###### Abstract

The coset construction is a tool for systematically building low energy
effective actions for Nambu-Goldstone modes. This technique is typically used
to compute time-ordered correlators appropriate for $S$-matrix computations
for systems in their ground state. In this paper, we extend this technique to
the Schwinger-Keldysh formalism, which enables one to calculate a wider
variety of correlators and applies also to systems in a mixed state. We focus
our attention on internal symmetries and demonstrate that, after identifying
the appropriate symmetry breaking pattern, Schwinger-Keldysh effective actions
for Nambu-Goldstone modes can be constructed using the standard rules of the
coset construction. Particular emphasis is placed on the thermal state and
ensuring that correlators satisfy the KMS relation. We also discuss explicitly
the power counting scheme underlying our effective actions. We comment on the
similarities and differences between our approach and others that have
previously appeared in the literature. In particular, our prescription does
not require the introduction of additional “diffusive” symmetries and retains
the full non-linear structure generated by the coset construction. We conclude
with a series of explicit examples, including a computation of the finite-
temperature two-point functions of conserved spin currents in non-relativistic
paramagnets, antiferromagnets, and ferromagnets. Along the way, we also
clarify the discrete symmetries that set antiferromagnets apart from
ferromagnets, and point out that the dynamical KMS symmetry must be
implemented in different ways in these two systems.

The Schwinger-Keldysh Coset Construction

Can Onur<EMAIL_ADDRESS>Garrett
<EMAIL_ADDRESS>and Riccardo<EMAIL_ADDRESS>
aDepartment of Physics, Carnegie Mellon University

5000 Forbes Ave, Pittsburgh, PA 15213, USA

bDetermined AI, Hewlett Packard Enterprise

###### Contents

  1. 1 Introduction
  2. 2 The Schwinger-Keldysh Formalism from an EFT perspective
    1. 2.1 Field Content
    2. 2.2 Effective Action
    3. 2.3 Locality and Expansion Parameters
    4. 2.4 Symmetries
    5. 2.5 Unitarity
    6. 2.6 A Convenient Field Redefinition: Keldysh Rotation
  3. 3 The Coset Construction
    1. 3.1 The Maurer-Cartan Form
    2. 3.2 Gauge Symmetries
  4. 4 Schwinger-Keldysh Effective Actions from a Coset Construction
    1. 4.1 Closed Systems in a Pure State
    2. 4.2 Closed Systems in a Thermal State
    3. 4.3 Implementing the dynamical KMS symmetry
    4. 4.4 Relation to other approaches in the literature
    5. 4.5 Lowest order DKMS-invariant combinations
    6. 4.6 Power counting rules
  5. 5 Examples: Paramagnets, Antiferromagnets, and Ferromagnets
    1. 5.1 Paramagnets
    2. 5.2 Antiferromagnets
    3. 5.3 Ferromagnets
  6. 6 Conclusions
  7. A Schwinger-Keldysh Path Integral for a Free System at Finite Temperature
  8. B Correlator Cheat Sheet

##  1 Introduction

The fundamental symmetries of the universe are not always easy to discern. For
instance, our immediate surroundings often obscure the underlying principle
that there is no preferred location.111Despite the claims of many New Yorkers.
Even when a symmetry is hidden by circumstances, i.e. it is spontaneously
broken, it still constrains the dynamics of a system and it leads to
observable consequences. One robust implication of spontaneous symmetry
breaking (SSB), encapsulated in Goldstone’s celebrated theorem, is the
existence of gapless modes which often dominate the behavior of the system at
large enough length scales.

Spontaneously broken symmetries act non-linearly on the Goldstone fields, and
for this reason it is often non-trivial to identify all possible invariant
operators that should be included in the low-energy effective action.
Fortunately, there exists a systematic procedure to achieve this, which goes
under the name of coset construction [1, 2, 3, 4]. In the simplest scenarios,
this method takes as its only input the symmetry breaking pattern $G\to H$,
where $G$ is the fundamental symmetry group, and $H$ is the subgroup of
symmetries that remain manifest and are realized linearly. The output of this
construction is a series of building blocks that can be easily combined to
write down the most general local effective action $S_{\rm EFT}[\vec{\pi}]$
describing the Goldstone modes $\vec{\pi}$. Zero-temperature, time-ordered
correlators and the corresponding $S$-matrix elements can then be calculated
using the standard path-integral representation of the generating functional:

$\displaystyle Z[J]$ $\displaystyle\equiv\int\mathcal{D}\vec{\pi}\,e^{iS_{\rm
EFT}[\vec{\pi}]+i\int{\rm d}^{d+1}x\,\vec{J}(x)\cdot\vec{\pi}(x)}.$ (1.1)

Generating functionals of the form (1.1) are appropriate for so-called in-out
calculations, in which the state of the system is specified both at early and
late times [5]. Scattering events are the prototypical example of this
scenario. There are however many other physical quantities which require
instead in-in calculations, in which the (pure or mixed) state of a system is
only specified at some initial time, and afterwards the system evolves
according to its own dynamics. The observables relevant for cosmology,
hydrodynamics, and generic dissipative systems, for example, are most
naturally phrased within this latter framework, and finite-temperature effects
can be included in a natural manner. In this setting, calculations are instead
often performed using the Schwinger-Keldysh generating functional, whose path-
integral representation requires a doubling of fields, the analogue of (1.1)
being

$\displaystyle Z[J_{1},J_{2}]$
$\displaystyle\equiv\int\mathcal{D}\vec{\pi}_{1}\mathcal{D}\vec{\pi}_{2}\,e^{iS_{\rm
EFT}[\vec{\pi}_{1},\vec{\pi}_{2}]+i\int{\rm
d}^{d+1}x\,\vec{J}_{1}(x)\cdot\vec{\pi}_{1}(x)-\vec{J}_{2}(x)\cdot\vec{\pi}_{2}(x)}\
.$ (1.2)

In this paper, we investigate to what extent the traditional coset
construction can be used to write down the doubled-field effective action
$S_{\rm EFT}[\vec{\pi}_{1},\vec{\pi}_{2}]$ that defines the Schwinger-Keldysh
generating functional (1.2). In the case of Goldstone fields, doubling the
field content would naively appear to be in tension with non-linear
realization of the spontaneously broken symmetries. We will explicitly address
this by providing a systematic prescription for writing down all possible
operators that involve two copies of the Goldstone fields and are compatible
with all the symmetries. Our results provide a complementary perspective on
the modern field-theoretic description of non-equilibrium systems, reviewed
recently in, e.g., [6], as the ingredients in both constructions are
intimately related. In particular, we study the additional constraints that a
finite temperature places on top of the traditional coset-construction rules,
as encapsulated by the discrete dynamical Kubo-Martin-Schwinger (DKMS)
symmetry argued for in [7]. Previous studies of the coset construction within
the Schwinger-Keldysh formalism can be found in [8, 9], for instance. See also
[10, 11, 12] for interesting discussions of Goldstone modes in out-of-
equilibrium systems.

The rest of this paper is organized as follows. In order to make our
discussion as self-contained as possible, we will start by reviewing the
basics of the Schwinger-Keldysh formalism in Sec. 2. We will focus in
particular on those subtle aspects that will be relevant for the remainder of
the paper. Readers that are already familiar with the Schwinger-Keldysh
formalism can skip most of this section, but might still benefit from reading
subsections in 2. We will then turn our attention to the coset construction in
Sec. 3. We will first review the basics of this method and then, in Sec. 4,
show how it can be applied to Schwinger-Keldysh effective actions, focusing
specifically on internal symmetries. This section introduces all the rules of
the game that one should follow to write down operators that are compatible
with all the symmetries. We will also discuss how turning on a finite
temperature impacts the symmetries of the system, and introduces new expansion
parameters in the effective theory. Finally, in Sec. 5 we illustrate our main
points with a variety of explicit examples. Some supplementary material is
provided in the Appendices.

Conventions: Unless otherwise specified, we work in units such that
$\hbar=k_{B}=1$. The vast majority of our results are independent of the
Minkowski metric, but wherever it matters we adopt a mostly plus signature. We
define Fourier transforms as $f(\omega,\vec{k})=\int dtd^{3}x\,e^{i\omega
t-i\vec{k}\cdot\vec{x}}f(t,\vec{x})$. Path integrals over spatial slices
versus those over all of spacetime are notationally distinguished via their
respective measures: $D\varphi\equiv\prod_{\vec{x}}d\varphi(t,\vec{x})$ versus
$\mathcal{D}\varphi\equiv\prod_{t,\vec{x}}d\varphi(t,\vec{x})$. States at
fixed asymptotic time slices are denoted as in $|\varphi,\pm\infty\rangle$.

##  2 The Schwinger-Keldysh Formalism from an EFT perspective

In this section, we review the Schwinger-Keldysh (or in-in) formalism from an
EFT perspective. We will discuss the main features of this formalism (doubling
of the field content, implementation of symmetries, power counting, etc.) and
define the notation that we will use throughout the paper. We will strive to
keep our remarks as general as possible, so that they equally apply to systems
with or without SSB.

###  2.1 Field Content

Consider a _closed_ quantum system described by some fields collectively
denoted by $\varphi$ and corresponding action $S[\varphi]$, and that is in a
state specified by the density matrix $\rho$.222We are working in the
Heisenberg picture, where operators and their eigenstates evolve in time,
while the state of the system does not. The most general observables of the
system are phrased in terms of correlation functions of the form

$\displaystyle\langle\mathcal{O}^{(n)}(x_{n})\ldots\mathcal{O}^{(1)}(x_{1})\rangle$
$\displaystyle\equiv\,{\rm
Tr}\,\rho\,\mathcal{O}^{(n)}(x_{n})\ldots\mathcal{O}^{(1)}(x_{1})\ ,$ (2.1)

where the $\mathcal{O}^{(i)}$’s are local operators built out of the fields
$\varphi$ and their derivatives evaluated at points $x_{i}$ that are not in
any particular order. In many applications, one is interested in (linear
combinations of) the restricted set of observables where the operator product
is of the following factorized form:

$\displaystyle\langle\bar{T}\left[\mathcal{O}^{(n)}(x_{n})\ldots\mathcal{O}^{(m+1)}(x_{m+1})\right]T\left[\mathcal{O}^{(m)}(x_{m})\ldots\mathcal{O}^{(1)}(x_{1})\right]\rangle\
,$ (2.2)

where $T$ and $\bar{T}$ denote time-ordered and anti-time-ordered products,
respectively. Specializing further to the case of a single operator
$\mathcal{O}(x)$ for clarity of presentation, correlators of this form can be
obtained systematically by differentiating a generating functional that
depends on two external currents:

$\displaystyle Z[J_{1},J_{2}]$ $\displaystyle\equiv\,{\rm
Tr}\,\left[\rho\,\bar{T}e^{-i\int J_{2}\mathcal{O}}\,Te^{i\int
J_{1}\mathcal{O}}\right].$ (2.3)

Generators for correlators of more than one operator or with more general
time-orderings can be similarly constructed, but the one in Eq. (2.3) will be
sufficient for our purposes.

Inserting resolutions of the identity ${\bf 1}=\int
D\varphi\,|\varphi,\pm\infty\rangle\langle\varphi,\pm\infty|$ between the
various factors inside the trace, we can rewrite the generating functional as

$\displaystyle Z[J_{1},J_{2}]$
$\displaystyle=\\!\int\\!D\varphi_{a}D\varphi_{b}D\varphi_{c}\,\langle\varphi_{a},-\infty|\rho|\varphi_{c},-\infty\rangle\langle\varphi_{c},-\infty|\bar{T}e^{-i\int
J_{2}\mathcal{O}}|\varphi_{b},+\infty\rangle$ (2.4)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\langle\varphi_{b},+\infty|Te^{i\int
J_{1}\mathcal{O}}|\varphi_{a},-\infty\rangle,$

where the last factor admits a path integral representation of the form

$\displaystyle\langle\varphi_{b},+\infty|Te^{i\int
J_{1}\mathcal{O}}|\varphi_{a},-\infty\rangle$
$\displaystyle=\int_{\varphi(-\infty)=\varphi_{a}}^{\varphi(+\infty)=\varphi_{b}}\mathcal{D}\varphi\,e^{iS[\varphi]+i\int{\rm
d}^{d+1}x\,J_{1}\mathcal{O}}\ ,$ (2.5)

and a similar, conjugated expression holds for the second factor.

It is customary for the preceding construction to be summarized in the compact
form

$\displaystyle Z[J_{1},J_{2}]$
$\displaystyle=\int_{\rho}\mathcal{D}\varphi_{1}\mathcal{D}\varphi_{2}\,\exp\left[iS[\varphi_{1}]-iS[\varphi_{2}]+i\int{\rm
d}^{d+1}x\,J_{1}\mathcal{O}(\varphi_{1})-J_{2}\mathcal{O}(\varphi_{2})\right]\
.$ (2.6)

We see therefore that calculating correlation functions of the form (2.2)
requires a doubling of the field content: this is a direct consequence of the
fact that expectation values, as opposed to transition amplitudes, require
time-evolving both the corresponding bras and kets, with one set of fields
performing each such evolution—see [5] for a related discussion.

###  2.2 Effective Action

The expression in Eq. (2.6) leaves the dependence on the density matrix $\rho$
and boundary conditions at $t=+\infty$ highly implicit. These are nonetheless
important features of the path integral formulation that will play an
important role in what follows.

The generator (2.6) corresponds to a path integral over a contour
$\mathcal{C}$ which extends from $t=-\infty$ to $t=+\infty$ and back, with
appropriate boundary conditions placed in the asymptotic regions; see for
instance Fig. 1 in appendix A for a depiction of this contour in the case
where $\rho$ is thermal. Consequently, $Z[J_{1},J_{2}]$ generates correlators
which are path-ordered along $\mathcal{C}$, for the usual reasons, with
operators labeled with a 2 always coming after those labeled with a 1. For
example, when $\mathcal{O}(\varphi)=\varphi$, the various permutations of the
two-point function are explicitly given by

$\displaystyle\langle\mathcal{P}\varphi_{i}(x)\varphi_{j}(x^{\prime})\rangle=\begin{cases}\langle
T\varphi(x)\varphi(x^{\prime})\rangle&\quad i=j=1\\\
\langle\bar{T}\varphi(x)\varphi(x^{\prime})\rangle&\quad i=j=2\\\
\langle\varphi(x)\varphi(x^{\prime})\rangle&\quad i=2,j=1\\\
\langle\varphi(x^{\prime})\varphi(x)\rangle&\quad i=1,j=2\end{cases}\ ,$ (2.7)

where $\mathcal{P}$ denotes path-ordering. For brevity, we will omit explicit
$\mathcal{P}$’s in subsequent expressions and path-ordering is always implied
unless noted otherwise.

The shorthand expression (2.6) is cryptic, at best. A particular shortcoming
is that the correlator (2.7) does not arise from simply inverting the kinetic
term of $S[\varphi_{1}]-S[\varphi_{2}]$, as the notation might suggest. Were
this true, all mixed correlators such as
$\langle\varphi_{1}(x)\varphi_{2}(x^{\prime})\rangle$ would necessarily be
vanishing. Instead, the boundary conditions implicit in (2.6) generate the
necessary, non-trivial cross-couplings—see App. A for an explicit example.

Nuisances such as this one motivate the use of an alternative description
based on an _effective action_ $S_{\rm EFT}[\varphi_{1},\varphi_{2}]$ in which
the boundary conditions implicit in (2.6) are made explicit and encoded into
$S_{\rm EFT}$ itself. These effective actions are of the functional form

$\displaystyle S_{\rm
EFT}[\varphi_{1},\varphi_{2}]=S[\varphi_{1}]-S[\varphi_{2}]+\Delta
S[\varphi_{1},\varphi_{2}]\ ,$ (2.8)

where the information about the state $\rho$ of the closed system as well as
the boundary conditions are encoded in the cross terms $\Delta
S[\varphi_{1},\varphi_{2}]$ [13, 6].333Note that the fields $\varphi$ that
appear in the effective action (2.8) generically do not coincide with the
fields $\varphi$ in Eq. (2.6)—a standard observation in more traditional in-
out effective field theories. Similarly, the functional $S[\varphi_{i}]$ in
Eq. (2.8) will generally differ from the functional $S[\varphi_{i}]$ in Eq.
(2.6); we use the same symbol in both contexts to avoid unnecessary additional
notation. We elaborate upon this point later in this section and will also
discuss the circumstances under which we expect $S_{\rm EFT}$ to be local.
Correlators are computed using $S_{\rm EFT}$ as in

$\displaystyle\langle\mathcal{O}^{(n)}(x_{n})\ldots\mathcal{O}^{(1)}(x_{1})\rangle=\int\mathcal{D}\varphi_{1}\mathcal{D}\varphi_{2}\,e^{iS_{\rm
EFT}[\varphi_{1},\varphi_{2}]}\mathcal{O}^{(n)}(x_{n})\ldots\mathcal{O}^{(1)}(x_{1})\
,$ (2.9)

where perturbative computations now proceed in the naive manner in which free
propagators are determined by the quadratic terms in $S_{\rm EFT}$ and non-
linearities are handled order by order.

A term of the form $\Delta S[\varphi_{1},\varphi_{2}]$ arises also in _open_
systems, albeit in a different way [14, 15, 16, 17]. To see this, imagine
decomposing our closed system in a subsystem of interest, with degrees of
freedom $\phi$, and an environment, with degrees of freedom $\Phi$. If we are
only interested in correlation functions of the subsystem, we can work with a
generating functional of the form

$\displaystyle Z[J_{1},J_{2}]$
$\displaystyle=\int_{\rho}\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\mathcal{D}\Phi_{1}\mathcal{D}\Phi_{2}\,e^{iS[\phi_{1},\Phi_{1}]-iS[\phi_{2},\Phi_{2}]+i\int{\rm
d}^{d+1}x\,J_{1}\mathcal{O}(\phi_{1})-J_{2}\mathcal{O}(\phi_{2})}\ ,$ (2.10)

where $S[\phi,\Phi]=S[\phi]+S_{\rm int}[\phi,\Phi]$, and $S_{\rm
int}[\phi,\Phi]$ captures the dynamics of the environment as well as its
interactions with the subsystem. Assuming for simplicity that the state of the
full system is factorized, i.e. $\rho=\rho_{\phi}\otimes\rho_{\Phi}$, we can
integrate out the fields $\Phi$ to obtain a correction to the action for the
subsystem given by

$\displaystyle e^{i\Delta
S[\phi_{1},\phi_{2}]}=\int_{\rho_{\Phi}}\mathcal{D}\Phi_{1}\mathcal{D}\Phi_{2}\,e^{iS_{\rm
int}[\phi_{1},\Phi_{1}]-iS_{\rm int}[\phi_{2},\Phi_{2}]}.$ (2.11)

The term $\Delta S[\phi_{1},\phi_{2}]$ that arises by integrating out the
environment is known as the _influence functional_ , and it generically
includes interactions between $\phi_{1}$ and $\phi_{2}$. The origin of these
interactions can be traced back to the “off-diagonal” correlation functions of
the environment degrees of freedom, e.g. $\langle\Phi_{i}\Phi_{j}\rangle$ with
$i\neq j$. Alternatively, one could also derive the influence functional using
the fact that, for open systems, the evolution between subsequent time slices
is not unitary and is performed by the Lindblad operator—see e.g. [16].

