Euler-scale dynamical fluctuations in non-equilibrium interacting integrable
systems

Gabriele Perfetto1 and Benjamin Doyon2

1 SISSA – International School for Advanced Studies and INFN, via Bonomea 265,
34136 Trieste, Italy.

2 Department of Mathematics, King’s College London, Strand, London WC2R 2LS,
U.K.

*<EMAIL_ADDRESS>
## Abstract

We derive an exact formula for the scaled cumulant generating function of the
time-integrated current associated to an arbitrary ballistically transported
conserved charge. Our results rely on the Euler-scale description of
interacting, many-body, integrable models out of equilibrium given by the
generalized hydrodynamics, and on the large deviation theory. Crucially, our
findings extend previous studies by accounting for inhomogeneous and dynamical
initial states in interacting systems. We present exact expressions for the
first three cumulants of the time-integrated current. Considering the non-
interacting limit of our general expression for the scaled cumulant generating
function, we further show that for the partitioning protocol initial state our
result coincides with previous results of the literature. Given the
universality of the generalized hydrodynamics, the expression obtained for the
scaled cumulant generating function is applicable to any interacting
integrable model obeying the hydrodynamic equations, both classical and
quantum.

###### Contents

  1. 1 Introduction
  2. 2 Introduction to the generalized hydrodynamics
    1. 2.1 The GGE and its thermodynamic Bethe ansatz description
    2. 2.2 The Euler-scaling limit and the GHD equations
    3. 2.3 Euler-scale dynamical correlation functions
  3. 3 Large deviation theory of ballistic transport
    1. 3.1 Large deviation principle and the scaled cumulant generating function
    2. 3.2 SCGF for homogeneous GGEs: review of the result
  4. 4 The Euler-scale SCGF for inhomogeneous states
    1. 4.1 SCGF for inhomogeneous GGEs: statement of the result
    2. 4.2 SCGF for inhomogeneous GGEs: derivation of the result
    3. 4.3 Analysis of the cumulants
    4. 4.4 The non-interacting limit
  5. 5 Conclusion
  6. A Indirect propagator for two-point correlation functions

##  1 Introduction

One-dimensional isolated many-body quantum systems have recently received a
great amount of attention thanks to ground-breaking advances in cold-atomic
experiments, see e.g., Refs. [1, 2] for excellent reviews on the subject. In
one dimension powerful analytical techniques are available for integrable
models, which possess an extensive number of conservation laws, see e.g., the
collection of reviews in Ref. [3]. In particular, it is by now well
established, that integrable systems do not relax to a thermal steady state
after an homogeneous quantum quench [4, 5, 6], where a global parameter of the
Hamiltonian is abruptly changed and the system undergoes unitary evolution
from a translationally invariant initial state. Relaxation occurs to the so-
called generalized Gibbs ensemble (GGE), first proposed in Ref. [7] with Refs.
[8, 6] reviews on the subject, which accounts for the presence of the infinite
number of conservation laws in the thermodynamic limit.

Out of equilibrium, the initial state of the system is, however, often
inhomogeneous and non-stationary. The study of the dynamics of interacting
integrable models from inhomogeneous states is much more difficult than the
one in homogeneous quantum quenches. In this context, the introduction of the
generalized hydrodynamics (GHD), in the independent works in Ref. [9] and Ref.
[10], has been a groundbreaking advancement. This theory is an extension of
hydrodynamics to interacting integrable systems: while Euler hydrodynamics
describes fluids with a finite number of conservation laws, the generalized
hydrodynamics is concerned with integrable systems, admitting an infinite
number of conserved quantities. As a consequence, the theory of generalized
hydrodynamics is based on the GGE, instead of the canonical Gibbs ensemble. As
GHD is a hydrodynamic theory, at its leading order, it applies in the so-
called Euler-scaling limit, where the scale of space and time at which
variations in the state of the system are observed, is taken to infinity.
Under this assumption, at any space-time point $(x,t)$ the many-body system
can be considered to locally relax to a GGE which depends on $(x,t)$, in the
spirit of a local-equilibrium approximation [11]. The dynamics is ruled by
hydrodynamic equations having a form analogous to the Euler equations. The
input of the method is the thermodynamic Bethe ansatz (TBA), see, e.g., Refs.
[12, 13]. Despite the name, the TBA is not restricted to quantum models: it
applies also to classical systems, such as the hard-rod gas [14], the
classical sinh-Gordon [15] and the Toda chain [16, 17]. The initial
application of the GHD formalism in Refs. [9, 10], and later in Refs. [14, 15,
17, 18, 19, 20, 21, 22, 23, 24, 25], has been to the study of ballistic
transport from the partitioning protocol initial inhomogeneous state, but the
versatility of GHD allows to study various inhomogeneous setups such as the
effect of confining potentials [26, 27], bump-release protocols [28, 29],
correlation functions [30, 31, 32] and entanglement spreading [33, 34, 35, 36,
37]. Remarkably, the formalism can also be used to study non-integrable
systems, provided the integrability breaking is weak enough so that on large
enough length and time scales, the conserved charges may be assumed not to be
broken [26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. We also note that GHD has
been experimentally verified in Ref. [48].

A complete understanding of the non-equilibrium processes where transport is
present requires, however, the knowledge not only of the mean values of the
ballistically transported conserved densities and the associated currents –
the natural quantities at the basis of hydrodynamics – but also of the
corresponding fluctuations. A relevant framework to account for fluctuations
is given by the large deviation theory, see e.g., Refs. [49, 50, 51]. This
framework builds on the computation of the large deviation function, which
provides the statistics of rare but significant fluctuations. It generalizes
to non-equilibrium conditions the concepts of entropy, thermodynamic phases
and phase transitions, that are canonical quantities for describing the
equilibrium statistical mechanics of many-body systems. The large deviation
function can be computed from the associated scaled cumulant generating
function (SCGF) [49], which is the non-equilibrium analogue of the free
energy. In the context of isolated systems, until recently only few results
for transport fluctuations were present, mostly regarding one-dimensional
critical systems [52, 53, 54] and non-interacting models, such as the Levitov-
Lesovik formula for free fermions [55, 56, 57, 58, 59, 60, 61, 62, 63, 64],
the free Klein-Gordon field theory [65] and harmonic chains [66]. Only
recently, results became available in interacting integrable systems for the
statistics of various observables not related to transport, see e.g., Refs.
[67, 68, 69, 70], and most importantly for the present paper, for ballistic
transport [71, 72]. In the latter, an exact expression for the current
fluctuations at the Euler scale has been derived combining GHD and the large-
deviation theory.

The results described above apply to homogeneous and stationary GGE states –
even though these states can be obtained from inhomogeneous settings, such as
the partitioning protocol for non-equilibrium steady states. At present, no
general formula is available for the Euler-scale fluctuations of current flows
within states affected by a nontrivial large-scale dynamics. The work [73]
provided an important first step, obtaining, for non-interacting fermionic and
bosonic systems, a formula for the scaled cumulant generating function
describing the energy current fluctuations far from the connection point in
the partitioning protocol, where a nontrivial Euler-scale dynamics occurs. The
derivation is based on stationary phase methods, which are characteristic of
free models, and therefore it is difficult to immediately generalize it to
interacting integrable systems. A full understanding of the Euler-scale
fluctuations of current flows in interacting integrable systems in states with
large-scale inhomogeneity and dynamics is therefore still missing.

In this manuscript we aim at filling this gap, by providing an exact
expression, in the Euler-scaling limit, for the SCGF of the time-integrated
current associated to an arbitrary ballistically transported conserved charge.
Our results apply to a broad class of inhomogeneous and dynamical states,
which include, e.g., the initial state of the partitioning protocol, as shown
in Fig. 1.

Figure 1: Schematic representation of the fluctuations of the time-integrated
current in the Euler-scaling limit. In the Figure we consider, as an example,
the partitioning protocol inhomogeneous state, where two GGEs, denoted as
$\vartheta_{l}$ (in red at $x<0$) and $\vartheta_{r}$ (in blue at $x>0$), are
joined at $x=0$ at the time $t=0$. The ensuing dynamics is described in terms
of a continuum of states, constant on the light rays parametrized by
$\xi=x/t$, which interpolate between $\vartheta_{l}$ and $\vartheta_{r}$. We
consider the time integral $\Delta
q_{i_{\ast}}(zx,zt)=\int_{0}^{zt}\mbox{d}\tau\,j_{i_{\ast}}(zx,\tau)$ of the
current $j_{i_{\ast}}$, associated to the conserved charge $q_{i_{\ast}}$,
flowing across the point $zx$ over the time interval $(0,zt)$. The
fluctuations of $\Delta q_{i_{\ast}}(zx,zt)$ in the Euler-scaling limit are
encoded in the SCGF
$G(s,x,t)=\lim_{z\rightarrow\infty}1/(zt)\,\mbox{ln}\,\langle\,\mbox{exp}({s\,\Delta
q_{i_{\ast}}}(zx,zt))\rangle_{\mathrm{inh},z}$. In the case depicted in the
Figure, the average $\langle\dots\rangle_{\mathrm{inh,z}}$ is taken over the
partitioning protocol inhomogeneous state. We emphasize, however, that the
formula for $G(s,x,t)$ derived in the manuscript applies more generally to a
wide class of inhomogeneous and dynamical states varying at large scales $z$.

The calculation of the SCGF is based on the biasing of the measure of the
initial state by the exponential of the time-integrated current, in a similar
way as the procedure followed in Refs. [71, 72]. The biasing of the measure is
shown to be fixed by the knowledge of two-point correlation functions in the
inhomogeneous initial state, derived in Ref. [30]. Our derivation accordingly
shows that the study of two-point correlation functions in GHD [30, 31, 32] is
tightly related to the large-deviation theory of current fluctuations. The
biasing technique developed in [72], generalised here to inhomogeneous,
dynamical situations, is in principle applicable to a wide class of
hydrodynamic systems, integrable or not. In this paper, however, we
concentrate on integrable systems, where all the necessary technical tools are
available for the technique to lead to calculable results.

