eta\sigma^{\mu\nu}+\mathcal{O}(\partial^{2})$ (2.19)

where $\eta$, the shear viscosity, is the last transport coefficient that
appears in standard relativistic hydrodynamics.

Had we chosen Eckart frame, we would have found the same expressions for
$\mathcal{T}^{\mu\nu}$ and $\mathcal{P}$, instead what we wrote for
$\mathcal{J}^{\mu}$ would now appear for $\mathcal{Q}^{\mu}$. Notice that
there are actually two ambiguity in hydrodynamics: one is related to the
choice of frame (in this case, Eckart or Landau), while the other is related
to fixing the base of independent coefficients used to write the constitutive
relations. For example, we have decided not to use the acceleration in writing
the constitutive relation for $\mathcal{J}^{\mu}$, however when working in the
Eckart frame it is more common to use the acceleration instead of
$\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}$. This choice might seem
innocuous, however one must be careful when computing quantities in
hydrodynamics to check that they are independent of these ambiguities by
working at the correct derivative order [74].

A comment on transport coefficients: these are quantities which cannot be
determined from hydrodynamics alone333This is not true for certain transport
coefficients related to quantum anomalies [34]., since they depend on the
microscopic details of the theory, while hydrodynamics knows only about the IR
macroscopic dynamics. Given a specific microscopic model one can compute these
coefficients via Kubo formulae (at least in principle), however from the point
of view of hydrodynamics they are just unknown functions of the scalar fields.

The Kubo formulae are independent of the choice of the frame and can be used
to match the microscopic theory to the EFT of hydrodynamics. This means that
the transport coefficients are frame invariant: they can appear in different
places in the constitutive relations, depending on the choice of the frame,
but their form in terms of the microscopic theory is unique and independent
this choice.

####  2.1.6 Entropy current

The constitutive relations we obtained above are based only on two consistency
requirements: Lorentz covariance and the formal derivative expansion.
Hydrodynamics however requires more conditions, in particular when developing
a theory of hydrodynamics one also imposes a local version of the second law
of thermodynamics, namely we will require that the production of entropy is
locally positive. Later on we will also discuss one last kind of constraint
that stems from the discrete symmetries of the microscopic theory.

In equilibrium, we can write the entropy current $S^{\mu}=su^{\mu}$ for
constant $u^{\mu}$, and we showed that the ideal fluid does not produce
entropy $\partial_{\mu}(su^{\mu})=0$. In first-order hydrodynamics we assume
that $S^{\mu}$ receives derivative corrections, consistent with what we would
expect by covariantizing the thermodynamics. In particular, we start from the
Euler relation $Ts=P+\varepsilon-\mu n$, and we write it in a covariant form
[168, 166]

$TS^{\mu}=Pu^{\mu}-T^{\mu\nu}u_{\nu}-\mu J^{\mu}$ (2.20)

From this expression, using the standard decomposition (2.4), we obtain for
the entropy current

$S^{\mu}=\left[s+\frac{1}{T}\left(\mathcal{E}-\varepsilon\right)-\frac{\mu}{T}\left(\mathcal{N}-n\right)\right]u^{\mu}+\frac{1}{T}\mathcal{Q}^{\mu}-\frac{\mu}{T}\mathcal{J}^{\mu}$
(2.21)

where the terms inside the square brackets is the rest frame entropy density,
while the other two terms represent the rest frame entropy flow. This is the
so-called canonical entropy current, and it is frame invariant (as always, the
explicit form of the current in terms of the gradients may vary depending on
the frame, but the value of the current does not). There are other
contributions to the entropy current, called non-canonical, which have
important roles when discussing hydrostatic or non-dissipative fluids.

In the specific case of the Landau frame the entropy current simplifies to
[14, 74, 7]

$S^{\mu}=su^{\mu}-\frac{\mu}{T}\mathcal{J}^{\nu}$ (2.22)

We must now use this expression, together with the Landau-frame constitutive
relations, to impose the local form of the second law of thermodynamics

$\partial_{\mu}S^{\mu}\geq 0$ (2.23)

on all possible flows that are solutions to the equations of motion (2.2a) and
(2.2b).

The steps are very straightforward, using the Gibbs–Duhem equation $\partial
P=s\partial T+n\partial\mu$ one arrives at

$\displaystyle\partial_{\mu}S^{\mu}$
$\displaystyle=\frac{\zeta}{T}\theta^{2}+\frac{\eta}{T}\sigma_{\mu\nu}\sigma^{\mu\nu}+\sigma\left(T\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}\right)^{2}-T\chi_{T}\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}\partial_{\mu}T\geq
0$ (2.24)

where we remind that $\theta$ is the expansion and $\sigma_{\mu\nu}$ the shear
tensor. Because the RHS must always be non-negative, we immediately find that

$\eta\geq 0\qquad\zeta\geq 0\qquad\sigma\geq 0\qquad\chi_{T}=0$ (2.25)

Hence, while Lorentz covariance leaves us with four transport coefficients at
first order in derivatives for relativistic hydrodynamics, positivity of
entropy production reduces this number to only three.

For future reference, we write here the complete first-order constitutive
relations in the Landau frame for a normal fluid that stem from the
requirement of Lorentz covariance, gradient expansion and positivity of
entropy production. They are

$\displaystyle T^{\mu\nu}$ $\displaystyle=\varepsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}-\eta\sigma^{\mu\nu}-\zeta\Delta^{\mu\nu}\partial_{\lambda}u^{\lambda}+\mathcal{O}(\partial^{2})$
(2.26a) $\displaystyle J^{\mu}$ $\displaystyle=nu^{\mu}-\sigma
T\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}+\mathcal{O}(\partial^{2})$ (2.26b)

There is a slightly different route to arrive at the same constitutive
relations that is sometimes quicker: after fixing the frame one can skip
completely the steps in Section 2.1.5 and instead compute directly the
divergence of the canonical entropy current. Then, by looking at the final
expression, one can infer the form of the constitutive relations. This is the
approach used e.g. in [7].

####  2.1.7 Hydrostatic generating functional

The hydrostatic generating functional [98, 99] is a powerful technique that
can be used to compute the $n$-point functions (hence also the constitutive
relations) from a path-integral approach in the hydrostatic regime, i.e. when
everything is time independent and, in the absence of external sources,
constant.

The hydrostatic flow, together with the dissipative sector described in the
previous Section 2.1.5 on Navier-Stokes fluids, are two of the eight possible
class of fluid transport classified in [47, 48] using the idea of adiabatic
flows.

Consider a charged fluid in the presence of spacetime and gauge sources
(2.58). The fluid is defined to be in (time-independent) hydrostatic
equilibrium if there exists a timelike Killing vector $V^{\mu}$ that
parametrizes time-translations such that all the thermodynamic quantities and
the external sources have zero Lie derivative $\mathcal{L}_{V}$ with respect
to the Killing field $V^{\mu}$. Physically, this means that the sources are
slowly varying and that the time derivative in the frame comoving with
$V^{\mu}$, when $V^{\mu}=(1,\boldsymbol{\mathbf{0}})$, is zero.