Eq. (2.11) shows explicitly that, in the case of an open system, the term
$\Delta S[\phi_{1},\phi_{2}]$ contains information about the state of the
environment $\rho_{\Phi}$ and its interactions with the subsystem. This is
conceptually different from the effective action of a closed system in (2.8),
which instead depends on the state of the system under consideration. Of
course, one could amend the influence functional so that it also captures the
state $\rho_{\phi}$ of the subsystem, following the same strategy we outlined
for a closed system. By doing so, $\Delta S[\phi_{1},\phi_{2}]$ would now
encode once again the state of a closed system,
$\rho=\rho_{\phi}\otimes\rho_{\Phi}$. This procedure would be equivalent to
first introducing the effective action for the total closed system, and then
integrating out the environment in the naive manner:

$\displaystyle\int_{\rho}\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\mathcal{D}\Phi_{1}\mathcal{D}\Phi_{2}e^{iS[\phi_{1},\Phi_{1}]-iS[\phi_{2},\Phi_{2}]}$
$\displaystyle=\int\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\mathcal{D}\Phi_{1}\mathcal{D}\Phi_{2}e^{iS[\phi_{1},\Phi_{1}]-iS[\phi_{2},\Phi_{2}]+i\Delta
S[\phi_{1},\phi_{2},\Phi_{1},\Phi_{2}]}$
$\displaystyle=\int\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\,e^{iS[\phi_{1}]-iS[\phi_{2}]+i\Delta
S[\phi_{1},\phi_{2}]}.$ (2.12)

From this perspective, the difference between an open and a closed system is
whether the degrees of freedom appearing in the path integral define a basis
for the Hilbert space on which the state $\rho$ is defined.

###  2.3 Locality and Expansion Parameters

Let’s consider an EFT described by an “in-out” effective action $S_{\rm
EFT}[\varphi]$ with energy cutoff $\Lambda$.444In non-relativistic EFTs one
should distinguish between the cutoffs for energy and momentum—as we will see
in the examples discussed in Sec. 5. In this section we focus our attention on
the energy cutoff to simplify the discussion.  This means that any operator in
the in-out effective action can be assigned a definite scaling in the ratio
$E/\Lambda$, with $E$ the typical scale of interest [18, 19]. Any state $\rho$
that is not the vacuum of the EFT will generically introduce additional scales
into the problem. We will collectively denote these scales with $M$; some
examples are: the temperature $T$ of a thermal state, the chemical potential
$\mu$ of a finite density state, the characteristic length scale $\ell$ of a
semi-classical field profile (converted to an energy scale using some
characteristic speed), etc. Only states such that $M\ll\Lambda$ can be
reliably described within the in-out EFT.

In the approach described in Sec. 2.2, these additional scales $M$ appear
explicitly in the Schwinger-Keldysh effective action $S_{\rm
EFT}[\varphi_{1},\varphi_{2}]$. While it is perhaps plausible that such an
action might be able to reproduce all correlators of the form (2.2), there is
certainly no expectation that this EFT should be local for all scales
$E\ll\Lambda$—and indeed, it generically won’t be. In order to work with a
local Schwinger-Keldysh effective action, we need to restrict ourselves to the
regime $E\ll M\ll\Lambda$, which will be the focus of this paper. Below the
scales $M$, all information about the state $\rho$ is encoded in the effective
action by an infinite tower of irrelevant local operators—the usual way in
which UV physics manifests itself at low energies. Thus, the Schwinger-Keldysh
effective action now contains a new expansion parameter, $E/M$.

This point is potentially confusing for thermal states: because EFTs at zero
temperature are usually described using a local action, one might expect that
the $T\ll E$ regime should also admit a local description by continuity.
However, thermality is generically encoded by $\sim e^{-E/T}$ factors [20, 21,
22] which only admit an expansion in powers of $E$ for $T\gg E$, the regime in
which being at finite temperature appears as a UV effect.555We are being
schematic here. More precisely, the scale $M$ is proportional to $T$ but does
not necessarily coincide with it. For weakly coupled UV completions, $M$ is
more accurately of order $g^{4}T\ln g^{-2}$, a scale associated with large-
angle scattering events. It is below this scale that the relevant degrees of
freedom become the hydrodynamic modes [23]. This situation is a manifestation
of the usual fact that, for weakly coupled UV completions, the cutoff of the
low-energy EFT can be parametrically smaller than the strong coupling scale
(in our case, $T$).

At scales $E\ll M$, the relevant degrees of freedom are often the Goldstone
modes associated with symmetries that are spontaneously broken by the state
$\rho$. In the following sections, we will uncover the principles that one
should follow to write down the most general Schwinger-Keldysh effective
action for such Goldstone modes.

Irrespective of the state $\rho$ of the system, Schwinger-Keldysh effective
actions are also naturally endowed with another expansion parameter, namely
$\hbar$, that controls the semi-classical expansion. In the absence of
Goldstone modes, fields usually transform linearly under all the symmetries,
and a systematic $\hbar$ expansion of the effective action is straightforward
to implement [6] (see also Sec. 2.6 for more details). In the presence of
Goldstone bosons, however, an expansion of the effective action in powers of
$\hbar$ requires some extra care, since a naive implementation would break
some of the symmetries that are realized non-linearly, as will become clear in
subsequent sections. For this reason, we will not be implementing this
expansion in what follows.

###  2.4 Symmetries

EFTs are specified not only by their field content and expansion parameter(s),
but also by their symmetries. It is therefore important to discuss which
symmetries one should impose when writing down the most general Schwinger-
Keldysh effective action $S_{\rm EFT}[\varphi_{1},\varphi_{2}]$. The symmetry
considerations that must inform the construction of the effective action for
closed systems are:

  * •

_Gauge symmetries_ : When the single field action $S[\varphi]$ enjoys a gauge
symmetry, the action
$S[\varphi_{1},\varphi_{2}]=S[\varphi_{1}]-S[\varphi_{2}]$ in Eq. (2.6) is
invariant under independent gauge transformations of the $\varphi_{i}$’s that
coincide at $t=\pm\infty$ [6]. As a result, the Schwinger-Keldysh effective
action $S_{\rm EFT}[\varphi_{1},\varphi_{2}]$ must be invariant under two
copies of the gauge group. This is consistent with the statement in the
previous section that one should double all the degrees of freedom—including
gauge fields.

  * •

_Continuous global symmetries_ : The fact that the two gauge transformations
must coincide at $t=\pm\infty$ implies that, in the global limit, the action
$S[\varphi_{1},\varphi_{2}]=S[\varphi_{1}]-S[\varphi_{2}]$ can only be
invariant under the diagonal symmetry group $G$ which transforms the
$\varphi_{i}$’s simultaneously. This fact is not manifest at the level of the
action, since $\varphi_{1}$ and $\varphi_{2}$ would appear to be decoupled; it
follows instead from the boundary conditions, and in this sense it can be
viewed as a non-local constraint on the symmetries of the system. These
boundary conditions, together with the state of the system, are built directly
into the Schwinger-Keldysh effective action $S_{\rm
EFT}[\varphi_{1},\varphi_{2}]$. Furthermore, gauge symmetries must become
physically indistinguishable from global symmetries in the limit of
vanishingly small gauge coupling.666See e.g. discussion in Sec. 21.4 of [24].
In the regime where the Schwinger-Keldysh effective action is local, this can
only be achieved if $S_{\rm EFT}[\varphi_{1},\varphi_{2}]$ is invariant under
two copies of all global symmetries, $G_{1}\times G_{2}$.777In the ground
state, $G_{1}\times G_{2}$ is explicitly broken down to its diagonal subgroup
by terms of $\mathcal{O}(\varepsilon)$ when the “$i\varepsilon$” prescription
is implemented at the level of the action [13]. This is another manifestation
of the fact that the state at $t=-\infty$ is only invariant under the diagonal
symmetry group. The factors of $i\varepsilon$ are crucial to reproduce the
correct $n$-point functions, but should not be taken into account when
discussing the symmetries of the Schwinger-Keldysh effective action. This is
standard EFT practice: for example, the in-out effective action for a $U(1)$
Goldstone is considered to be shift-invariant even though implementing the
“$i\varepsilon$” prescription in the action would break the shift symmetry
explicitly. This enhancement of global symmetries is a direct consequence of
the requirement that the Schwinger-Keldysh effective action be local and leads
to the existence of two separate Ward identities, which are required to
reproduce the same information of a single Ward identity defined on the two
segments of the Schwinger-Keldysh contour (see Fig. 1 in App A). The symmetry
properties of the matrix elements of $\rho$ in the past infinity determine to
what extent $G_{1}\times G_{2}$ is spontaneously broken down to a subgroup—we
will discuss this more in depth in Sec. 4.

  * •

_Discrete symmetries:_ The difference between past and future boundary
conditions breaks explicitly time reversal, which therefore is not a symmetry
of the effective action $S_{\rm EFT}[\varphi_{1},\varphi_{2}]$ even when it is
a symmetry of the single-field action $S[\varphi]$. All other discrete
symmetries of the single-field action are realized twice in the Schwinger-
Keldysh effective action, as is the case for continuous global symmetries.
This is required for consistency, since discrete subgroups of continuous
symmetries are always realized twice.

  * •

_Emergent symmetries:_ The state $\rho$ not only determines whether some of
the symmetries are spontaneously broken, but can also give rise to additional
“emergent” symmetries in the EFT. For instance, the homogeneity and isotropy
of a state would be encoded by an emergent internal $ISO(d)$ symmetry [25,
26]. A more generic example is provided by the thermal state, which endows the
effective theory with an additional discrete symmetry—the _dynamical KMS
symmetry_ (DKMS) [27, 28, 6]—that, in the simplest case,888The existence of
the symmetry (2.13) relies on the invariance of the underlying dynamics under
time reversal. More in general, the DKMS symmetry can be implemented by
combining the transformations (2.13) with additional discrete symmetries, e.g.
parity and charge conjugation [6]. The important point is that the ground
state of the system must be invariant under the combined action of these
discrete symmetries and time reversal. As we will see in Sec. 5.3, this
requirement plays an important role for ferromagnets.  can be implemented on
the two copies of the fields as follows:

$\displaystyle\varphi_{1}^{\,\prime}(t,\vec{x})$
$\displaystyle=\varphi_{1}(-t+i\beta/2,\vec{x})=\varphi_{1}(-t,\vec{x})-\frac{i\beta}{2}\partial_{t}\varphi_{1}(-t,\vec{x})+\mathcal{O}(\beta^{2}),$
(2.13a) $\displaystyle\varphi_{2}^{\,\prime}(t,\vec{x})$
$\displaystyle=\varphi_{2}(-t-i\beta/2,\vec{x})=\varphi_{2}(-t,\vec{x})+\frac{i\beta}{2}\partial_{t}\varphi_{2}(-t,\vec{x})+\mathcal{O}(\beta^{2}),$
(2.13b)

where $\beta=1/T$ (In this paper we are aiming for an effective action up to
leading order in an expansion in $E/T$, which is why we expanded the DKMS
symmetry in powers of $\partial_{t}/T$).999We are assuming here that all
continuous global symmetries act linearly on the $\varphi_{i}$ fields, i.e.
that there is no spontaneous symmetry breaking. Furthermore, we have made an
additional imaginary time-translation to bring (2.13) to a convenient form—see
e.g. [6] for a more general form of these transformation rules. This symmetry
ensures that correlation functions of the system satisfy the KMS condition
[29, 30], which e.g. for the 2-point correlator of any operator $\mathcal{O}$
reads

$\displaystyle\langle\mathcal{O}(t,\vec{x})\mathcal{O}(t^{\prime},\vec{x}^{\,\prime})\rangle=\langle\mathcal{O}(t^{\prime}-i\beta/2,\vec{x}^{\,\prime})\mathcal{O}(t+i\beta/2,\vec{x})\rangle.$
(2.14)

Furthermore, when combined with the unitarity conditions discussed below in
Sec. 2.5, the DKMS symmetry leads to the existence of an entropy current with
non-negative divergence [31]. It is worth mentioning that Eq. (2.13) is not
the only possible way of implementing the KMS condition as a symmetry of the
effective action—see for instance [32, 33], and [34] for a detailed comparison
with the approach put forward in [6] and adopted in this paper.

###  2.5 Unitarity

In the standard in-out path integral, unitarity implies that the single field
effective action $S_{\rm EFT}[\varphi]$ must be real. Similarly, unitarity
also restricts the form of the Schwinger-Keldysh effective action $S_{\rm
EFT}[\varphi_{1},\varphi_{2}]$, and requires that the following conditions be
satisfied [6]:

  1. 1.

The action must vanish when $\varphi_{1}=\varphi_{2}$: $S_{\rm
EFT}[\varphi,\varphi]=0$.

  2. 2.

Unlike in the usual in-out path integral, the effective action that appears in
the Schwinger-Keldysh generating functional is allowed to be complex. However,
under conjugation we must have $S_{\rm
EFT}[\varphi_{1},\varphi_{2}]^{*}=-S_{\rm EFT}[\varphi_{2},\varphi_{1}]$.

  3. 3.

Furthermore, the imaginary part of the action must be non-negative: $\,{\rm
Im}\,S_{\rm EFT}[\varphi_{1},\varphi_{2}]\geq 0$.

In terms of the decomposition (2.8) $S_{\rm
EFT}[\varphi_{1},\varphi_{2}]=S[\varphi_{1}]-S[\varphi_{2}]+\Delta
S[\varphi_{1},\varphi_{2}]$, the first unitarity constraint implies that
$\Delta S[\varphi,\varphi]=0$; the second condition requires $S[\varphi]$ to
be purely real and the mixing term to obey $\Delta
S[\varphi_{1},\varphi_{2}]^{*}=-\Delta S[\varphi_{2},\varphi_{1}]$; and the
third unitarity condition further imposes $\,{\rm Im}\,\Delta
S[\varphi_{1},\varphi_{2}]\geq 0$.

###  2.6 A Convenient Field Redefinition: Keldysh Rotation

In order to simplify calculations and make the causality properties of various
quantities manifest, it is convenient to perform a field redefinition and
switch to the so-called _Keldysh basis_. Assuming for now that all symmetries
are linearly realized on our fields, this is done by performing a Keldysh
rotation to introduce the following new degrees of freedom:101010See [35] for
a detailed discussion of subtleties associated with the Keldysh basis.

$\displaystyle\varphi_{a}(x)\equiv\varphi_{1}(x)-\varphi_{2}(x)\
,\qquad\quad\varphi_{r}(x)\equiv\tfrac{1}{2}\left[\varphi_{1}(x)+\varphi_{2}(x)\right].$
(2.15)

The Keldysh basis has several advantages. First, even though the 2-point
functions of $\varphi_{1}$ and $\varphi_{2}$ are all generically non-zero, as
shown in Eq. (2.7), they are actually not all independent of each other.
Writing out the time-orderings, one can easily show that the following
identity holds [13]:

$\displaystyle\langle\varphi_{1}(x)\varphi_{1}(x^{\prime})\rangle+\langle\varphi_{2}(x)\varphi_{2}(x^{\prime})\rangle-\langle\varphi_{1}(x)\varphi_{2}(x^{\prime})\rangle-\langle\varphi_{2}(x)\varphi_{1}(x^{\prime})\rangle=0\
.$ (2.16)

In the Keldysh basis, this redundancy is made explicit through the fact that
$\langle\varphi_{a}(x)\varphi_{a}(x^{\prime})\rangle$ vanishes:111111This
pattern generalizes quite widely: any correlator involving a string of $a$\-
and $r$-operators $\sim\langle\varphi_{r}^{m}\varphi_{a}^{n}\rangle$ vanishes
if among all operators, the one with the largest temporal argument is an
$a$-operator. This non-perturbative result is known as the largest-time
equation [36] and it implies, in particular, that
$\langle\varphi_{a}^{n}\rangle$ is vanishing for all $n$.121212Our notation
for various correlation functions and some of their properties are summarized
in App. B.

$\displaystyle\langle\varphi_{i}(x)\varphi_{j}(x^{\prime})\rangle=\begin{cases}\tfrac{1}{2}\langle\left\\{\varphi(x),\varphi(x^{\prime})\right\\}\rangle&\quad
i=j=r\\\ 0&\quad i=j=a\\\
\theta(t-t^{\prime})\langle[\varphi(x),\varphi(x^{\prime})]\rangle&\quad
i=r,j=a\\\
\theta(t^{\prime}-t)\langle[\varphi(x^{\prime}),\varphi(x)]\rangle&\quad
i=a,j=r\end{cases}\ .$ (2.17)

Another technical advantage to the Keldysh basis also becomes apparent in Eq.
(2.17): mixed $a$-$r$ two-point correlators have manifest causal properties.
The retarded Green’s function is proportional to
$\langle\varphi_{r}(x)\varphi_{a}(x^{\prime})\rangle$, while the advanced one
is proportional to $\langle\varphi_{a}(x)\varphi_{r}(x^{\prime})\rangle$. One
can think of the mixed correlators as containing information about the
system’s fundamental dynamics, while $r$-$r$ correlators encode information
about the state of the system.131313For instance,
$\langle\varphi_{r}(\omega,\vec{k})\varphi_{r}(-\omega,-\vec{k})\rangle\propto\frac{1}{2}+n_{\rm
BE}(\beta\omega)$ in a thermal state, where $n_{\rm BE}(\beta\omega)$ is the
Bose-Einstein distribution (assuming $\mathcal{O}$ is bosonic)—see App. A for
more details.