We provide exact expressions for the first three cumulants of the time-
integrated current and we further show that in the non-interacting limit, for
the partitioning protocol initial state, our general formula for the SCGF
reduces to the result of Ref. [73]. Our study of the SCGF thereby deeply
generalizes the analysis of Refs. [71, 72] and of Ref. [73] by accounting for
inhomogeneous situations and interactions, respectively. The general
expression presented for the SCGF relies solely on the large deviation theory
and on the generalized hydrodynamics. It is therefore applicable to any
integrable model described by the GHD, including the classical hard-rod gas
[74, 75, 14, 32] and the Toda chain [16, 17], the classical sinh-Gordon field
theory [15] , the quantum Lieb-Liniger model [27, 28, 29, 30], which describes
the one-dimensional Bose gas, and quantum spin chains [9, 18, 19, 20, 21, 23,
24], such as the XXZ model.

The rest of the paper is organized as follows. In Sec. 2 we review the basic
concepts underlying the generalized hydrodynamics theory of integrable systems
that will be needed for the presentation of our results. In Sec. 3 we briefly
review the large-deviation theory for the rare fluctuations of ballistic
current flows. Sec. 4 contains our main result for the exact expression of the
SCGF in inhogeneous and dynamical initial states together with the
applications of the result to the calculation of the cumulants and to the non-
interacting limit. In Sec. 5 we draw our final conclusions. Some technical
aspects are reported in the Appendix A.

##  2 Introduction to the generalized hydrodynamics

In this Section we review integrable models and their generalized
hydrodynamics theory. The results that will follow in Sec. 4 apply to this
class of systems. Reference [26] and the lecture notes in Ref. [76] provide a
very good review on the subject. In Subsec. 2.1 we recall the basic identities
regarding the description of the GGE in the thermodynamic Bethe ansatz
formalism. In Subsec. 2.2, we first define the Euler-scaling limit for a
generic local observable, and then we report the main GHD hydrodynamic
equations. In Sec. 2.3 we briefly review the expressions of the dynamical two-
point functions in inhomogeneous and non-stationary states, derived in Ref.
[30] and later numerically evaluated in Ref. [37], which constitutes a central
building block of the results presented in Sec. 4.

###  2.1 The GGE and its thermodynamic Bethe ansatz description

We consider interacting integrable models in one spatial dimension, where, in
the thermodynamic limit, an infinite number of local (or quasi-local)
conserved charges $\\{Q_{i}\\}$ exists, with $i,j=1,2\dots\infty$. For
Hamiltonian time evolution one of the conserved charges is the Hamiltonian of
the system $H$; henceforth we further assume that the conserved charges
commute among themselves $[Q_{i},Q_{j}]=0$, for any $i$ and $j$. The charges
can be written as $Q_{i}=\int\mbox{d}x\,q_{i}(x)$, where the density $q_{i}$
associated to $Q_{i}$ obeys a continuity equation with the current density
$j_{i}$, corresponding to the current $J_{i}=\int\mbox{d}x\,j_{i}(x)$, written
in the form111We use in this Section the continuum space notation just for
simplicity, but the discussion carries over to discrete lattice models, e.g.,
spin chains.

$\partial_{t}q_{i}(x)+\partial_{x}j_{i}=0,\quad\mbox{with}\quad
i=1,2\dots\infty.$ (1)

We will indicate by $q_{0}$, $q_{1}$ and $q_{2}$, respectively, the number,
momentum and energy charges. In non-integrable models the latter are the only
conserved quantities. The thermodynamic properties of integrable systems are
described by the so-called generalized Gibbs ensemble (GGE) [7, 8, 6], which
generalizes the canonical Gibbs ensemble to the case where an infinite number
of conserved charges is present

$\rho_{GGE}=\frac{\mbox{e}^{-W}}{Z_{GGE}},\qquad\mbox{where}\qquad
W=\sum_{i}\beta^{i}\,Q_{i},$ (2)

and $Z_{GGE}$ is the normalization constant. The parameters $\\{\beta^{i}\\}$
are Lagrange multipliers whose knowledge uniquely determines the GGE. Their
values are fixed by the initial values of the conserved charges $\\{Q_{i}\\}$.
In this manuscript we are using the superscript notation $\\{\beta^{i}\\}$
following the notation of Refs. [71, 72], but the superscript $i$ should not
be confused with an exponent. Notice that the present discussion applies both
to classical and quantum models. In the former case $\rho_{GGE}$ is a
statistical distribution in the phase-space, while in the latter it is a
density matrix.

The thermodynamic Bethe ansatz (TBA) formalism, see, e.g., Refs. [77, 12],
provides an efficient method to represent the GGE. The system is described in
terms of quasi-particles with rapidity $\lambda$, where each quasi-particle
carries a bare charge $h_{i}(\lambda)$ corresponding to the single-particle
eigenvalue of $Q_{i}$ in Eq. (1). The statistics of the quasi-particles is
encoded into the free energy function $F(\varepsilon)$. The latter is given by
$F(\varepsilon)=-\mbox{ln}(1+e^{-\varepsilon})$ for fermions,
$F(\varepsilon)=\mbox{ln}(1-e^{-\varepsilon})$, for bosons, and
$F(\varepsilon)=-\mbox{e}^{-\varepsilon}$ for classical particles. The GGE in
Eq. (2) is then fixed by the pseudo-energy function $\varepsilon(\lambda)$,
which, in turn, is determined by solving the non-linear integral equation

$\varepsilon(\lambda)=w(\lambda)+\int\mbox{d}\mu\,T(\lambda,\mu)\,F(\varepsilon(\mu))\,,$
(3)

where $T(\lambda,\mu)=\Theta(\lambda,\mu)/2\pi$ and $\Theta(\lambda,\mu)$ is
the differential two-body scattering phase. The term $w(\lambda)$ on the right
hand side of Eq. (3) is named source term and it is defined as

$w(\lambda)=\sum_{i}\beta^{i}\,h_{i}(\lambda),$ (4)

where the $\\{\beta^{i}\\}$ are the parameters of the GGE in Eq. (2). For a
Galilean model, for example, $h_{0}(\lambda)=1$, $h_{1}(\lambda)=\lambda$ and
$h_{2}(\lambda)=\lambda^{2}/2$. The only model-specic quantities in Eqs. (3)
and (4) are the single-particle eigenvalues $h_{i}(\lambda)$, the integral
kernel $T(\lambda,\mu)$, the free energy function $F(\varepsilon)$, and the
spectral space $\int\mbox{d}\mu$. The latter, in the cases where multiple
quasi-particle species $n$ are present, has to be enlarged to
$\sum_{n}\int\mbox{d}\mu$. In this work we will drop the summation over the
different quasi-particle species for brevity, but the results of Sec. 4
straightforwardly generalize to those cases.

From Eq. (3) it is customary to introduce the mode-occupation (or filling)
function $\vartheta(\lambda)$, the root density $\rho_{p}(\lambda)$ and the
state density $\rho_{s}(\lambda)$ as

$\vartheta(\lambda)=\frac{\mbox{d}F(\varepsilon)}{\mbox{d}\varepsilon}\Bigr{|}_{\varepsilon(\lambda)},\quad\rho_{p}(\lambda)=\vartheta(\lambda)\rho_{s}(\lambda),\quad\mbox{with}\quad\rho_{s}(\lambda)=\frac{1}{2\pi}1^{\mathrm{dr}},$
(5)

where the dressing of a generic function $h^{\mathrm{dr}}(\lambda)$ of
$\lambda$ is defined by the linear integral equation

$h^{\mathrm{dr}}(\lambda)=h(\lambda)+\int\mbox{d}\mu\,T(\lambda,\mu)\vartheta(\mu)h^{\mathrm{dr}}(\mu).$
(6)

The GGE may be identified in various equivalent ways. The root density
$\rho_{p}(\lambda)$ completely specifies it, as well as the mode occupation
function $\vartheta(\lambda)$; since the knowledge of either determines the
average values of all the conserved charges $\\{Q_{i}\\}$ of the model. As a
consequence, in the following, we will the denote the average of a local
observable $\mathcal{O}(x,t)$ at point $x$ and time $t$ over the GGE in Eq.
(2) as

$\langle\mathcal{O}(x,t)\rangle_{\vartheta}=\langle\mathcal{O}(0,0)\rangle_{\vartheta}=\mbox{Tr}\left[\rho_{GGE}\,\mathcal{O}\right],$
(7)

where the second equality follows from the fact that the GGE is homogeneous
and stationary. The dependence of $\vartheta$ on $\lambda$ is omitted in this
notation. In Eq. (7) we have used for convenience the quantum-mechanical trace
notation, for classical systems it has to be replaced by an integral in phase
space.