Because we are dealing with equilibrium configurations (in the presence of
sources that are adiabatically turned on) we can formally obtain the Green
functions from a generating functional by differentiating with respect to the
sources $g_{\mu\nu}$ and $A_{\mu}$.

In the frame where $V^{\mu}=(1,\boldsymbol{\mathbf{0}})$, consider the set of
zero-frequency $n$-point functions truncated up to order $m$ in powers of the
wavevector $\boldsymbol{\mathbf{k}}$. We can inverse-Fourier transform to
obtain a set of correlators in position space which are approximately valid on
scales much larger than the microscopic correlation length of the system. By
formally integrating this set of approximate Green functions we arrive to the
equilibrium generating functional for the truncated correlators that is a
local function of the sources

$W_{m}[g,A]=\int\mathop{}\\!\mathrm{d}^{d}x\ \mathcal{W}[\text{sources}(x)]$
(2.27)

where the function $\mathcal{W}$ contains terms up to order $m$ in
derivatives. Because this expression is covariant, it holds also for frames
which are not comoving with $V^{\mu}$.

In order for $W_{m}$ to reproduce the equations of motion (2.58) it must be
diffeomorphism and gauge invariant, hence $\mathcal{W}$ is a function of all
the possible diffeomorphism and gauge invariant scalars of the theory. These
scalar observables should be local in space, but can be non-local in Euclidean
time, in order to describe finite temperature systems.

Consider then the two quantities: the invariant length $L$ of the time circle
in the Euclidean theory and the Polyakov loop $P_{A}$ of the gauge field
$\mathrm{U(1)}$ around the same direction. The length $L$ can be computed
simply as

$L=\int_{0}^{\beta}\mathop{}\\!\mathrm{d}\tau\sqrt{g_{\tau\tau}}$ (2.28)

in the frame comoving with $V^{\mu}$, where $\beta=T_{0}^{-1}$ is the period
of the coordinate in the time direction. Then, covariantizing the expression
and rotating back to real time, we find that $L$ and $P_{A}$ are written as

$L=\beta\sqrt{-V^{2}}\qquad\qquad\ln P_{A}=\beta V^{\mu}A_{\mu}$ (2.29)

We can now identify the temperature, chemical potential and fluid velocity
from these invariant quantities as

$\displaystyle T$ $\displaystyle=\frac{1}{L}=\frac{T_{0}}{\sqrt{-V^{2}}}$
(2.30a) $\displaystyle\mu$ $\displaystyle=\frac{\ln
P_{A}}{L}=\frac{V^{\mu}A_{\mu}-\Lambda_{\beta}}{\sqrt{-V^{2}}}$ (2.30b)
$\displaystyle u^{\mu}$ $\displaystyle=\frac{V^{\mu}}{\sqrt{-V^{2}}}$ (2.30c)

where $\Lambda_{\beta}$ is a gauge parameter that must be included to preserve
gauge invariance. These expressions can be understood as the thermal twist and
thermal vector, in the presence of sources, that appear in the equilibrium
statistical mechanics perspective introduced in (2.3). Notice that, contrary
to (2.3), these expressions for $T$, $\mu$ and $u^{\mu}$ have a dependence on
spacetime coordinates inherited from the sources $g_{\mu\nu}$ and $A_{\mu}$.

For a given derivative order $m$ the theory has $N_{m}$ scalars quantities
$s_{m,1},s_{m,2},\dots,s_{m,N_{m}}$ constructed from the invariants (2.30) and
their derivatives. Therefore, the most general hydrostatic generating
functional at order $m$ takes the form

$W_{m}[g,A]=\int\mathop{}\\!\mathrm{d}^{d}x\sqrt{-g}\left[P(s_{0})+\sum_{n=1}^{m}\sum_{i=1}^{N_{n}}F_{n,i}(s_{0})s_{n,i}\right]$
(2.31)

In this expression: $s_{0}$ are the order-zero scalars and for a simple fluid
they are $s_{0}=\\{T,\mu\\}$, $F_{n,i}$ are unknown functions of the order-
zero scalars, while $P$ can be identified with the pressure. Indeed, at zero
sources, when all the derivative corrections to $W_{m}$ vanish, the functional
$W$ corresponds to the free energy.

Given this expression we can now compute the hydrostatic constitutive
relations for the stress-energy tensor and the current, the one-point
functions, by varying $W_{m}$ with respect to the metric and gauge field
respectively

$\langle T^{\mu\nu}\rangle=\frac{2}{\sqrt{-g}}\frac{\delta W_{m}}{\delta
g_{\mu\nu}}\quad\qquad\langle J^{\mu}\rangle=\frac{1}{\sqrt{-g}}\frac{\delta
W_{m}}{\delta A_{\mu}}$ (2.32)

Thanks to the requirement of gauge and diffeomorphism invariance of $W_{m}$
the hydrostatic constitutive relations obtain from the above variations will
identically satisfy the hydrodynamic equations of motion.

The constitutive relations obtained from the hydrostatic generating functional
appear in a specific hydrodynamic frame called thermodynamic frame. Contrary
to the Landau frame, defined by the matching conditions (2.14), the
thermodynamic frame physically means that the definitions of temperature,
chemical potential and fluid velocity are not modified on shell with respect
to their equilibrium value (2.30). Notice that, while it is common to use the
term thermodynamic frame, this is not a true frame, but rather a class of
frame. This is because the thermodynamic frame, by definition, is a
prescription for how to deal with the hydrostatic sector of the constitutive
relations, but it says nothing about the dissipative parts, for which one
needs further frame fixing conditions.

One might also wonder what happens to the entropy current in the hydrostatic
sector. The hydrostatic solutions are, as expected, non-dissipative, hence
$\partial_{\mu}S^{\mu}=0$, however in order to achieve this results the
entropy current itself must be modified compared to its canonical value. The
total entropy current is

$S^{\mu}=S^{\mu}_{\text{c.}}+S^{\mu}_{\text{n.c.}}$ (2.33)

where the first term is the canonical entropy current discussed in (2.21),
while the latter is the so-called non-canonical entropy current and its role
is to ensure that the hydrostatic derivative corrections do not produce
entropy [173, 174].