The effective action $S_{\rm EFT}[\varphi_{a},\varphi_{r}]$ is generally
organized as an expansion in $\varphi_{a}$ fields which, due to the form of
the action in (2.8), can be written as

$\displaystyle S_{\rm EFT}[\varphi_{a},\varphi_{r}]$
$\displaystyle=\sum_{n\in\\{1,3,5,\ldots\\}}^{\infty}\left(\prod_{i=1}^{n}\int{\rm
d}^{d+1}x_{i}\right)\frac{\delta^{n}S[\varphi_{r}]}{\delta\varphi_{r}(x_{1})\ldots\delta\varphi_{r}(x_{n})}\frac{1}{2^{n-1}n!}\varphi_{a}(x_{1})\ldots\varphi_{a}(x_{n})+\Delta
S[\varphi_{a},\varphi_{r}]\ ,$ (2.18)

where only odd powers of $\varphi_{a}$ can arise from the expansion of
$S[\varphi_{1}]-S[\varphi_{2}]$. In these variables, the unitarity constraints
previously discussed in Sec. 2.5 now imply the conditions

  1. 1.

$\Delta S[\varphi_{a}=0,\varphi_{r}]=0$,

  2. 2.

$\Delta S[\varphi_{a},\varphi_{r}]^{*}=-\Delta S[-\varphi_{a},\varphi_{r}]$,

  3. 3.

$\,{\rm Im}\,\Delta S[\varphi_{a},\varphi_{r}]\geq 0$.

The first condition further implies that $\varphi_{a}=0$ is always a
consistent solution of the equations of motion that follow from varying the
effective action with respect to $\varphi_{r}$. Variation with respect to
$\varphi_{a}$ yields instead, at lowest order in $\varphi_{a}$, the classical
equations of motion for $\varphi_{r}$. As a result, the $a$\- and $r$-fields
have also the conceptual advantage of admitting a natural physical
interpretation. The $\varphi_{r}$ operator can be identified with a classical
field in the $\hbar\to 0$ limit, whereas $\varphi_{a}$ is responsible for
quantum and stochastic effects [13]. In the absence of spontaneous symmetry
breaking, this can be made explicit by rescaling
$\varphi_{a}\to\hbar\varphi_{a}$, so that the effective action admits a
straightforward expansion in powers of $\hbar$. This procedure is consistent
with semi-classical expansion of in-out effective actions in terms of some
dimensionless combination of coupling constants and $\hbar$ [37] in the limit
of high occupation number [38]. In the presence of spontaneous symmetry
breaking, however, an expansion in powers of $a-$fields becomes more subtle
because it obscures (i.e. break explicitly) some of the symmetries realized
non-linearly. Understanding how this relates to the semi-classical expansion
of in-out effective theories with Goldstones [39] is an open problem that we
leave for future study. In the present paper we will not truncate our results
at a finite order in $a-$fields.

For future use, we also report here the action of the DKMS symmetry on the
fields in the Keldysh basis. Working at lowest order in $E/T$, we immediately
find from the transformation rules (2.13):

$\displaystyle\varphi_{r}^{\,\prime}(t,\vec{x})$
$\displaystyle\simeq\varphi_{r}(-t,\vec{x})-\tfrac{i}{4}\beta\partial_{t}\varphi_{a}(-t,\vec{x}),$
(2.19a) $\displaystyle\varphi_{a}^{\,\prime}(t,\vec{x})$
$\displaystyle\simeq\varphi_{a}(-t,\vec{x})-i\beta\partial_{t}\varphi_{r}(-t,\vec{x}).$
(2.19b)

Note that, for linearly realized symmetries, the change in $\varphi_{r}$ in
Eq. (2.19a) is of $\mathcal{O}(\hbar)$, and is therefore often neglected—see
e.g. [6].141414Note that, when restoring all factors of $\hbar$, one should
replace $\beta\to\hbar\beta$ in order for the temperature to have units of
energy. We are keeping this correction here since for our purposes it will be
important to distinguish between expansions in powers of $E/T$ and $\hbar$.

In the body of the paper, we focus on constructing low-energy Schwinger-
Keldysh effective actions for systems in which one or more symmetries are
spontanµeously broken. As we will see, working in terms of $r$\- and
$a$-fields becomes especially natural under these circumstances. This is
because the Goldstone fields in the Keldysh basis have non-linear
transformation properties that resemble those in a more traditional in-out
effective theory. This will allow us, in particular, to implement the DKMS
symmetry on the Goldstone fields in a way that is consistent with all the non-
linearly realized symmetries. Because of the relations (2.15), instead, the
Goldstone fields in the $(1,2)$ basis would satisfy much more complicated
transformation rules.

##  3 The Coset Construction

One of the tenets of modern EFTs is the principle that all the terms
compatible with the symmetries of a physical system should appear in its
corresponding effective action. In the presence of spontaneous symmetry
breaking the identification of all such operators can be technically
challenging, as the symmetries are non-linearly realized on the Goldstone
modes of the system, and the standard framework for organizing the
classification is the _coset construction_ [2, 3, 4]. Essential elements of
the technique are reviewed below in the particular case of spontaneously
broken _internal_ symmetries; a more detailed discussion can be found for
instance in [19].

###  3.1 The Maurer-Cartan Form

The first step is to distinguish those symmetries that are spontaneously
broken from those that are not. Letting the breaking pattern be $G\to H$ and
denoting the generators of spontaneously broken symmetries by $X_{i}$, the
coset construction analysis starts by forming the _coset parametrization_
$\Omega\in G/H$ which can be canonically written as

$\displaystyle\Omega=e^{i\pi^{j}(x)X_{j}}\ .$ (3.1)

Every broken generator151515The broken generators are of course only specified
up to the addition of unbroken generators—and in fact this ambiguity can often
be leveraged to simplify explicit calculations. One should include in the
coset parametrization only those broken generators that are not equivalent up
to an unbroken transformation. appears with an associated Goldstone field
$\pi^{i}(x)$. The transformation properties of the Goldstone fields under the
action of a generic symmetry transformation $g$ are defined by the relation

$g\,\Omega(\vec{\pi})=\Omega(\vec{\pi}^{\,\prime})\,h(g,\vec{\pi}),$ (3.2)

where $h$ is some element of the unbroken subgroup which could in principle
depend on $g$ and the fields $\pi^{i}(x)$.

The transformation laws for $\pi^{i}$ encapsulated by (3.2) are in general
non-linear and contain infinitely many terms when expanding in fields.
Therefore, the operators which respect the requisite symmetries are severely
constrained. The central building block for constructing such operators is the
_Maurer-Cartan 1-form_ , $\boldsymbol{\omega}=\Omega^{-1}{\rm d}\Omega$, which
is a Lie-algebra valued 1-form that can be conveniently written as

$\displaystyle\boldsymbol{\omega}=\Omega^{-1}\partial_{\mu}\Omega\,{\rm
d}x^{\mu}\equiv
i\,\left(D_{\mu}\pi^{i}X_{i}+\mathcal{A}_{\mu}^{B}T_{B}\right){\rm d}x^{\mu}\
,$ (3.3)

where we have denoted unbroken generators with $T_{B}$.

The Maurer-Cartan form is useful because the above coefficients enjoy
relatively simple transformation properties under the relevant $G$-symmetries
[4]:

  * •

The $D_{\mu}\pi^{i}$ factors are non-linear combinations of the Goldstone
fields of the form $D_{\mu}\pi^{i}=\partial_{\mu}\pi^{i}+\mathcal{O}(\pi^{2})$
which transform linearly under (3.2):

$\displaystyle D_{\mu}\pi^{i}\quad\longrightarrow\quad
h(g,\vec{\pi})^{i}{}_{j}D_{\mu}\pi^{j}\ ,$ (3.4)

where $h(g,\vec{\pi})^{i}{}_{j}$ is a (possibly reducible) representation of
the unbroken transformation $h(g,\vec{\pi})$. Because of $D_{\mu}\pi^{i}$’s
linear transformation property (3.4), it is conventional to refer to this
building block as a “covariant derivative”, but we wish to emphasize that this
terminology is somewhat misleading: because of $\pi^{i}(x)$’s highly non-
linear transformation laws (3.2) one cannot interpret $D_{\mu}\pi^{i}(x)$ as a
combination of a partial derivative and a connection acting on $\vec{\pi}(x)$
in any standard manner.

  * •

The $\mathcal{A}_{\mu}^{B}$ coefficients transform like connections under
$h(g,\vec{\pi})$ and can be used to define the following covariant derivative:

$\displaystyle\nabla_{\mu}=(\partial_{\mu}+i\mathcal{A}_{\mu}^{B}T_{B})\ .$
(3.5)

We use the $\nabla_{\mu}$ to symbol to represent proper covariant derivatives,
which distinguish them from the $D_{\mu}$ notation used for Goldstone
“covariant derivative” defined above. This derivative can act on any operator
$\mathcal{O}^{I}$ in a linear representation of $h(g,\vec{\pi})$—be that a
Goldstone covariant derivative, some matter field, or a combination
thereof—and it yields a quantity that once again transforms linearly:

$\displaystyle\nabla_{\mu}\mathcal{O}^{I}\quad\longrightarrow\quad
h(g,\vec{\pi})^{J}{}_{I}\nabla_{\mu}\mathcal{O}^{I}.$ (3.6)

The most general $G$-symmetric in-out effective action describing Goldstone
modes and their interactions with matter fields $\Psi^{I}$ is then of the
functional form

$\displaystyle S=\int{\rm
d}^{d+1}x\,\mathcal{L}(D_{\mu}\pi^{i},\Psi^{I},\nabla_{\mu}),$ (3.7)

where all indices are contracted using the appropriate invariant tensors
associated with the unbroken group $H$.

###  3.2 Gauge Symmetries

One can also extend the previous construction to the case in which some or all
the symmetries are gauged. This requires the introduction of a gauge field
$A_{\mu}=\vec{A}_{\mu}\cdot\vec{Q}$, where the generators $\vec{Q}$ stand for
some or all the $T_{A}$’s and $X_{i}$’s. The gauged Maurer-Cartan form then
reads:

$\displaystyle\boldsymbol{\omega}=\Omega^{-1}\left(\partial_{\mu}+iA_{\mu}\right)\Omega\,{\rm
d}x^{\mu}\ ,$ (3.8)

where $A_{\mu}\longrightarrow
g(x)\left(A_{\mu}-i\partial_{\mu}\right)g^{-1}(x)$ under a gauge
transformation, as usual. The various components of $\boldsymbol{\omega}$
defined in Eq. (3.3) transform as before, except that now the building blocks
$\mathcal{D}_{\alpha}\pi^{i}$ and $\mathcal{A}_{\mu}^{B}$ also depend on the
gauge field $A_{\mu}$.

The gauged coset construction is, of course, natural to consider in the
context of spontaneously broken gauge theories, in which case $A_{\mu}$ is a
dynamical field. Even in theories with spontaneously broken global symmetries,
though, it can be fruitful to introduce non-dynamical gauge fields to aid in
the path-integral description of such systems. More precisely, treating the
$A_{\mu}$ as an external field, one can define a generating functional for
correlators of the conserved currents $\mathcal{J}^{\mu}$ associated with the
symmetries that have been gauged, as in:

$\displaystyle Z[A]$ $\displaystyle=\int\mathcal{D}\vec{\pi}\,e^{iS_{\rm
gauged}}\ ,\qquad i^{n}\frac{\delta^{n}Z[A]}{\delta
A_{\mu_{1}}(x_{1})\ldots\delta A_{\mu_{n}}(x_{n})}\Big{|}_{A=0}\equiv\langle
T\mathcal{J}^{\mu_{1}}(x_{1})\ldots\mathcal{J}^{\mu_{n}}(x_{n})\rangle.$ (3.9)

It is straightforward to show that $Z[A]$ is gauge-invariant (in the absence
of anomalies), from which the conservation of the corresponding currents
$\mathcal{J}^{\mu}$ (and Ward-identities, more generally) follows. See e.g.
[40, 41, 42] for reviews of this construction.

##  4 Schwinger-Keldysh Effective Actions from a Coset Construction

We will now discuss how the coset construction reviewed in the previous
section can also be used to construct Schwinger-Keldysh effective actions. In
Sec. 2.4 we argued that, if the single-field effective action is invariant
under the global symmetry group $G$, the corresponding Schwinger-Keldysh
effective action for a closed system must enjoy twice as many symmetries—i.e.
be invariant under the group $G_{1}\times G_{2}$—in a regime where the action
is local. How many of these symmetries are realized linearly will depend on
the state of the system $\rho$. Inspired by the form of the generating
functional in Eq. (2.4), we will _postulate_ that the symmetry breaking
pattern should be determined by acting with a $G_{1}\times G_{2}$
transformation on the state $\rho$ as follows:161616We should stress that Eq.
(4.1) is not in contradiction with the familiar statement that symmetry
transformations must act on density matrices as $\rho\to U\rho\,U^{-1}$ in
order to preserve the trace. When the state of the system is encoded using a
local Schwinger-Keldysh effective action, one may wonder what part of
$G_{1}\times G_{2}$ is preserved by the ground state in the Hilbert space of
this effective theory. Eq. (4.1) provides a way of answering this question. We
will support the validity of this criterion in the hydrodynamic regime by
discussing a series of non-trivial examples in Sec. 5. Note however that this
criterion does not apply to free fields in their ground state—see App. A—for
which a hydrodynamic regime does not exist.

$\rho\,\to\,U_{1}\rho\,U_{2}^{-1},$ (4.1)

where $U_{1}$ $(U_{2})$ is an element of $G_{1}$ $(G_{2})$. Symmetry
transformations of this form that do not leave $\rho$ invariant are
spontaneously broken. As we will see, this rule of thumb implies different
symmetry breaking patterns depending on whether the state $\rho$ is pure or
thermal.

###  4.1 Closed Systems in a Pure State

Let us start by considering the case of pure states:
$\rho=|\psi\rangle\langle\psi|$. Spontaneous symmetry breaking occurs if there
exists a local operator $\mathcal{O}$ whose expectation value on
$|\psi\rangle$ is not invariant under some symmetry transformations. More
precisely, the symmetry generator $X$ is spontaneously broken if

$\displaystyle\text{Tr}(\rho[X,\mathcal{O}])=\langle\psi|[X,\mathcal{O}]|\psi\rangle\neq
0.$ (4.2)

This condition can be satisfied only if the state $|\psi\rangle$ is an
eigenstate of some but not all of the generators of the symmetry group
$G$—say, those spanning a subgroup $H$. As a result, the state $\rho$ remains
invariant under a transformation (4.1) only when $U_{1}\in H_{1}$ and
$U_{2}\in H_{2}$. In other words, the pure state $\rho$ effectively gives rise
to a symmetry breaking pattern $G_{1}\times G_{2}\to H_{1}\times H_{2}$.
Denoting with $X_{1}^{i}\,(X_{2}^{i})$ the broken generators of
$G_{1}\,(G_{2})$, we introduce for later convenience the linear combinations
of generators

$\displaystyle X_{r}^{i}=X_{1}^{i}+X_{2}^{i},\qquad\quad
X_{a}^{i}=X_{1}^{i}-X_{2}^{i}.$ (4.3)

Then, the coset parametrization in the case of pure states can be chosen to be
of the form

$\displaystyle\Omega=e^{i\pi_{r}\cdot X_{r}}e^{i\pi_{a}\cdot
X_{a}}\qquad\qquad\quad\text{(closed system in a pure state)}.$ (4.4)

For the purposes of calculating the corresponding Maurer-Cartan form, it is
important to keep in mind that, while $[X_{1}^{i},X_{2}^{j}]=0$, the $X_{r}$’s
and $X_{a}$’s do not commute with each other. Let us now turn our attention to
thermal states.

###  4.2 Closed Systems in a Thermal State

It would appear at first sight that the matrix elements of a thermal state
$\rho\sim e^{-\mathcal{H}/T}$ should always be invariant under the diagonal
subgroup of $G_{1}\times G_{2}$ based on the very definition of symmetry, i.e.
$U\mathcal{H}\,U^{-1}=\mathcal{H}$. This conclusion however would be too hasty
[43, 16, 44]. In order to formulate more precisely the criterion for SSB
around a thermal state, we can follow [45] and deform the Hamiltonian by
adding an operator $\Delta\mathcal{H}$ that explicitly breaks the global
symmetry under consideration: $\mathcal{H}\to\mathcal{H}+\Delta\mathcal{H}$.
We will denote the resulting canonical ensemble with
$\rho_{\Delta\mathcal{H}}$. Then, SSB occurs whenever there is an order
parameter $\mathcal{O}$ whose expectation value is not invariant even in the
limit $\Delta H\to 0$. Denoting a symmetry transformation as $U=e^{i\alpha X}$
with $X$ a symmetry generator, this is equivalent to the statement that

$\displaystyle\lim_{\Delta\mathcal{H}\to 0}\,{\rm
tr}\,\\!\left(\rho_{\Delta\mathcal{H}}[X,\mathcal{O}]\right)\neq 0.$ (4.5)

We can now diagonalize the maximum number of generators of $G$ that commute
simultaneously with each other and with the modified Hamiltonian
$\mathcal{H}+\Delta\mathcal{H}$, and write the thermal state more explicitly
as171717Working only with eigenstates of those generators that commute with
$\mathcal{H}+\Delta\mathcal{H}$ rather than $\mathcal{H}$ alone is formally
equivalent to modifying $e^{-\mathcal{H}/T}$ by adding a projection operator
[45].

$\displaystyle\rho\sim\sum_{E,\vec{Q},\alpha}e^{-E/T}|E,\vec{Q},\alpha\rangle\langle
E,\vec{Q},\alpha|,$ (4.6)

where $E$ is the eigenvalue of $\mathcal{H}+\Delta\mathcal{H}$, $\vec{Q}$ are
the charges associated with the commuting generators, and $\alpha$ is an
additional collective index that accounts for all possible degeneracies. The
states $|E,\vec{Q},\alpha\rangle$ form all possible representations of the
unbroken group $H$, and when we act on the thermal state with an $H_{1}\times
H_{2}$ transformation as in (4.1), only the elements of the diagonal subgroup
$H_{\rm diag}$ will leave $\rho$ invariant.

