# Hydrodynamics as the effective field theory of strong-to-weak spontaneous
symmetry breaking

Xiaoyang Huang  Department of Physics and Center for Theory of Quantum Matter,
University of Colorado, Boulder, CO 80309, USA  Kavli Institute for
Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Marvin Qi  Department of Physics and Center for Theory of Quantum Matter,
University of Colorado, Boulder, CO 80309, USA  Jian-Hao Zhang  Department of
Physics and Center for Theory of Quantum Matter, University of Colorado,
Boulder, CO 80309, USA  Andrew Lucas<EMAIL_ADDRESS>Department
of Physics and Center for Theory of Quantum Matter, University of Colorado,
Boulder, CO 80309, USA

###### Abstract

Inspired by the hunt for new phases of matter in quantum mixed states, it has
recently been proposed that the equivalence of microcanonical and canonical
ensembles in statistical mechanics is a manifestation of strong-to-weak
spontaneous symmetry breaking (SWSSB) in an underlying many-body quantum
description. Here, we build an effective field theory for SWSSB of a global
U(1) symmetry; the answer exactly reproduces the Schwinger-Keldysh effective
field theory of diffusion for the conserved charge. We conclude that
hydrodynamics can be understood as a theory of “superfluidity” for the broken
strong symmetry: a non-vanishing susceptibility is a measurable order
parameter for SWSSB, the diffusion mode is the Goldstone boson of the
spontaneously broken continuous symmetry, and a generalization of Goldstone’s
Theorem implies that the diffusion mode is always long-lived. This perspective
provides a transparent physical explanation for the unusual
“reparameterization” symmetries which are a necessary ingredient of Schwinger-
Keldysh effective field theories for “normal fluids”.

##  1 Introduction

Hydrodynamics describes the dynamics of generic interacting many-body systems
on long time and length scales. The degrees of freedom in hydrodynamics are
conventionally said to be associated to all of the conserved quantities in the
system, such as charge, energy or momentum, together with Goldstone bosons for
any spontaneously broken symmetries. The past decade has seen a resurgence of
work Crossley  _et al._ (2017); Glorioso  _et al._ (2017); Haehl  _et al._
(2016); Jensen  _et al._ (2018); Liu and Glorioso (2018) on hydrodynamics as a
formal effective field theory (EFT), built systematically from a Lagrangian
made out of symmetry-invariant building blocks. For the simplest theories of
hydrodynamics, such as Fick’s Law for the diffusion of a single conserved U(1)
charge, this EFT reproduces well-established physics. In more exotic theories
Landry (2020); Gromov  _et al._ (2020); Huang and Lucas (2022); Farrell  _et
al._ (2022); Glorioso  _et al._ (2023); Jain  _et al._ (2023), such EFT
principles have played a critical role in carefully analyzing and
understanding the possible hydrodynamic phenomena consistent with nontrivial
spacetime symmetries and their spontaneous breaking.

Let us review, at a high level, the way that the EFT of hydrodynamics works,
for the simplest theory of a conserved U(1) charge; we follow the discussion
of Crossley  _et al._ (2017). The concrete goal of the EFT is to calculate
correlation functions of the conserved U(1) current $J^{\mu}=(n,\mathbf{J})$,
which obeys $\partial_{\mu}J^{\mu}=0$. Our goal is to calculate a generating
function, which takes the schematic form

$Z[A_{1},A_{2}]=\mathrm{tr}\left[\rho\mathrm{e}^{\mathrm{i}\int J^{\mu}A_{\mu
1}}\mathrm{e}^{-\mathrm{i}\int J^{\mu}A_{\mu 2}}\right],$ (1)

where $\rho$ represents the thermal density matrix, and we have introduced two
probe fields $A_{1,2}$ because (_1_) hydrodynamic correlation functions are
often probed using retarded Green’s functions, which are defined in terms of
unequal time commutators which cannot be calculated in a time-ordered path
integral, and (_2_) mixed state $\rho$ involves both a bra and a ket, which
each are sourced separately. This is the standard Schwinger-Keldysh formalism.
The key insight of the EFT is that the presence of gapless hydrodynamic modes
renders this generating functional nonlocal in space and time. To obtain a
local EFT, one integrates in two scalar fields $\phi_{1,2}$:

$A_{\mu\alpha}\rightarrow
A_{\mu\alpha}+\partial_{\mu}\phi_{\alpha}\;\;\;(\alpha=1,2).$ (2)

The EFT asserts that

$Z[A_{1},A_{2}]=\int\mathrm{D}\phi_{\alpha}\;\mathrm{e}^{\mathrm{i}S[A_{\alpha}+\partial\phi_{\alpha}]},$
(3)

where $S$ is now a local functional that includes all allowed terms subject to
various symmetries and/or constraints. Most of these constraints are
relatively well-motivated; here we will focus on the one which, in our view,
is not. The standard prescription in the literature has been to demand that

$S[\phi_{1},\phi_{2}]=S[\phi_{1}+c_{1}+f(x),\phi_{2}+c_{2}+f(x)]$ (4)

where $c_{1,2}$ are constants, but $f(x)$ is an arbitrary time-independent
function. The $f(x)$-shift symmetry, which we will refer to as a
reparameterization symmetry, is to be imposed so long as the U(1) symmetry is
not spontaneously broken – i.e., in a normal fluid phase. In contrast, if we
study a U(1) superfluid phase, then we demand only the constant $c_{1,2}$
shift symmetries. It is clear from the form of the EFT, which we will review
later, that this prescription is the right one, but its physical motivation
has appeared relatively ad hoc.

The purpose of this paper is to explain a clear physical origin for the
reparameterization symmetry. Our key insight comes from recent literature de
Groot  _et al._ (2022); Ma and Wang (2023); Zhang  _et al._ (2024); Lee  _et
al._ (2024); Ma  _et al._ (2024); Zhang  _et al._ (2023); Chen and Grover
(2024a, b); Sang  _et al._ (2023); Sang and Hsieh (2024); Ma and Turzillo
(2024); Xue  _et al._ (2024); Guo  _et al._ (2024a); Lee  _et al._ (2023);
Sala  _et al._ (2024); Lessa  _et al._ (2024) which uses insight from quantum
information theory and (symmetry-protected) topological phases of quantum
matter to classify mixed states. A first crucial observation is that in a
mixed state $\rho$ with symmetry $G$, there are two different ways to
implement that symmetry. Letting $U$ represent a unitary matrix representing
one of the elements of $G$, we say that $\rho$ is strongly symmetric if

$U\rho=\mathrm{e}^{\mathrm{i}\theta}\rho,\;\;\;\rho
U^{-1}=\rho\mathrm{e}^{-\mathrm{i}\theta},$ (5)

and any choice of $U$ at most transforms the state $\rho$ by a phase. In
contrast, we could also have a weak symmetry:

$U\rho U^{-1}=\rho,$ (6)

but (5) does not hold. If we study quantum dynamics on a pure state
$|\psi\rangle$, generated by symmetric Hamiltonian $H$ obeying $[H,U]=0$, then
the generator of time evolution on the mixed state $\rho$ is strongly
symmetric.

The hydrodynamic EFT for the conserved U(1), sketched above, describes the
relaxation to a thermal state, such as

$\rho_{\mu}=\mathcal{Z}^{-1}\mathrm{e}^{-\mu Q},$ (7)

where $\mathcal{Z}$ is the partition function and $Q$ represents a conserved
U(1) charge, i.e. the generator of the global U(1) symmetry. It is easy to see
that $\rho_{\mu}$ is weakly symmetric: $\mathrm{e}^{\mathrm{i}\alpha
Q}\rho_{\mu}\mathrm{e}^{-\mathrm{i}\alpha Q}=\rho_{\mu}$; it is not, however,
strongly symmetric: $\mathrm{e}^{\mathrm{i}\alpha
Q}\rho_{\mu}\neq\mathrm{e}^{\mathrm{i}\theta}\rho_{\mu}$, since the grand
canonical ensemble averages over different charge sectors.