####  2.1.8 Hydrostatic conditions

For the fluid to be in hydrostatic flows, its variables must obey certain
conditions which in a sense define equilibrium. These constraints are obtained
by requiring that the Lie derivatives of the sources and of the thermodynamic
quantities (2.30) with respect to the Killing vector $V^{\mu}$ vanish. To
proceed, we first decompose the $\mathrm{U(1)}$ electromagnetic field strength
by projecting along the fluid velocity in ($3+1$)-dimensions

$F_{\mu\nu}=u_{\mu}E_{\nu}-u_{\nu}E_{\mu}-\epsilon_{\mu\nu\rho\sigma}u^{\rho}B^{\sigma}$
(2.34)

where we defined the covariant electric and magnetic fields as

$E_{\mu}=F_{\mu\nu}u^{\nu}\qquad
B_{\mu}=-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}u_{\nu}F_{\alpha\beta}$ (2.35)

Notice that both $E^{\mu}$ and $B^{\mu}$ are transverse to $u^{\mu}$. Then,
requiring $\mathcal{L}_{V}T=\mathcal{L}_{V}\mu=0$ give use the first two
hydrostatic constraints [98]

$\displaystyle\partial_{\mu}T$ $\displaystyle=-Ta_{\mu}$ (2.36a)
$\displaystyle\partial_{\mu}\mu$ $\displaystyle=E_{\mu}-\mu a_{\mu}$ (2.36b)
where $a^{\mu}=u^{\lambda}\nabla_{\lambda}u^{\mu}$ is the acceleration. The
other constraints come from studying the Lie derivative of the fluid velocity,
for which we find that an equilibrium fluid must have vanishing expansion and
shear tensor $\theta=\nabla_{\mu}u^{\mu}=0\qquad\sigma^{\mu\nu}=0$ (2.36c)

Finally, we will require the Bianchi identity of the gauge field,
$\epsilon^{\mu\nu\alpha\beta}\nabla_{\nu}F_{\alpha\beta}=0$.

These are equality-type constraints that the fluid variables must obey for the
fluid to be in equilibrium in the presence of external sources. In particular,
any dissipative correction to the constitutive relations must be understood as
fluctuating away from these constraints. For example, in equilibrium we find
that the fluid must have zero expansion $\theta=0$, which agrees with the fact
that the bulk viscosity is a dissipative transport coefficient and the fluid
produce entropy when expansion is not zero. For the vector sector, second law
of thermodynamics tells us that the two constraints (2.36) do not give rise to
two independent dissipative transport coefficients, but to a single
dissipative transverse vector

$E^{\mu}-\Delta^{\mu\nu}\partial_{\nu}\frac{\mu}{T}=0$ (2.37)

that appears in the current constitutive relation (2.2.2).

Notice that the opposite is not true: there exist transport coefficients which
are non-dissipative (they drop out of the equation when computing the
positivity of entropy production), but nonetheless they vanish on hydrostatic
flows. These are called non-dissipative non-hydrostatic terms and can be
obtained from a generalization of the above generating functional [47].

###  2.2 Linear Response Theory

Hydrodynamics is a highly non-linear theory, meaning that small differences in
the initial conditions can lead to drastically different evolutions of the
fluid dynamics. This makes solving the full non-linear equations of
hydrodynamics notoriously hard, with entire careers dedicated to numerically
solving fluid flows.

As we discussed in the introduction, the problems that appear in hydrodynamics
are often related to trusting the theory far beyond its regime of validity.
Hydrodynamics is a low-energy EFT that works only when $\omega$ and
$\boldsymbol{\mathbf{k}}$ are small, nonetheless if one wishes to use the
equations of hydrodynamics as a system of non-linear PDEs with predictive
power, one needs to solve the equations even at high momenta, when we expect
hydrodynamics to fail.

Things are instead better behaved if we focus on linearized hydrodynamics.
This means that we trust the theory only in its regime of validity: we
consider a uniform global thermodynamic equilibrium state given by a constant
temperature $T=\text{const}$, chemical potential $\mu=\text{const}$ and in the
rest frame $u^{\mu}=(1,\boldsymbol{\mathbf{0}})$ and study the small
fluctuations above this solution. Furthermore, we will only care about the
small-wavevector/small-frequency expansion, which is where hydrodynamics is
expected to hold.

Linearized hydrodynamics still gives plenty of information, in particular: (i)
it gives access to the retarded correlators, which can then be matched against
the same two-point functions obtained from other approaches (holography,
kinetic theory, QFT, …), leading to the Kubo formulae, (ii) the same
correlators give physical information about the transport properties of the
system, like the thermoelectric conductivities, (iii) it allows us to find new
constraints on the constitutive relations from the microscopic discrete
symmetries, and (iv) it gives information about the stability of the theory to
small perturbations.

For most of this section instead of working with $T$, $\mu$ and $u^{\mu}$ we
will consider their thermodynamically conjugate charges as independent
variables, namely energy, charge and momentum density. The only reason to do
so is that the latter quantities have a microscopic definition in terms of
operators $\hat{T}^{0\mu}$ and $\hat{J}^{0}$, which allows us to study how
they appear in the Hamiltonian and obtain the correlators associated to these
charges.

####  2.2.1 Martin-Kadanoff method

In this section we will quickly resume the main idea behind linear response
theory, that is to describe the response of a system in thermal equilibrium to
small perturbations, based on the classical paper by Martin and Kadanoff [175]
from which this method takes its name. We consider a set of hydrodynamic
fields $\phi_{a}(t,\boldsymbol{\mathbf{x}})$ (that are microscopically well-
defined) and we couple them to their (weak slowly-varying) sources
$\lambda_{a}(t,\boldsymbol{\mathbf{x}})$. The sources are turned on at
$t=-\infty$ and are adiabatically increased; at $t=0$ they are turned off and
the system is let free to evolve. In linearized hydrodynamics the classical
fields $\phi_{a}$ are described by a set of linear equations that, in momentum
space, read444Here the fields $\phi_{a}$ whose dynamics is described by
linearized hydrodynamics are actually $\delta\phi_{a}$, that is the difference
with respect to the equilibrium
$\delta\phi_{a}(t,\boldsymbol{\mathbf{x}})=\phi_{a}(t,\boldsymbol{\mathbf{x}})-\phi_{a}$.

$\partial_{t}\phi_{a}(t,\boldsymbol{\mathbf{k}})+M_{ab}(\boldsymbol{\mathbf{k}})\phi_{b}(t,\boldsymbol{\mathbf{k}})=0$
(2.38)

where $M_{ab}$ is a matrix determined by the conservation laws of
hydrodynamics and the constitutive relations. This set of equations is valid
in the hydrodynamic regime (long wavelength
$\boldsymbol{\mathbf{k}}\rightarrow 0$ and small frequency $\omega\rightarrow
0$), and describes the free evolution of the fields at $t>0$. To study the
solutions we perform a Laplace transform in time

$\phi_{a}(z,\boldsymbol{\mathbf{k}})=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}t\
e^{izt}\phi_{a}(t,\boldsymbol{\mathbf{k}})$ (2.39)

with $z$ that has to lie on the upper half of the complex plane for the
integral to converge. Transforming the set of equation we find

$(-iz\delta_{ab}+M_{ab})\phi_{b}(z,\boldsymbol{\mathbf{k}})=\phi_{a}^{0}(\boldsymbol{\mathbf{k}})$
(2.40)

where we defined
$\phi^{0}_{a}(\boldsymbol{\mathbf{k}})=\phi_{a}(t=0,\boldsymbol{\mathbf{k}})$,
the value of the field just when the source is turned off. The values at $t=0$
of the hydrodynamic variables are related to the initial values of the
sources, that is just before we turn the sources off we have a relation that
is local in space. For small fluctuations we can write

$\phi_{a}^{0}(\boldsymbol{\mathbf{k}}\rightarrow
0)=\chi_{ab}\lambda_{b}^{0}(\boldsymbol{\mathbf{k}}\rightarrow
0)\qquad\Longrightarrow\qquad\chi_{ab}=\left(\frac{\partial\phi_{a}}{\partial\lambda_{b}}\right)$
(2.41)

$\chi_{ab}$ is called static thermodynamic susceptibility, it is not a
dynamical response, but rather a thermodynamic quantity that depends on the
system. With this definition the hydrodynamic equations can be formally solved
in terms of the initial conditions

$\phi_{a}(z,\boldsymbol{\mathbf{k}})=(K^{-1})_{ab}\chi_{bc}\lambda_{c}^{0}(\boldsymbol{\mathbf{k}})$
(2.42)

with $K_{ab}=-iz\delta_{ab}+M_{ab}(\boldsymbol{\mathbf{k}})$.