It might be helpful to illustrate this point with a simple example. Consider a
system where an internal $SO(4)$ is spontaneously broken down to $SO(3)$.
Then, the deformation $\Delta\mathcal{H}$ breaks $SO(4)$ explicitly down to
$SO(3)$. We can diagonalize at most one generator of $SO(3)$ together with
$\mathcal{H}+\Delta\mathcal{H}$, and as a result the thermal state takes the
form

$\displaystyle\rho\sim\sum_{E,m,j}e^{-E/T}|E,m,j\rangle\langle E,m,j|,$ (4.7)

with $m$ and $j$ the usual $SO(3)$ quantum numbers, playing respectively the
role of charge $Q$ and degeneracy parameter $\alpha$. Let us now act with a
$SO(3)_{1}\times SO(3)_{2}$ transformation on Eq. (4.7), and we obtain

$\displaystyle\rho\to$
$\displaystyle\sum_{E,m,j}\sum_{m^{\prime},m^{\prime\prime}}e^{-E/T}D^{(j)}_{m^{\prime},m}(U_{1})|E,m^{\prime},j\rangle\langle
E,m^{\prime\prime},j|D^{(j)\,*}_{m^{\prime\prime},m}(U_{2})$
$\displaystyle=\sum_{E,m^{\prime},m^{\prime\prime},j}e^{-E/T}D^{(j)}_{m^{\prime},m^{\prime\prime}}(U_{1}U_{2}^{-1})|E,m^{\prime},j\rangle\langle
E,m^{\prime\prime},j|,$ (4.8)

where we obtained the second line by using standard properties of the
$D^{(j)}_{m^{\prime},m}$ matrices, namely
$D^{(j)}_{m^{\prime},m}(U^{-1})=D^{(j)\,*}_{m,m^{\prime}}(U)$ and
$\sum_{m}D^{(j)}_{m^{\prime},m}(U_{1})D^{(j)}_{m,m^{\prime\prime}}(U_{2})=D^{(j)}_{m^{\prime},m^{\prime\prime}}(U_{1}U_{2})$
[46]. Eq. (4.8) shows that $\rho$ is only invariant under the diagonal
subgroup of $SO(3)_{1}\times SO(3)_{2}$, i.e. when $U_{1}=U_{2}$, while the
off-diagonal combination of generators is spontaneously broken.

We conclude therefore that a thermal state realizes the symmetry breaking
pattern $G_{1}\times G_{2}\to H_{\rm diag}$, which differs from the pattern of
a pure state. Denoting with $T_{1}^{A}\,(T_{2}^{A})$ the generators of
$H_{1}\,(H_{2})$, we introduce the linear combinations of generators

$\displaystyle T_{r}^{A}=T_{1}^{A}+T_{2}^{A},\qquad\quad
T_{a}^{A}=T_{1}^{A}-T_{2}^{A},$ (4.9)

with $T_{r}^{A}$ the generators of $H_{\rm diag}$. We are therefore led to the
conclusion that the appropriate coset parametrization for a thermal state
should be

$\displaystyle\Omega=e^{i\pi_{r}\cdot X_{r}}e^{i\pi_{a}\cdot
X_{a}}e^{i\varphi_{a}\cdot T_{a}},\qquad\qquad\quad\text{(closed system in a
thermal state),}$ (4.10)

where the specific ordering of the various factors has been chosen for later
convenience. However, the field content in Eq. (4.10) is by itself
incompatible with the principle that Schwinger-Keldysh effective actions
should contain twice the number of degrees of freedom as regular in-out
effective actions—see discussion in Sec. 2.1. In other words, the fields
$\varphi_{a}$ are missing their “$r$” partners which, among other things, are
necessary to implement the DKMS symmetry.

To remedy this situation, we will add to our field content a set of “matter
fields” $\rho_{r}^{A}$ in the adjoint representation of the unbroken group
$H_{\rm diag}$. In accordance with the standard rules of the coset
construction [24], these matter fields will transform under a generic
transformation $g\in G_{1}\times G_{2}$ as

$\displaystyle\rho_{r}^{A}\to
h(g,\pi_{r},\pi_{a},\varphi_{a},x)^{A}{}_{B}\,\rho_{r}^{B},$ (4.11)

with $h$ some element of $H_{\rm diag}$. As we will see, these fields admit a
simple physical interpretation: in the classical limit, they are related to
the densities of unbroken charges.

###  4.3 Implementing the dynamical KMS symmetry

When a system is in a thermal state, correlators must satisfy the KMS
condition. This, in turn, imposes some restrictions on the form of the
Schwinger-Keldysh effective action. When all the symmetries are linearly
realized, these restrictions are enforced by invariance under a DKMS symmetry
transformation of the form (2.19). We will now discuss how this symmetry
should be implemented on the Goldstone fields and the matter fields
$\rho_{r}$.

To this end, it is helpful to discuss how the KMS condition would affect, say,
the 2-point correlation functions of the conserved Noether currents in the
effective theory:

$\displaystyle\langle\mathcal{J}_{\mu}(t,\vec{x})\mathcal{J}_{\nu}(t^{\prime},\vec{x}^{\,\prime})\rangle=\langle\mathcal{J}_{\nu}(t^{\prime}-i\beta/2,\vec{x}^{\,\prime})\mathcal{J}_{\mu}(t+i\beta/2,\vec{x})\rangle,$
(4.12)

where we have suppressed the index labeling the corresponding symmetry
generators to streamline the notation. We can introduce a generating
functional similar to the one in Eq. (2.3) that allows us to systematically
calculate such correlators. In this case, the external sources for the
$\mathcal{J}_{\mu}$’s are gauge fields $A_{1,\mu}$ and $A_{2,\mu}$, and the
KMS condition (4.12) implies that the generating functional must satisfy the
following property:181818We are implicitly assuming here that DKMS
transformations are implemented without resorting to additional discrete
symmetries—see comment in footnote 8. Eq. (4.13) could be generalized by
allowing such discrete symmetries to act on the gauge fields on the right-hand
side. It is easy to work out how the following results would need to be
modified.

$\displaystyle Z[A_{1}(t),A_{2}(t)]=Z[A_{1}(-t+i\beta/2),A_{2}(-t-i\beta/2)]\
.$ (4.13)

Switching to the $(a,r)$ basis of fields, and expanding in powers of $E/T$,
this means that $Z$ should be invariant under the following transformation of
sources:

$\displaystyle A_{r}(t)$ $\displaystyle\to
A_{r}(-t)-\frac{i\beta}{4}\partial_{t}A_{a}(-t)+\mathcal{O}(E^{2}/T^{2})\ ,$
(4.14a) $\displaystyle A_{a}(t)$ $\displaystyle\to
A_{a}(-t)-i\beta\partial_{t}A_{r}(-t)+\mathcal{O}(E^{2}/T^{2}).$ (4.14b)

Within the context of the coset construction, external gauge fields can be
introduced by gauging the Maurer-Cartan form as discussed in Sec. 3.2:

$\displaystyle\Omega^{-1}\left(\partial_{\mu}+iA_{\mu}^{r}\cdot
X_{r}+iA_{\mu}^{a}\cdot X_{a}+iA_{\mu}^{a}\cdot T_{a}+iA_{\mu}^{r}\cdot
T_{r}\right)\Omega$ (4.15)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\equiv
i\left(D_{\mu}\pi_{r}\cdot X_{r}+D_{\mu}\pi_{a}\cdot
X_{a}+D_{\mu}\varphi_{a}\cdot T_{a}+\mathcal{A}_{\mu}\cdot T_{r}\right),$

where we have denoted with a dot the contraction of indices labeling the
various generators. The coset covariant derivatives and connection now depend
both on the Goldstone fields and the external gauge fields.

Our goal is to derive how the DKMS symmetry should be implemented on the
Goldstone fields and the matter fields $\rho_{r}$ to ensure that the
generating functional is invariant under (4.14). To this end, we will start by
considering a situation where all the generators are spontaneously broken.
This is not the physical symmetry breaking pattern we are actually interested
in: it is just a convenient trick we will use to figure out how our Goldstone
and matter fields should transform. If all the symmetries were spontaneously
broken, the coset parametrization would read

$\displaystyle\tilde{\Omega}=e^{i\pi_{r}\cdot X_{r}}e^{i\pi_{a}\cdot
X_{a}}e^{i\varphi_{a}\cdot T_{a}}e^{i\varphi_{r}\cdot T_{r}}\ ,$ (4.16)

and the components of the associated Maurer-Cartan form would be

$\displaystyle\tilde{\Omega}^{-1}$
$\displaystyle\left(\partial_{\mu}+iA_{\mu}^{r}\cdot X_{r}+iA_{\mu}^{a}\cdot
X_{a}+iA_{\mu}^{a}\cdot T_{a}+iA_{\mu}^{r}\cdot T_{r}\right)\tilde{\Omega}$
$\displaystyle\equiv i\,(\tilde{D}_{\mu}\pi_{r}\cdot
X_{r}+\tilde{D}_{\mu}\pi_{a}\cdot X_{a}+\tilde{D}_{\mu}\varphi_{a}\cdot
T_{a}+\tilde{D}_{\mu}\varphi_{r}\cdot T_{r})$ (4.17)
$\displaystyle=i\,e^{-i\varphi_{r}\cdot T_{r}}\left[D_{\mu}\pi_{r}\cdot
X_{r}+D_{\mu}\pi_{a}\cdot X_{a}+D_{\mu}\varphi_{a}\cdot
T_{a}+\left(ie^{i\varphi_{r}\cdot T_{r}}\partial_{\mu}e^{-i\varphi_{r}\cdot
T_{r}}+\mathcal{A}_{\mu}\right)\cdot T_{r}\right]e^{i\varphi_{r}\cdot T_{r}}\
.$

In the second line, we are showing explicitly how the building blocks of our
new coset are related to the “physical” ones defined in Eq. (4.15).

The advantage of considering an auxiliary symmetry breaking pattern where all
the symmetries are spontaneously broken is twofold: first, all symmetries are
now treated on equal footing and, in particular, all components of our
external gauge fields can be obtained from a covariant derivative by turning
off the Goldstone fields:

$\displaystyle\tilde{D}_{\mu}\pi^{i}_{r}\to
A^{ri}_{\mu},\qquad\quad\tilde{D}_{\mu}\pi^{i}_{a}\to
A^{ai}_{\mu},\qquad\quad\tilde{D}_{\mu}\varphi^{B}_{a}\to
A^{aB}_{\mu},\qquad\quad\tilde{D}_{\mu}\varphi^{B}_{r}\to A^{rB}_{\mu}\ .$
(4.18)

Therefore, if this was the symmetry breaking pattern that we were interested
in, we could ensure that the generating functional is invariant under the
transformations (4.14) by demanding that the effective action be symmetric
under191919An important comment on our notation: when $\mu=t$, equations
(4.19) reduce to
$\tilde{D}_{t}\pi_{r}(t)\to\tilde{D}_{t}\pi_{r}(-t)+...=-\tilde{D}_{-t}\pi_{r}(-t)+...$,
and so on. The same goes for Eqs. (4.21).

$\displaystyle\tilde{D}_{\mu}\pi_{a}(t)\to\tilde{D}_{\mu}\pi_{a}(-t)-i\beta\partial_{t}\tilde{D}_{\mu}\pi_{r}(-t)+\mathcal{O}(E^{2}/T^{2})\
,$ (4.19a)
$\displaystyle\tilde{D}_{\mu}\pi_{r}(t)\to\tilde{D}_{\mu}\pi_{r}(-t)-\frac{i\beta}{4}\partial_{t}\tilde{D}_{\mu}\pi_{a}(-t)+\mathcal{O}(E^{2}/T^{2})\
,$ (4.19b)
$\displaystyle\tilde{D}_{\mu}\varphi_{a}(t)\to\tilde{D}_{\mu}\varphi_{a}(-t)-i\beta\partial_{t}\tilde{D}_{\mu}\varphi_{r}(-t)+\mathcal{O}(E^{2}/T^{2})\
,$ (4.19c)
$\displaystyle\tilde{D}_{\mu}\varphi_{r}(t)\to\tilde{D}_{\mu}\varphi_{r}(-t)-\frac{i\beta}{4}\partial_{t}\tilde{D}_{\mu}\varphi_{a}(-t)+\mathcal{O}(E^{2}/T^{2})\
.$ (4.19d)

The second advantage is that the coset connection is now trivial, and
therefore higher covariant derivatives of the Goldstone fields can be obtained
by acting with regular partial derivatives on the Maurer-Cartan components;
hence, the symmetry transformations (4.19) are covariant under all the non-
linearly realized symmetries.

Based on the second line of (4.3), they can be expressed equivalently in terms
of the physical covariant derivatives
$\mathcal{D}_{\mu}\pi_{r},\mathcal{D}_{\mu}\pi_{a},\mathcal{D}_{\mu}\varphi_{a},\nabla_{\mu}$
and the combination

$\displaystyle D_{\mu}\varphi_{r}\equiv i\,e^{i\varphi_{r}\cdot
T_{r}}\partial_{\mu}e^{-i\varphi_{r}\cdot T_{r}}+\mathcal{A}_{\mu}$ (4.20)

as follows:202020To derive these expressions, we assumed that the structure
constants are totally antisymmetric, which is ensured whenever the symmetry
group is compact [24].

$\displaystyle D_{\mu}\pi_{a}(t)\to
D_{\mu}\pi_{a}(-t)-i\beta\nabla_{t}D_{\mu}\pi_{r}(-t)-\beta\left[D_{t}\varphi_{r}(-t),D_{\mu}\pi_{r}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (4.21a) $\displaystyle D_{\mu}\pi_{r}(t)\to
D_{\mu}\pi_{r}(-t)-\frac{i\beta}{4}\nabla_{t}D_{\mu}\pi_{a}(-t)-\frac{\beta}{4}\left[D_{t}\varphi_{r}(-t),D_{\mu}\pi_{a}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (4.21b) $\displaystyle D_{\mu}\varphi_{a}(t)\to
D_{\mu}\varphi_{a}(-t)-i\beta\nabla_{t}D_{\mu}\varphi_{r}(-t)-\beta\left[D_{t}\varphi_{r}(-t),D_{\mu}\varphi_{r}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (4.21c) $\displaystyle D_{\mu}\varphi_{r}(t)\to
D_{\mu}\varphi_{r}(-t)-\frac{i\beta}{4}\nabla_{t}D_{\mu}\varphi_{a}(-t)-\frac{\beta}{4}\left[D_{t}\varphi_{r}(-t),D_{\mu}\varphi_{a}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
.$ (4.21d)

where, on the right hand side of these equations,
$\nabla_{t}=\partial_{t}+i\mathcal{A}_{t}^{B}(-t)T_{B}$, and we have
streamlined our notation by defining commutators between covariant
derivatives, e.g.

$\displaystyle\left[D_{t}\varphi_{r}(-t),D_{\mu}\pi_{a}(-t)\right]_{k}\equiv
if_{Ajk}D_{t}\varphi_{r}^{A}(-t)D_{\mu}\pi_{a}^{j}(-t)\ ,$ (4.22)

and similarly for the other commutators.212121Note that all four components
$X_{r},X_{a},T_{a}$ and $T_{r}$ are in independent representations of $T_{r}$
and therefore do not mix. This is due to the structure of commutation
relations, e.g.the term $e^{-i\varphi_{r}\cdot T_{r}}D_{\mu}\pi_{r}\cdot
X_{r}e^{i\varphi_{r}\cdot T_{r}}$ is a linear combination of $X_{r}$
generators due to $[X_{r},T_{r}]\sim X_{r}$. For this reason, (4.19) is
related to (4.21) through conjugation with $e^{-i\varphi_{r}\cdot T_{r}}$ Once
again, the transformations (4.21) are manifestly covariant under all the non-
linearly realized symmetries.

At this point, we notice that the quantity $D_{t}\varphi_{r}$ has exactly the
same transformation properties as our matter fields $\rho_{r}$ (we are
focusing on internal symmetries, and therefore boosts are irrelevant—i.e.
explicitly broken—as far as we are concerned). Therefore, if Eqs. (4.21)
involved only $D_{t}\varphi_{r}$ and its derivatives, we could simply replace
$D_{t}\varphi_{r}(t)\to\rho_{r}(t)$ everywhere and obtain DKMS symmetry
transformations involving the Goldstones $\pi_{r},\pi_{a},\varphi_{a}$ and the
matter fields $\rho_{r}$. Unfortunately, Eq. (4.21c) depends on all components
of $D_{\mu}\varphi_{r}$, but this can be remedied by “commuting” the covariant
derivatives of $\nabla_{t}D_{\mu}\varphi_{r}(-t)$: tedious but straightforward
manipulations show that

$\displaystyle\nabla_{t}D_{\mu}\varphi_{r}(-t)-i[D_{t}\varphi_{r}(-t),D_{\mu}\varphi_{r}(-t)]=\mathcal{F}_{t\mu}(-t)+\nabla_{\mu}D_{t}\varphi_{r}(-t)\
,$ (4.23)

where $\mathcal{F}_{\mu\nu}$ is the usual field strength tensor associated
with $\mathcal{A}_{\mu}$. Using this identity, and replacing
$D_{t}\varphi_{r}(t)\to\rho_{r}(t)$ everywhere,222222This also means replacing
$D_{t}\varphi_{r}(-t)=-D_{-t}\varphi_{r}(-t)\to-\rho_{r}(-t)$. we finally
obtain the desired form of the DKMS transformation rules for our Goldstone and
matter fields (we only need to consider the $\mu=t$ component of Eq. (4.21d)):

$\displaystyle D_{\mu}\pi_{a}(t)\to
D_{\mu}\pi_{a}(-t)-i\beta\nabla_{t}D_{\mu}\pi_{r}(-t)+\beta\left[\rho_{r}(-t),D_{\mu}\pi_{r}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (4.24a) $\displaystyle D_{\mu}\pi_{r}(t)\to
D_{\mu}\pi_{r}(-t)-\frac{i\beta}{4}\nabla_{t}D_{\mu}\pi_{a}(-t)+\frac{\beta}{4}\left[\rho_{r}(-t),D_{\mu}\pi_{a}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (4.24b) $\displaystyle D_{\mu}\varphi_{a}(t)\to
D_{\mu}\varphi_{a}(-t)+i\beta\nabla_{\mu}\rho_{r}(-t)-i\beta\mathcal{F}_{t\mu}(-t)+\mathcal{O}(E^{2}/T^{2})\
,$ (4.24c)
$\displaystyle\rho_{r}(t)\to-\rho_{r}(-t)-\frac{i\beta}{4}\nabla_{t}D_{t}\varphi_{a}(-t)+\frac{\beta}{4}\left[\rho_{r}(-t),D_{t}\varphi_{a}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
.$ (4.24d)

These transformation rules are the main result of this subsection. They act
non-locally at the level of the fields, but to the best of our knowledge there
is no fundamental obstruction to having non-local discrete symmetries.
Furthermore, because the Goldstone fields enter the effective action only
through their covariant derivatives, and these transform locally, it is not
hard to impose the DKMS symmetry in practice. In this paper, we will be
concerned with the implementation of DKMS relations up to first order in
$E/T$; the systematics of higher order corrections are still an open question
that we hope to explore in the near future. We will derive the lowest-order
invariant combinations in Sec. (4.5).