###  2.2 The Euler-scaling limit and the GHD equations

The TBA formalism and the GGE describe integrable systems in homogeneous and
stationary states. The generalized hydrodynamics (GHD) theory accounts for
situations where the state of the many-body system is inhomogeneous and non-
stationary. In order to introduce this formalism, we first define the Euler-
scaling limit, as the latter will be a central concept that will be used
throughout the rest of the manuscript. Let us consider the case in which an
integrable system is initialized in some generic state $\rho_{0}$ which is
inhomogeneous and non-stationary. The average of a local operator
$\mathcal{O}(x,t)$ over a generic inhomogeneous state $\rho_{0}$ will be
denoted as $\langle\mathcal{O}(x,t)\rangle$. Upon considering inhomogeneities
which slowly vary in space-time, one can use the local relaxation assumption
or “local maximization of entropy principle” (an excellent book on this is
Ref. [11]), which asserts that the system locally relaxes to a GGE:

$\langle\mathcal{O}(x,t)\rangle\simeq\langle\mathcal{O}(0,0)\rangle_{\vartheta(x,t)}.$
(8)

The space-time dependence is recovered, in the spirit of a local-density
approximation, by promoting the GGE Lagrange multipliers
$\\{\beta^{i}\\}\rightarrow\\{\beta^{i}(x,t)\\}$, and, equivalently, the mode
occupation function $\vartheta\rightarrow\vartheta(x,t)$, to be dependent on
the space-time point $(x,t)$ where the operator is located. Equation (8)
becomes exact in the limit of infinite length scale of the variation of the
inhomogeneity. In the case of large, but finite, length scales for the
variation in space and time of the inhomogeneity, instead, Eq. (8) is only
approximate. The limit where Eq. (8) becomes exact is usually named “Euler-
scaling limit” or simply “Euler scale”. Henceforth we will use both names
interchangeably. For one-point functions at the space-time point $(x,t)$, we
will use the subscript $\vartheta(x,t)$ to denote the averages over the local
homogeneous GGE in Eq. (2) at $(x,t)$, as done in Eq. (8) consistently with
the notation introduced in the previous Subsec. 2.1 in Eq. (7).

To make the discussion more concrete, here we introduce a particular class of
inhomogeneous and dynamical initial states, which generalize the GGE in Eq.
(2), which are given by

$\rho_{0}=\frac{1}{Z}\,\mbox{exp}\left(-\sum_{i}\int_{\mathbb{R}}\mbox{d}x\,\beta^{i}(x/z,0)\,q_{i}(x,0)\right).$
(9)

In this expression $z$ is a scale parameter which has to be large enough such
that the resulting Lagrange parameters $\\{\beta^{i}(x/z)\\}$ are functions of
the position $x$ that vary only on large scales. If the multipliers
$\\{\beta^{i}\\}$ do not depend on space, the state $\rho_{0}$ is a GGE as in
Eq. (2). For this reason, we will refer henceforth to $\rho_{0}$ as
inhomogeneous GGE.

In this manuscript, we will denote $\langle\dots\rangle_{\mathrm{inh},z}$ the
averages over $\rho_{0}$ in Eq. (9). In the case of $\rho_{0}$ in Eq. (9) the
Euler-scaling limit can be achieved by taking the limit where the scale of the
inhomogeneity is sent to infinity $z\rightarrow\infty$, simultaneously with
the space-time observation point $(x,t)$ of the operator $\mathcal{O}$. In
formulas

$\displaystyle\langle\mathcal{O}(0,0)\rangle_{\vartheta(x,t)}$
$\displaystyle=\lim_{z\rightarrow\infty}\langle\mathcal{O}(zx,zt)\rangle_{\mathrm{inh},z}$
$\displaystyle=\lim_{z\rightarrow\infty}\frac{1}{Z}\mbox{Tr}\left[\mathcal{O}(zx,zt)\,\mbox{exp}\left(-\sum_{i}\int_{\mathbb{R}}\mbox{d}x\beta^{i}(x/z,0)q_{i}(x,0)\right)\right].$
(10)

A comment is in order here: in fact, sharp jumps in the state are also allowed
and describable by Euler hydrodynamics. These are interpreted as “weak
solutions” to the hydrodynamic equations. Therefore, $\beta^{i}(x,0)$ do not
need to be smooth functions: jumps are allowed, and the limit in Eq. (10) is
still described by Euler hydrodynamics.

A particular initial inhomogeneity of the form of $\rho_{0}$ in Eq. (9), which
we will discuss, is the partitioning protocol state, where indeed a sharp jump
occurs. In this state two homogeneous GGEs, conventionally denoted as right
($r$ at $x>0$) and left ($l$ at $x<0$) are joined at the point $x=0$ at time
$t=0$, see Fig. 1. Accordingly, the generalized inverse temperatures are
chosen as

$\beta^{i}(x,0)=\beta^{i}_{r}\,\Theta(x)+\beta^{i}_{l}\,\Theta(-x),$ (11)

where $\Theta(x)$ is the Heaviside step function. The initial filling function
$\vartheta(x,0,\lambda)$ corresponding to Eq. (11) is given by

$\vartheta(x,0,\lambda)=\vartheta_{r}(\lambda)\,\Theta(x)+\vartheta_{l}(\lambda)\,\Theta(-x),$
(12)

where $\vartheta_{l,r}(\lambda)$ are fixed by the boundary conditions one
imposes on the left $\\{\beta^{i}_{l}\\}$ and right $\\{\beta^{i}_{r}\\}$
reservoirs. For our results in Sec. 4 we will consider initial inhomogeneous
and dynamical states of the form of Eq. (9) in the Eulerian limit
$z\rightarrow\infty$.

Once the Euler-scaling limit in Eqs. (8) and (10) is assumed, it is rather
simple to derive the hydrodynamic equations ruling the evolution of the
system. Starting from Eq. (1), one has

$\partial_{t}\langle q_{i}(0,0)\rangle_{\vartheta(x,t)}+\partial_{x}\langle
j_{i}(0,0)\rangle_{\vartheta(x,t)}=0.$ (13)

The mean value of the charge density in Eq. (1) is given by the TBA
description

$\langle
q_{i}(0,0)\rangle_{\vartheta(x,t)}=\int\frac{\mbox{d}\lambda}{2\pi}\,h_{i}(\lambda)\vartheta(x,t)1^{\mathrm{dr}}(x,t,\lambda),$
(14)

where the dressing operation has been defined in Eqs. (5) and (6). In
particular, all dressed quantities become functions of space and time, being
determined by $\vartheta(x,t)$ via Eq. (6). The expression of the current in
Eq. (1) has been first discovered in Refs. [9, 10] and it reads as

$\displaystyle\langle
j_{i}(0,0)\rangle_{\vartheta(x,t)}=\int\frac{\mbox{d}\lambda}{2\pi}\,(E^{\prime})^{\mathrm{dr}}(x,t,\lambda)\vartheta(x,t)h_{i}(\lambda),$
(15)

where $h_{2}(\lambda)=E(\lambda)$ is the bare single-particle energy
eigenvalue and the effective velocity $v^{\mathrm{eff}}$ is given by

$v^{\mathrm{eff}}(x,t,\lambda)=\frac{(E^{\prime})^{\mathrm{dr}}(x,t,\lambda)}{1^{\mathrm{dr}}(x,t,\lambda)}=\frac{(E^{\prime})^{\mathrm{dr}}(x,t,\lambda)}{(P^{\prime})^{\mathrm{dr}}(x,t,\lambda)},$
(16)

where in the last equality we used that for non-relativistic models
$h_{1}(\lambda)=P(\lambda)=\lambda$ (even though the second relation in Eq.
(16) holds also for more general parametrizations of the momentum as a
function of the rapidity). The effective velocity $v^{\mathrm{eff}}$ is a
generalization of the group velocity $v_{g}$, which takes into account the
interactions among the quasi-particles via the dressing operation.
$v^{\mathrm{eff}}$ has been first defined in Ref. [78] in the context of the
light-cone spreading of correlations in homogeneous quantum quenches. We
stress that Eq. (15) goes beyond the results available from the TBA and it has
been first proved in relativistically invariant quantum field theories in Ref.
[10] and numerically tested in quantum spin chains in Ref. [9]. Later, it has
been proved in a variety of contexts including quantum spin chains and
classical models [79, 80, 81, 82, 83, 84, 85]. Inserting Eqs. (14) and (15)
into the continuity equation (13), and assuming the completeness of the set of
local (or quasi-local) charges $h_{i}(\lambda)$, the final GHD equation for
the filling function, see Refs. [9, 10] for the original derivation, turns out
to be

$\partial_{t}\vartheta(x,t,\lambda)+v^{\mathrm{eff}}(x,t,\lambda)\partial_{x}\vartheta(x,t,\lambda)=0.$
(17)

Equation (17) express the fact that the occupation function
$\vartheta(x,t,\lambda)$ represents the normal modes of the hydrodynamics,
which, in integrable systems, form a continuum labelled by the rapidity
$\lambda$. At the Euler scale, the normal modes are convectively transported
with velocity $v^{\mathrm{eff}}(x,t,\lambda)$. Generalizations of Eq. (17) to
account for the presence of trapping potentials [26], diffusive corrections
[86, 87, 88, 89, 43], space-time variations of the interaction terms of the
Hamiltonian [38] and Markovian coupling to an external bath [40] have been
further developed. For the derivation of our results in Sec. 4 the form in Eq.
(17) will be sufficient.