We now try to relate the expectation values of the hydrodynamic fields to the
correlation functions of the system. Imagine perturbing the fields using
slowly-varying (with respect to both $\boldsymbol{\mathbf{x}}$ and $t$)
sources that are adiabatically turned on at $t=-\infty$ and then switched off
at $t=0$, for example with $\lambda(t,\boldsymbol{\mathbf{x}})=e^{\epsilon
t}\lambda^{0}(\boldsymbol{\mathbf{x}})\theta(-t)$ ($\epsilon>0$ is a small
adiabatic parameter and $\theta(t)$ is the step function); we can then link
the evolution of the system at $t>0$ to the result we found in equation
(2.42). To do so we perturb the Hamiltonian with a term that couples the
hydrodynamic fields to their sources

$\delta\hat{H}(t)=-\int\mathop{}\\!\mathrm{d}^{d}\boldsymbol{\mathbf{x}}\
\lambda_{a}(t,\boldsymbol{\mathbf{x}})\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}})$
(2.43)

From first-order time-dependent perturbation theory in Quantum Mechanics we
can find the response of an operator
$\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}})$ in the Heisenberg picture. If the
system has a time-independent Hamiltonian, and we add the small perturbation
above the expected value of the observable changes as

$\delta\langle\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}})\rangle=-i\int_{-\infty}^{t}\mathop{}\\!\mathrm{d}t^{\prime}\langle[\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}}),\delta\hat{H}(t^{\prime})]\rangle$
(2.44)

where $\langle\dots\rangle$ stands for the thermal average
$\langle\hat{O}\rangle=\text{Tr}(\hat{\rho}\hat{O})$ with $\hat{\rho}$ the
density matrix (we work in the grand canonical ensemble, so the Heisenberg
operators are defined with the Hamiltonian
$\hat{H}^{\prime}=\hat{H}-\mu\hat{N}$, with the chemical potential $\mu$ as
the Lagrange multiplier of the charge number operator $\hat{N}$). Plugging
together the two equations we find

$\delta\langle\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}})\rangle=-\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}t^{\prime}\int\mathop{}\\!\mathrm{d}^{d}x^{\prime}\
G^{R}_{ab}(t-t^{\prime},\boldsymbol{\mathbf{x}}-\boldsymbol{\mathbf{x}}^{\prime})\lambda_{b}(t,\boldsymbol{\mathbf{x}})$
(2.45)

where $G^{R}_{ab}$ is called retarded response function, and it is defined as

$G^{R}_{ab}(t-t^{\prime},\boldsymbol{\mathbf{x}}-\boldsymbol{\mathbf{x}}^{\prime})=-i\theta(t-t^{\prime})\langle[\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}}),\hat{\phi}_{b}(t^{\prime},\boldsymbol{\mathbf{x}}^{\prime})]\rangle$
(2.46)

It depends only on the differences of the coordinates, due to its invariance
under spacetime translations. In Fourier space the expectation value
simplifies and the convolution gives

$\delta\langle\hat{\phi}_{a}(\omega,\boldsymbol{\mathbf{k}})\rangle=-G^{R}_{ab}(\omega,\boldsymbol{\mathbf{k}})\lambda_{b}(\omega,\boldsymbol{\mathbf{k}})$
(2.47)

We still have to Laplace-Fourier transform equation (2.45) to link this result
with the one obtained from linearized hydrodynamics.

We start by performing the Fourier transform in space; with our assumption on
the form of the sources we find

$\langle\hat{\phi}_{a}(t,\boldsymbol{\mathbf{k}})\rangle=-\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}t^{\prime}e^{\epsilon
t^{\prime}}G^{R}_{ab}(t-t^{\prime},\boldsymbol{\mathbf{k}})\lambda_{b}^{0}(\boldsymbol{\mathbf{k}})$
(2.48)

We now Fourier transform only the retarded function in time

$G^{R}(t-t^{\prime},\boldsymbol{\mathbf{k}})=\int_{-\infty}^{\infty}\frac{\mathop{}\\!\mathrm{d}\omega}{2\pi}G^{R}(\omega,\boldsymbol{\mathbf{k}})e^{-i\omega(t-t^{\prime})}$
(2.49)

Because of the step function in the definition of
$G^{R}(t,\boldsymbol{\mathbf{k}})$, this function is identically zero for
$t<0$, this in turn means that $G^{R}(\omega,\boldsymbol{\mathbf{k}})$ is an
analytic function in the upper half-plane of complex $\omega$, and we are free
to analytically continue $G^{R}(\omega,\boldsymbol{\mathbf{k}})$ to the whole
plane. After performing the $t^{\prime}$ integral we are left with

$\langle\hat{\phi}_{a}(t,\boldsymbol{\mathbf{k}})\rangle=-\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})\int\frac{\mathop{}\\!\mathrm{d}\omega}{2\pi}G^{R}_{ab}(\omega,\boldsymbol{\mathbf{k}})\frac{e^{-i\omega
t}}{i\omega+\epsilon}$ (2.50)

where $\epsilon$ is needed for the convergence of the integral. We can now
multiply both sides by $e^{izt}$ (with the prescription that $\text{Im}(z)>0$
for the integral to convergence) and integrate over $t$ from $0$ to $\infty$
(that is, we perform the same Laplace transform we applied earlier to our
linearized hydrodynamic equations)

$\langle\hat{\phi}_{a}(z,\boldsymbol{\mathbf{k}})\rangle=-\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})\int\frac{\mathop{}\\!\mathrm{d}\omega}{2\pi}\frac{G^{R}_{ab}(\omega,\boldsymbol{\mathbf{k}})}{(i\omega+\epsilon)(i(\omega-z)+\epsilon)}$
(2.51)

As we mentioned, $G^{R}(\omega)$ is analytic in the upper half-plane, this
allows us to perform the integral by closing the contour with
$\text{Im}(\omega)>0$. There are two poles inside the region, one at
$\omega=i\epsilon$ and one at $\omega=z+i\epsilon$. The residue theorem gives
us

$\langle\hat{\phi}_{a}(z,\boldsymbol{\mathbf{k}})\rangle=-\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})\frac{G^{R}_{ab}(z,\boldsymbol{\mathbf{k}})-G^{R}_{ab}(z=0,\boldsymbol{\mathbf{k}})}{iz}$
(2.52)

where the argument $z=0$ is meant to be taken slightly above the real axis. We
still need to find what $G^{R}(z=0,\boldsymbol{\mathbf{k}})$ is, so we look at
equation (2.48) evaluated at $t=0$. The argument of the integral is the
Laplace transform evaluated at $z=0$ (it is again meant to be slightly above
the real axis, from $z=i\epsilon$).