Statistical mechanics tells us that the hydrodynamic EFT generically applies
even if we start in an initial state with known global charge: namely, the
initial state does have strong symmetry. This can be understood as the
statement that physics is insensitive to the microcanonical vs. (grand)
canonical ensemble in the thermodynamic limit, which has a nice interpretation
in quantum many-body systems via the eigenstate thermalization hypothesis
Deutsch (1991); Srednicki (1994). In any pure state where the diffusive
hydrodynamic EFT applies, we must have $[H,U]=0$ for any $U\in\mathrm{U}(1)$,
so we conclude that: (_1_) the mixed state dynamics must be strongly
symmetric; (_2_) the state to which we locally relax (i.e. which captures
local correlation functions) is not invariant under strong symmetry. Taken
together, these two facts imply that a symmetry has spontaneously broken, and
we will call this phenomenon strong-to-weak spontaneous symmetry breaking
(SWSSB) Sala  _et al._ (2024); Lessa  _et al._ (2024).

This paper synthesizes the story of SWSSB with the EFT of hydrodynamics,
focusing on the simplest case of diffusion for clarity, although our punchline
should directly generalize to other hydrodynamic models. Our key observation,
which we will carefully and pedagogically motivate, is that SWSSB
quantitatively justifies the reparameterization symmetry of the EFT, providing
a physical justification for the one “mysterious” aspect of the prescription
of Crossley  _et al._ (2017). Aspects of this idea, in particular an
identification of broken strong symmetry in hydrodynamics, can be found in
earlier work Akyuz  _et al._ (2024); Ogunnaike  _et al._ (2023). Our paper
will start by deducing the most general effective field theory for
“superfluidity” in a theory with SWSSB, and derive the corresponding EFT,
which exactly reproduces hydrodynamics Crossley  _et al._ (2017). Along the
way, we will also propose a physical interpretation for the integrated-in
$\phi$ fields as quantum channels, which may be of independent interest.

Our work also provides a concrete answer to a puzzle raised in Sala  _et al._
(2024); Lessa  _et al._ (2024): if a continuous symmetry has SWSSB, is there
an analogue of Goldstone’s Theorem? Our answer is affirmative: we will see
that the hydrodynamic diffusion mode of the EFT is precisely the Nambu-
Goldstone boson of SWSSB. The “gaplessness” of this mode is mandated by an
appropriate generalization of Goldstone’s Theorem to SWSSB.

For the rest of the paper, we have aimed to present a pedagogical
understanding of these conclusions. In Section 2, we first review the EFT of
spontaneous symmetry breaking (SSB) in pure states. Section 4 then uses these
reviewed methods to define and interpret SWSSB in mixed states. Section 5 then
builds an EFT for SWSSB, which is easily seen to reproduce diffusion, along
with the hydrodynamic EFT of Crossley  _et al._ (2017), while Section 6
briefly remarks on the derivation of dissipative superfluid hydrodynamics when
both strong and weak symmetries spontaneously break.

##  2 Review: SSB in pure states

We first review the established theory of SSB in a pure state of a quantum
system in $d$ spatial dimensions. For simplicity we will assume rotational
symmetry, although it could be straightforwardly relaxed. Suppose that we have
a Hamiltonian $H$ which is invariant under symmetry group $G$: the unitary
$U(g)$ obeys $[H,U(g)]=0$ for all $g\in G$. If $|\psi\rangle$ is an eigenstate
of $H$, we say that $|\psi\rangle$ exhibits SSB if and only if

$U(g)|\psi\rangle\neq\mathrm{e}^{\mathrm{i}\theta}|\psi\rangle$ (8)

for some real $\theta$. In words, the state $|\psi\rangle$ is not invariant
under the symmetry group of the dynamics (i.e. Hamiltonian).

Of interest to us will be SSB of a continuous symmetry. The simplest choice is
then to consider $G=\mathrm{U}(1)$, with

$Q=\int\mathrm{d}^{d}x\;n(x)$ (9)

the generator of the symmetry group: $U(\alpha)=\mathrm{e}^{\mathrm{i}\alpha
Q}$ represents $G$ on the Hilbert space. Here $Q$ denotes total charge, while
$n$ denotes charge density. If

$|\psi\rangle\neq|\psi_{\alpha}\rangle:=U(\alpha)|\psi\rangle,$ (10)

then there should be some physical observable $A$ that distinguishes the two
states:

$\left.\frac{\mathrm{d}}{\mathrm{d}\alpha}\langle\psi|\mathrm{e}^{-\mathrm{i}\alpha
Q}A\mathrm{e}^{\mathrm{i}\alpha
Q}|\psi\rangle\right|_{\alpha=0}=-\mathrm{i}\langle\psi|[Q,A]|\psi\rangle\neq
0.$ (11)

We will take $A=\mathrm{e}^{\mathrm{i}\phi}$, where $\phi$ is a canonical
conjugate of the charge density,

$[\phi(x),n(y)]=\mathrm{i}\delta^{(d)}(x-y).$ (12)

Since U(1) is compact, $\phi\sim\phi+2\pi$, so only $A$ is well-defined –
strictly speaking, (12) is not. From (9), (11) and (12),

$\langle\psi_{\alpha}|\mathrm{e}^{\mathrm{i}\phi(x)}|\psi_{\alpha}\rangle=C\mathrm{e}^{-\mathrm{i}(\alpha-\alpha_{0})}.$
(13)

up to an integration constant $\alpha_{0}$. The constant here $C>0$ when there
is SSB, while if $C=0$ the state is symmetric.

We can now deduce an EFT for these degrees of freedom. From (12), $n$ and
$\phi$ are canonical conjugates, so we can write a “Hamiltonian” effective
theory by choosing Lagrangian

$\mathcal{L}=n\partial_{t}\phi-\mathcal{H}(n,\nabla n,\nabla\phi,\ldots).$
(14)

Due to the freedom to shift integration constant in (13), $\mathcal{L}$ must
be invariant under the shift symmetry

$\phi\rightarrow\phi+\alpha_{0},$ (15)

and this is why $\mathcal{H}$ cannot depend on $\phi$. The shift symmetry (15)
is the nonlinear realization of U(1) in the EFT.

If $C>0$ in (13), then (assuming translation symmetry of the states
$|\psi_{\alpha}\rangle$) the integration constant $\alpha_{0}$ can only be
chosen once, and must be the same for all $x$. At lowest order in derivatives,
the effective Hamiltonian is

$\mathcal{H}=\epsilon(n)+\frac{K(n)}{2}(\nabla\phi)^{2}+\cdots.$ (16)

and only depends on $\nabla\phi$. Alternatively, we note that $\mathcal{H}$
must be invariant under the U(1) transformation (15) in order for the charge
density $n$ to be the corresponding Noether charge density. Having fixed
$\mathcal{H}$, it is conventional to then integrate out $n$, which enforces

$\partial_{t}\phi=\mu\equiv\frac{\mathrm{d}\epsilon}{\mathrm{d}n}.$ (17)

Here $\mu$ is the chemical potential; (17) is the first law of thermodynamics
(at zero temperature). Presuming that we can invert the thermodynamic relation
to express $n$ in terms of $\mu$, we then obtain an EFT for a superfluid
phase. Expanding around the background $\phi=\mu_{0}t+\tilde{\phi}$,

$\mathcal{L}=\frac{\chi}{2}(\partial_{t}\tilde{\phi})^{2}-\frac{K_{0}}{2}(\nabla\tilde{\phi})^{2}+\cdots.$
(18)

where we defined the positive constants $K_{0}=K(\mu_{0})$, and
$\chi^{-1}=\epsilon^{\prime\prime}(\mu_{0})$ is the charge susceptibility.