As a final piece of introduction, we mention that Eq. (17) admits a solution
by the characteristic function, $\mathcal{U}(x,t,\lambda,t_{0})$, encoding the
position at time $t_{0}$ of the characteristic curve
$x(\mathcal{U},t,\lambda,t_{0})$ of the quasi-particle with rapidity $\lambda$
that at time $t$ is in $x$ (therefore $\mathcal{U}(x,t_{0},\lambda,t_{0})=x$).
The characteristic curve $x(\mathcal{U},t,\lambda,t_{0})$ is defined as the
curve tangent to the effective velocity $v^{\mathrm{eff}}$ in Eq. (16). The
function $\mathcal{U}(x,t,\lambda,t_{0})$ is defined by inverting
$x(\mathcal{U},t,\lambda,t_{0})$ with the respect to the initial position
$\mathcal{U}$, see Refs. [29, 36] for a detailed discussion. From Eq. (17), it
is simple to see that the filling function $\vartheta(x,t,\lambda)$ is
constant along the characteristic $\mathcal{U}(x,t,\lambda,t_{0})$

$\vartheta(x,t,\lambda)=\vartheta(\mathcal{U}(x,t,\lambda,t_{0}),t_{0},\lambda)\;.$
(18)

Remarkably, in Ref. [29], it has been proved that
$\mathcal{U}(x,t,\lambda,t_{0})$ can be determined by solving the following
integral equation

$\int_{x_{0}}^{x}\mbox{d}y\,\rho_{s}(y,t,\lambda)=\int_{x_{0}}^{\mathcal{U}(x,t,\lambda,t_{0})}\mbox{d}y\,\rho_{s}(y,t_{0},\lambda)+v^{\mathrm{eff}}(x_{0},t_{0},\lambda)\rho_{s}(x_{0},t_{0},\lambda)(t-t_{0}),$
(19)

where $x_{0}$ is an asymptotically stationary point (see the discussion after
Eq. (LABEL:eq:SourceTerm) in Appendix A). Equations (18) and (19) provide an
efficient way to solve the main GHD equation (17) initial value problem, as
shown in Ref. [29]. The integral equation (19) for
$\mathcal{U}(x,t,\lambda,t_{0})$, furthermore, is fundamental for the
derivation of the exact expression of the Euler-scale dynamical correlation
functions, which will be the object of the next Subsection.

###  2.3 Euler-scale dynamical correlation functions

The discussion of Subsec. 2.2 focused on the Euler scaling of one-point
functions (mean values) of local observables. Since for weak inhomogeneities
the system is composed by locally homogeneous space-time fluid cells, one-
point functions at the space-time point $(x,t)$ can be computed based on the
knowledge of their expression
$\langle\mathcal{O}(0,0)\rangle_{\vartheta(x,t)}$ in the local homogeneous GGE
in Eq. (2) at $(x,t)$ according to Eq. (10). Euler-scale connected correlation
functions are, instead, harder to compute as they depend on the whole
inhomogeneous state $\rho_{0}$ characterizing the initial state of the system,
and not only on the homogeneous GGE on a specific fluid cell. Considering two-
point functions at the space-time points $(0,0)$ and $(x,t)$, the approach for
their calculation [11, 90, 32] consists in studying the linear response of the
system at the space-time point $(x,t)$ to a local perturbation at the point
$(0,0)$. Correlations, at the Euler scale, are then determined by the stable
normal modes propagating between the two space-time points. In this way, exact
expressions for the two-point functions of generic local observables in the
large-scale limit can be obtained. However, the expressions obtained in this
way in Refs. [11, 90, 32] assume that background fluid state at the initial
space-time point $(0,0)$ is homogeneous and stationary. In Ref. [30], this
approach has been extended to account for two-point correlation functions from
inhomogeneous initial states of the form in Eq. (9), under homogeneous
evolution (for a dynamics with an homogeneous and time-independent Hamiltonian
– that is, without spatially or temporally dependent external force fields).
We report just the main ideas behind the derivation together with the final
expression of the the dynamical two-point function, which will be essential
for the derivation of the scaled cumulant generating function detailed in Sec.
4.

Consider a small perturbation of one Lagrange parameter
$\beta_{j}(y,0)\rightarrow\beta_{j}(y,0)+\delta\beta_{j}(y,0)$ in the initial
state $\rho_{0}$ in Eq. (9). The response of the system to the small
perturbation of $\beta_{j}(y,0)$ is related to the connected correlation
function involving the associated density $q_{j}(y,0)$. Consider the average
of some other density $q_{i}(x,t)$ over $\rho_{0}$ in Eq. (9): the functional
derivative of this average with respect to $\beta_{i}(y,0)$ is

$-\frac{\delta}{\delta\beta_{j}(y,0)}\langle
q_{i}(x,t)\rangle_{\vartheta_{0}}^{\mathrm{Eul}}=\langle
q_{i}(x,t)q_{j}(y,0)\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}},$ (20)

with the Euler-scaling limit for the connected correlator defined as

$\langle
q_{i}(x,t)q_{j}(y,0)\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}}=\lim_{z\to\infty}z\Big{(}\left\langle
q_{i}(z\,x,z\,t)q_{j}(zy,0)\right\rangle_{\mathrm{inh},z}-\left\langle
q_{i}(z\,x,z\,t)\right\rangle_{\mathrm{inh},z}\left\langle
q_{j}(z\,y,0)\right\rangle_{\mathrm{inh},z}\Big{)}.$ (21)

In $\langle q_{i}(x,t)q_{j}(y,0)\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}}$ the
subscript $\vartheta_{0}$ denotes the filling function which characterizes
globally, as a function of space, the inhomogeneous state $\rho_{0}$ in Eq.
(9) at the time slice $t=0$ in the Euler-scaling limit $z\rightarrow\infty$.
For the partitioning protocol initial state, for example, $\vartheta_{0}$ is
given in Eq. (12). This notation, where time appears as a lower index,
stresses the fact that two-point correlation functions depend on the whole
initial inhomogeneous state $\vartheta_{0}$ of the system and not only on the
homogeneous GGE $\vartheta(x,t)$ at a specific space-time fluid cell $(x,t)$.
For one-point functions, instead, we have denoted with $\langle
q_{i}(x,t)\rangle_{\vartheta_{0}}^{\mathrm{Eul}}=\langle
q_{i}(0,0)\rangle_{\vartheta(x,t)}$ the average over $\rho_{0}$ in Eq. (9) in
the limit $z\rightarrow\infty$ according to Eq. (10) and in agreement with the
notation introduced in Subsecs. 2.1 and 2.2. [One-point functions of the
charge densities and of the associated currents are given in Eqs. (14) and
(15), respectively.]

We emphasize that, for some specific models, the Euler-scale limit of two-
point correlation functions requires additional care and $q_{i}(zx,zt)$,
$q_{j}(zy,0)$ in the r.h.s. of Eq. (21) have to be averaged over space-time
fluid-cells, as explained in Ref. [30] and shown in Ref. [15] for the
classical sinh-Gordon field theory. For the classical hard-rod gas considered
in Ref. [37], instead, fluid-cell averaging is not necessary and the Euler-
scale predictions for the two-point function are simply obtained by averaging
in space.

Equation (20) represents an extension of the fluctuation-dissipation theorem
of statistical mechanics [91] to the framework of GHD, where infinitely many
conserved charges are present. From Eq. (20) an expression for the two-point
connected correlation function $\langle
q_{i}(x,t)q_{j}(y,0)\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}}$ can then be
derived exploiting the fact that the expression of the one-point function on
the l.h.s is known in Eq. (14), as shown in Ref. [30].

These results for correlation functions of conserved densities can then be
extended to the connected two-point function of any pair of local observables
$\left\langle\mathcal{O}(x,t)\mathcal{O}^{\prime}(y,0)\right\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}$,
via hydrodynamic projections [32]. The main idea of the latter is that at the
Euler scale correlations are determined by the ballistically propagating
conserved charges. Correlations can then be determined by “expanding” the
operator $\mathcal{O}(x,t)$ of interest in the basis of the local (or quasi-
local) conserved charges

$\mathcal{O}(x,t)=\sum_{j,k}q_{j}(x,t)\left(C_{[\vartheta(x,t)]}\right)^{-1}_{jk}\langle
q_{k}|\mathcal{O}\rangle_{\vartheta(x,t)},$ (22)

where the scalar product $\langle q_{k}|\mathcal{O}\rangle_{\vartheta(x,t)}$
is computed at the space-time point $(x,t)$ of the observable as

$\langle\mathcal{O}|q_{k}\rangle_{\vartheta(x,t)}=-\frac{\partial}{\partial\beta_{k}(x,t)}\langle\mathcal{O}\rangle_{\vartheta(x,t)}=\int\mbox{d}\lambda\>\rho_{p}(x,t,\lambda)f(x,t,\lambda)V^{\mathcal{O}}(x,t,\lambda)h_{k}^{\mathrm{dr}}(x,t,\lambda)\;.$
(23)

$f(x,t,\lambda)$ is dubbed statistical factor of the model. For models with
fermionic quasi-particle statistics $f(x,t,\lambda)=1-\vartheta(x,t,\lambda)$,
for bosonicic quasi-particles $f(x,t,\lambda)=1+\vartheta(x,t,\lambda)$, while
for classical particle models $f(x,t,\lambda)=1$. $V^{\mathcal{O}}$ in Eq.
(23) is named one-particle-hole form factor of the operator $\mathcal{O}$, it
a functional of the filling function defined such that Eq. (23) holds. It must
be worked out for every operator individually. For charge densities and the
associated currents they are simply related to dressed single-particle
eigenvalues $h_{i}^{\mathrm{dr}}$ in Eq. (6) [32] as

$V^{q_{i}}=h_{i}^{\mathrm{dr}}\quad\text{and}\quad
V^{j_{i}}=v^{\mathrm{eff}}h_{i}^{\mathrm{dr}}\;.$ (24)

$C_{\vartheta(x,t)}^{-1}$ in Eq. (23) is the inverse of the correlation matrix
(cf. Eq. (23) and Eq. (24) for $V^{q_{i}}$)

$(C_{\vartheta(x,t)})_{ij}=\langle
q_{i}|q_{j}\rangle_{\vartheta(x,t)}=\int\mbox{d}\lambda\>\rho_{p}(x,t,\lambda)f(x,t,\lambda)h_{i}^{\mathrm{dr}}(x,t,\lambda)h_{j}^{\mathrm{dr}}(x,t,\lambda).$
(25)

The factor $C_{\vartheta(x,t)}^{-1}$ is introduced in Eq. (22) because the
densities $\\{q_{i}\\}$ are not orthonormal under the scalar product
$(C_{\vartheta(x,t)})_{ij}=\langle q_{i}|q_{j}\rangle_{\vartheta(x,t)}$ given
in Eqs. (22) and (23). A detailed justification of the hydrodynamic projection
identity in Eq. (22) is provided in Refs. [32, 92].