$\langle\hat{\phi}_{a}(t=0,\boldsymbol{\mathbf{k}})\rangle=-\int_{0}^{\infty}\mathop{}\\!\mathrm{d}t^{\prime}\
e^{-\epsilon
t^{\prime}}G^{R}_{ab}(t^{\prime},\boldsymbol{\mathbf{k}})\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})=-G^{R}_{ab}(z=0,\boldsymbol{\mathbf{k}})\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})$
(2.53)

From this equation we find that, in the long wavelength limit,
$G^{R}(z=0,\boldsymbol{\mathbf{k}})$ is minus the static susceptibility
$\chi(\boldsymbol{\mathbf{k}})$ defined above (2.41),
$G^{R}(z=0,\boldsymbol{\mathbf{k}})=-\chi(\boldsymbol{\mathbf{k}})$.

We can finally compare this result with our earlier expression from linearized
hydrodynamics obtaining

$-\frac{1}{iz}\bigl{(}G^{R}_{ab}(z,\boldsymbol{\mathbf{k}})+\chi_{ab}(\boldsymbol{\mathbf{k}})\bigr{)}\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})=(K^{-1})_{ac}\chi_{cb}\lambda^{0}_{b}(\boldsymbol{\mathbf{k}})$
(2.54)

and we find an expression for the retarded response function matrix

$G^{R}_{ab}(z,\boldsymbol{\mathbf{k}})=-(\delta_{ac}+iz(K^{-1})_{ac})\chi_{cb}$
(2.55)

This function is always analytic in the upper half-plane of complex $z$, hence
we can define $G^{R}(\omega,\boldsymbol{\mathbf{k}})$ in the whole complex
plane as the analytical continuation of $G^{R}(z,\boldsymbol{\mathbf{k}})$
from the upper half-plane. $G^{R}$ has many analytic properties and can be
related to other Green functions (advanced and symmetric) [74], in particular
it is possible to show that

$-\Im G^{R}_{aa}(\omega,\boldsymbol{\mathbf{k}})\geq
0\qquad\text{for}\quad\omega\geq 0$ (2.56)

This result, when applied to the hydrodynamic response functions, implies that
the transport coefficients are all non-negative: $\sigma\geq 0$, $\zeta\geq 0$
and $\eta\geq 0$, without the need to define an entropy current.

####  2.2.2 Variational method

In the previous section we presented the classical approach to compute the
retarded Green functions from hydrodynamics using linear response theory, by
introducing sources for the conserved charge densities that follow from
equilibrium thermodynamics. This method has the advantage of being very
transparent from a physical perspective, however it has a couple of drawbacks,
the most important one being that it is not possible to obtain all the
correlators associated with the currents [74, 68].

There is a second method to obtain the same quantities from a field theory
perspective, which introduces sources that couple directly to the covariant
form of the currents $T^{\mu\nu}$ and $J^{\mu}$. Given a generic theory we can
always add to it external sources, in particular we will consider a curved
metric $g_{\mu\nu}$ and a gauge field $A_{\mu}$. Then we can formally
construct a generating functional $W[A,g]$ whose variations give us all the
connected $n$-point functions of the theory.

By definition, varying with respect to the metric or the gauge field gives us
the one-point functions associated with the stress-energy tensor and the
current

$\delta W[A,g]=\int\mathop{}\\!\mathrm{d}x\sqrt{-g}\left(T^{\mu\nu}\delta
g_{\mu\nu}+J^{\mu}\delta A_{\mu}\right)$ (2.57)

Furthermore, when the metric and gauge fields variations are associated to
diffeomorphism or gauge symmetry, the invariance of $W[A,g]$ under these
operations gives us the conservation equations of hydrodynamics (assuming
there are no anomalies)

$\displaystyle\nabla_{\mu}T^{\mu\nu}$
$\displaystyle=F^{\nu\lambda}J_{\lambda}$ (2.58a)
$\displaystyle\nabla_{\mu}J^{\mu}$ $\displaystyle=0$ (2.58b)

These are the obvious generalization of (2.2a) and (2.2b) in the presence of
external curvature and electromagnetic fields.

Although in hydrodynamics we do not have access to a microscopic generating
functional for the full dissipative theory, we can still construct the
generators of the retarded Green functions in the presence of background
sources as

$\mathcal{T}^{\mu\nu}[A,g]=\sqrt{-g}\langle\hat{T}^{\mu\nu}\rangle\qquad\mathcal{J}^{\mu}[A,g]=\sqrt{-g}\langle\hat{J}^{\mu}\rangle$
(2.59)

where $\langle\hat{T}^{\mu\nu}\rangle$ and $\langle\hat{J}^{\mu}\rangle$ are
the solutions to the equations of hydrodynamics in the presence of non-trivial
$A_{\mu}$ and $g_{\mu\nu}$. Finally, we can obtain the equilibrium correlators
at zero sources as

$\displaystyle G^{R}_{J^{\mu}J^{\nu}}(x)$
$\displaystyle=-\frac{\delta\mathcal{J}^{\mu}(x)}{\delta
A_{\nu}(0)}\biggr{\rvert}_{A=h=0}$ $\displaystyle
G^{R}_{T^{\mu\nu}J^{\sigma}}(x)$
$\displaystyle=-\frac{\delta\mathcal{T}^{\mu\nu}(x)}{\delta
A_{\sigma}(0)}\biggr{\rvert}_{A=h=0}$ (2.60a) $\displaystyle
G^{R}_{J^{\sigma}T^{\mu\nu}}(x)$
$\displaystyle=-2\frac{\delta\mathcal{J}^{\sigma}(x)}{\delta
h_{\mu\nu}(0)}\biggr{\rvert}_{A=h=0}$ $\displaystyle
G^{R}_{T^{\mu\nu}T^{\sigma\rho}}(x)$
$\displaystyle=-2\frac{\delta\mathcal{T}^{\mu\nu}(x)}{\delta
h_{\rho\sigma}(0)}\biggr{\rvert}_{A=h=0}$ (2.60b)

where we defined the metric fluctuation as
$h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$.