In contrast, when $C=0$ and we do not have SSB, then we must set $K_{0}=0$.
After all, if $C=0$, then $\alpha-\alpha_{0}$ in (13) is ill-defined and can
take on any value. When we build an EFT around such a normal fluid state with
$C=0$, which does not have SSB, we must impose a stronger symmetry than (15):
the reparameterization symmetry

$\phi\rightarrow\phi+f(x)$ (19)

for any $t$-independent function $f$. This enforces that
$\mathcal{H}=\mathcal{H}(n)$ does not depend at all on $\phi$. The resulting
Hamiltonian EFT is therefore trivially integrable: $\partial_{t}n=0$ and
$\partial_{t}\phi$ is independent of $t$.

The Mermin-Wagner Theorem Mermin and Wagner (1966) tells us that this EFT of
superfluidity is, at zero temperature, only valid in $d>2$ spatial dimensions.
Using the leading order EFT, we calculate

$\int\mathrm{d}t\;\langle\tilde{\phi}(x,t)\tilde{\phi}(0)\rangle\propto\int\mathrm{d}^{d}k\;\frac{\mathrm{e}^{\mathrm{i}k\cdot
x}}{k^{2}}\sim|x|^{-(d-2)},$ (20)

with $\log|x|$ in $d=2$, which implies that if $d\leq 2$, correlation
functions of $\phi$ grow with distance and hence $\phi$ cannot be long-range
ordered at different points in space. While this is commonly taken to
demonstrate the absence of SSB, we note that there do exist EFTs of SSB phases
without long-range order Glorioso  _et al._ (2023); Jain  _et al._ (2023), and
so this “lore” is incorrect in general.

It is not accidental that (18) describes a gapless degree of freedom – the
Nambu-Goldstone boson, with dispersion relation

$\omega\approx\pm\sqrt{\frac{K_{0}}{\chi}}|k|.$ (21)

The existence of this mode is guaranteed by Goldstone’s Theorem, which can be
understood as follows. Using (12),

$\displaystyle\int\mathrm{d}^{d}x\;G^{\mathrm{R}}_{nA}(x,t)$
$\displaystyle=\mathrm{i}\mathrm{\Theta}(t)\int\mathrm{d}^{d}x\left\langle[n(x,t),A(0,0)]\right\rangle$
$\displaystyle=\mathrm{i}\mathrm{\Theta}(t)\langle[Q,A(0)]\rangle=\mathrm{i}\mathrm{\Theta}(t)\langle
A\rangle.$ (22)

Here $A=\mathrm{e}^{\mathrm{i}\phi}$, and expectation value is taken in the
state $|\psi_{\alpha=0}\rangle$, for example, with real $\langle A\rangle$.
This implies that

$G^{\mathrm{R}}_{nA}(k=0,\omega)=\left(\mathrm{i}\pi\delta(\omega)-\frac{1}{\omega}\right)\langle
A\rangle.$ (23)

The singular structure in the spectral weight of this Green’s function at
$\omega=0$ necessitates the existence of a gapless mode (21) in the spectrum
independently of microscopic details.

##  3 SWSSB from mixed states

We now turn to the problem of SWSSB in mixed states. In this section, we first
describe a short argument for SWSSB from a single-copy Hilbert space.

Suppose our system undergoes dynamics (e.g. time-evolution by a Lindbladian)
which is invariant under a strong continuous symmetry $G$. In the case of
interest where $G=\mathrm{U}(1)$, this corresponds to dynamics where the
particle number is separately conserved in both the system and environment. We
will restrict to $G=\mathrm{U}(1)$ in the remainder of the paper. If $\rho$ is
a steady state of the dynamics, we say that $\rho$ exhibits SWSSB if and only
if the weak symmetry is preserved,

$U(\alpha)\rho U(\alpha)^{-1}=\rho,$ (24)

and the strong symmetry is broken,

$U(\alpha)\rho\neq\mathrm{e}^{\mathrm{i}\theta}\rho,$ (25)

with $U(\alpha)=\mathrm{e}^{\mathrm{i}\alpha Q}$ and $Q$ defined as in (9).

For any steady state density matrix $\rho$, the family of states

$\rho_{\alpha}=\frac{1}{\mathrm{Tr}(\mathrm{e}^{\alpha Q}\rho)}e^{\alpha
Q}\rho$ (26)

are also steady state density matrices. If $\rho_{\alpha}\neq\rho$, then there
should be a local operator $A$ whose expectation value distinguishes the two
states:

$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}\alpha}\langle
A\rangle_{\alpha}\right|_{\alpha=0}$
$\displaystyle=\left.\frac{\mathrm{d}}{\mathrm{d}\alpha}\frac{\mathrm{Tr}(A\rho_{\alpha})}{\mathrm{Tr}\rho_{\alpha}}\right|_{\alpha=0}$
(27) $\displaystyle=\mathrm{Tr}(AQ\rho)-\mathrm{Tr}{Q\rho}\mathrm{Tr}{A\rho}$
$\displaystyle=\langle AQ\rangle-\langle A\rangle\langle Q\rangle$
$\displaystyle=\frac{1}{2}\langle\\{A,Q\\}\rangle-\langle A\rangle\langle
Q\rangle$

where the last expression follows because $[Q,\rho]=0$, i.e. the weak symmetry
is unbroken. If we take $A=n$ to be the local charge density, we see that this
quantity can be understood as the charge susceptibility $\chi$. The charge
susceptibility can therefore be identified with the order parameter of SWSSB:
a nonvanishing charge susceptibility implies that $\rho_{\alpha}\neq\rho$,
indicating that the strong symmetry is spontaneously broken.

We now deduce the analogue of Goldstone’s Theorem for SWSSB. Using (9) and the
fact that $Q=Q(t)$ is conserved, let us rewrite the susceptibility as

$\displaystyle\chi$
$\displaystyle=\frac{1}{2}\langle\\{n(0),Q(t)\\}\rangle-\langle
n\rangle\langle Q(t)\rangle$ (28)
$\displaystyle=\int\mathrm{d}^{d}x\;\frac{1}{2}\langle\\{n(0),n(x,t)\\}\rangle-\langle
n\rangle^{2}$
$\displaystyle=\int\mathrm{d}^{d}x\;\frac{1}{2}\langle\\{n(0)-\bar{n},n(x,t)-\bar{n}\\}\rangle$
$\displaystyle=\int\mathrm{d}^{d}x\;G^{\mathrm{S}}_{nn}(x,t)$

where $G^{\mathrm{S}}_{nn}$ is the symmetric Green’s function. Fourier
transforming in time, we obtain

$2\pi\chi_{nn}\delta(\omega)=G^{\mathrm{S}}_{nn}(k=0,\omega).$ (29)

We see that a nonvanishing susceptibility implies a nontrivial pole structure
for the symmetric Green’s function $G^{S}_{nn}$. In Section 5, we will
demonstrate how this structure is reflected in the effective field theory.

##  4 SWSSB in a “doubled” Hilbert space

To make contact with the effective field theory and identify the Goldstone
mode, it will be useful to pass from mixed state density matrices to pure
states on the “doubled Hilbert space”:

$|a\rangle\langle b|\rightarrow|ab)$ (30)

The parentheses bra-ket notation emphasizes that we are in the doubled Hilbert
space. The inner product on the doubled Hilbert space is

$(\rho_{1}|\rho_{2})=\mathrm{tr}\left(\rho_{1}^{\dagger}\rho_{2}\right).$ (31)

We will also define left (L) and right (R) operators that act as follows:

$\displaystyle A_{\mathrm{L}}|ab)$ $\displaystyle:=A|a\rangle\langle b|,$
(32a) $\displaystyle B_{\mathrm{R}}|ab)$ $\displaystyle:=|a\rangle\langle
b|B^{\mathsf{T}}.$ (32b)

Physical dynamics in the doubled Hilbert space, in continuous time, is
generated by Lindbladian $\mathcal{H}$ (this is not the standard letter used,
but will avoid clashes in Section 5), which is a completely-positive trace-
preserving (CPTP) map. CP does not provide a linear constraint on
$\mathcal{H}$, but TP does: letting $|I)$ denote the identity operator,

$\mathcal{H}^{\dagger}|I)=0.$ (33)

From the doubled Hilbert space perspective, we will need to build
$\mathcal{H}$ consistent with (33), which we will see generally leads to a
non-Hermitian $\mathcal{H}$. Physical correlation functions are measured with
$(I|\cdots|\rho)$, which has many consequences that we will later unpack.