For our purpose, it is sufficient to say that an exact formula for two-point
correlations of generic local observables
$\left\langle\mathcal{O}(x,t)\mathcal{O}^{\prime}(y,0)\right\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}$
can be derived using Eq. (22) and the result for the two-point correlator of
charge densities obtained from Eq. (20). The formula reads as

$\left\langle\mathcal{O}(x,t)\mathcal{O}^{\prime}(y,0)\right\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}\\!=\\!\int\mbox{d}\lambda\,\rho_{p}(x,t,\lambda)f(x,t,\lambda)V^{\mathcal{O}}(x,t,\lambda)\\!\left[\Gamma_{(y,0)\rightarrow(x,t)}V^{\mathcal{O}^{\prime}}(y,0)\right]\\!\\!(\lambda).$
(26)

The square brackets in Eq. (26) denote the integral-operator notation

$\left[\Gamma_{(y,0)\rightarrow(x,t)}h\right](\lambda)=\int\mbox{d}\mu\;\Gamma_{(y,0)\rightarrow(x,t)}(\lambda,\mu)h(\mu)\;.$
(27)

The propagator, $\Gamma_{(y,0)\rightarrow(x,t)}(\lambda,\mu)$, describes how
the local perturbation at the space-time point $(y,0)$ is transported by the
normal modes of the hydrodynamics on given characteristic curve
$\mathcal{U}(x,y,\lambda,0)$, until it reaches the space-time point $(x,t)$.
In particular, $\Gamma_{(y,0)\rightarrow(x,t)}$ can be split into two parts

$\Gamma_{(y,0)\rightarrow(x,t)}(\lambda,\mu)=\delta(y-\mathcal{U}(x,t,\lambda,0))\>\delta(\lambda-\mu)+\Delta_{(y,0)\rightarrow(x,t)}(\lambda,\mu)\;.$
(28)

The first term is dubbed direct propagator, it describes the normal-modes
propagation along the curved characteristic curve $\mathcal{U}(x,t,\lambda,0)$
(see Eqs. (18) and (19) in Subsec. 2.2) within the inhomogeneous fluid
background. Therefore, only the quasi-particles with the suitable rapidity
$\lambda$ to propagate from $(y,0)$ to $(x,t)$ enter in this term. On the
other hand, $\Delta_{(y,0)\rightarrow(x,t)}$ is named indirect propagator and
it encodes a more subtle effect. $\Delta_{(y,0)\rightarrow(x,t)}$ describes
the perturbation to the trajectory of the rapidity $\lambda$ due to the
interaction with other rapidities $\lambda^{\prime}$. Quasi-particles with
arbitrary rapidities $\lambda^{\prime}$, not necessarily connecting the two
observation points $(y,0)$ and $(x,t)$, are therefore involved into the
definition of the indirect propagator. Inserting Eq. (28) into Eq. (26) an
analogous decomposition for the two-point correlation formula applies

$\displaystyle\left\langle\mathcal{O}(x,t)\mathcal{O}^{\prime}(y,0)\right\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}$
$\displaystyle=\\!\\!\\!\sum_{\gamma\in\lambda_{\star}(x,t,y,0)}\\!\\!\frac{\rho_{s}(x,t,\gamma)\vartheta(y,0,\gamma)f(y,0,\gamma)}{\left|\partial_{\lambda}\mathcal{U}(x,t,\gamma,0)\right|}V^{\mathcal{O}}(x,t,\gamma)V^{\mathcal{O}^{\prime}}(y,0,\gamma)$
$\displaystyle+\int\mbox{d}\lambda\>\rho_{p}(x,t,\lambda)f(x,t,\lambda)V^{\mathcal{O}}(x,t,\lambda)\left[\Delta_{(y,0)\rightarrow(x,t)}V^{\mathcal{O}^{\prime}}(x,t)\right]\\!(\lambda),$
(29)

where the term on the first line of the r.h.s of Eq. (29) is named direct
correlator, while the term on the second line is dubbed indirect correlator.
The set $\lambda_{\star}(x,t,y,0)=\\{\lambda:\mathcal{U}(x,t,\lambda,0)=y\\}$
over which the sum on the first line runs expresses the fact that direct
correlations are determined solely by the propagation of the quasi-particles
with the right rapidity $\lambda_{\star}$ to connect the space-time point
$(y,0)$ with $(x,t)$.

Note that in Eqs. (26)-(29) we have placed for convenience the operator
$\mathcal{O}^{\prime}(y,0)$ at the space-time point $(y,0)$. However, the very
same formulas apply for the operator $\mathcal{O}^{\prime}(y,t_{0})$ at a
generic time $t_{0}\neq 0$ upon replacing $\mathcal{U}(x,t,\lambda,0)$ with
$\mathcal{U}(x,t,\lambda,t_{0})$, according to Eqs. (18) and (19), and
similarly for all TBA quantities evaluated at the time $0$. In particular,
Eqs. (26)-(29) apply also in the case $t<0$, where the operator
$\mathcal{O}(x,t)$ is computed at a time earlier than
$\mathcal{O^{\prime}}(y,0)$, and time ordering is not needed. This is a
consequence of the fact that the Euler-scale equation (17), on which Eqs.
(26)-(29) are based, is time reversible. The indirect propagator
$\Delta_{(y,0)\rightarrow(x,t)}$ can be obtained by solving a linear integral
equation, which we report in the Appendix A for completeness.

Only very recently, in Ref. [37], $\Delta_{(y,0)\rightarrow(x,t)}$ and the
formula in Eq. (29) have been numerically evaluated for various interacting
integrable models (Lieb-Liniger, sinh-Gordon and the classical hard-rod gas)
in different non-equilibrium protocols involving inhomogeneous initial states.
Moreover, in Ref. [37], the first numerical demonstration of the validity of
the Euler-scale formulas in Eq. (29) has been given by comparing them with
Monte Carlo simulations of the microscopic hard-rod dynamics. An excellent
agreement has been, in particular, found for long times and large values of
the scale parameter $z$ of the inhomogeneity in Eq. (9), consistently with the
expectation that the Euler-scale predictions apply to weakly inhomogeneous
settings.

##  3 Large deviation theory of ballistic transport

The large deviation theory, together with the generalized hydrodynamics, forms
the basis of the results presented in Sec. 4 for the calculation of the SCGF
in inhomogeneous initial states. In Subsec. 3.1, we accordingly briefly
discuss the large deviation theory of ballistically transported conserved
quantities. Then, in Subsec. 3.2, we briefly recall the results from Refs.
[71, 72], which will be important for comparison with our analysis.

###  3.1 Large deviation principle and the scaled cumulant generating function

Considering the continuity equation in Eq. (1), each conserved density
satisfies, for a particular density $q_{i_{\ast}}$ denoted by the subscript
$i_{\ast}$ we define the time-integrated current

$\Delta q_{i_{\ast}}(x,t)=\int_{0}^{t}\mbox{d}\tau\,j_{i_{\ast}}(x,\tau).$
(30)

$q_{i_{\ast}}$ could denote, for example, the energy, the particle or any
other density of the model. The GHD equation in Eq. (17) describes ballistic
transport of the hydrodynamic modes. Ballistic transport, indeed, is relevant
in integrable models as a consequence of the presence of an infinite number of
conserved charges. For ballistic motion, the time-integrated current is
expected to depend extensively on the time $t$ as $\Delta
q_{i_{\ast}}(x,t)\propto t$ for large times $t$, and it is therefore
convenient to focus on the intensive variable $J_{i_{\ast}}=\Delta
q_{i_{\ast}}(x,t)/t$. According to the large deviation theory, see, e.g.,
Refs. [50, 51, 49], the probability density function of $J_{i_{\ast}}$ for
large times $t$ peaks exponentially around the mean, and most-probable, value
$\langle J_{i_{\ast}}\rangle$ as

$p(J_{i_{\ast}})\asymp\mbox{exp}(-tI(J_{i_{\ast}})).$ (31)

In Eq. (31) the notation “$\asymp$”, taken from Ref. [49], denotes equality of
the right and the left hand side in logarithmic scale in the long-time limit
$t\rightarrow\infty$. Equation (31) is named large-deviation principle and it
expresses the leading asymptotic dependence on $t$ of the probability density
$p(J_{i_{\ast}})$. The function $I(J_{i_{\ast}})$ is dubbed large-deviation or
rate function; it is convex, non-negative, with a unique zero around the
average and most probable value $I(J_{i_{\ast}}=\langle
J_{i_{\ast}}\rangle)=0$, implying that large fluctuations of $J_{i_{\ast}}$
far from the mean value $\langle J_{i_{\ast}}\rangle$ are exponentially
suppressed according to Eq. (31). The Legendre-Fenchel transform [49] of
$I(J_{i_{\ast}})$ is the scaled cumulant generating function (SCGF) $G(s,x,t)$
of the time-integrated current in Eq. (30).

Here, as we are interested in the Euler-scaling limit, the SCGF is defined as

$G(s,x,t)=\lim_{z\rightarrow\infty}\frac{1}{zt}\,\mbox{ln}\,\langle\,\mbox{exp}({s\,\Delta
q_{i_{\ast}}}(zx,zt))\rangle_{\mathrm{inh},z}=\sum_{k=1}^{\infty}\frac{s^{k}}{k!}c_{k}(x,t).$
(32)

The average is taken over the rescaled inhomogeneous state $\rho_{0}$ in Eq.
(9). Equation (32) serves to define the meaning of the probability
$p(J_{i_{\ast}})$ in (31), which depends on the initial state, and therefore
in particular on $z$; it is the scaling in $z$ as per Eq. (32) that gives rise
to the asymptotic equality represented by “$\asymp$” in Eq. (31).