The way to compute these quantities is as follows: the constitutive relations
in the presence of background sources in the Landau frame, at order one in
derivatives, are

$\displaystyle T^{\mu\nu}$ $\displaystyle=\varepsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}+$
$\displaystyle\quad-\eta\Delta^{\mu\alpha}\Delta^{\nu\beta}\left(\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{\alpha}-\frac{2}{d}g_{\alpha\beta}\nabla_{\lambda}u^{\lambda}\right)-\zeta\Delta^{\mu\nu}\nabla_{\lambda}u^{\lambda}$
$\displaystyle J^{\mu}$
$\displaystyle=nu^{\mu}+\sigma\left(E^{\mu}-T\Delta^{\mu\nu}\nabla_{\nu}\frac{\mu}{T}\right)$
(2.61)

where $E_{\mu}=F_{\mu\nu}u^{\nu}$ is the covariant electric field and appears
naturally from the study of the entropy current in the presence of sources.
Then we can solve the equations of hydrodynamics in Fourier space by
linearizing around global thermodynamic equilibrium with $T=\text{const}$,
$\mu=\text{const}$ and $u^{\mu}=(1,\boldsymbol{\mathbf{0}})$ in the presence
of small fluctuating sources $\delta A_{\mu}$ and $\delta h_{\mu\nu}$.
Inserting the result in the generators (2.59) and taking the functional
derivatives (2.60) gives access to all the Green functions of the theory.

The Green functions obtained by this approach might differ from the ones
obtain via Martin-Kadanoff by contact terms that appear because of the
$\sqrt{-g}$ term in the generators (2.59), see [176].

####  2.2.3 Discrete symmetries: Onsager relations

The retarded functions have to be consistent with the symmetries of the
theory, in particular time-reversal turns out to impose powerful constraints
on the transport coefficients. Consider a Hermitian field
$\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}})$ that transform under time reversal
as
$\hat{\Theta}\hat{\phi}_{a}(t,\boldsymbol{\mathbf{x}})\hat{\Theta}^{-1}=\eta_{a}\hat{\phi}_{a}(-t,\boldsymbol{\mathbf{x}})$,
where $\hat{\Theta}$ is the anti-unitary time-reversal operator and
$\eta_{a}=\pm 1$ is the eigenvalue of $\hat{\phi}_{a}$. If the microscopic
system is time-reversal invariant, such that $[\hat{H},\hat{\Theta}]=0$, then
the retarded response functions must obey

$G^{R}_{ab}(t,\boldsymbol{\mathbf{x}})=G^{R}_{ba}(t,-\boldsymbol{\mathbf{x}})\eta_{a}\eta_{b}$
(2.62)

On the other hand, if time-reversal is not a symmetry of the microscopic
system, for example if there is an external magnetic field $B$ that breaks the
symmetry, the Hamiltonian satisfies
$\hat{\Theta}\hat{H}(B)\hat{\Theta}^{-1}=\hat{H}(-B)$. Then the above equation
is modified to

$G^{R}_{ab}(\omega,\boldsymbol{\mathbf{k}};B)=\eta_{a}\eta_{b}G^{R}_{ba}(\omega,-\boldsymbol{\mathbf{k}};-B)$
(2.63)

This equation is at the core of the so-called Onsager reciprocal relations
[177, 178], although older works use an approach based on thermodynamics and
the relations between fluxes and forces [169, 75].

This condition on $G^{R}$ imposed by time-reversal is not automatically
satisfied by linearized hydrodynamics, instead it should be interpreted as a
constraint on the possible terms in the constitutive equations: in the limit
$\boldsymbol{\mathbf{k}}\rightarrow 0$, at $\omega=0$, this condition implies
that $\chi_{T}=0$ in first-order hydrodynamics (2.18). This is another way to
constraint the transport coefficients without resorting to the entropy
current.

From (2.55) and (2.63) it follows that the matrix $M_{ab}$ must obey the
following condition

$\chi(B)SM^{T}(-\boldsymbol{\mathbf{k}};-B)=M(\boldsymbol{\mathbf{k}};B)\chi(B)S$
(2.64)

where $S=\text{diag}\left(\eta_{1},\eta_{2},\dots\right)$ is the matrix of the
time-reversal eigenvalues and the susceptibilities obey

$S\chi(B)S=\chi^{T}(-B)$ (2.65)

where $T$ stands for transpose matrix.

One could also apply the same procedure to other discrete symmetries, such as
parity, however in these other cases the procedure is not as useful, since it
is usually enough to keep track of the parity breaking parameters of the
theory [179]. This is related to the fact that only time-reversal is
associated with an anti-unitary operator.

####  2.2.4 Thermoelectric transport

In this thesis we are often interested in finding the thermoelectric transport
coefficients in a linear-response-theory framework. Usually the charge and
heat currents couple respectively with the electric field and a temperature
gradient, but there are also thermoelectric effects where the sources create
mixed responses described by the full thermoelectric matrix. In linear
response we can define [169]

$\begin{pmatrix}\delta J_{i}\\\ \delta
Q_{i}\end{pmatrix}=\begin{pmatrix}\sigma_{ij}&\alpha_{ij}\\\
T\bar{\alpha}_{ij}&\bar{\kappa}_{ij}\end{pmatrix}\begin{pmatrix}\delta\mathbb{E}_{j}\\\
-\partial_{j}\delta T\end{pmatrix}$ (2.66)

where $\sigma$ is the electrical conductivity tensor, $\bar{\kappa}$ is the
thermal conductivity tensor, $\alpha$ and $\bar{\alpha}$ are the
thermoelectric tensors.

The $\sigma$ that appears in the constitutive relations is not to be confused
with $\sigma_{ij}$: $\sigma$ is a transport coefficient, cannot be determined
by hydrodynamics, and depends on the microscopic theory, on the other hand
$\sigma_{ij}$ describes the macroscopic response of the current with respect
to an external electric field and will contain a contribution due to $\sigma$.

It should be noted that $\bar{\kappa}$ is not the usual thermal conductivity
measured in experiments: in the laboratory measurements of the thermal
conductivity are usually performed with the boundary condition
$\boldsymbol{\mathbf{J}}=0$, while in the above definition the boundary
condition is $\boldsymbol{\mathbf{E}}=0$. The relation between $\kappa_{ij}$
and $\bar{\kappa}_{ij}$ is

$\kappa_{ij}=\bar{\kappa}_{ij}-T\bar{\alpha}_{ik}\sigma_{kl}^{-1}\alpha_{lj}$
(2.67)

Transport coefficients have to obey Onsager relations, this implies that the
matrices $\alpha_{ij}$ and $\bar{\alpha}_{ij}$ are related by
$\alpha_{ij}(B)=\bar{\alpha}_{ij}(B)$, with $B$ an external magnetic field
that breaks time-reversal invariance [169, 14].

In linear response theory we would like to perturb the system with a small
electric field and temperature gradient, from there, find how the electric and
heat currents react upon these perturbations. To do so, we have to first
figure out the correct sources: for example the electric current does not
couple to the electric field in the Hamiltonian, but instead it couples to the
four-potential $A^{\mu}$.