With this notation in mind, let us consider dynamics that protects the mixed
state of a system with a conserved U(1) symmetry in the grand canonical
ensemble (7). For concreteness in the discussion that follows, it helps to
consider the explicit setting of a lattice model with quantum rotor degrees of
freedom $n_{i}$, $\phi_{i}$ on lattice sites, with
$[\phi_{i},n_{j}]=\mathrm{i}\delta_{ij}$. As before,
$\phi_{i}\sim\phi_{i}+2\pi$, so $n_{i}\in\mathbb{Z}$. A useful basis for the
Hilbert space on each site is simply $|n\rangle$, and
$\mathrm{e}^{\mathrm{i}p\phi}|n\rangle=|n+p\rangle$, with
$\mathrm{e}^{\mathrm{i}p\phi}$ only defined for integer $p$. The generator of
the conserved U(1) on the single-copy Hilbert space is

$Q=\sum_{i}n_{i}.$ (34)

Suppose that the dynamics drives the quantum system to the steady-state

$\rho=\frac{1}{\mathcal{Z}}\mathrm{e}^{-\Phi}=\frac{1}{\mathcal{Z}}\exp\left[-\sum_{i}\varphi(n_{i})\right]$
(35)

with $\mathcal{Z}=\mathrm{tr}(\mathrm{e}^{-\Phi})$ and $\varphi$ a function
which tends to $+\infty$ as $n\rightarrow\pm\infty$. In the doubled Hilbert
space picture,

$\rho=\frac{1}{\mathcal{Z}}\bigotimes_{i}\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\varphi(n)}|nn)_{i}.$
(36)

We will find it useful to define a “chemical potential” operator:

$\displaystyle\mu_{i}$
$\displaystyle=\mathrm{e}^{-\mathrm{i}\phi}\varphi(n_{i})\mathrm{e}^{\mathrm{i}\phi}-\varphi(n_{i})$
$\displaystyle=\varphi(n_{i}+1)-\varphi(n_{i})\approx\frac{\partial\Phi}{\partial
n_{i}}.$ (37)

The last approximation holds if $\varphi$ is slowly-varying and $n_{i}$ is
relatively large; this semiclassical limit will connect very concretely to our
EFT later.

We consider strongly-symmetric dynamics generated by a Lindbladian
$\mathcal{H}$, which has a strong $\mathrm{U}(1)$ symmetry, meaning that in
the doubled Hilbert space,

$[\mathcal{H},Q_{\mathrm{L}}]=[\mathcal{H},Q_{\mathrm{R}}]=0.$ (38)

Notice that the steady state $|\rho)$ is _not_ invariant under the strong
symmetry:

$\mathrm{e}^{\alpha Q_{\mathrm{L}}}|\rho)=\mathrm{e}^{\alpha
Q_{\mathrm{R}}}|\rho)\propto|\rho_{\alpha}),$ (39)

where $\rho_{\alpha}$ has $\varphi\rightarrow\varphi-\alpha n_{i}$. This
equation is not an equality because the density matrix does not stay
normalized. On the other hand, $|\rho)$ _is_ invariant under the weak U(1)
symmetry generated by $Q_{\mathrm{L}}-Q_{\mathrm{R}}$:

$(Q_{\mathrm{L}}-Q_{\mathrm{R}})|\rho)=0.$ (40)

Therefore, we have SWSSB of
$\mathrm{U}(1)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{R}}$ being broken to
the weak $\mathrm{U}(1)_{\mathrm{r}}$ symmetry, where we have defined a
convenient basis of r (weak) and a (strong) operators

$\displaystyle Q_{\mathrm{a}}$
$\displaystyle:=\frac{Q_{\mathrm{L}}+Q_{\mathrm{R}}}{2},$ (41a) $\displaystyle
Q_{\mathrm{r}}$ $\displaystyle:=Q_{\mathrm{L}}-Q_{\mathrm{R}}.$ (41b)

The labels a/r refer to strong/weak symmetries for later comparison with the
KMS-invariant EFT (79).

Strictly speaking, to justify the existence of SSB in the thermodynamic limit,
we should replace the grand canonical $\rho$ from (35) with

$\tilde{\rho}\propto\delta_{\sum n_{i},N}\rho.$ (42)

Notice that $\rho$ and $\delta_{\sum n_{i},N}$ commute, and that
$\mathrm{e}^{\mathrm{i}\alpha
Q_{\mathrm{L}}}|\tilde{\rho})=\mathrm{e}^{\mathrm{i}\alpha N}|\tilde{\rho})$,
meaning that $|\tilde{\rho})$ is strongly symmetric as it shifts only by a
phase. We will later carefully explain why $|\tilde{\rho})$ indeed exhibits
SWSSB, but intuitively, we notice that statistical mechanics suggests that
there is no physical way to tell $|\rho)$ and $|\tilde{\rho})$ apart when
studying local observables. For this reason, it makes sense to work with the
original state $|\rho)$ which is not strongly symmetric.

Ultimately our goal is to build a Hamiltonian EFT of this SWSSB “phase”.
Following Section 2, we will build an EFT in terms of the local densities
$n_{\mathrm{a}}$ and $n_{\mathrm{r}}$ along with their canonical conjugates
$\phi_{\mathrm{a}}$ and $\phi_{\mathrm{r}}$. More precisely, since
$\phi_{\mathrm{a}}$ and $\phi_{\mathrm{r}}$ are not well-defined, we will work
with the operators

$\displaystyle\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}$
$\displaystyle:=\left(\mathrm{e}^{\mathrm{i}\phi}\right)_{\mathrm{L}}\left(\mathrm{e}^{\mathrm{i}\phi}\right)_{\mathrm{R}},$
(43a) $\displaystyle\mathrm{e}^{-\mathrm{i}\phi_{\mathrm{a}}}$
$\displaystyle:=\left(\mathrm{e}^{-\mathrm{i}\phi}\right)_{\mathrm{L}}\left(\mathrm{e}^{-\mathrm{i}\phi}\right)_{\mathrm{R}},$
(43b) $\displaystyle\mathrm{e}^{2\mathrm{i}\phi_{\mathrm{r}}}$
$\displaystyle:=\left(\mathrm{e}^{\mathrm{i}\phi}\right)_{\mathrm{L}}\left(\mathrm{e}^{-\mathrm{i}\phi}\right)_{\mathrm{R}},$
(43c) $\displaystyle\mathrm{e}^{-2\mathrm{i}\phi_{\mathrm{r}}}$
$\displaystyle:=\left(\mathrm{e}^{-\mathrm{i}\phi}\right)_{\mathrm{L}}\left(\mathrm{e}^{\mathrm{i}\phi}\right)_{\mathrm{R}}.$
(43d)

Notice that $\mathrm{e}^{\pm\mathrm{i}\phi_{\mathrm{a}}}|nn)=|(n\pm 1)(n\pm
1))$ and that these definitions are for single site operators. Normalizations
are chosen so that a/r variables obey the same rotor algebra as before. We
also have the useful identity

$(I|\mathrm{e}^{\pm\mathrm{i}\phi_{\mathrm{a}}}=(I|,$ (44)

which is why we have chosen these definitions for $\phi_{\mathrm{a,r}}$ over
others that would have obeyed the same algebra:

$\displaystyle[n_{\mathrm{a}},\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}]$
$\displaystyle=\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}},$ (45a)
$\displaystyle[n_{\mathrm{r}},\mathrm{e}^{2\mathrm{i}\phi_{\mathrm{r}}}]$
$\displaystyle=2\mathrm{e}^{2\mathrm{i}\phi_{\mathrm{r}}},$ (45b)
$\displaystyle[n_{\mathrm{a}},\mathrm{e}^{2\mathrm{i}\phi_{\mathrm{r}}}]=[n_{\mathrm{r}},\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}]$
$\displaystyle=0.$ (45c)

Crucially, (44) implies that $\mathrm{e}^{\pm\mathrm{i}\phi_{\mathrm{a}}}$ is
a quantum channel, not an operator acting on the single-copy Hilbert space.
Lastly, explicit calculation confirms that (for a single site)

$\displaystyle(I|n_{\mathrm{a}}|\rho)$ $\displaystyle=\mathrm{tr}(n\rho),$
(46a) $\displaystyle(I|\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}|\rho)$
$\displaystyle=(I|\rho)=1,$ (46b) $\displaystyle(I|n_{i\mathrm{r}}|\rho)$
$\displaystyle=0.$ (46c)

$\phi_{\mathrm{r}}$ does not have a well-defined value in our steady state, as
discussed around (19).