From the knowledge of $G(\lambda,x,t)$ in Eq. (32), one can obtain by Taylor
expansion the cumulants $\\{c_{k}\\}$ of the time-integrated current, which
are defined as

$c_{k}(x,t)=\lim_{z\rightarrow\infty}\frac{1}{zt}\langle[\Delta
q_{i_{\ast}}(zx,zt)]^{k}\rangle_{\mathrm{inh,z}}^{c}=\frac{1}{t}\int_{0}^{t}\mbox{d}t_{1}\dots\int_{0}^{t}\mbox{d}t_{k}\,\langle
j_{i_{\ast}}(x,t_{1})\dots
j_{i_{\ast}}(x,t_{k})\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}},$ (33)

where the Euler-scaling limit of the $k$-point connected correlation function
$\langle\mathcal{O}(x_{1},t_{1})\dots\mathcal{O}(x_{k},t_{k})\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}$
of a local observable $\mathcal{O}$ is defined in a similar way as in Eq. (21)
for the two-point function [30]222As in the case of two-point functions, at
the Euler scale, time-ordering is not necessary in the definition of $k$-point
connected correlations. Fluid-cell averaging might, instead, be needed in Eq.
(34), as discussed after Eq. (21).:

$\langle\mathcal{O}(x_{1},t_{1})\dots\mathcal{O}(x_{k},t_{k})\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}=\lim_{z\rightarrow\infty}z^{k-1}\langle\mathcal{O}(zx_{1},zt_{1})\dots\mathcal{O}(zx_{k},zt_{k})\rangle^{c}_{\mathrm{inh,z}}.$
(34)

The validity of the large deviation principle in Eq. (31) implies that all
that the connected correlation functions $\\{\langle[\Delta
q_{i_{\ast}}(zx,zt)]^{k}\rangle_{\mathrm{inh},z}^{c}\\}$ scale as $z$, with
the cumulants $\\{c_{k}\\}$ being therefore finite and the series expansion in
Eq. (32) being valid in some interval of $s$ around the origin. The main
result of this manuscript is the derivation of an exact expression for
$G(s,x,t)$ in Eq. (32) valid for interacting integrable models, both classical
and quantum, which are described by GHD.

###  3.2 SCGF for homogeneous GGEs: review of the result

In Refs. [71, 72] a general approach has been developed to compute $G(s,x,t)$
in the case where the average in Eq. (32) is taken w.r.t. the homogeneous GGE
$\vartheta$ as in Eqs. (2) and (7) (and not w.r.t. the inhomogeneous state
$\rho_{0}$ in Eq. (9), as it will be the case in Sec. 4). In this case,
because of the homogeneity of the GGE, $G(s,x,t)=G(s)$. Let us review this
approach.

The analysis of Refs. [71, 72] relies on the GHD description of the Euler-
scale hydrodynamics. In particular, we need to introduce the so-called flux
jacobian matrix, $A_{i}^{j}$

$A_{i}^{j}=\frac{\partial\langle j_{i}\rangle}{\partial\langle q_{j}\rangle},$
(35)

with $i$ the row index and $j$ the column one, which will play an important
role in the analysis of this Subsection. This matrix describes how the average
currents depends on the average densities and therefore it depends on the
equation of state of the model. In the case of integrable systems, where the
averages in Eq. (35) are computed over the homogeneous GGE $\vartheta$ in Eqs.
(2) and (7) as per Eqs. (14) and (15), an expression for the matrix elements
of $A_{i}^{j}$ can be given by means of the hydrodynamic projection formalism,
as shown in Ref. [32]. In the basis of the single-particle eigenvalues
$\\{h_{i}(\lambda)\\}$ the matrix $A_{i}^{j}$ reads

$A_{i}^{j}=\int\mbox{d}\lambda\,h_{i}^{\mathrm{dr}}(\lambda)\,v^{\mathrm{eff}}(\lambda)\,h^{j}_{\mathrm{dr}}(\lambda),$
(36)

with the effective velocity $v^{\mathrm{eff}}(\lambda)$ given in Eq. (16). The
function $h^{i}_{\mathrm{dr}}(\lambda)$ denotes the orthonormal conjugate of
$h_{i}^{\mathrm{dr}}(\lambda)$, which satisfies the orthogonality and
completeness relations

$\displaystyle\int\mbox{d}\lambda\,h^{j}_{\mathrm{dr}}(\lambda)\,h_{i}^{\mathrm{dr}}(\lambda)$
$\displaystyle=\delta^{j}_{i},$
$\displaystyle\sum_{j}h^{j}_{\mathrm{dr}}(\lambda)\,h_{j}^{\mathrm{dr}}(\lambda^{\prime})$
$\displaystyle=\delta(\lambda-\lambda^{\prime}),$ (37)

respectively. Notice that the matrix $A_{i}^{j}$ in integrable systems is
infinite-dimensional, yet it can be formally defined. From Eqs. (36) and (37)
the eigenvalue equation for $A_{i}^{j}$ readily follows as

$\sum_{j}A_{i}^{j}\,h_{j}^{\mathrm{dr}}(\lambda)=v^{\mathrm{eff}}(\lambda)\,h_{i}^{\mathrm{dr}}(\lambda),$
(38)

which shows that the flux jacobian has a continuous spectrum indexed by the
rapidity $\lambda$ with eigenvalue the effective velocity
$v^{\mathrm{eff}}(\lambda)$.

In order to compute $G(s)$ one then biases the GGE measure in Eq. (2) by
multiplying it by the exponential of the time-integrated current appearing in
Eq. (32). Averages over this tilted measure become dependent on the parameter
$s$ conjugate to the time-integrated current
$\langle\mathcal{O}\rangle_{\vartheta(s)}$. Note that we are here extending
the notation introduced in Eq. (7) for the average over the homogeneous GGE by
promoting $\vartheta(s)$ to be dependent on $s$ (in addition to the rapidity
$\lambda$ whose dependence is not reported for brevity). In Subsec. 4.2 we
will explain this procedure in more details, for the moment we just report the
main result for the flow equation from Ref. [71, 72].

The flow equation describes how the homogeneous GGE $\vartheta$ in Eq. (2) is
modified by the insertion of the exponential of the time-integrated current.
Fundamentally, the $s$ modified state $\vartheta(s)$ is still an homogeneous
GGE, yet with Lagrange parameters $\\{\beta^{n}\\}$ dependent on $s$. This
dependence is captured by the flow equation, which can be written as

$\frac{\partial\beta^{n}(s)}{\partial
s}=-\mbox{sgn}\left(A(s)\right)^{n}_{i_{\ast}},$ (39)

where the sign of the flux jacobian is obtained by diagonalizing the latter
and by taking the sign of its eigenvalues. For interacting integrable systems,
from Eq. (36), this implies

$\mbox{sgn}\left(A\right)_{i}^{j}=\int\mbox{d}\lambda\,h_{i}^{\mathrm{dr}}(\lambda)\,\mbox{sgn}(v^{\mathrm{eff}}(\lambda))\,h^{j}_{\mathrm{dr}}(\lambda).$
(40)

By inserting Eq. (39) into Eq. (3) and by taking the derivative with respect
to $s$, the flow equation can be recast in a form where the dependence of the
pseudo-energy $\varepsilon(\lambda;s)$ on $s$ is exposed

$\frac{\partial\varepsilon(\lambda;s)}{\partial
s}=-\mbox{sgn}(v^{\mathrm{eff}}(\lambda;s))h_{i_{\ast}}^{\mathrm{dr}}(\lambda;s).$
(41)

Notice that, as a consequence of Eq. (39), all the dressed quantities, which
depend on the GGE state, acquire an additional dependence on the parameter
$s$. Exploiting Eqs. (39), and equivalently (41), the SCGF $G(s)$ can be
eventually computed as

$G(s)=\int_{0}^{s}\mbox{d}s^{\prime}\langle
j_{i_{\ast}}(0,0)\rangle_{\vartheta(0,0;s^{\prime})},$ (42)

where we stress that the current expectation on the r.h.s. is taken over the
homogeneous GGE and it is thereby given by Eq. (15). In Eq. (42) we have
extended the notation introduced after (8) for the average over the local,
homogeneous GGE $\vartheta(0,0;s^{\prime})$ at the space-time point $(0,0)$ by
including the additional dependence on $s$ because of the biasing of the
measure.

Equations (39) (or equivalently Eq. (41)) and (42) are the two main results of
Refs. [71, 72] and they are the necessary equations to compute $G(s)$. First
one has to solve numerically Eq. (41) to fix the dependence of the state on
$s$, and then the result must be plugged into Eq. (42) to get $G(s)$. This
procedure has been carried on in Ref. [71] for the classical hard-rod gas and
for the quantum Lieb-Liniger model, where $G(s)$ has been computed in the
homogeneous steady state developing at long times from the partitioning
protocol initial state in Eqs. (11) and (12). In the classical hard-rod fluid,
moreover, the cumulants obtained from the series expansion of $G(s)$ according
to Eq. (33) have been compared against Monte-Carlo simulations, finding an
excellent agreement and thus corroborating the validity of the approach. In
concluding this Subsection, we mention that the flow equation in Eq. (39) can
be proved, as shown in Ref. [72], solely on the basis of linear fluctuating
hydrodynamics and hydrodynamics projections [11, 90] without the need of using
any tool coming from integrability and Bethe ansatz. Only in deriving the form
in Eq. (41) the TBA machinery is exploited. The expression for $G(s)$ in Eq.
(42) together with Eq. (39) is therefore expected to apply more generically to
any system, integrable or not integrable, displaying ballistic transport.