###  4.4 Relation to other approaches in the literature

We should briefly comment on the relation between our approach and previous
results in the literature on out-of-equilibrium effective actions. It was
previously proposed that, in the hydrodynamic limit, the Schwinger-Keldysh
effective action should contain one Goldstone field for each continuous
symmetry, regardless of whether it is spontaneously broken or not (see e.g.
[6, 9, 42]). The effective action then must be invariant under an additional
local symmetry, which only depends on the spatial coordinates and acts on the
fields associated with unbroken generators as follows:

$\displaystyle e^{i\varphi_{r}(t,\vec{x})\cdot T_{r}}\quad\to\quad
e^{i\varphi_{r}(t,\vec{x})\cdot T_{r}}h_{r}(\vec{x})\ ,\,\qquad\qquad\qquad
h_{r}(\vec{x})\equiv e^{ic(\vec{x})\cdot T_{r}}\ .$ (4.25)

In the simplest, abelian case, this symmetry reduces to a local shift,
$\varphi_{r}(t,\vec{x})\to\varphi_{r}(t,\vec{x})+c(\vec{x})$. The symmetry
(4.25) is equivalent to the transformation
$\tilde{\Omega}\to\tilde{\Omega}\,h_{r}(\vec{x})$, which in turn implies the
following transformation rules for the coefficients of the Maurer-Cartan form:

$\displaystyle\tilde{D}_{\mu}\pi_{a}\cdot X_{a}$
$\displaystyle\to\tilde{D}_{\mu}\pi_{a}\cdot
h_{r}^{-1}(\vec{x})X_{a}h_{r}(\vec{x})\ ,$ (4.26a)
$\displaystyle\tilde{D}_{\mu}\pi_{r}\cdot X_{r}$
$\displaystyle\to\tilde{D}_{\mu}\pi_{r}\cdot
h_{r}^{-1}(\vec{x})X_{r}h_{r}(\vec{x})\ ,$ (4.26b)
$\displaystyle\tilde{D}_{\mu}\varphi_{a}\cdot T_{a}$
$\displaystyle\to\tilde{D}_{\mu}\varphi_{a}\cdot
h_{r}^{-1}(\vec{x})T_{a}h_{r}(\vec{x})\ ,$ (4.26c)
$\displaystyle\tilde{D}_{\mu}\varphi_{r}\cdot T_{r}$
$\displaystyle\to\tilde{D}_{\mu}\varphi_{r}\cdot
h_{r}^{-1}(\vec{x})T_{r}h_{r}(\vec{x})-i\delta_{\mu}^{j}h_{r}^{-1}(\vec{x})\partial_{j}h_{r}(\vec{x})\
.$ (4.26d)

This symmetry plays two important roles: (1) it effectively forces us to
contract all the indices in a way that is invariant under $H_{\rm diag}$, even
though $H_{\rm diag}$ is formally broken; and (2) it ensures that the fields
$\varphi_{r}$ appear in the effective action only through
$\tilde{D}_{t}\varphi_{r}$ and its covariant derivatives—the spatial
components $\tilde{D}_{i}\varphi_{r}$’s are not allowed building blocks. This,
in turn, gives rise to a diffusive behavior for the unbroken currents, and for
this reason we’ll also refer to invariance under (4.26) as _diffusive
symmetry_. This additional symmetry is to be regarded as emergent at low
energies, and its physical origin is not particularly clear.232323See however
[47, 48] for a holographic interpretation of this symmetry. Furthermore, its
implications have so far been explored mostly in the classical limit, i.e. by
working only up to quadratic order in the $a-$type fields. This approach has
the advantage that the DKMS symmetry becomes easier to implement [28]—which is
why we started this section by considering a similar symmetry breaking
pattern. However, in practice one is actually more interested in the
properties of the charge density $\rho_{r}$ rather than the field
$\varphi_{r}$, and some authors even resort to an explicit change of variable
from the latter to the former—see e.g. [49], which performs precise numerical
tests of the EFT approach to diffusion.

This alternative approach yields exactly the same correlation functions for
conserved currents as the one developed in this paper. However, given the
different number of time derivatives at play, we don’t expect these two
approaches to be fully equivalent, and ultimately expect the number of
propagating degrees of freedom to be different—at least based on our
experience with more conventional EFTs. We plan to further investigate this
question in the near future, but in the meantime we find the conceptual
simplicity of our approach—which relies on the standard rules of the coset
construction and doesn’t require additional symmetries—particularly
compelling.

###  4.5 Lowest order DKMS-invariant combinations

Now that we understand how the DKMS symmetry acts on our Goldstone and matter
fields up to first order in $E/T$, we can try to build combinations that are
invariant under these symmetries at this order. To this end, it is once again
convenient to work at first with our “fictitious” covariant derivatives
$\tilde{D}_{\mu}\pi_{r},\dots$ because the DKMS transformation rules (4.19)
are simpler than the ones for the physical fields given in (4.24). We will
eventually rewrite the combinations we derive purely in terms of the physical
fields, but to do so we’ll need to impose the additional diffusive symmetry
introduced in the previous section.

It is easy to see that the following combination changes by a total derivative
up to first order in $E/T$,

$\displaystyle\tilde{D}_{t}\pi_{a}\cdot\tilde{D}_{t}\pi_{r}\quad\to\quad\tilde{D}_{t}\pi_{a}\cdot\tilde{D}_{t}\pi_{r}-\frac{i\beta}{2}\partial_{t}\left[\tilde{D}_{t}\pi_{r}\cdot\tilde{D}_{t}\pi_{r}-\frac{1}{4}\tilde{D}_{t}\pi_{a}\cdot\tilde{D}_{t}\pi_{a}\right]+\mathcal{O}(E^{2}/T^{2}),$
(4.27)

and thus provides an invariant contribution to the effective action. The dot
in this equation stands for the most general symmetric contraction that is
invariant under the diffusive symmetry. An analogous statement can be made
about $\tilde{D}_{i}\pi_{a}\cdot\tilde{D}^{i}\pi_{r}$ and
$\tilde{D}_{t}\varphi_{a}\cdot\tilde{D}_{t}\varphi_{r}$. Using Eq. (4.3), it
is easy to see that these combinations are respectively equal to
$D_{t}\pi_{a}\cdot D_{t}\pi_{r}$, $D_{i}\pi_{a}\cdot D^{i}\pi_{r}$, and
$D_{t}\varphi_{a}\cdot D_{t}\varphi_{r}\to D_{t}\varphi_{a}\cdot\rho_{r}$,
which are therefore invariant under the DKMS symmetry up to a total
derivative.

The last combination that would be natural to consider,
$\tilde{D}_{i}\varphi_{a}\cdot\tilde{D}^{i}\varphi_{r}$, would also be
invariant under DKMS but, alas, not under the diffusive symmetry. This
suggests that we should act with at least one time derivative on
$\tilde{D}^{i}\varphi_{r}$, but the contraction
$\tilde{D}_{i}\varphi_{a}\cdot\partial_{t}\tilde{D}^{i}\varphi_{r}$, albeit
now invariant under the diffusive symmetry, would now no longer be invariant
under DKMS. However, this can be easily remedied by adding a term quadratic in
$\tilde{D}_{i}\varphi_{a}$ to form the combination

$\displaystyle\tilde{D}^{j}\varphi_{a}\cdot\left(\partial_{t}\tilde{D}_{j}\varphi_{r}-\frac{i}{\beta}\tilde{D}_{j}\varphi_{a}\right)\
,$ (4.28)

which is exactly invariant under a DKMS transformation up to
$\mathcal{O}(E^{2}/T^{2})$. Performing manipulations analogous to those that
took us from Eqs. (4.19) to Eqs. (4.24), we can rewrite this combination in
terms of our “physical” fields as follows:

$\displaystyle
D^{j}\varphi_{a}\cdot\left(\nabla_{j}\rho_{r}+\mathcal{F}_{tj}-\frac{i}{\beta}D_{j}\varphi_{a}\right)\
.$ (4.29)

The combinations we have identified above provide the leading kinetic terms
for our fields. Of course, the DKMS symmetry can also be imposed at higher
orders. Examples of such higher order terms that will play a role in our
discussion of antiferromagnets (see Sec. 5.2) are

$\displaystyle
D^{j}\pi_{a}\cdot\left(-\nabla_{t}D_{j}\pi_{r}+i\left[\rho_{r},D_{j}\pi_{r}\right]+\frac{i}{\beta}D_{j}\pi_{a}\right)\
,$ (4.30)

or an equivalent expression with spatial derivatives replaced by time
derivatives.

###  4.6 Power counting rules

As we discussed in Sec. 2.3, in order for our EFT to be well defined we must
be able to assign to each term in the effective action a definite scaling in
terms of our expansion parameters. This, in turn, requires us to specify how
covariant derivatives and matter fields scale with energy and momentum. The
scaling of covariant derivatives is the conventional one—time derivatives
scale like energy, spatial derivatives like momentum. The scaling of matter
fields, instead, is determined by the requirement that the DKMS
transformations (4.24) are organized in an expansion of $E/T$. As a result, we
have that

$\displaystyle\rho_{r},\,D_{t}\pi_{r},\,D_{t}\pi_{a},\,D_{t}\varphi_{a},\,\nabla_{t}\,\sim
E,\qquad\qquad\quad
D_{i}\pi_{r},\,D_{i}\pi_{a},\,D_{i}\varphi_{a},\,\nabla_{i}\,\sim k.$ (4.31)

In the next section we will discuss a few concrete examples, and show how
these power counting rules can be used to estimate the size of various
operators in the effective action.

##  5 Examples: Paramagnets, Antiferromagnets, and Ferromagnets

Non-relativistic magnetic systems at finite temperature are endowed with an
internal $SO(3)$ symmetry that corresponds to global rotations of all the
spins, and thus provide a non-trivial testing ground for our formalism. In the
case of paramagnets this $SO(3)$ symmetry remains unbroken, while in
(anti-)ferromagnets it is spontaneously broken down to an $SO(2)$ subgroup.
The feature that sets ferromagnets apart is that they have a non-zero density
of unbroken $SO(2)$ charge—i.e., a non-zero magnetization density. In this
section, we will discuss separately these three possibilities, restricting our
attention to the internal $SO(3)$ symmetry and neglect the space-time
symmetries that would also be broken by a finite temperature state.242424This
amounts to neglecting phonon excitations by working in the incompressible
limit and, in particular, treating boosts as if they were explicitly broken.
An EFT treatment of the Goldstone modes arising at zero temperature from the
simultaneous breaking of $SO(3)$ and spacetime symmetries was recently
discussed in [50]. These examples will illustrate how to power count terms in
the effective action and how to calculate correlation functions of Noether
currents.252525Classic studies of the high-$T$ Noether current correlator for
these systems include for instance [51, 45].

###  5.1 Paramagnets

Paramagnets are systems where the internal $SO(3)$ symmetry remains
unbroken.262626See [42] for a more general study of non-Abelian hydrodynamics
in the absence of SSB and in the classical limit. As a result, the fields
$\pi_{r}$ and $\pi_{a}$ are absent, the only fields that enter the low energy
EFT are the triplet of Goldstone fields $\vec{\varphi}_{a}$ corresponding to
the breaking of $SO(3)_{1}\times SO(3)_{2}$ down to the diagonal subgroup
$SO(3)_{\rm diag}$, and the three associated matter fields $\vec{\rho}_{r}$.
Using the invariant building blocks we have identified in Sec. 4.5, we can
write the following leading-order effective action for a paramagnet:

$\displaystyle S_{\rm para}=\int
dt\,d^{3}x\left[\frac{k_{*}^{3}}{E_{*}}\,\rho_{r}\cdot
D_{t}\varphi_{a}-k_{*}\,D^{j}\varphi_{a}\cdot\left(\nabla_{j}\rho_{r}+\mathcal{F}_{tj}^{r}-\frac{i}{\beta}D_{j}\varphi_{a}\right)\right]\
,$ (5.1)

where the dot stands for a contraction of the internal indices with a
3-dimensional Kronecker delta. Note that this action is not invariant under
time reversal since this is never a symmetry of Schwinger-Keldysh effective
actions, as we have discussed alread in Sec. 2.4. Physically, this makes it
possible to reproduce a diffusive behavior, as we will see below.

We have chosen to parametrize the two free coefficients in the effective
action (5.1) in terms of some microscopic momentum and energy scales, denoted
respectively with $k_{*}$ and $E_{*}$, so that our effective action will be
organized in powers of $k/k_{*}$, $E/E_{*}$, and $E/T$. In fact, there are
only two independent expansion parameters, because the first two ratios are
related to each other by the free equations of motion for $\rho_{r}$, which
can be obtained by varying the action with respect to $\varphi_{a}$:272727In
the simplest case where $E_{*}=T$, then there is just one independent
expansion parameter.

$\displaystyle\left.\frac{\delta S_{\rm
para}}{\delta\varphi_{a}}\right|_{\varphi_{a}=A_{a}=A_{r}=0}=-\frac{k_{*}^{3}}{E_{*}}\partial_{t}\rho_{r}+k_{*}\partial_{j}\partial^{j}\rho_{r}=0\
,\qquad\rightarrow\qquad\frac{E}{E_{*}}\sim\frac{k^{2}}{k_{*}^{2}}\ .$ (5.2)

Combining this relation with the scaling rules (4.31) and the fact that
$d^{4}x\sim E^{-1}k^{-3}$, we see that the effective action (5.1) contains all
the terms of $\mathcal{O}(k_{*}/k)$, with the term quadratic in $\varphi_{a}$
further enhanced by a factor of $T/E$. In other words, our action is accurate
up to leading order in $k/k_{*}$, and up to the first subleading order in
$E/T$. Notice also that our expression in (5.1) does not rely on a classical
approximation: the covariant derivative $D_{t}\varphi_{a}$ is generically a
non-linear combination of the fields $\varphi_{a}$, and as such contains terms
of all orders in $\hbar$ (see discussion in Sec. 2.6).

Varying instead our action with respect to the gauge fields $A_{\mu}^{a}$ and
$A_{\mu}^{r}$ yields respectively the conserved currents
$\mathcal{J}^{\mu}_{r}$ and $\mathcal{J}^{\mu}_{a}$ expressed in terms of the
external gauge fields, $\rho_{r},\varphi_{a}$, and their derivatives. In
particular, setting $A_{\mu}^{a}=\varphi_{a}=0$, the current
$\mathcal{J}_{r}^{\mu}$ reduces to the _classical_ conserved current in the
presence of external gauge fields $A_{\mu}^{r}$:

$\displaystyle\left.\mathcal{J}^{\mu}_{r}\right|_{\varphi_{a}=A_{a}=0}=\left(\frac{k_{*}^{3}}{E_{*}}\,\rho_{r}\
,-k_{*}\nabla^{j}\rho_{r}+k_{*}F^{tj}_{r}\right)\ .$ (5.3)

From this, we see that the fields $\rho_{r}$ are equal to the conserved charge
densities up to an overall normalization. In the absence of external gauge
fields, we recover the standard constitutive relation
$\mathcal{J}_{r}^{i}=-\mathcal{D}\partial^{i}\mathcal{J}_{r}^{0}$ with a
diffusion coefficient $\mathcal{D}\equiv E_{*}/k_{*}^{2}$. In fact, the
equations of motion (5.2) are just a set of diffusion equations for the charge
densities $\mathcal{J}_{r}^{0}$. When $\rho_{r}=0$, instead, the second term
in the classical current densities reproduces Ohm’s law,
$\mathcal{J}_{r}^{i}=\sigma E^{i}_{r}$, with $E^{i}_{r}$ the electric
component of the field strength and $\sigma\equiv k_{*}$ the conductivity.

More in general, we can calculate correlation functions of the currents
$\mathcal{J}^{\mu}_{r}$ and $\mathcal{J}^{\mu}_{a}$ by taking functional
derivatives with respect to $A_{\mu}^{a}$ and $A_{\mu}^{r}$ of the generating
functional

$\displaystyle
Z[A_{\mu}^{a},A_{\mu}^{r}]=\int\mathcal{D}\rho_{r}\mathcal{D}\varphi_{a}e^{iS_{\rm
para}}\ .$ (5.4)

A single derivative with respect to $A_{\mu}^{a}$ yields the expectation value
of (5.3), which coincides with the expectation value of the physical Noether
current, $\langle\mathcal{J}^{\mu}\rangle$. The requirement that paramagnets
preserve the $SO(3)$ symmetry in the absence of external fields implies that
$\langle\rho_{r}\rangle=0$.

Similarly, by taking two functional derivatives we can calculate two point
functions:

$\displaystyle\langle\mathcal{J}^{\mu}_{rA}(p)\mathcal{J}^{\nu}_{rB}(k)\rangle=-\frac{\delta^{2}Z}{\delta
A^{A}_{\mu a}(-p)\delta A^{B}_{\nu a}(-k)}\
,\qquad\langle\mathcal{J}^{\mu}_{rA}(p)\mathcal{J}^{\nu}_{aB}(k)\rangle=-\frac{\delta^{2}Z}{\delta
A^{A}_{\mu a}(-p)\delta A^{B}_{\nu r}(-k)}\ .$ (5.5)

These current correlators can be inferred from the correlators of $\rho_{r}$
and $\varphi_{a}$, and correspond to different Green’s functions of the
physical Noether current $\mathcal{J}^{\mu}$, along the lines of what we
discussed in Sec. 2.6). For example,

$\displaystyle\langle\mathcal{J}^{\mu}_{r}(x)\mathcal{J}^{\nu}_{r}(x^{\prime})\rangle=\tfrac{1}{2}\langle\\{\mathcal{J}^{\mu}(x),\mathcal{J}^{\nu}(x^{\prime})\\}\rangle\
,\qquad\langle\mathcal{J}^{\mu}_{r}(x)\mathcal{J}^{\nu}_{a}(x^{\prime})\rangle=\theta(t-t^{\prime})\langle[\mathcal{J}^{\mu}(x),\mathcal{J}^{\nu}(x^{\prime})]\rangle\
.$ (5.6)

For simplicity we will calculate these correlators with vanishing external
sources, i.e. setting $A_{\mu}^{r}=A_{\mu}^{a}=0$ after taking the appropriate
functional derivatives.