A disturbance in temperature couples to the Hamiltonian, and since we are
working in the grand canonical ensemble we find

$\delta H=-\int\mathop{}\\!\mathrm{d}^{d}x\left(\frac{\delta
T(t,\boldsymbol{\mathbf{x}})}{T}(\varepsilon(t,\boldsymbol{\mathbf{x}})-\mu
n(t,\boldsymbol{\mathbf{x}}))+\delta
A^{\mu}(t,\boldsymbol{\mathbf{x}})J_{\mu}(t,\boldsymbol{\mathbf{x}})\right)$
(2.68)

We can fix the gauge by choosing the electric field to be minus the gradient
of $A^{0}$, $\mathbb{E}_{i}=-\partial_{i}A^{0}$, so we can ignore the $\delta
A^{i}J_{i}$ terms and focus on $\delta A^{0}n$

$\delta H=-\int\mathop{}\\!\mathrm{d}^{d}x\left(\frac{\delta
T(t,\boldsymbol{\mathbf{x}})}{T}(\varepsilon(t,\boldsymbol{\mathbf{x}})-\mu
n(t,\boldsymbol{\mathbf{x}}))+\delta
A^{0}(t,\boldsymbol{\mathbf{x}})n(t,\boldsymbol{\mathbf{x}})\right)$ (2.69)

In classic thermodynamics the heat current is defined as [169]
$\boldsymbol{\mathbf{J}}_{Q}=T\boldsymbol{\mathbf{J}}_{S}=\boldsymbol{\mathbf{J}}_{U}-\mu\boldsymbol{\mathbf{J}}_{N}$
where $\boldsymbol{\mathbf{J}}_{S}$ is the entropy current while
$\boldsymbol{\mathbf{J}}_{U}$ and $\boldsymbol{\mathbf{J}}_{N}$ are
respectively the energy and charge current. In the relativistic case
$\varepsilon-\mu n$ is the time component of the heat current four-vector.
Hence, the relativistic generalization to the heat current, in linear response
theory, is simply

$\delta Q^{i}=\delta T^{0i}-\mu\delta J^{i}$ (2.70)

It is also possible to arrive at the same result from a variational approach
perspective, by studying temperature fluctuations as changes in the Euclidean
time component of the metric. This is the method followed in [180, 176] to
arrive at the form of the canonical heat current, without resorting to
thermodynamics.

The Kubo formulae for the conductivities in terms of the retarded Green
functions are

$\displaystyle\sigma_{ij}(\omega)$
$\displaystyle=-\frac{G^{R}_{J_{i}J_{j}}(\omega)-G^{R}_{J_{i}J_{j}}(0)}{i\omega}$
(2.71a) $\displaystyle\alpha_{ij}(\omega)$
$\displaystyle=-\frac{G^{R}_{J_{i}Q_{j}}(\omega)-G^{R}_{J_{i}Q_{j}}(0)}{i\omega
T}$ (2.71b) $\displaystyle\bar{\alpha}_{ij}(\omega)$
$\displaystyle=-\frac{G^{R}_{Q_{i}J_{j}}(\omega)-G^{R}_{Q_{i}J_{j}}(0)}{i\omega
T}$ (2.71c) $\displaystyle\bar{\kappa}_{ij}(\omega)$
$\displaystyle=-\frac{G^{R}_{Q_{i}Q_{j}}(\omega)-G^{R}_{Q_{i}Q_{j}}(0)}{i\omega
T}$ (2.71d)

Then, from a practical perspective, if we are interested in the conductivities
we can simply perturb the system with linear sources

$T\rightarrow T+\delta T-Tx^{i}\delta\zeta_{i}\qquad\qquad
F^{0i}\rightarrow\delta\mathbb{E}^{i}$ (2.72)

where $\zeta_{i}=\partial_{i}T/T$ is the heat source, and solve the equations
of hydrodynamics in Laplace-Fourier space at $\boldsymbol{\mathbf{k}}=0$ to
arrive at (2.52), from which we can read off the conductivities (2.71).

For future reference, we report here the optical conductivities in order-one
hydrodynamics without background magnetic field. These are

$\displaystyle\sigma(\omega)$
$\displaystyle=\sigma+\frac{in^{2}}{\omega(\varepsilon+P)}$ (2.73a)
$\displaystyle\alpha(\omega)$
$\displaystyle=-\frac{\mu}{T}\sigma+\frac{ins}{\omega(\varepsilon+P)}$ (2.73b)
$\displaystyle\bar{\kappa}(\omega)$
$\displaystyle=\frac{\mu^{2}}{T}\sigma+\frac{is^{2}T}{\omega(\varepsilon+P)}$
(2.73c)

and Onsager relations constraint $\bar{\alpha}=\alpha$. We notice that the
conductivities diverge as $\omega$ goes to zero, in particular the imaginary
part has a pole, hence Kramers–Kronig relations imply that the real part has a
delta function at $\omega=0$ [24]. This is because momentum is a conserved
operator, which means that in the presence of a background constant electric
field a charged fluid accelerates without bound. Finally, the Green functions
of conserved quantities obey certain Ward identities [176, 74, 181] that are
reflected in relations between the conductivities. In this case it is easy to
check that

$\displaystyle\alpha(\omega)$
$\displaystyle=-\frac{\mu}{T}\sigma(\omega)+\frac{in}{T\omega}$ (2.74a)
$\displaystyle\bar{\kappa}(\omega)$
$\displaystyle=\frac{\mu^{2}}{T}\sigma(\omega)+\frac{i\left(\varepsilon+P-2\mu
n\right)}{T\omega}$ (2.74b)

####  2.2.5 Modes and linear stability

The first and easiest thing one can wish to compute when he is handed a theory
of hydrodynamics are the modes. These give information about how small
collective fluctuations near thermal equilibrium propagate inside the fluid,
and furthermore it gives important clues about the stability of the theory.

From a formal perspective, the modes $\omega(\boldsymbol{\mathbf{k}})$ are
eigenfrequencies of the linearized equations of motion. Starting from the
conservation laws and the constitutive relations, we can consider fluctuations
on a global thermodynamic equilibrium state of the form

$\phi_{a}(t,\boldsymbol{\mathbf{x}})=e^{i\omega
t-i\boldsymbol{\mathbf{k}}\cdot\boldsymbol{\mathbf{x}}}\phi_{a}$ (2.75)

The equations of hydrodynamics can be rewritten in Fourier space as

$M_{ab}(\boldsymbol{\mathbf{k}})\phi_{b}=\omega\phi_{a}$ (2.76)

and host non-trivial solutions only if $\text{det}(M-\mathds{1}\omega)=0$,
from which we obtain the modes of the system.

The requirement of causality, when applied to the analytic structure of the
modes for complex frequency $\omega$, tells us that the imaginary part of the
modes must be non-positive. This is related to the discussion on the
analyticity of the retarded correlators $G^{R}$ discussed in Section 2.2.1,
because the poles of the Green functions are the modes themselves (this can be
seen by the fact that to obtain the Green functions we need to invert the
hydrodynamic matrix, which in turn leaves us with the determinant of
$M-\mathds{1}\omega$ in the denominator), and the poles must lay in the lower-
half of the complex $\omega$ plane.

One final comment on the terminology: by definitions the hydrodynamic modes
should be such that $\omega(\boldsymbol{\mathbf{k}}=0)=0$, however when there
are impurities, magnetic fields or other slow variables that enter the
hydrodynamic regime this might not be true [50]. It is thus common to call a
mode _hydrodynamic_ if it is the lowest lying mode (the closest one to the
real axis at $\boldsymbol{\mathbf{k}}=0$), irrespective of its actual value,
and provided that the non-hydrodynamic modes are deep enough in the bottom-
half of the complex plane.