##  4 The Euler-scale SCGF for inhomogeneous states

This Section contains the main result of this work, which is the exact
expression of $G(s,x,t)$ in the Euler-scaling limit $z\rightarrow\infty$ as
per Eq. (32), with the initial state $\rho_{0}$ the inhomogeneous GGE in Eq.
(9). In Subsec. 4.1 we state for the reader’s convenience the result and the
equations necessary for the numerical computation of $G(s,x,t)$. In Subsec.
4.2 we report the derivation of the results. In Subsec. 4.3 the results for
the first three cumulants of the time-integrated current, according to Eq.
(33), are reported. In Subsec. 4.4 the non-interacting limit of our general
expression is studied.

###  4.1 SCGF for inhomogeneous GGEs: statement of the result

For the inhomogeneous and non-stationary GGEs $\rho_{0}$, as in Eq. (9), the
evaluation of $G(s,x,t)$ is much more difficult and has not been considered
before. In this Section we provide the first expression for the SCGF for
inhomogeneous and non-stationary states in the form of $\rho_{0}$ in (9) in
the limit $z\rightarrow\infty$ according to Eq. (32).

Consider the inhomogeneous state $\rho_{0}$ in Eq. (9). Then $G(s,x,t)$
defined in Eq. (32) can be computed as

$G(s,x,t)=\frac{1}{t}\int_{0}^{s}\mbox{d}s^{\prime}\int_{0}^{t}\mbox{d}\tau\,\langle
j_{i_{\ast}}(0,0)\rangle_{\vartheta(x,\tau;s^{\prime})}\,,$ (43)

where the average current is evaluated on the homogeneous GGE
$\vartheta(x,\tau;s^{\prime})$ in Eq. (2) and it is therefore given by Eq.
(15). The fundamental difference with respect to the homogeneous case, using
the same extension of the notation introduced for Eq. (42), is that the GGE
$\vartheta(x,\tau;s^{\prime})$ depends not only on the parameter $s^{\prime}$
as a consequence of the biasing of the measure in Eq. (9) via the exponential
of the time-integrated current, but also on the (scaled) space-time
coordinates $(x,\tau)$ of the fluid cell, because, for every deformation
parameter $s^{\prime}$, the state is inhomogeneous and non-stationary.

To be accurate, the bias of the measure by the exponential of the time
integral of the current as in Eq. (32), depends not only on $s$, but also on
the parameters $x$, $t$ characterising the (scaled) space-time position of the
integration interval (see, for instance, Fig. 1). Therefore, an average at the
fluid cell $(x^{\prime},t^{\prime})$ in the deformed state should be denoted
as $\langle\dots\rangle_{\vartheta(x^{\prime},t^{\prime};x,t,s)}$. For
lightness of notation, we keep implicit the $x,t$ dependence of the bias
itself in the fluid-cell average notation; these can be considered as fixed
parameters throughout. The dependence on $s$ is important, as for instance
this is integrated over in Eq. (43).

As in the homogeneous case, the $s$ dependence is described by a flow equation
for an $s$-dependent state, which in the inhomogeneous case is however
significantly more complex than Eq. (39). In terms of the fluid-cell Lagrange
parameters described by $\beta^{n}(x^{\prime},t^{\prime};s)$, it takes the
following form:

$\frac{\partial\beta^{n}(x^{\prime},t^{\prime};s)}{\partial
s}=-\int_{0}^{t}\mbox{d}\tau\,\mbox{d}\lambda\,\left[\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}V^{j_{i_{\ast}}}(x,\tau;s)\right](\lambda)\,\,h^{n}_{\mathrm{dr}}(x^{\prime},t^{\prime},\lambda;s),$
(44)

where the propagator $\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}$ has
been defined in Eq. (28) with $\tau$ as initial time according to the
discussion after Eq. (29). In Eq. (44) we are further using the integral-
operator notation introduced in Eq. (27). The one-particle form factor
$V^{j_{i_{\ast}}}$ of the current $j_{i_{\ast}}$ is given in Eq. (24).

It is convenient to re-write this as a flow equation for the pseudo-energy
$\varepsilon$. This directly follows upon differentiating the left and the
right hand side of Eq. (3) with respect to $s$

$\displaystyle\frac{\partial\varepsilon(x^{\prime},t^{\prime},\lambda;s)}{\partial
s}$
$\displaystyle=\sum_{n}\frac{\partial\beta^{n}(x^{\prime},t^{\prime};s)}{\partial
s}h_{n}(\lambda)+\int\mbox{d}\mu\,T(\lambda,\mu)\vartheta(x^{\prime},t^{\prime},\mu;s)\frac{\partial\varepsilon(x^{\prime},t^{\prime},\mu;s)}{\partial
s}$
$\displaystyle=\sum_{n}\frac{\partial\beta^{n}(x^{\prime},t^{\prime};s)}{\partial
s}\left(h_{n}(\lambda)+\int\mbox{d}\mu\,T(\lambda,\mu)\vartheta(x^{\prime},t^{\prime},\mu;s)h_{n}^{\mathrm{dr}}(x^{\prime},t^{\prime},\mu;s)\\!\right)$
$\displaystyle=\sum_{n}\frac{\partial\beta^{n}(x^{\prime},t^{\prime};s)}{\partial
s}h_{n}^{\mathrm{dr}}(x^{\prime},t^{\prime},\lambda;s),$ (45)

where in the first line we have used the chain rule and the relation
$\vartheta(\varepsilon)=\mbox{d}F(\varepsilon)/\mbox{d}\varepsilon$. In the
second line we have again used the chain rule and the identity
$h^{\mathrm{dr}}_{n}(\lambda)=\partial\varepsilon/\partial\beta^{n}$, which
follows upon differentiating with respect to $\beta^{n}$ Eq. (3) and by
recognizing the integral equation in Eq. (6) defining the dressing operation.
Upon inserting Eq. (44) into the last equality in Eq. (45) one obtains the
flow equation for the pseudo-energy $\varepsilon$:

$\frac{\partial\varepsilon(x^{\prime},t^{\prime},\lambda;s)}{\partial
s}=-\int_{0}^{t}\mbox{d}\tau\left[\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}V^{j_{i_{\ast}}}(x,\tau;s)\right](\lambda),$
(46)

where we have used the completeness relation in Eq. (37). Using the
decomposition of the propagator $\Gamma$ in Eq. (28), the last equation can be
written as

$\displaystyle\frac{\partial\varepsilon(x^{\prime},t^{\prime},\lambda;s)}{\partial
s}=-$
$\displaystyle\sum_{\delta\in\tau_{\star}(x^{\prime},t^{\prime},\lambda,x)}\mbox{sgn}(v^{\mathrm{eff}}(x,\delta,\lambda;s))h_{i_{\ast}}^{\mathrm{dr}}(x,\delta,\lambda;s)$
$\displaystyle-$
$\displaystyle\int_{0}^{t}\mbox{d}\tau\left[\Delta_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}V^{j_{i_{\ast}}}(x,\tau;s)\right](\lambda),$
(47)

where we have defined the set of times
$\tau_{\star}(x^{\prime},t^{\prime},\lambda,x)=\\{\tau:\mathcal{U}(x^{\prime},t^{\prime},\lambda,\tau)=x\\}$,
with the characterstic curve defined in Eqs. (18) and (19).

Notice that for the homogeneous GGE $\vartheta$, the indirect propagator
$\Delta$ vanishes, as detailed after Eq. (LABEL:eq:SourceTerm) in Appendix A.
The characteristic curve, from Eq. (19), in this case is simply given by
$\mathcal{U}(x^{\prime},t^{\prime},\lambda,\tau)=x^{\prime}-v^{\mathrm{eff}}(\lambda)(t^{\prime}-\tau)$,
the set $\tau_{\star}$ is composed by one element only and Eq. (47) reduces to
Eq. (41). In the same limit, the time-integral in Eq. (43) trivializes, as the
current average value is independent on time, and Eq. (42) is re-obtained. One
therefore sees that Eqs. (44) and (47) generalize the results of Refs. [71,
72], recalled in Subsec. 3.2, by including non-stationarity and inhomogeneous
situations, when motion occurs at the Euler scale of hydrodynamics. Equations
(43), (44) and (47) are the main results of this paper.

Equations (43) and (47) involve the propagator
$\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}$ which, as we have seen
in Sec. 2.3, describes the motion of the quasi-particles between $(x,\tau)$
and $(x^{\prime},t^{\prime})$ in the inhomogeneous fluid background (under the
homogeneous evolution Hamiltonian of the model considered). It is consequently
clear that the Euler-scale expression for $G(s,x,t)$ is expected to apply in
the same limit where the expression for the two-point correlator of Subsec.
2.3 does. In particular, based on the findings of Ref. [37], where the
formulas for two-point functions have been tested again numerical simulations
of the microscopic hard-rod dynamics, we expect the expression for $G(s,x,t)$
in Eqs. (43) and (47) to apply for smooth initial inhomogeneities, large
enough $z$ in Eq. (9), and long times. The actual verification of this
statement by comparing the predictions coming from Eqs. (43) and (47) against
simulations of the hard-rod gas will be addressed in a future publication,
together with the evaluation of $G(s,x,t)$ for some specific quantum models,
e.g., the Lieb-Liniger and the sinh-Gordon. In this work, we will solely
present the general theory based on Eqs. (43) and (47) describing the large
deviation theory of ballistically transported conserved quantities.