At leading order in $k/k_{*}$ the current correlators can be calculated by
approximating the currents $\mathcal{J}^{\mu}_{r}$ and $\mathcal{J}^{\mu}_{a}$
up to linear order in the fields $\rho_{r}$ and $\varphi_{a}$; thus, we only
need to know the 2-point functions of these fields, which can be obtained
simply by inverting the quadratic term in (5.1)—this is in fact one of the
main advantages of working with the effective action:

$\displaystyle\langle\rho_{r}^{A}(\omega,\vec{k})\rho_{r}^{B}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{2}{\beta}\frac{\mathcal{D}}{\sigma}\,\frac{\mathcal{D}k^{2}\delta^{AB}}{\omega^{2}+\mathcal{D}^{2}k^{4}}\,$
(5.7a)
$\displaystyle\langle\rho_{r}^{A}(\omega,\vec{k})\varphi_{a}^{B}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{\mathcal{D}}{\sigma}\,\frac{\delta^{AB}}{\omega+i\mathcal{D}k^{2}}\
,$ (5.7b)
$\displaystyle\langle\varphi_{a}^{A}(\omega,\vec{k})\rho_{r}^{B}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{\mathcal{D}}{\sigma}\,\frac{\delta^{AB}}{-\omega+i\mathcal{D}k^{2}}\
,$ (5.7c)

where the primes on the left-hand side denote the fact that we have dropped
the delta functions imposing energy and momentum conservation. Combining these
2-point functions we can easily calculate all the components of the $r$-$r$
correlator,

$\displaystyle\langle\mathcal{J}^{tA}_{r}(\omega,\vec{k})\mathcal{J}^{tB}_{r}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{2\sigma}{\beta}\frac{k^{2}\delta^{AB}}{\omega^{2}+\mathcal{D}^{2}k^{4}}\
,$ (5.8a)
$\displaystyle\langle\mathcal{J}^{tA}_{r}(\omega,\vec{k})\mathcal{J}^{jB}_{r}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{2\sigma}{\beta}\frac{\omega
k^{j}\delta^{AB}}{\omega^{2}+\mathcal{D}^{2}k^{4}}\ ,$ (5.8b)
$\displaystyle\langle\mathcal{J}^{iA}_{r}(\omega,\vec{k})\mathcal{J}^{jB}_{r}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{2\sigma}{\beta}\left(\delta^{ij}-\frac{\mathcal{D}^{2}k^{2}k^{i}k^{j}}{\omega^{2}+\mathcal{D}^{2}k^{4}}\right)\delta^{AB}\
,$ (5.8c)

which are precisely of the form needed to ensure current conservation,
$k^{\mu}\langle J^{A}_{r\mu}J^{B}_{r\nu}\rangle^{\prime}=0$.

The retarded $r$-$a$ correlator is computed in a similar way, and we have
verified that it is also conserved. For completeness, we list here its
components:

$\displaystyle\langle\mathcal{J}^{tA}_{r}(\omega,\vec{k})\mathcal{J}^{tB}_{a}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{\sigma k^{2}\delta^{AB}}{\omega+i\mathcal{D}k^{2}}\ ,$
(5.9a)
$\displaystyle\langle\mathcal{J}^{tA}_{r}(\omega,\vec{k})\mathcal{J}^{jB}_{a}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\langle\mathcal{J}^{jA}_{r}(\omega,\vec{k})\mathcal{J}^{tB}_{a}(-\omega,-\vec{k})\rangle^{\prime}=\frac{\sigma\omega
k^{j}\delta^{AB}}{\omega+i\mathcal{D}k^{2}}\ ,$ (5.9b)
$\displaystyle\langle\mathcal{J}^{iA}_{r}(\omega,\vec{k})\mathcal{J}^{jB}_{a}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\sigma\omega\left(\delta^{ij}-\frac{i\mathcal{D}k^{i}k^{j}}{\omega+i\mathcal{D}k^{2}}\right)\delta^{AB}\
,$ (5.9c)

displaying the usual diffusive pole. All of the above correlators agree with
previous well-known results in the literature, see e.g. [51, 45]. Note that
the correlators above receive contributions from contact terms—i.e. terms in
the effective action (5.1) that are quadratic in the external gauge fields—and
these are crucial to ensure current conservation.

###  5.2 Antiferromagnets

Antiferromagnets are systems where the internal $SO(3)$ symmetry is
spontaneously broken down to $SO(2)$ by a non-trivial staggered-magnetization
order parameter which, without loss of generality, we take to be along the
3-direction. Consequently, the relevant degrees of freedom at low energies are
the two doublets of Goldstones $\pi_{r}$ and $\pi_{a}$, a single Goldstone
$\varphi_{a}$ and its associated matter field $\rho_{r}$.

The antiferromagnetic ground state differs from the ferromagnetic one in that
the magnetization density vanishes. This distinction is often captured by the
statement that ferromagnets break time reversal whereas antiferromagnets do
not—see e.g. [52, 19]. This is actually only part of the story, because
otherwise the Schwinger-Keldysh effective actions for these two systems would
be identical given that time reversal is always broken. The staggered
magnetization of the antiferromagnetic ground state picks a preferred
direction in spin space but not an orientation. Therefore, it breaks rotations
by generic angles around the 1- and 2-directions, but is still invariant under
a residual discrete subgroup that consist of 180°-rotations around these same
axes. These discrete symmetries act on our fields as

$\displaystyle\rho_{r}\to-\rho_{r},\qquad\quad\varphi_{a}\to-\varphi_{a},\qquad\quad\pi_{a,r}^{1}\to-\pi_{a,r}^{1},\qquad\quad\pi_{a,r}^{2}\to\pi_{a,r}^{2},$
(5.10)

in the case of rotations around the 2-axis; for rotations around the 1-axis,
it would be the fields $\pi_{a,r}^{1}$ that are left invariant.282828Rotations
around the 2-direction flip the sign of the generators of rotations around the
1- and 3- directions. In our notation, this means that
$X^{1}_{a,r}\to-X^{1}_{a,r}$ and $T_{a,r}\to-T_{a,r}$. The transformation
properties of the Goldstone modes are determined as usual by demanding that
coset parametrization $\Omega$ remains invariant when the generators are
transformed this way. The transformation rule for $\rho_{r}$ should be the
same as that of its partner field, $\varphi_{a}$. This is also consistent with
that fact that, as we will see, $\rho_{r}$ ir related to the density of spin
along the 3-direction. The Schwinger-Keldysh effective action for
antiferromagnets is invariant under these discrete transformations, whereas
the one for ferromagnets is not. This statement applies equally to zero-
temperature effective actions. In that context, antiferromagnets are also
separately invariant under time reversal, whereas ferromagnets are only
invariant under a combination of discrete rotations and time reversal. This
state of affairs is summarized in Table 1.

| AF $T=0$ | AF $T\neq 0$ | F $T=0$ | F $T\neq 0$
---|---|---|---|---
$T$ | ✓ | ✗ | ✗ | ✗
$D$ | ✓ | ✓ | ✗ | ✗
$T+D$ | ✓ | ✗ | ✓ | ✗
Table 1: Discrete symmetries ferromagnets (F) and antiferromagnets (AF) in
ordinary ($T=0$) and Schwinger-Keldysh ($T\neq 0$) effective actions. We have
denoted with $D$ the residual discrete transformations of the form (5.10). The
symbol ✓(✗) indicates unbroken (broken) symmetries.

Our Goldstone covariant derivatives transform in a simple way under the
discrete symmetry (5.10):

$\displaystyle D_{\mu}\varphi_{a}\to-D_{\mu}\varphi_{a},\qquad\quad
D_{\mu}\pi_{a,r}^{1}\to-D_{\mu}\pi_{a,r}^{1},\qquad\quad
D_{\mu}\pi_{a,r}^{2}\to D_{\mu}\pi_{a,r}^{2}\ .$ (5.11)

Keeping also in mind the requirement of DKMS invariance, the low energy
effective action for antiferromagnets up to first subleading order in $E/T$
turns out to be:

$\displaystyle S_{\rm anti}$ $\displaystyle=\int
dt\,d^{3}x\bigg{[}\frac{\Lambda^{2}}{c_{s}^{3}}\left(D_{t}\pi_{a}\cdot
D_{t}\pi_{r}-c_{s}^{2}D_{i}\pi_{a}\cdot D^{i}\pi_{r}\right)$
$\displaystyle\quad\qquad\qquad\quad+\frac{\Lambda}{c_{s}^{3}}\,\sigma^{\mu\nu}D_{\mu}\pi_{a}\cdot\left(-\nabla_{t}D_{\nu}\pi_{r}+i\left[\rho_{r},D_{\nu}\pi_{r}\right]+\frac{i}{\beta}D_{\nu}\pi_{a}\right)$
(5.12)
$\displaystyle\quad\qquad\qquad\qquad\qquad+\frac{\sigma}{\mathcal{D}}\,\rho_{r}D_{t}\varphi_{a}-\sigma
D^{i}\varphi_{a}\left(\partial_{i}\rho_{r}+\mathcal{F}^{r}_{ti}-\frac{i}{\beta}D_{i}\varphi_{a}\right)\bigg{]},$

where the dot now stands for a contraction of the internal indices with a
2-dimensional Kronecker delta, $\sigma^{\mu\nu}={\rm
diag}(\Sigma_{\pi},c_{s}^{2}\sigma_{\pi},c_{s}^{2}\sigma_{\pi},c_{s}^{2}\sigma_{\pi})$
with $\Sigma_{\pi},\sigma_{\pi}$ both non-negative and dimensionless, and
$\Lambda$ is the energy scale at which spontaneous symmetry breaking occurs.
In light of what we learned in the context of paramagnets, we have already
parametrized the diffusive sector in terms of the conductivity and the
diffusion coefficient. Our action doesn’t contain a tadpole for the external
field $A_{0}^{a}$, which is consistent with the fact that the expectation
value of the $SO(3)$ Noether current must vanish for antiferromagnets. A
discussion of the diffusive sector would be very similar to the analysis we
carried out for paramagnets, and for this reason we’ll mostly focus our
attention to the $\pi$-sector in what follows.

Based on our power counting rules, the second line of the action (5.12) is
suppressed by one power of $E/\Lambda$ compared to the first line.292929We are
relying on the free equation for $\pi_{r}$ that follows from the first line of
(5.12) to set $E\sim c_{s}k$. However, we have included it because thermal
effects enter first at subleading order in $E/\Lambda$. More precisely, the
first two lines of our action contain terms that schematically scale as
follows:

$\displaystyle S_{\rm
anti,\pi}\sim\frac{\Lambda^{2}}{E^{2}}+\frac{\Lambda}{E}\left(\frac{T}{E}+1\right)\
.$ (5.13)

This should be contrasted with the scaling of the terms in the leading
Lagrangian for paramagnets (or the diffusive sector of antiferromagnets),
where thermal effects appear already at leading order in $k/k_{*}$:

$\displaystyle S_{\rm para}\sim\frac{k_{*}}{k}\left(\frac{T}{E}+1\right)\ .$
(5.14)

In both cases we see that at any given order in the temperature-independent
expansion parameter $E/\Lambda$ or $k/k_{*}$ there is an additional expansion
in powers of $E/T$, which we are carrying out only up to first subleading
order.

Once again, varying the generating functional with respect to the external
gauge fields we can calculate correlators of the currents
$\mathcal{J}_{r}^{\mu}$ and $\mathcal{J}_{a}^{\mu}$. For instance, the 2-point
function of $\mathcal{J}_{r}^{t}$ along the broken directions are

$\displaystyle\langle\mathcal{J}^{tI}_{r}(\omega,\vec{k})\mathcal{J}^{tJ}_{r}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{2\Lambda}{\beta
c_{s}}\frac{k^{2}(\sigma_{\pi}\omega^{2}+\Sigma_{\pi}c_{s}^{2}k^{2})}{(\omega^{2}-c_{s}^{2}k^{2})^{2}}\,\delta^{IJ}$
(5.15)

where $I,J=1,2$ are here the $SO(2)$ subgroup indices, not to be confused with
spatial indices. Note that, in deriving Eq. (5.15) we have neglected
corrections of $\mathcal{O}(E^{2}/\Lambda^{2})$ that cannot be trusted at the
order we are working.

The retarded density-density correlator along the broken directions takes
instead the form

$\displaystyle\langle\mathcal{J}^{tI}_{r}(\omega,\vec{k})\mathcal{J}^{tJ}_{a}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{\Lambda^{2}}{c_{s}}\frac{k^{2}\left[1+i(\Sigma_{\pi}-\sigma_{\pi})\omega/\Lambda\right]}{\omega^{2}-c_{s}^{2}k^{2}+i\left(\Sigma_{\pi}\omega^{2}+\sigma_{\pi}c_{s}^{2}k^{2}\right)\omega/\Lambda}\,\delta^{IJ}\
.$ (5.16)

The poles of this propagator are at

$\displaystyle\omega\approx\pm
c_{s}k-\frac{ic_{s}^{2}k^{2}}{2\Lambda}(\Sigma_{\pi}+\sigma_{\pi})\ ,$ (5.17)

showing that sound modes decay at a rate $\Gamma\sim k^{2}$. The correlators
above are in agreement with classic results in the literature up to the order
in $E/\Lambda$ we are considering [51, 53, 45].

Our treatment of the antiferromagnet is notably different from the standard
analyses, which start from the derivative expansion of the equations of motion
for the total magnetization $\vec{M}$ and the staggered-magnetization
$\vec{N}$ (see e.g. [45]). The effective theory (5.12) was constructed to
compute correlators of the conserved currents associated with $\vec{M}$ alone,
and the presence of a non-trivial $\vec{N}$ expectation value is encoded in
the assumption that $SO(3)$ is spontaneously broken down to $SO(2)$ even
though the magnetization density vanishes. This last property is what
distinguishes antiferromagnets from ferromagnets, which we will now turn our
attention to.

###  5.3 Ferromagnets

Ferromagnets spontaneously break $SO(3)\to SO(2)$ because the temporal
component of the Noether current acquires a non-zero expectation value, which
we again assume to be along the 3-direction. As we discussed at the beginning
of the previous section, what distinguishes ferromagnets from antiferromagnets
is the lack of separate invariance under time reversal and discrete
transformations of the form (5.10). Only a combination of these symmetries
leaves the ground state invariant, and therefore our DKMS transformations must
be amended by acting also with (5.10) on the right-hand side, thus obtaining

$\displaystyle D_{\mu}\pi_{a}^{1}(t)\to-
D_{\mu}\pi_{a}^{1}(-t)+i\beta\nabla_{t}D_{\mu}\pi_{r}^{1}(-t)+\beta\left[\rho_{r}(-t),D_{\mu}\pi_{r}^{1}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (5.18a) $\displaystyle D_{\mu}\pi_{a}^{2}(t)\to
D_{\mu}\pi_{a}^{2}(-t)-i\beta\nabla_{t}D_{\mu}\pi_{r}^{2}(-t)-\beta\left[\rho_{r}(-t),D_{\mu}\pi_{r}^{2}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (5.18b) $\displaystyle D_{\mu}\pi_{r}^{1}(t)\to-
D_{\mu}\pi_{r}^{1}(-t)+\frac{i\beta}{4}\nabla_{t}D_{\mu}\pi_{a}^{1}(-t)+\frac{\beta}{4}\left[\rho_{r}(-t),D_{\mu}\pi_{a}^{1}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (5.18c) $\displaystyle D_{\mu}\pi_{r}^{2}(t)\to
D_{\mu}\pi_{r}^{2}(-t)-\frac{i\beta}{4}\nabla_{t}D_{\mu}\pi_{a}^{2}(-t)-\frac{\beta}{4}\left[\rho_{r}(-t),D_{\mu}\pi_{a}^{2}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
,$ (5.18d) $\displaystyle D_{\mu}\varphi_{a}(t)\to-
D_{\mu}\varphi_{a}(-t)-i\beta\nabla_{\mu}\rho_{r}(-t)-i\beta\mathcal{F}_{t\mu}(-t)+\mathcal{O}(E^{2}/T^{2})\
,$ (5.18e)
$\displaystyle\rho_{r}(t)\to\rho_{r}(-t)+\frac{i\beta}{4}\nabla_{t}D_{t}\varphi_{a}(-t)+\frac{\beta}{4}\left[\rho_{r}(-t),D_{t}\varphi_{a}(-t)\right]+\mathcal{O}(E^{2}/T^{2})\
.$ (5.18f)

As a result, the part of the action that describes diffusion of the unbroken
$SO(2)$ current now admits one additional invariant, i.e. $D_{t}\varphi_{a}$:

$\displaystyle S_{\rm diff}$ $\displaystyle=\int
dt\,d^{3}x\bigg{[}k_{\star}^{3}\,D_{t}\varphi_{a}+\frac{\sigma}{\mathcal{D}}\,\rho_{r}D_{t}\varphi_{a}-\sigma
D^{i}\varphi_{a}\left(\partial_{i}\rho_{r}+\mathcal{F}^{r}_{ti}-\frac{i}{\beta}D_{i}\varphi_{a}\right)\bigg{]}.$
(5.19)

This additional term gives rise to a tadpole term for $A_{0}^{a}$, so that the
expectation value of the $SO(3)$ Noether current is now no longer zero:

$\displaystyle\langle\mathcal{J}^{0}_{A}(x)\rangle=k_{\star}^{3}\,\delta_{A}^{3}\
.$ (5.20)

This new term also gives rise to a kinetic term for the $\pi$’s with a single
time derivative, since up to quadratic order in the Goldstones we have
$D_{t}\varphi_{a}=\partial_{t}\varphi_{a}+\epsilon_{IJ}\pi_{a}^{I}\partial_{t}\pi_{r}^{J}+\dots$.
This changes the dispersion relation of the Goldstone modes, and therefore the
power counting scheme. As a result, the effective action for ferromagnets is
now $S_{\rm ferro}=S_{\rm diff}+S_{\pi}$ with303030A term of the form
$\epsilon_{IJ}D_{i}\pi_{a}^{I}D^{i}\pi_{r}^{J}$ is not allowed even in
ferromagnets because it is not DKMS-invariant.