It will be important to understand the time-reversal transformation in our
system. Following Guo  _et al._ (2024b), notice that given any physical
observable, we may always write

$(I|\mathcal{A}|\rho)^{*}=(\rho|\mathcal{A}^{\dagger}|I)=(I|\mathcal{S}\mathcal{A}^{\dagger}\mathcal{S}^{-1}|\rho)$
(47)

where

$\mathcal{S}|A):=|\sqrt{\rho}A\sqrt{\rho}).$ (48)

Define the time-reversed superoperator

$\tilde{\mathcal{A}}:=\mathcal{S}\mathcal{A}^{\dagger}\mathcal{S}^{-1}.$ (49)

Notice that this generalized time-reversal squares to the identity, so it is a
sensible candidate for time-reversal. We can explicitly calculate that
$(A_{\mathrm{L}})^{\dagger}=(A^{\dagger})_{\mathrm{L}}$ and
$(A_{\mathrm{R}})^{\dagger}=(A^{\dagger})_{\mathrm{R}}$, and
$\mathcal{S}(\mathrm{e}^{\pm\mathrm{i}\phi})_{\mathrm{L,R}}\mathcal{S}^{-1}\approx\mathrm{e}^{\mp\mu/2}(\mathrm{e}^{\pm\mathrm{i}\phi})_{\mathrm{L,R}}$,
with the latter approximation reasonable so long as $\mu$ is a slowly varying
function. Hence,

$\displaystyle\tilde{n}_{i\mathrm{a}}$
$\displaystyle=\tilde{n}_{i\mathrm{a}},$ (50a)
$\displaystyle\tilde{n}_{i\mathrm{r}}$
$\displaystyle=\tilde{n}_{i\mathrm{r}},$ (50b)
$\displaystyle\mathrm{e}^{\pm\mathrm{i}\tilde{\phi}_{i\mathrm{a}}}$
$\displaystyle=\mathrm{e}^{\pm\mathrm{i}(-\phi_{i\mathrm{a}}-\mathrm{i}\mu_{i})},$
(50c) $\displaystyle\mathrm{e}^{\pm\mathrm{i}\tilde{\phi}_{i\mathrm{r}}}$
$\displaystyle=\mathrm{e}^{\pm\mathrm{i}(-\phi_{i\mathrm{r}})}.$ (50d)

Now, let us explain why there is no Mermin-Wagner Theorem Mermin and Wagner
(1966) for SWSSB. Roughly speaking, from the eigenstate thermalization
hypothesis and statistical mechanics, insofar as local correlation functions
are concerned, the density matrix in our mixed steady state $\rho$ is well-
approximated by a grand canonical ensemble (7), which exhibits SWSSB. Since
(7) is a tensor product, there is no long-range order acccessible in any local
correlators, without using multiple copies of the density matrix. (This
renders the probes of SSB unmeasurable in current experiments.)

Let us now be a little more careful. To see the emergence of a “long-range
order” which can be associated to SWSSB, we should start with a strongly
symmetric state, and a useful choice will be (42). In the thermodynamic limit,
if $N=\langle n\rangle V$ (where $V$ is the number of lattice sites) is chosen
such that $\langle n\rangle$ is the expectation value in the grand canonical
ensemble $\rho$, defined in (35). To see the presence of long-range order, we
must look at more complicated correlation functions involving multiple copies
of a density matrix, such as

$\displaystyle F_{xy}:=\mathrm{tr}$
$\displaystyle\left(\sqrt{\tilde{\rho}}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)\sqrt{\tilde{\rho}}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)^{\dagger}\right).$
(51)

for some integer $p\neq 0$. Let us now argue that

$F_{xy}=\delta_{xy}+a(1-\delta_{xy}),$ (52)

where

$a=\left(\dfrac{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{-[\varphi(n)+\varphi(n+p)]/2}}{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\varphi(n)}}\right)^{2}.$
(53)

From the form of $F_{xy}$, and because $\delta_{\sum n,N}$ commutes with both
$\sqrt{\rho}$ and the product
$\mathrm{e}^{\mathrm{i}\phi_{x}}\mathrm{e}^{-\mathrm{i}\phi_{y}}$, we may
write

$\displaystyle F_{xy}$ $\displaystyle=\mathrm{tr}\left(\delta_{\sum
n,N}\sqrt{\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)\sqrt{\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)^{\dagger}\right)$
$\displaystyle=\int\limits_{0}^{2\pi}\frac{\mathrm{d}\theta}{2\pi}\mathrm{tr}\left(\mathrm{e}^{\mathrm{i}\theta(\sum
n-N)}\sqrt{\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)\sqrt{\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)^{\dagger}\right)$
$\displaystyle=\int\limits_{0}^{2\pi}\frac{\mathrm{d}\theta}{2\pi}\mathcal{Y}(\theta)^{V-2}|\tilde{\mathcal{Y}}(\theta)|^{2}$
(54)

where

$\displaystyle\mathcal{Y}(\theta)$
$\displaystyle=\dfrac{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}\theta(n-\langle
n\rangle)-\varphi(n)}}{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\varphi(n)}},$
(55a) $\displaystyle\tilde{\mathcal{Y}}(\theta)$
$\displaystyle=\dfrac{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}\theta(n-\langle
n\rangle)-[\varphi(n)+\varphi(n+p)]/2}}{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\varphi(n)}}.$
(55b)

In the thermodynamic limit $V\rightarrow\infty$, the integral in (54) can be
done asymptotically by saddle point which will collapse onto $\theta\approx
0$. This leads to (52). In contrast, the “disconnected” correlator

$\mathrm{tr}\left(\sqrt{\tilde{\rho}}\mathrm{e}^{\mathrm{i}p\phi_{x}}\sqrt{\tilde{\rho}}\mathrm{e}^{-\mathrm{i}p\phi_{x}}\right)=0$
(56)

since
$\mathrm{e}^{\mathrm{i}p\phi_{x}}\sqrt{\tilde{\rho}}\mathrm{e}^{-\mathrm{i}p\phi_{x}}$
lies in the charge sector $N+1$ which is orthogonal to $N$. Hence we have
long-range order in $F_{xy}$. The result does not depend on spatial dimension.
Since $\tilde{\rho}$ is not the ground state of a spatially-local Hermitian
Hamiltonian, the conventional Mermin-Wagner Theorem has no bearing on
$F_{xy}$. Since $\rho$ is a diagonal density matrix from (35) and remains
diagonal in the same basis if we conjugate $\rho$ by the operator
$\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}$, (51) is
exactly the fidelity correlator defined in Lessa  _et al._ (2024) to be the
universal order parameter of SWSSB:

$\displaystyle F$
$\displaystyle\left(\rho,\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right){\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)^{\dagger}\right)$
$\displaystyle=\mathrm{tr}\left(\sqrt{\sqrt{\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right){\rho}\left(\mathrm{e}^{\mathrm{i}p\phi_{x}}\mathrm{e}^{-\mathrm{i}p\phi_{y}}\right)^{\dagger}\sqrt{\rho}}\right).$
(57)

In general, $a=0$ only when $\Phi$ is concentrated on a single value of $n$,
which describes an insulating state without SWSSB; all other states have
SWSSB. As this correlation function is not measurable in an actual many-body
system, and the existence of long-range order is a property of the state
$\rho$, which is a physical steady state in any spatial dimension, we again
see that the Mermin-Wagner Theorem is not related to SWSSB.