##  Chapter 3 Hydrodynamic and holographic Charge Density Waves

> “Reality exists in the human mind, and nowhere else.”
>
> George Orwell, _1984_

###  3.1 Introduction

As we have already discussed in the introduction, High-Temperature
superconductors, of which cuprates are typical examples, are believed to host
strongly coupled phases of matter that do not allow a microscopic description
in terms of quasiparticles, see [182, 183] for some reviews. This is usually
argued based on the peculiar features of the cuprates, such as the electric
resistivity that scales linear in $T$ on a very broad range of temperature
(from as low as 10, up to 300), implying that the standard description in
terms of phonons and Fermi-liquid cannot be accurate [147, 184, 14, 185, 186].
On the other hand holography, which is not based on a description in terms of
quasiparticles, together with the existence of a quantum critical point, might
be able to capture the Planckian transport many-body dynamics of these
strongly coupled systems [187, 180, 188, 146].

Figure 3.1: Typical phase diagram of cuprates, taken from [189]. The strange
metal phase above the superconducting dome is non-Fermi liquid and presents
the typical linear-in-$T$ resistivity. Close to optimal doping there is often
a coexistence of charge density wave and superconducting phase.

One common feature of cuprates is the presence of Charge Density Wave order in
the phase diagram for certain values of the parameters. Charge Density Waves
are the collective behaviour of charges that can sometimes arise in the
presence of an ionic lattice [190]. In metals usually the charge density is
uniform throughout the system, however in the presence of electron-phonon
interactions the lattice is distorted to a new periodicity $\lambda_{c}$, and
a gap opens up that leads to Charge Density Wave modulation of the electron
fluid, see Figure 3.2. The long-lived CDW collective modes due to the
spontaneous breaking of translations in the electron fluids, because they are
(almost) gapless, can be important for the low-energy dynamics in the
hydrodynamic regime, on par with other conserved quantities [116]. Wigner
crystals have a different physical origin, but nonetheless they can give rise
to a low-energy dynamics which is very similar. The main difference is that
Wigner crystals always break translations in all spatial directions, while CDW
can break the symmetry only in certain directions, producing smectic phases.
In this chapter, however, we will only focus on the case in which translations
are broken in all directions.

Finally, one last ingredient we need to consider is that condensed matter
experiments are usually performed in the presence of an external magnetic
field. This is both to suppress superconductivity and enhance the strongly-
coupled strange metal regimes, but also to study other transport coefficients
such as the Hall conductivities.

Figure 3.2: Figure taken from [191]. Without interactions the charge density
is uniform in equilibrium. The presence of electron-phonon interactions
induces a charge modulation order, while thermal fluctuations work to destroy
this effect.

For these reasons, many holographic models in which translation symmetry is
broken either spontaneously or pseudo-spontaneously have been intensively
studied to better understand the low energy physics of these systems and, at
the same time, holography itself. This has led to many different approaches to
translations breaking, from massive gravity [161, 162, 162, 192, 193, 194,
195, 196, 197, 198], to spatially modulated charge density [199, 200, 201] and
Q-lattices [202, 203, 204, 205, 206, 207, 208, 209, 210, 211].

At the same time, holography sparkled a new interest in the low energy EFT and
hydrodynamic description of these phases of matter [212, 184, 213, 39, 214,
41, 40]. However, it is now known that in the context of broken-translation
symmetry, holography and hydrodynamics do not naively match. This is because
the holographic Q-lattice description is metastable, namely the vacuum does
not minimize the free energy, and thus the hydrodynamic description has to be
modified to include the effect of the so-called lattice pressure $P_{l}$ [214,
41, 215, 216]. In stable systems the lattice pressure vanishes and its
thermodynamic derivatives $\partial_{T}P_{l}$, $\partial_{\mu}P_{l}$ can be
absorbed by redefining certain transport coefficients.

In this chapter we will consider a model of CDW, both from a hydrodynamic and
holographic perspective, in the presence of a strong magnetic field which is
order zero in derivatives $B\sim\mathcal{O}(1)$. To match the two approaches,
we need to also include the lattice pressure in the thermodynamic of our
hydrodynamic model. We consider both the case in which translation symmetry is
broken spontaneously, thus leading to a gapless Goldstone mode, and pseudo-
spontaneously, i.e. in which the Goldstone acquires a parametrically-small
mass. We compute the AC conductivities analytically from hydrodynamics and
numerically from holography; subsequently, using a method developed in [217]
based on Ward identities, we express the AC results in terms of the DC values
of the conductivities, which can be computed analytically from holography in
terms of horizon data. This allows us to match the correlators obtained from
the two approaches to high precision, and we observe a very good agreement
between the two descriptions.

In Section 3.2 we discuss general properties of systems in which translations
are broken by a scalar field. Then, in Section 3.3 and 3.4, we develop the
hydrodynamic theory for the cases of spontaneous and pseudo-spontaneous
breaking of translations respectively. Finally, in Section 3.5 we briefly
review our holographic model and match the conductivities obtained from
numerics to the analytic hydrodynamic ones.

###  3.2 Breaking the translation symmetry and Ward identities

The equations of motion for the hydrodynamic model are still the conservation
equations for the stress-energy tensor and a $\mathrm{U(1)}$ current, however
this time we must also include the presence of an extra scalar due to the
symmetry breaking

$\displaystyle\partial_{\mu}\langle T^{\mu\nu}\rangle$
$\displaystyle=F^{\nu\lambda}\langle
J_{\lambda}\rangle-\partial^{\nu}\Phi^{I}\langle O_{I}\rangle$ (3.1a)
$\displaystyle\partial_{\mu}\langle J^{\mu}\rangle$ $\displaystyle=0$ (3.1b)

where $O_{I}$, with $I=1,\dots,d$, is a set of scalar operators that break
spatial translations and can be identified with the Goldstone modes for the
broken symmetry [214], while $\Phi^{I}$ are their conjugate sources. The
scalars $O_{I}$ are dynamical, which means that we need other equations of
motion that will take the form of Josephson-like relations. However, because
these equations are not constrained by symmetries and are obtained order by
order in derivative expansion, they do not lead to Ward identities.

To achieve the spontaneous breaking of translations we consider scalars
operators that take a non-zero vev proportional to the spatial coordinates in
the background $\langle O_{I}\rangle\propto x^{i}\delta_{iI}$, while for the
explicit and pseudo-spontaneous case instead we take the source fields to be
proportional to the spatial coordinates $\Phi_{I}=\varphi x^{i}\delta_{iI}$
where $\varphi$ is some parameter, such that the spacetime derivatives of
$\Phi_{I}$ are constants.

Tuning the value of $\varphi$ allows us to better identify two different
regimes: the pseudo-spontaneous case, which is of interest for this chapter,
is when $\varphi\ll\lvert\partial_{i}\langle O_{I}\rangle\rvert$. Then the
Goldstone acquires a small mass, denoted by $\omega_{0}^{2}$ and called