$\displaystyle S_{\pi}$ $\displaystyle=\int
dt\,d^{3}x\bigg{[}-E_{\star}k_{\star}D_{i}\pi_{a}\cdot
D^{i}\pi_{r}+k_{\star}\,\sigma_{\pi}D^{j}\pi_{a}\cdot\left(-\nabla_{t}D_{j}\pi_{r}+i\left[\rho_{r},D_{j}\pi_{r}\right]+\frac{i}{\beta}D_{j}\pi_{a}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\mathcal{O}(k_{\star}/k)\bigg{]}.$
(5.21)

This part of the action introduces a characteristic energy scale, $E_{\star}$,
and a dimensionless coupling $\sigma_{\pi}$. The leading equations of motion
for $\pi_{r}$ now imply a quadratic dispersion relation of the form

$\displaystyle\frac{E}{E_{\star}}\sim\frac{k^{2}}{k_{\star}^{2}}\ ,$ (5.22)

and therefore the size of the terms shown in (5.21) can be estimated to be

$\displaystyle
S_{\pi}\sim\frac{k_{\star}^{3}}{k^{3}}+\frac{k_{\star}}{k}\left(1+\frac{T}{E}\frac{k_{\star}^{2}}{k^{2}}\right)+\mathcal{O}(k_{\star}/k)\
.$ (5.23)

This scaling should be contrasted with the one for the Goldstone sector of
antiferromagnets given in Eq (5.13). In that case, terms that appear in DKMS
invariant combinations where of the same order in $E/\Lambda$, but thermal
effects only appeared a subleading order in this expansion. The power counting
for ferromagnets is instead more subtle, with DKMS invariant combinations
containing terms of different order in $k/k_{\star}$, but with thermal effects
now appearing at leading order. The terms of $\mathcal{O}(k_{\star}/k)$ that
we didn’t write out explicitly (e.g. $D_{t}\pi_{r}\cdot D_{t}\pi_{a}$) have
been omitted to simplify our analysis, and because they would only give a
subleading correction to the real part of the magnon dispersion relation. On
the contrary, the terms that we included at this order because they were
forced upon us by DKMS invariance provide the leading contribution to the
imaginary part of the dispersion relation.

Varying the action with respect to the external gauge fields we can first
derive the Noether currents in terms of our fields, and then calculate their
correlators. For brevity we are going to report here only the correlation
functions of the time components of our currents:

$\displaystyle\langle\mathcal{J}^{t3}_{r}(\omega,\vec{k})\mathcal{J}^{t3}_{a}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{\sigma k^{2}}{\omega+i\mathcal{D}k^{2}}\ ,$ (5.24a)
$\displaystyle\langle\mathcal{J}^{t3}_{r}(\omega,\vec{k})\mathcal{J}^{t3}_{r}(-\omega,-\vec{k})\rangle$
$\displaystyle=k_{\star}^{6}+\frac{2\sigma}{\beta}\frac{k^{2}}{\omega^{2}+\mathcal{D}^{2}k^{4}}\
,$ (5.24b)
$\displaystyle\langle\mathcal{J}^{tI}_{r}(\omega,\vec{k})\mathcal{J}^{tJ}_{a}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{ik^{2}k_{\star}^{5}(E_{\star}-i\sigma_{\pi}\omega)\delta^{IJ}-k_{\star}^{7}\omega\epsilon^{IJ}}{\omega^{2}k_{\star}^{4}-E_{\star}^{2}k^{4}+2i\sigma_{\pi}E_{\star}\omega
k^{4}}\ ,$ (5.24c)
$\displaystyle\langle\mathcal{J}^{tI}_{r}(\omega,\vec{k})\mathcal{J}^{tJ}_{r}(-\omega,-\vec{k})\rangle^{\prime}$
$\displaystyle=\frac{2\sigma_{\pi}}{\beta}\frac{k^{2}k_{\star}^{5}\left[\delta^{IJ}\left(E_{\star}^{2}k^{4}+\omega^{2}k_{\star}^{4}\right)+2iE_{\star}\omega
k_{\star}^{2}k^{2}\epsilon^{IJ}\right]}{\left(\omega^{2}k_{\star}^{4}-E_{\star}^{2}k^{4}\right)^{2}}\
.$ (5.24d)

In the above expressions, we have only kept the terms in the numerators and
denominators that can be trusted given the order in $k/k_{\star}$ we are
working at. Note also that the Eqs. (5.24a) and (5.24b) are consistent with
the results we found in the paramagnet section once we take into account that
our Noether current now has an expectation value (5.20). The poles in the
retarded correlator (5.24c) yield once again the dispersion relations for
magnon excitations, which now display a quadratic dispersion relation with a
decay rate $\Gamma\sim k^{4}$:

$\displaystyle\omega\simeq
E_{\star}\left(\pm\frac{k^{2}}{k_{\star}^{2}}-i\sigma_{\pi}\frac{k^{4}}{k_{\star}^{4}}\right)\
.$ (5.25)

##  6 Conclusions

The coset construction is heralded for its general applicability, ranging from
its origins in nuclear physics [2] to more recent applications to systems at
finite density [25, 54], conformal field theories [55, 56] and gravity [57,
58], just to name a few. However, so far this technique has largely been
applied to the limited case of regular effective actions for the purposes of
computing scattering amplitudes or time-ordered correlators around pure
states. In the present paper, we have extended the construction to Schwinger-
Keldysh effective actions, which can more naturally incorporate the effects of
non-trivial density matrices and facilitate the computation of a more diverse
set of correlators. We focused on spontaneously broken internal symmetries,
with particular emphasis on thermal states. We also took great care in making
our power counting parameters explicit, and distinguish in particular between
the classical ($\hbar$) and high temperature ($E/T$) expansions. Our main
conclusion is that, once the correct symmetry breaking pattern has been
properly identified, the standard rules of the coset construction can be
brought to bear to write down Schwinger-Keldysh effective actions. We would
like to highlight in particular the advantages of the framework developed in
this paper:

  * •

We retain the full non-linear structure inherent to the coset construction,
thus preserving all the symmetries realized non-linearly. This should be
contrasted with the common practice of linearizing Schwinger-Keldysh actions
in the $a$-fields, focusing on the classical regime. Our methodology in
principle allows for a systematic computation in powers of $\hbar$.

  * •

In previous work on the effective Schwinger-Keldysh field theory of thermal
systems, a mysterious, diffusive symmetry was needed to differentiate the
normal phase from a spontaneously broken one [6]. In our approach, no such
symmetry is needed: the symmetry breaking pattern together with basic
principles such as unitarity dictate the relevant degrees of freedom and their
transformation properties under all the symmetries.

In order to illustrate our framework we calculated 2-point functions of
conserved spin currents for paramagnets, anti-ferromagnets, and ferromagnets
in Sec. 5. Our analysis generalizes the classic work in [51, 45], reproducing
their results in the appropriate limits.

There are various avenues along which to extend the present work. First, it
would be interesting to generalize our framework to include spontaneously
broken space-time symmetries. This could be used to extend the coset-based
approach to condensed matter systems put forward in [25, 59], and would
provide a different viewpoint on recent developments surrounding EFTs for
dissipative hydrodynamics [33, 6]. Second, by eschewing the classical limit,
our approach could also shed a new light on the quantum properties of perfect
fluids [60, 61]. Third, it would be interesting to further explore how to
systematically build DKMS-invariants at higher order in $E/T$. And, finally,
we would like to investigate the symmetry breaking pattern associated with
finite density non-thermal density matrices, and understand how the properties
of such states can be encoded in a Schwinger-Keldysh effective action. We
leave all of this for future work.

Acknowledgements: We thank Paolo Creminelli, Luca Delacretaz, Michael Landry,
Mehrdad Mirbabayi, Alberto Nicolis, and Ira Rothstein for helpful discussions,
and in particular Luca Delacretaz for extensive feedback on the draft. The
Mathematica packages xAct [62] and xTras [63] were used extensively in the
course of this work. The work of GG and RP was partially supported by the
National Science Foundation under Grant No. PHY-1915611. RP is now supported
by the Depatment of Energy Award DE-SC0010118. He would also like to
acknowledge the hospitality of the Abdus Salam International Centre for
Theoretical Physics (ICTP), where part of this work was carried out.

##  Appendix A Schwinger-Keldysh Path Integral for a Free System at Finite
Temperature

In this Appendix, we calculate explicitly the Schwinger-Keldysh generating
functional for a free, massive, relativistic $SO(N)$ scalar described by an
in-out action.

$\displaystyle S=-\frac{1}{2}\int
d^{4}x\left(\partial_{\mu}\Phi^{A}\partial^{\mu}\Phi^{A}+m^{2}\Phi^{A}\Phi^{A}\right)\
,\qquad\quad A=1,\ldots,N\ .$ (A.1)

The goal of this appendix is twofold. First, we show explicitly how the
boundary conditions give rise to off-diagonal correlation functions shown e.g.
in Eq. (2.7). Second, we show how the $E\ll T$ limit of the 2-point functions
can be reproduced using an effective action. This is by now textbook material
[13], but we discuss it here for completeness.

Our starting point is the generating functional in Eq. (2.4) with
$\varphi=\mathcal{O}=\vec{\Phi}$:

$\displaystyle Z[J_{1},J_{2}]$
$\displaystyle=\\!\int\\!D\vec{\Phi}_{a}D\vec{\Phi}_{b}D\vec{\Phi}_{c}\,\langle\vec{\Phi}_{a},-\infty|\rho|\vec{\Phi}_{c},-\infty\rangle\langle\vec{\Phi}_{c},-\infty|\bar{T}e^{-i\int\vec{J}_{2}\cdot\vec{\Phi}}|\vec{\Phi}_{b},+\infty\rangle$
(A.2)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\langle\vec{\Phi}_{b},+\infty|Te^{i\int\vec{J}_{1}\cdot\vec{\Phi}}|\vec{\Phi}_{a},-\infty\rangle,$

When the density matrix is thermal, $\rho\propto e^{-\beta H}$, this
functional can be computed from the knowledge of the amplitude

$\displaystyle\langle\vec{\Phi}_{f}({\bf x}),+t_{\star}|Te^{i\int{\rm
d}^{4}x\,\vec{J}\cdot\vec{\Phi}}|\vec{\Phi}_{i}({\bf x}),-t_{\star}\rangle$
$\displaystyle\equiv\int_{\vec{\Phi}(-t_{\star})=\vec{\Phi}_{i}({\bf
x})}^{\vec{\Phi}(+t_{\star})=\vec{\Phi}_{f}({\bf
x})}\mathcal{D}\Phi\,e^{iS[\vec{\Phi}]+i\int{\rm
d}^{4}x\,\vec{J}\cdot\vec{\Phi}}$
$\displaystyle\equiv\langle\vec{\Phi}_{f},+t_{\star}|\vec{\Phi}_{i},-t_{\star}\rangle_{\vec{J}}\
,$ (A.3)

since each factor on the right-hand side of (A.2) is a special case of the
quantity above. In particular, the thermal density matrix factor comes from
setting the source $\vec{J}$ to zero and properly Wick rotating. Combining all
three factors together, we obtain a single path integral expression for the
generating functional, with fields defined along a time-contour $\mathcal{C}$
in the complex plane shown in Fig. 1.

$\,{\rm Re}\,t$$\,{\rm
Im}\,t$$t=+\infty$$t=-\infty$$\phi_{2},\mathcal{O}_{2},J_{2}$$\phi_{1},\mathcal{O}_{1},J_{1}$$-i\beta$

Figure 1: The contour $\mathcal{C}$ used for the Schwinger-Keldysh generator
(2.4) when $\rho\propto e^{-\beta H}$. The dotted line represents the
completion of the trace; there is no evolution along this segment. Other
contours for finite-$T$ systems can be found in the literature; see [64] for a
review.

Due to the free nature of (A.1), the amplitude in (A.3) can be computed
exactly by substituting the classical solution obeying the appropriate
boundary conditions in the presence of the source $\vec{J}$ into the action.
The explicit form of such a classical solution is313131Note that in this
appendix we are denoting spatial vectors in boldface to reserve the vector
symbol for objects in the fundamental representation of the internal $SO(N)$
group.

$\displaystyle\vec{\Phi}_{\rm cl}(t,{\bf k})=K(t,{\bf
k},t_{\star})\cdot\vec{\Phi}_{b}({\bf k})+K(-t,{\bf
k},t_{\star})\cdot\vec{\Phi}_{a}({\bf k})+\int_{-t_{\star}}^{t_{\star}}{\rm
d}t^{\prime}\,G(t,t^{\prime},{\bf k},t_{\star})\cdot\vec{J}(t^{\prime},{\bf
k})\ ,$ (A.4a) $\displaystyle K^{AB}(t,{\bf
k},t_{\star})\equiv\delta^{AB}\frac{\sin(\omega_{{\bf
k}}(t+t_{\star}))}{\sin(2\omega_{{\bf k}}t_{\star})}\ ,$ (A.4b) $\displaystyle
G^{AB}(t,t^{\prime},{\bf
k},t_{\star})\equiv\delta^{AB}\times\Big{[}\frac{\sin(\omega_{{\bf
k}}(t-t^{\prime}))}{2\omega_{{\bf
k}}}\left[\theta(t-t^{\prime})-\theta(t^{\prime}-t)\right]$
$\displaystyle\qquad\qquad\qquad\qquad\quad-\frac{2\cos(\omega_{{\bf
k}}(t+t^{\prime}))-\cos(\omega_{{\bf
k}}(2t_{\star}+t-t^{\prime}))-\cos(\omega_{{\bf
k}}(2t_{\star}+t^{\prime}-t))}{4\omega_{{\bf k}}\sin(2\omega_{{\bf
k}}t_{\star})}\Big{]}\ ,$ (A.4c)

where $\omega_{{\bf k}}^{2}={\bf k}^{2}+m^{2}$ and where, for now, we are
imposing boundary conditions at $\pm t_{\star}$. The $G^{AB}(t,t^{\prime},{\bf
k},t_{\star})$ propagator obeys
$(\square_{x}-m^{2})G^{AB}(x,z,t_{\star})=-\delta^{AB}\delta^{4}(x-z)$ in
position-space and is symmetric under $t\to t^{\prime}$ and vanishes at the
boundaries where either $t^{\prime}$ or $t$ equals $\pm t_{\star}$ while
$K^{AB}(x,{\bf z},t_{\star})$ solves the equation of motion identically,
reduces to the identity at the $t=t_{\star}$ boundary, and vanishes at
$t=-t_{\star}$.

Plugging the expression above into the action, the amplitude (A.3) reduces to

$\displaystyle\langle\vec{\Phi}_{f},+t_{\star}|\vec{\Phi}_{i},-t_{\star}\rangle_{\vec{J}}$
$\displaystyle=\exp\Big{[}\frac{i}{2}\int\frac{{\rm
d}^{3}k}{(2\pi)^{3}}\,\omega_{{\bf k}}\cot(2\omega_{{\bf
k}}t_{\star})\left(\vec{\Phi}_{i}\cdot\vec{\Phi}_{i}+\vec{\Phi}_{f}\cdot\vec{\Phi}_{f}\right)-2\omega_{{\bf
k}}\csc(2\omega_{{\bf k}}t_{\star})\vec{\Phi}_{i}\cdot\vec{\Phi}_{f}$
$\displaystyle\quad+i\int_{-t_{\star}}^{t_{\star}}{\rm d}t\,\int\frac{{\rm
d}^{3}k}{(2\pi)^{3}}\,\vec{J}(t,{\bf k})\cdot K(t,{\bf
k},t_{\star})\cdot\vec{\Phi}_{f}+\vec{J}(t,{\bf k})\cdot K(-t,{\bf
k},t_{\star})\cdot\vec{\Phi}_{i}$
$\displaystyle\quad+\frac{i}{2}\int_{-t_{\star}}^{t_{\star}}{\rm d}t{\rm
d}w\int\frac{{\rm d}^{3}k}{(2\pi)^{3}}\,\vec{J}(t,-{\bf k})\cdot G(t,w,{\bf
k})\cdot\vec{J}(w,{\bf k})\Big{]}$ (A.5)

where $\vec{\Phi}_{i}=\vec{\Phi}_{i}({\bf k})$ everywhere. Note that this
result is manifestly invariant under $SO(N)$. The density matrix components
come from setting $t_{\star}=i\beta/2$ and $\vec{J}\to 0$ in the above.

After calculating the three amplitude factors in (A.2), it is straightforward
to stitch them all together by computing the remaining Gaussian path integrals
over $\vec{\Phi}_{a},\vec{\Phi}_{b}$, and $\vec{\Phi}_{c}$. It is this last
step that gives rise to the cross terms $\sim J_{1}\times J_{2}$. The ultimate
expression is written most compactly in terms of the $\vec{J}_{a},\vec{J}_{r}$
Keldysh basis sources:

$\displaystyle\quad\ln Z[\vec{J}_{a},\vec{J}_{r}]$
$\displaystyle=-\frac{1}{2}\int\frac{{\rm d}\omega{\rm
d}^{3}k}{(2\pi)^{4}}\,\begin{pmatrix}\vec{J}_{r}(-k)&\vec{J}_{a}(-k)\end{pmatrix}\cdot\begin{pmatrix}0&\frac{i}{(\omega+i\epsilon)^{2}-\omega_{{\bf
k}}^{2}}\\\ \frac{i}{(\omega-i\epsilon)^{2}-\omega_{{\bf
k}}^{2}}&\left(\frac{1}{2}+\frac{1}{e^{\beta|\omega|}-1}\right)2\pi\delta(\omega^{2}-\omega_{{\bf
k}}^{2})\end{pmatrix}\cdot\begin{pmatrix}\vec{J}_{r}(k)\\\
\vec{J}_{a}(k)\end{pmatrix}\ .$ (A.6)

In this paper we considered an alternative representation of the generating
functional that relies on effective fields323232In what follows, we denote
these effective fields with the same symbol, $\vec{\Phi}$, to streamline the
notation. and for which the usual, naive rules of Gaussian integration can be
used. Such a representation allows us to avoid the complicated, multi-step
process above. From this viewpoint, the generator (A.6) is instead constructed
as

$\displaystyle Z[J_{a},J_{r}]$
$\displaystyle=\int\mathcal{D}\vec{\Phi}_{a}\mathcal{D}\vec{\Phi}_{r}\,e^{iS_{\rm
EFT}[\vec{\Phi}_{a},\vec{\Phi}_{r}]+i\int{\rm
d}^{d+1}x\,\vec{J}_{r}\cdot\vec{\Phi}_{a}+\vec{J}_{a}\cdot\vec{\Phi}_{r}}$
(A.7)

for some $S_{\rm EFT}$. Finding the appropriate effective action which
reproduces (A.6) is a simple exercise in reverse engineering via standard
Gaussian integral formulas, and a convenient form is333333Equivalence follows
from the identity $\lim_{\varepsilon\to
0}\frac{\varepsilon}{x^{2}+\varepsilon^{2}}=\pi\delta(x)$. The fact that the
imaginary terms are $\mathcal{O}(\varepsilon)$ is an artifact of the free
limit [13]; they are finite in realistic, interacting systems, see e.g. [65].