Although the fidelity correlator that detected SWSSB is not measurable in
experiment, the non-vanishing of $F_{xy}$ is detectable via a very simple
thermodynamic quantity: the compressibility $\chi=\partial\langle
n\rangle/\partial\mu$, where $\mu$ is the thermodynamic chemical potential.
Imagine shifting $\mu\rightarrow\mu+\delta\mu$, which would adjust

$\varphi(n)\rightarrow\varphi(n)+n\cdot\delta\mu.$ (58)

Then

$\chi=\frac{\partial\langle
n\rangle}{\partial\delta\mu}=\frac{\partial}{\partial\delta\mu}\dfrac{\displaystyle\sum_{n\in\mathbb{Z}}n\mathrm{e}^{-\varphi(n)-\delta\mu
n}}{\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\varphi(n)-\delta\mu
n}}=\left\langle n^{2}\right\rangle-\langle n\rangle^{2},$ (59)

with averages taken in the ensemble $\rho$. $\chi>0$ if and only if
$\varphi(n)<\infty$ on more than one value of $n$, which occurs if and only if
$F_{xy}\neq 0$ for some $p$.

If we consider the toy model

$\varphi(n)\approx\frac{(n-\langle n\rangle)^{2}}{2\chi}$ (60)

where $\chi\ll 1$, such that the discretization in $n$ is not important, then
we notice that (on a single lattice site)

$\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}|\rho_{\langle
n\rangle})=|\rho_{\langle
n\rangle+1})\propto\mathrm{e}^{n_{\mathrm{a}}/\chi}|\rho_{\langle n\rangle}).$
(61)

shifts the steady state to one with a higher value of $\langle n\rangle$,
which from (58) can be interpreted as adjusting the chemical potential. This
implies that in a state with SWSSB, we can mirror (11) by studying

$1=(I|[n_{\mathrm{a}},\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}]|\rho)=\mathrm{tr}(n\rho_{\langle
n\rangle+1})-\mathrm{tr}(n\rho_{\langle n\rangle}).$ (62)

The latter equality above only holds for states with SWSSB, while the former
equality holds in all states. To return to the many-body problem, $|\rho)$ in
(62) should be understood as the reduced density matrix on a single site.
Still, the former equality will underlie our generalization of Goldstone’s
Theorem to SWSSB in Section 5, whenever $\chi>0$.

##  5 Effective field theory

At long last, we are ready to put together the EFT of SWSSB, following Section
2 while being careful about the subtleties discussed in Section 4. The fields
in the Hamiltonian EFT are
$n_{\mathrm{a}},n_{\mathrm{r}},\phi_{\mathrm{a}},\phi_{\mathrm{r}}$, with
Lagrangian

$\mathcal{L}=n_{\mathrm{a}}\partial_{t}\phi_{\mathrm{a}}+n_{\mathrm{r}}\partial_{t}\phi_{\mathrm{r}}-\mathcal{H}(n_{\mathrm{a,r}},\phi_{\mathrm{a,r}},\nabla
n_{\mathrm{a,r}}\ldots).$ (63)

So far, this is not the standard Schwinger-Keldysh (SK) formalism Crossley
_et al._ (2017), although there is a transparent way to derive the SK EFT
which we will come back to compare to soon. There are a number of constraints
on $\mathcal{H}$, which we now enumerate.

(_1_) For simplicity, we assume that the EFT is time-reversal symmetric. Time-
reversal acts on the fields as in (50): $n_{\mathrm{a,r}}$ are even, while
$\phi_{\mathrm{r}}\rightarrow-\phi_{\mathrm{r}}$ and

$\phi_{\mathrm{a}}\rightarrow-\phi_{\mathrm{a}}-\mathrm{i}\mu(n_{\mathrm{a}}).$
(64)

If $\mathcal{L}$ is invariant up to a total derivative, we must have

$\mathcal{H}(n_{\mathrm{a}},n_{\mathrm{r}},\phi_{\mathrm{a}},\phi_{\mathrm{r}})=\mathcal{H}(n_{\mathrm{a}},n_{\mathrm{r}},-\phi_{\mathrm{a}}-\mathrm{i}\mu(n_{\mathrm{a}}),-\phi_{\mathrm{r}})$
(65)

where $\mu(n)$ is the chemical potential (37).

(_2_) Generalizing the discussion below (16), we must have a shift symmetry
associated to $\mathrm{U}(1)_{\mathrm{a}}$ and $\mathrm{U}(1)_{\mathrm{r}}$;
after all, neither symmetry is explicitly broken. Thus, $\mathcal{H}$ can only
depend on $\nabla\phi_{\mathrm{a,r}}$. But if we are studying dynamics in the
grand canonical ensemble $|\rho)$, $\mathrm{U}(1)_{\mathrm{a}}$ is SSB while
$\mathrm{U}(1)_{\mathrm{r}}$ is not. Following (19), we then expect to have a
reparameterization symmetry for $\phi_{\mathrm{r}}$. Hence,

$\mathcal{H}(n_{\mathrm{a}},n_{\mathrm{r}},\phi_{\mathrm{a}},\phi_{\mathrm{r}})=\mathcal{H}(n_{\mathrm{a}},n_{\mathrm{r}},\phi_{\mathrm{a}}+\alpha,\phi_{\mathrm{r}}+f(x))$
(66)

for constant $\alpha$ and function $f$.

(_3_) Suppose that

$\frac{\mathrm{d}}{\mathrm{d}t}|\rho)=-\mathrm{i}\mathcal{H}|\rho);$ (67)

as we explained in Section 4, $\mathcal{H}$ need not be Hermitian. In the EFT,
therefore, $\mathcal{H}$ will not be real-valued. Convergence of a path
integral over $\exp[\mathrm{i}\int\mathrm{d}^{d}x\mathrm{d}t\;\mathcal{L}]$
evidently requires

$\mathrm{Im}(\mathcal{H})\leq 0.$ (68)

(_4_) Lastly, we must account for the constraint (33). To see what this does,
let us consider a special case where

$\mathcal{H}=H(n_{\mathrm{L}})-H(n_{\mathrm{R}}),$ (69)

corresponding to purely unitary quantum dynamics generated by a density-
dependent Hamiltonian $H$, appropriate to a normal fluid phase. Notice that
$\mathcal{H}^{\dagger}|I)=0$, which follows from
$\mathcal{H}|\rho)=|[H,\rho])$. If we had chosen $\mathcal{H}$ to depend only
on $n_{\mathrm{a}}$, then $\mathcal{H}|\rho)$ would be a series of
anticommutators, which would not be guaranteed to have vanishing trace.
Meanwhile, no $\phi_{\mathrm{r}}$-only terms are allowed since in (43),
$\mathrm{e}^{\pm\mathrm{i}\phi_{\mathrm{r}}}$ changes the relative charge in
the L/R Hilbert spaces, meaning that there is a vanishing inner product
between $|\rho)$ and $(I|$ under dynamics. Therefore, we demand

$\mathcal{H}(n_{\mathrm{a}},0,0,\phi_{\mathrm{r}})=0,$ (70)

such that every term in $\mathcal{H}$ needs at least one power of either
$\nabla\phi_{\mathrm{a}}$ or $n_{\mathrm{r}}$.