$\displaystyle S_{\rm EFT}[\vec{\Phi}_{a},\vec{\Phi}_{r}]$
$\displaystyle\equiv\int{\rm
d}^{4}x\,\left(-\partial\vec{\Phi}_{a}\cdot\partial\vec{\Phi}_{r}-m^{2}\vec{\Phi}_{a}\cdot\vec{\Phi}_{r}-2\varepsilon\vec{\Phi}_{a}(x)\cdot\partial_{t}\vec{\Phi}_{r}(x)\right)$
$\displaystyle\quad\qquad\qquad\qquad+i\varepsilon\int\frac{{\rm d}\omega{\rm
d}^{3}k}{(2\pi)^{4}}\,\omega\coth(\beta\omega/2)\vec{\Phi}_{a}(k)\cdot\vec{\Phi}_{a}(-k)\
,$ (A.8)

where we take $\varepsilon\to 0^{+}$ at the end of any given computation, as
usual.

For emphasis, in contrast to the path integral considered at the start of this
section, when using (A.8) there is no need to carefully consider boundary
conditions on fields at $t\to\pm\infty$ or the presence of a non-trivial
density matrix. Such features are already encoded in $S_{\rm EFT}$ itself. In
particular, the explicit factors of $\beta$ in (A.8) reflect the thermal
nature of the system which is also a consequence of the dynamical KMS symmetry
[6] of the effective $\vec{\Phi}$ fields. Explicitly, this acts linearly on
the fields in frequency space as

$\displaystyle\vec{\Phi}_{a}(\omega)$
$\displaystyle\to\cosh(\beta\omega/2)\vec{\Phi}_{a}(-\omega)-2\sinh(\beta\omega/2)\vec{\Phi}_{r}(-\omega)$
(A.9a) $\displaystyle\vec{\Phi}_{r}(\omega)$
$\displaystyle\to\cosh(\beta\omega/2)\vec{\Phi}_{r}(-\omega)-\frac{1}{2}\sinh(\beta\omega/2)\vec{\Phi}_{a}(-\omega)\
,$ (A.9b)

and it can be checked that (A.8) is precisely invariant under the above.

For general values of $\beta\omega$, the representation of the generating
functional via an effective action (A.8) is not obviously advantageous. Though
this construction allows us to more easily use familiar path-integral methods,
its convenience is offset by the fact that the terms $\sim\vec{\Phi}_{a}^{2}$
are non-local. However, $S_{\rm EFT}$ becomes approximately local in the low-
energy limit, $\beta\omega\ll 1$:

$\displaystyle S_{\rm EFT}$ $\displaystyle\simeq\int{\rm
d}^{4}x\,\left\\{-\partial\vec{\Phi}_{a}\cdot\partial\vec{\Phi}_{r}-m^{2}\vec{\Phi}_{a}\cdot\vec{\Phi}_{r}+2\varepsilon\left(-\vec{\Phi}_{a}\cdot\partial_{t}\vec{\Phi}_{r}+\frac{i}{\beta}\vec{\Phi}_{a}(x)\cdot\vec{\Phi}_{a}(x)\right)\right\\}\
.$ (A.10)

The structure of the $\mathcal{O}(\varepsilon)$ terms is dictated by the
$\beta$-expansion of the dynamical KMS symmetry (A.9) in which the symmetry
acts on the fields in momentum space as in

$\displaystyle\vec{\Phi}_{a}(\omega,{\bf k})\to\vec{\Phi}_{a}(-\omega,{\bf
k})-\beta\omega\vec{\Phi}_{r}(-\omega,{\bf k})\
,\qquad\vec{\Phi}_{r}(\omega,{\bf k})\to\vec{\Phi}_{r}(-\omega,{\bf
k})-\frac{\beta\omega}{4}\vec{\Phi}_{a}(-\omega,{\bf k})\ ,$ (A.11)

under which the $\mathcal{O}(\varepsilon)$ terms in (A.10) are strictly
invariant.

Note that the effective action is invariant under two copies of the global
$SO(N)$ symmetries provided we disregard the terms proportional to
$\varepsilon$, as is customary when discussing the symmetries of effective
actions. However, it is easy to show that both copies of the symmetries are
realized linearly on our effective fields, meaning that the off-diagonal
symmetry is not spontaneously broken in this very simple case. This is an
artifact of the free limit, which prevents the existence of a hydrodynamic
regime at low energies.

##  Appendix B Correlator Cheat Sheet

A wide variety of conventions and notations exist for the possible correlators
in quantum field theory. Ours are found below, along with various useful
relations they satisfy. Given a set of operators $\mathcal{O}^{i}$ with $i$
some general indices, some of the correlators one might be interested in
calculating are

$\displaystyle\Delta^{ij}(x,x^{\prime})$
$\displaystyle\equiv\langle\left[\mathcal{O}^{i}(x),\mathcal{O}^{j}(x^{\prime})\right]\rangle$
$\displaystyle G^{ij}_{W}(x,x^{\prime})$
$\displaystyle\equiv\langle\mathcal{O}^{i}(x)\mathcal{O}^{j}(x^{\prime})\rangle$
$\displaystyle G^{ij}_{R}(x,x^{\prime})$ $\displaystyle\equiv
i\theta(t-t^{\prime})\langle\left[\mathcal{O}^{i}(x),\mathcal{O}^{j}(x^{\prime})\right]\rangle=i\langle\mathcal{O}_{r}^{i}(x)\mathcal{O}_{a}^{j}(x^{\prime})\rangle$
$\displaystyle G^{ij}_{A}(x,x^{\prime})$
$\displaystyle\equiv-i\theta(t^{\prime}-t)\langle\left[\mathcal{O}^{i}(x),\mathcal{O}^{j}(x^{\prime})\right]\rangle=-i\langle\mathcal{O}_{a}^{i}(x)\mathcal{O}_{r}^{j}(x^{\prime})\rangle$
$\displaystyle G^{ij}_{S}(x,x^{\prime})$
$\displaystyle\equiv\frac{1}{2}\langle\\{\mathcal{O}^{i}(x),\mathcal{O}^{j}(x^{\prime})\\}\rangle=\langle\mathcal{O}_{r}^{i}(x)\mathcal{O}_{r}^{j}(x^{\prime})\rangle\
,$ (B.1)

where for the last three correlators we have also provided an expression in
terms of fields in the Keldysh basis introduced in Sec. 2.6. The momentum-
space versions of the above are defined in the usual way, with
$G^{ij}_{X}(\omega,\vec{k})$ determined from a correlator of
$\mathcal{O}^{i}(\omega,\vec{k})$ and $\mathcal{O}^{j}(-\omega,-\vec{k})$.

The commutator $\Delta^{ij}$ and the retarded correlator $G_{R}^{ij}$ are
related in momentum space by

$\displaystyle G_{R}^{ij}(\omega,\vec{k})$ $\displaystyle=\lim_{\varepsilon\to
0^{+}}\int\frac{{\rm
d}\omega^{\prime}}{2\pi}\,\frac{\Delta^{ij}(\omega^{\prime},\vec{k})}{\omega^{\prime}-\omega-i\varepsilon}\
,$ (B.2)

as follows from the Fourier-representation of the Heaviside function. The
zero-frequency limit of this relation determines the static susceptibilities
$\chi^{ij}(\vec{k})$,

$\displaystyle\chi^{ij}(\vec{k})\equiv\lim_{\omega\to
0}G_{R}^{ij}(\omega,\vec{k}\neq 0)=\lim_{\varepsilon\to 0^{+}}\int\frac{{\rm
d}\omega^{\prime}}{2\pi}\,\frac{\Delta^{ij}(\omega^{\prime},\vec{k})}{\omega^{\prime}-i\varepsilon}\
.$ (B.3)

At finite temperature, the KMS conditions relate various correlators to each
other. These can be straightforwardly derived starting from inserting ${\bf
1}=e^{-\beta H}e^{\beta H}$ judiciously into the Wightman function
$G_{W}^{ij}$ (also called the dynamical structure factor) and using the
cyclicity of the trace to get (highlighting only the temporal and frequency
dependence)

$\displaystyle G^{ij}_{W}(t)$
$\displaystyle=G^{ji}_{W}(-i\beta-t)=e^{i\beta\partial_{t}}G^{ji}_{W}(-t)\qquad\implies\qquad
G^{ij}_{W}(\omega)=e^{\beta\omega}G^{ji}_{W}(-\omega)\ .$ (B.4)

In particular, in the limit $\beta\omega\ll 1$ that is the focus of our paper,
this implies a form of the Fluctuation-Dissipation Theorem

$\displaystyle G_{S}^{ij}(\omega)\simeq
G_{W}^{ij}(\omega)\approx\frac{\Delta^{ij}(\omega)}{\beta\omega}\ ,$ (B.5)

up to corrections of higher order in $\beta\omega$. The first relation above
implies that $\mathcal{O}^{i}$ and $\mathcal{O}^{j}$ approximately commute,
thereby motivating the “classical limit” terminology.

Using the effective actions discussed in this paper it is straightforward to
calculate the symmetric $G_{S}^{ij}$ viewed as an $r$-$r$ correlator in the
language of the Keldysh rotation of Sec. 2.6. The low energy expressions for
$G_{R}^{ij},\Delta^{ij}$, and $G_{W}^{ij}$ then follow from $G_{S}^{ij}$using
(B.2) and (B.5).

## References

  * [1] S. R. Coleman, J. Wess, and B. Zumino, “Structure of phenomenological Lagrangians. 1.,” Phys.Rev. 177 (1969) 2239–2247.
  * [2] J. Callan, Curtis G., S. R. Coleman, J. Wess, and B. Zumino, “Structure of phenomenological Lagrangians. 2.,” Phys.Rev. 177 (1969) 2247–2250.
  * [3] D. V. Volkov, “Phenomenological Lagrangians,” Fiz. Elem. Chast. Atom. Yadra 4 (1973) 3–41.
  * [4] V. I. Ogievetsky, “ Nonlinear Realizations of Internal and Space-time Symmetries ,” Proc. of. X-th Winter. School of Theoretical Physics in Karpacz, Vol. 1, Wroclaw 227 (1974) .
  * [5] C. R. Galley, “Classical Mechanics of Nonconservative Systems,” Phys. Rev. Lett. 110 (2013) no. 17, 174301, arXiv:1210.2745 [gr-qc].
  * [6] H. Liu and P. Glorioso, “Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,” PoS TASI2017 (2018) 008, arXiv:1805.09331 [hep-th].
  * [7] M. Crossley, P. Glorioso, and H. Liu, “Effective field theory of dissipative fluids,” JHEP 09 (2017) 095, arXiv:1511.03646 [hep-th].
  * [8] M. Hongo, S. Kim, T. Noumi, and A. Ota, “Effective Lagrangian for Nambu-Goldstone modes in nonequilibrium open systems,” arXiv:1907.08609 [hep-th].
  * [9] M. J. Landry, “The coset construction for non-equilibrium systems,” arXiv:1912.12301 [hep-th].
  * [10] Y. Minami and Y. Hidaka, “Spontaneous symmetry breaking and Nambu-Goldstone modes in dissipative systems,” Phys. Rev. E97 (2018) no. 1, 012130, arXiv:1509.05042 [cond-mat.stat-mech].
  * [11] Y. Hidaka and Y. Minami, “Spontaneous symmetry breaking and Nambu-Goldstone modes in open classical and quantum systems,” arXiv:1907.08241 [hep-th].
  * [12] Y. Minami, H. Nakano, and Y. Hidaka, “Rainbow Nambu-Goldstone modes under a nonequilibrium steady flow,” arXiv:2009.10357 [cond-mat.stat-mech].
  * [13] A. Kamenev, Field Theory of Non-Equilibrium Systems.  Cambridge University Press, 2011.
  * [14] R. P. Feynman and F. L. Vernon, Jr., “The Theory of a general quantum system interacting with a linear dissipative system,” Annals Phys. 24 (1963) 118–173.
  * [15] E. A. Calzetta and B.-L. B. Hu, Nonequilibrium Quantum Field Theory.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 9, 2008.
  * [16] L. M. Sieberer, M. Buchhold, and S. Diehl, “Keldysh Field Theory for Driven Open Quantum Systems,” Rept. Prog. Phys. 79 (2016) no. 9, 096001, arXiv:1512.00637 [cond-mat.quant-gas].
  * [17] F. M. Haehl, R. Loganayagam, and M. Rangamani, “Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace,” JHEP 06 (2017) 069, arXiv:1610.01940 [hep-th].
  * [18] I. Z. Rothstein, “TASI lectures on effective field theories,” arXiv:hep-ph/0308266 [hep-ph].
  * [19] R. Penco, “An Introduction to Effective Field Theories,” arXiv:2006.16285 [hep-th].
  * [20] A. J. Niemi and G. W. Semenoff, “Thermodynamic Calculations in Relativistic Finite Temperature Quantum Field Theories,” Nucl. Phys. B 230 (1984) 181–221.
  * [21] A. Niemi and G. Semenoff, “Finite Temperature Quantum Field Theory in Minkowski Space,” Annals Phys. 152 (1984) 105.
  * [22] N. Landsman and C. van Weert, “Real and Imaginary Time Field Theory at Finite Temperature and Density,” Phys. Rept. 145 (1987) 141.
  * [23] P. Arnold and L. G. Yaffe, “Effective theories for real-time correlations in hot plasmas,” Physical Review D 57 (1998) no. 2, 1178.
  * [24] S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications.  Cambridge University Press, 1996.
  * [25] A. Nicolis, R. Penco, and R. A. Rosen, “Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction,” Phys.Rev. D89 (2014) no. 4, 045002, arXiv:1307.0517 [hep-th].
  * [26] A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi, “Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff,” JHEP 06 (2015) 155, arXiv:1501.03845 [hep-th].
  * [27] L. M. Sieberer, A. Chiocchetta, A. Gambassi, U. C. Täuber, and S. Diehl, “Thermodynamic Equilibrium as a Symmetry of the Schwinger-Keldysh Action,” Phys. Rev. B 92 (2015) no. 13, 134307, arXiv:1505.00912 [cond-mat.stat-mech].
  * [28] P. Glorioso, M. Crossley, and H. Liu, “Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current,” JHEP 09 (2017) 096, arXiv:1701.07817 [hep-th].
  * [29] R. Kubo, “Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems,” J. Phys. Soc. Jap. 12 (1957) 570–586.
  * [30] P. C. Martin and J. S. Schwinger, “Theory of many particle systems. 1.,” Phys. Rev. 115 (1959) 1342–1373.
  * [31] P. Glorioso and H. Liu, “The second law of thermodynamics from symmetry and unitarity,” arXiv:1612.07705 [hep-th].
  * [32] F. M. Haehl, R. Loganayagam, and M. Rangamani, “The eightfold way to dissipation,” Phys. Rev. Lett. 114 (2015) 201601, arXiv:1412.1090 [hep-th].
  * [33] F. M. Haehl, R. Loganayagam, and M. Rangamani, “Adiabatic hydrodynamics: The eightfold way to dissipation,” JHEP 05 (2015) 060, arXiv:1502.00636 [hep-th].
  * [34] F. M. Haehl, R. Loganayagam, and M. Rangamani, “Two roads to hydrodynamic effective actions: a comparison,” arXiv:1701.07896 [hep-th].
  * [35] K. Jensen, N. Pinzani-Fokeeva, and A. Yarom, “Dissipative hydrodynamics in superspace,” JHEP 09 (2018) 127, arXiv:1701.07436 [hep-th].
  * [36] M. J. G. Veltman, “Unitarity and causality in a renormalizable field theory with unstable particles,” Physica 29 (1963) 186–207.
  * [37] S. J. Brodsky and P. Hoyer, “The $\bar{h}$ Expansion in Quantum Field Theory,” Phys. Rev. D 83 (2011) 045026, arXiv:1009.2313 [hep-ph].
  * [38] A. A. Radovskaya and A. G. Semenov, “Semiclassical approximation meets Keldysh–Schwinger diagrammatic technique: scalar $\varphi^{4}$,” Eur. Phys. J. C 81 (2021) no. 8, 704, arXiv:2003.06395 [hep-ph].
  * [39] G. Panico and A. Wulzer, The Composite Nambu-Goldstone Higgs, vol. 913.  Springer, 2016.  arXiv:1506.01961 [hep-ph].
  * [40] H. Leutwyler, “On the foundations of chiral perturbation theory,” Annals Phys. 235 (1994) 165–203, arXiv:hep-ph/9311274 [hep-ph].
  * [41] H. Leutwyler, “Nonrelativistic effective Lagrangians,” Phys. Rev. D 49 (1994) 3033–3043, arXiv:hep-ph/9311264.
  * [42] P. Glorioso, L. V. Delacrétaz, X. Chen, R. M. Nandkishore, and A. Lucas, “Hydrodynamics in lattice models with continuous non-Abelian symmetries,” arXiv:2007.13753 [cond-mat.stat-mech].
  * [43] Y. Takahashi and H. Umezawa, “Thermo field dynamics,” Int. J. Mod. Phys. B10 (1996) no. 13n14, 1755–1805.
  * [44] A. J. Beekman, L. Rademaker, and J. van Wezel, “An Introduction to Spontaneous Symmetry Breaking,” SciPost Phys. Lect. Notes 11 (2019) 1, arXiv:1909.01820 [hep-th].
  * [45] D. Forster, “Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions,”. https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19760560918.