We have now enumerated the symmetries and constraints on the EFT of SWSSB. The
most general expression at leading order in fields with vanishing expectation
value in equilibrium ($\nabla\phi_{\mathrm{a}}$, $n_{\mathrm{r}}$) is

$\mathcal{H}=\hat{\mu}(n_{\mathrm{a}})n_{\mathrm{r}}-\mathrm{i}\sigma(n_{\mathrm{a}})\nabla\phi_{\mathrm{a}}\cdot\nabla(\phi_{\mathrm{a}}+\mathrm{i}\mu(n_{\mathrm{a}}))+\cdots.$
(71)

where $\sigma\geq 0$ due to (68). We emphasize that the coefficient
$\hat{\mu}$ above does not formally need to equal $\mu$ from (37), at this
point. Combining (63) and (71), we notice that $n_{\mathrm{r}}$ is a Lagrange
multiplier that fixes

$\partial_{t}\phi_{\mathrm{r}}=\hat{\mu}(n_{\mathrm{a}})+\cdots,$ (72)

where $\cdots$ denote irrelevant corrections. Hence, we can integrate out two
of our four fields. One choice is to integrate out the r-fields, which leads
to the Martin-Siggia-Rose (MSR) Martin  _et al._ (1973) Lagrangian: at leading
order,

$\mathcal{L}=n_{\mathrm{a}}\partial_{t}\phi_{\mathrm{a}}+\mathrm{i}\sigma(n_{\mathrm{a}})\nabla\phi_{\mathrm{a}}\cdot\nabla(\phi_{\mathrm{a}}+\mathrm{i}\mu(n_{\mathrm{a}}))$
(73)

Varying with respect to $\phi_{\mathrm{a}}$, we obtain the diffusion equation
for the conserved charge:

$\partial_{t}n_{\mathrm{a}}=\nabla\cdot(D\nabla n_{\mathrm{a}})+\cdots.$ (74)

with $D=\sigma/\chi$ the diffusion constant, and the thermodynamic stability
constraint that susceptibility $\chi=\partial n_{\mathrm{a}}/\partial\mu\geq
0$. The $\cdots$ include $\phi_{\mathrm{a}}$-dependent terms, which are
interpreted as stochastic fluctuations Huang  _et al._ (2023). To translate
(73) to the formalism of Akyuz  _et al._ (2024); Huang  _et al._ (2023), one
sets $n_{\mathrm{a}}\rightarrow n$ (the density of the U(1) charge), and
$\phi_{\mathrm{a}}\rightarrow-\pi$, where $\pi$ is the MSR noise field
conjugate to $n$.

The more popular choice in the recent literature, however, has been to
integrate out $n_{\mathrm{a}}$ and $n_{\mathrm{r}}$ instead Crossley  _et al._
(2017). This choice becomes particularly elegant when we assume that
$\hat{\mu}\propto\mu$, with the former defined in (71) governing the steady
state to which the dynamics approaches, as is the case for systems in a
thermal steady state $\rho\propto\mathrm{e}^{-\beta H}$, where the same
(single-copy Hilbert space) Hamiltonian $H$ generates both time evolution and
controls the steady state. In this setting, we indeed would see from (69) and
$\Phi=\beta H$ that

$\mu=\beta\hat{\mu}.$ (75)

Strictly speaking, systems with conserved energy would also have a diffusing
energy density mode, which we are neglecting in our EFT; still, this
complication does not change the essential physics we are focusing on here.
Integrating out $n_{\mathrm{a}}$ and $n_{\mathrm{r}}$, we could then write
down an EFT subject to the reparameterization symmetries

$\mathcal{L}(\phi_{\mathrm{a}},\phi_{\mathrm{r}})=\mathcal{L}(\phi_{\mathrm{a}}+\alpha,\phi_{\mathrm{r}}+f(x))$
(76)

positivity (68), the modification of (70)

$\mathcal{L}(\phi_{\mathrm{a}}=0,\phi_{\mathrm{r}})=0,$ (77)

and lastly the Kubo-Martin-Schwinger (KMS) symmetry

$\mathcal{L}(-\phi_{\mathrm{a}}-\mathrm{i}\beta\partial_{t}\phi_{\mathrm{r}},-\phi_{\mathrm{r}},-t)=\mathcal{L}(\phi_{\mathrm{a}},\phi_{\mathrm{r}},t)+\partial(\cdots)$
(78)

which generalizes (64), and ensures that up to total derivatives the EFT is
time-reversal symmetric. Notice that the KMS transformation is particularly
simple due to (75). These are precisely the postulates of Crossley  _et al._
(2017) which have led to a modern renaissance in our understanding of the EFT
of hydrodynamics. Their methods led to the following Lagrangian describing
diffusion:

$\mathcal{L}=n\partial_{t}\phi_{\mathrm{a}}+\mathrm{i}\sigma\nabla\phi_{\mathrm{a}}\cdot\nabla(\phi_{\mathrm{a}}+\mathrm{i}\beta\partial_{t}\phi_{\mathrm{r}})+\cdots.$
(79)

where $\cdots$ denote irrelevant corrections, and the coefficients $n$ and
$\sigma$ are functions of $\mu=\beta\partial_{t}\phi_{\mathrm{r}}$ subject to
the positivity constraint $\sigma\geq 0$. The main result of our paper is an
intuitive understanding of the postulates which led to (79) – chiefly the
unusual reparameterization symmetry (76) – which is a clear manifestation of
SWSSB, as explained around (66). It is easy to see that assuming (75),
integrating out the $n$-fields from the EFT (63) and (71) leads to (79); in
particular, the quantum/classical field
$\phi_{\mathrm{a}}$/$\phi_{\mathrm{r}}$ introduced in the SK formalism arose
naturally as canonical conjugates to the strong/weak charge densities
$n_{\mathrm{a}}$/$n_{\mathrm{r}}$.

We already discussed various probes and consistency checks for SWSSB coming
entirely from equilibrium considerations in Section 4. The last crucial
property of SSB in pure states was the presence of Goldstone’s Theorem.
Generalizing the discussion around (22), we now define
$A_{\mathrm{a}}=\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}}$ and consider

$\displaystyle\int\mathrm{d}^{d}x\;G^{\mathrm{R}}_{n_{\mathrm{a}}A_{\mathrm{a}}}(x,t)$
$\displaystyle:=\mathrm{i}\mathrm{\Theta}(t)\int\mathrm{d}^{d}x(I|[n_{\mathrm{a}}(x,t),A_{\mathrm{a}}(0)]|\rho)$
$\displaystyle=\mathrm{i}\mathrm{\Theta}(t)(I|\rho)=\mathrm{i}\mathrm{\Theta}(t),$
(80)

where the commutator is evaluated analytically as in (12), and we are
neglecting the compactness of $\phi$ in the hydrodynamic (long-wavelength)
limit. We remind the reader that $n_{\mathrm{a}}$ and $\phi_{\mathrm{a}}$ are
genuine linear operators in the doubled Hilbert space, and that (as stated
before) $\phi_{\mathrm{a}}$ is not defined on a single-copy Hilbert space.
Similar to the discussion around (23), the imaginary part of the Green’s
function $G^{\mathrm{R}}_{n_{\mathrm{a}}A_{\mathrm{a}}}(k=0,\omega)$ defined
in (5) necessarily has a $\delta(\omega)$ singularity, which signals the
presence of gapless modes in the EFT. These gapless modes are the Nambu-
Goldstone bosons of SWSSB, and we now argue that they are precisely the
diffusion quasinormal mode (74) in hydrodynamics. Taylor expanding
$A_{\mathrm{a}}\approx 1+\mathrm{i}\phi_{\mathrm{a}}+\cdots$, and calculating
the Green’s function

$G^{\mathrm{R}}_{n_{\mathrm{a}}A_{\mathrm{a}}}(k,\omega)\approx\mathrm{i}G^{\mathrm{R}}_{n_{\mathrm{a}}\phi_{\mathrm{a}}}(k,\omega)=\frac{-1}{\omega+\mathrm{i}Dk^{2}},$
(81)

we see that the spectral weight

$\mathrm{Im}G^{\mathrm{R}}_{n_{\mathrm{a}}A_{\mathrm{a}}}(k,\omega)=\frac{1}{2\chi}G^{\mathrm{S}}_{nn}(k,\omega)=\frac{Dk^{2}}{\omega^{2}+(Dk^{2})^{2}}.$
(82)

where $G^{\mathrm{S}}_{nn}(x,t)=\langle\\{n(x,t),n(0)\\}\rangle$ is the
symmetric Green’s function in the single-copy Hilbert space, whose value in
the hydrodynamic limit is provided in e.g. Crossley  _et al._ (2017).

The singularity in (82),

$G^{\mathrm{S}}_{nn}(k=0,\omega)=2\pi\chi\times\delta(\omega),$ (83)

precisely matches the singularity in the Fourier transform of (5), as
described in (23). A physical understanding of this result follows from (61),
which implies that

$(I|[n_{\mathrm{a}}(x,t),\mathrm{e}^{\mathrm{i}\phi_{\mathrm{a}}(0)}]|\rho)\approx\frac{1}{2\chi}G^{\mathrm{S}}_{nn}(x,t).$
(84)

Therefore, so long as $\chi>0$ and we have SWSSB, the Goldstone boson is
simply the diffusion mode, which can (and has, routinely) been detected in
experiment. The spectral weight in (5) as $\omega\rightarrow 0$ ensures that
the diffusion mode has arbitrarily long lifetimes in the thermodynamic limit
(on sufficiently long length scales).

Lastly, we note that (82) is a direct consequence of time-reversal (KMS)
symmetry, which further implies a fluctuation-dissipation theorem Crossley
_et al._ (2017); Huang  _et al._ (2023).

##  6 Breaking the weak symmetry

We can also use our formalism to transparently understand how to build the EFT
for a dissipative superfluid. Such a prescription is well-known in the
literature Crossley  _et al._ (2017), so we will be brief; our point here is
just to stress that we can now transparently understand the prescription we
summarize below. To go from a normal fluid to a superfluid, we should clearly
relax the reparameterization symmetry (66) down to

$\mathcal{H}(n_{\mathrm{a}},n_{\mathrm{r}},\phi_{\mathrm{a}},\phi_{\mathrm{r}})=\mathcal{H}(n_{\mathrm{a}},n_{\mathrm{r}},\phi_{\mathrm{a}}+\alpha,\phi_{\mathrm{r}}+\alpha^{\prime})$
(85)

where $\alpha,\alpha^{\prime}$ are constants, since now both the strong
$\mathrm{U}(1)_{\mathrm{a}}$ and weak $\mathrm{U}(1)_{\mathrm{r}}$ undergo
SSB. We still require (70), such that each term in $\mathcal{H}$ requires
either a copy of $\nabla\phi_{\mathrm{a}}$ or $n_{\mathrm{r}}$.

The most relevant terms in $\mathcal{H}$ are

$\displaystyle\mathcal{H}$
$\displaystyle=\hat{\mu}(n_{\mathrm{a}})n_{\mathrm{r}}+K(n_{\mathrm{a}})\nabla\phi_{\mathrm{r}}\cdot\nabla\phi_{\mathrm{a}}$
$\displaystyle\quad\quad-\mathrm{i}\sigma(n_{\mathrm{a}})\nabla\phi_{\mathrm{a}}\cdot\nabla(\phi_{\mathrm{a}}+\mathrm{i}\mu(n_{\mathrm{a}}))+\cdots.$
(86)

After integrating out $n_{\mathrm{r}}$ with the constraint (72), we arrive at
the Lagrangian

$\displaystyle\mathcal{L}$
$\displaystyle=n_{\mathrm{a}}\partial_{t}\phi_{\mathrm{a}}-K(n_{\mathrm{a}})\nabla\phi_{\mathrm{r}}\cdot\nabla\phi_{\mathrm{a}}$
$\displaystyle+\mathrm{i}\sigma(n_{\mathrm{a}})\nabla\phi_{\mathrm{a}}\cdot\nabla(\phi_{\mathrm{a}}+\mathrm{i}\mu(n_{\mathrm{a}})).$
(87)

Varying with respect to $\phi_{\mathrm{a}}$, we obtain the dissipative
equation for the superfluid:

$\partial_{t}^{2}\phi_{\mathrm{r}}-v_{\mathrm{s}}^{2}\nabla^{2}\phi_{\mathrm{r}}=\nabla\cdot(D\nabla\partial_{t}\phi_{\mathrm{r}})+\cdots,$
(88)

where we used (72) to write
$n_{\mathrm{a}}=\hat{\chi}\partial_{t}\phi_{\mathrm{r}}$ with
$\hat{\chi}\equiv\partial n_{a}/\partial\hat{\mu}$, and
$v_{\mathrm{s}}=\sqrt{K/\hat{\chi}}$ is the superfluid velocity. The normal
modes corresponding to (88) are given by

$\displaystyle\omega=\pm v_{s}k-\mathrm{i}\frac{D}{2}k^{2}+\cdots,$ (89)

which features a pair of superfluid “second sound” modes – the Nambu-Goldstone
bosons for the SSB $\mathrm{U}(1)$ in the single-copy Hilbert space – damped
by diffusion.

##  7 Outlook

In this paper, we studied a minimal model of SWSSB for a continuous symmetry –
the emergence of hydrodynamic diffusion in a mixed state. We showed that the
hydrodynamic EFT of diffusion can be derived as a kind of “superfluid” EFT for
SWSSB, which provides a transparent understanding of the origins of all
symmetries of hydrodynamic EFTs Crossley  _et al._ (2017). Moreover, the
gapless hydrodynamic quasinormal modes were the Nambu-Goldstone bosons of
SWSSB, which provides a profound new understanding of why hydrodynamic modes
must always be long-lived. Our work also therefore demonstrates the physical
insight gained from using new frameworks, such as SWSSB, to understand old
problems in many-body physics. Although we have focused on a minimal model, we
do not see any reason why our conclusions do not generalize to more
complicated hydrodynamic theories.

One interesting thing that arose in our analysis is that it appeared possible
to write down a hydrodynamic EFT for a quantum system which was not in thermal
equilibrium. To do this, one simply does not require (75). One can still
integrate out $n_{\mathrm{a}}$ and $n_{\mathrm{r}}$ to obtain a hydrodynamic
EFT, but the KMS transformation (78) will be, in general, more complicated:

$\phi_{\mathrm{a}}\rightarrow-\phi_{\mathrm{a}}-\mathrm{i}\mu\left(\hat{\mu}^{-1}\left(\partial_{t}\phi_{\mathrm{r}}\right)\right)+\cdots.$
(90)

This suggests that following Huang  _et al._ (2023), one can systematically
build field theories for nonthermal quantum many-body systems, such as random
unitary circuits Ogunnaike  _et al._ (2023); Khemani  _et al._ (2018);
Rakovszky  _et al._ (2018) where hydrodynamic phenomena have been extensively
studied. It is an interesting open question whether such field theories can
capture any of the nonclassical phenomena, such as dynamics with measurement
and feedback Skinner  _et al._ (2019); Li  _et al._ (2018); Choi  _et al._
(2020); Jian  _et al._ (2020); Roy  _et al._ (2020); Lee  _et al._ (2022);
Friedman  _et al._ (2023), described in the systematic framework of Guo  _et
al._ (2024b).

_Note added._ — As we were completing this manuscript, another manuscript Gu
_et al._ (2024), which also includes discussions of diffusion and SWSSB,
appeared.

## Acknowledgements

We thank Drew Potter, Chong Wang, Cenke Xu, and Yichen Xu for useful
discussions.

This work was supported by the Alfred P. Sloan Foundation under Grant
FG-2020-13795 (AL), the National Science Foundation under CAREER Grant
DMR-2145544 (AL), the Department of Energy under Basic Energy Sciences Award
DE-SC0014415 (MQ) and Quantum Pathfinder Award DE-SC0024324 (JHZ), and the
Heising-Simons Foundation under Grant 2024-4848 (XH, AL). XH and JHZ thank the
Kavli Institute for Theoretical Physics for hospitality; KITP is supported in
part by the Heising-Simons Foundation, the Simons Foundation, and the National
Science Foundation under Grant PHY-2309135.

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