We conclude this Subsection by noting that, if the state $\vartheta_{0}$ is
invariant under simultaneous rescaling of space and time
$(x,t)\rightarrow(ax,at)$, and therefore
$\beta^{n}(ax^{\prime},at^{\prime};0)=\beta^{n}(x^{\prime},t^{\prime};0)$ with
$a>0$ an arbitrary positive constant (e.g., the partitioning protocol initial
state in Eqs. (11) and (12)), then the expression in Eq. (43) becomes a
function of the scaling variable $\xi=x/t$. As a matter of fact, using the
property of the propagator $\Gamma$, valid if
$\beta^{n}(ax^{\prime},at^{\prime};0)=\beta^{n}(x^{\prime},t^{\prime};0)$ for
every $n$, $x^{\prime}$ and $t^{\prime}$,

$\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}=a\,\Gamma_{(ax,a\tau)\rightarrow(ax^{\prime},at^{\prime})},$
(48)

which directly follows from Eqs. (28) and (81), the following scaling property
is obtained for the Lagrange multipliers $\beta^{n}(x^{\prime},t^{\prime};s)$
for arbitrary values of $s$ by using Eq. (48) in Eq. (44)

$\beta^{n}(ax^{\prime},at^{\prime};s)=\beta^{n}(x^{\prime},t^{\prime};s).$
(49)

Equation (49) in turn implies that the same scaling property holds also for
the pseudo-energy $\varepsilon(x^{\prime},t^{\prime},\lambda;s)$ and any
dressed quantity $h_{i}^{\mathrm{dr}}(x^{\prime},t^{\prime},\lambda;s)$:

$\displaystyle\varepsilon(ax^{\prime},at^{\prime},\lambda;s)$ $\displaystyle=$
$\displaystyle\varepsilon(x^{\prime},t^{\prime},\lambda;s)$ $\displaystyle
h_{i}^{\mathrm{dr}}(ax^{\prime},at^{\prime},\lambda;s)$ $\displaystyle=$
$\displaystyle h_{i}^{\mathrm{dr}}(x^{\prime},t^{\prime},\lambda;s).$ (50)

Exploiting the last identity, the expression for $G(s,x,t)$ can be eventually
written in the form

$G(s,\xi)=\int_{0}^{s}\mbox{d}s^{\prime}\int_{0}^{1}\mbox{d}t^{\prime}\,\langle
j_{i_{\ast}}(0,0)\rangle_{\vartheta(\xi,t^{\prime};\,s^{\prime})}.$ (51)

In the next Subsection, we provide the derivation of the main results Eqs.
(43), (44) and (47).

###  4.2 SCGF for inhomogeneous GGEs: derivation of the result

We start by defining the $s$-tilted ensemble as that obtained by biasing the
measure of the inhomogeneous GGE in Eq. (9) with the time-integrated current
in Eq. (30). This procedure is analogous to the one introduced in Refs. [52,
53, 54, 93] for one-dimensional critical systems described by conformal field
theory, and later extended in Refs. [62, 73] to non-interacting systems. We
will further comment in Subsec. 4.4 about the relation between the present
approach and how it simplifies to the case of non-interacting systems.

Averages over the $s$-tilted ensemble will be denoted as
$\langle\dots\rangle^{(s)}_{\mathrm{inh},z}$, with
$\langle\dots\rangle^{(0)}_{\mathrm{inh},z}=\langle\dots\rangle_{\mathrm{inh},z}$
by construction. Namely, for a local operator
$\mathcal{O}(zx^{\prime},zt^{\prime})$ at the space-time point
$(zx^{\prime},zt^{\prime})$ the $s$-tilted ensemble is defined as

$\langle\mathcal{O}(zx^{\prime},zt^{\prime})\rangle^{(s)}_{\mathrm{inh},z}=\frac{\langle\mathcal{O}(zx^{\prime},zt^{\prime})\,\mbox{exp}(s\Delta
q_{i_{\ast}}(zx,zt))\rangle_{\mathrm{inh},z}}{\langle\mbox{exp}(s\Delta
q_{i_{\ast}}(zx,zt))\rangle_{\mathrm{inh,z}}},$ (52)

where $\Delta q_{i_{\ast}}$ is given in Eq. (30). From the definition (52) one
also has

$\frac{\partial\langle\mathcal{O}(zx^{\prime},zt^{\prime})\rangle^{(s)}_{\mathrm{inh},z}}{\partial
s}=\int_{0}^{t}\mbox{d}\tau\,z\langle\mathcal{O}(zx^{\prime},zt^{\prime})j_{i_{\ast}}(zx,z\tau)\rangle^{c,(s)}_{\mathrm{inh},z},$
(53)

with the integrand on the r.h.s. denoting the two-point connected correlation
function over the $s$-tilted ensemble with the notation of Eq. (52).

It is crucial to note that, although the $s$-tilting involves an integral over
time, this does not affect the dynamics. The bias is to be understood as a
modification of the measure, that is, of the distribution of states at time 0,
or on any chosen time slice. According to (52), the measure is modified by a
weight which is evaluated by evolving the observables in time (or equivalenty,
evolving the distribution of states in time), and evaluating the exponential
of the time-integrated current. The dynamics is still given by the original,
homogeneous and time-independent Hamiltonian of the model under consideration,
and thus, at the Euler scale, time slices are related to each other by the
original GHD equation (17) of the model. This is important, as we can then
use, below, the results for correlations in inhomogeneous and dynamical states
reviewed in Subsec. 2.3, which, as emphasised, are based on the assumption of
an homogeneous and time-independent dynamics.

The definition of the $s$-tilted measure in Eq. (53) is widely known in the
context of large deviations in classical stochastic systems, see, e.g., Refs.
[94, 95], and in open quantum systems. In the latter framework it is named “s
ensemble”, see, e.g., Refs. [96, 97]. There, the approach consists in relating
the bias in $s$, as an exponential of the time-integrated current, to a
modification of the Lindbladian ruling the evolution of the system. This
operation can be done via the so-called “quantum Doob” transformation, as
shown in Refs. [98, 99], and it amounts to a biasing of the probability
measures which makes rare events typical. The approach we pursue here is
complementary to the quantum Doob transform, in the sense that here we relate
the insertion of the exponential of the time-integrated current to a change of
the (inhomogeneous) GGE measure. Fundamentally, the latter modification still
produces a (inhomogeneous) GGE, whose $s$-dependence can be determined exactly
in terms of the flow equation in Eqs. (44) and (47).

We now show that the biasing procedure in Eqs. (52) and (53) defines a flow in
the manifold of the inhomogeneous GGEs. Since, by construction, the dynamics
is unchanged, we may think of the manifold of inhomogenenous GGEs as described
by a space-dependent GGE on any given time slice,
$\\{\beta^{n}(x^{\prime},t^{\prime};s)\\}_{x^{\prime},n}$. Let us choose the
time slice $t^{\prime}=0$. One can adapt the argument developed in Ref. [72]
for the homogeneous case, summarized in Subsec. 3.2. Namely, one defines the
Lie derivative $\mathcal{L}$ at the point
$\langle...\rangle_{\vartheta_{0}}^{\mathrm{Eul}}$ in the manifold of
inhomogeneous GGEs as

$\mathcal{L}\langle\mathcal{O}(x^{\prime},0)\rangle_{\vartheta_{0}}^{\mathrm{Eul}}=\int_{0}^{t}\mbox{d}\tau\,\langle\mathcal{O}(x^{\prime},0)j_{i_{\ast}}(x,\tau)\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}}.$
(54)

Using Eq. (26) with $\tau$ as initial time for the propagator
$\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}$, and Eqs. (23) and (24),
the Lie derivative can be written as

$\mathcal{L}\langle\mathcal{O}(x^{\prime},0)\rangle_{\vartheta_{0}}^{\mathrm{Eul}}=\\!-\sum_{n}\frac{\partial\langle\mathcal{O}\rangle_{\vartheta(x^{\prime},0)}}{\partial\beta^{n}(x^{\prime},0)}\\!\int_{0}^{t}\\!\mbox{d}\tau\,\mbox{d}\lambda\left[\Gamma_{(x,\tau)\rightarrow(x^{\prime},0)}V^{j_{i_{\ast}}}(x,\tau)\right]\\!\\!(\lambda)h^{n}_{\mathrm{dr}}(x^{\prime},0,\lambda).$
(55)

Equation (55) shows that the Lie derivative lies on the tangent space to the
manifold of inhomogeneous GGEs identified by the Lagrange multipliers
$\\{\beta^{n}(x^{\prime},0)\\}_{x^{\prime},n}$. Comparing Eqs. (54) and (55)
with (53), and remembering the definition of the Euler-scaling limit of two-
point correlation functions in Eq. (21), one realizes that the $s$-tilted
measure defines a flow directed along the Lie derivative as
$z\rightarrow\infty$. Accordingly, from Eq. (53), infinitesimal $s$
modifications lie on the tangent space to that manifold, and given that at
$s=0$ the state $\langle\dots\rangle^{(0)}_{\mathrm{inh},z}$ is an
inhomogeneous GGE and lies on that manifold, then it remains so even after the
$s$-tilting. As a consequence, one has for the Euler-scaling limit
$z\rightarrow\infty$ of Eq. (52) that

$\lim_{z\rightarrow\infty}\langle\mathcal{O}(zx^{\prime},zt^{\prime})\rangle^{(s)}_{\mathrm{inh},z}=\langle\mathcal{O}(x^{\prime},t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{Eul}}=\langle\mathcal{O}(0,0)\rangle_{\vartheta(x^{\prime},t^{\prime};s)},$
(56)

where in the second equality we have used the local relaxation assumption (8)
thereby expressing the average of the local operator $\mathcal{O}$ over the
local homogeneous GGE in Eq. (2) at the fluid cell
$(x^{\prime},t^{\prime};s)$. It is also worth to remark that in the second
equality of (56) we have extended the notation introduced after (21) for the
mode-occupation function $\vartheta_{0}(s)$, corresponding to the state
$\rho_{0}$ in Eq. (9) as $z\rightarrow\infty$, by exposing the additional
dependence on $s$ due to the biasing of the measure.

In order to fix the additional dependence on $s$ due to the tilting in Eq.
(52) one further needs to specify the flow equation. To do this we start from
(53) together with Eq. (56) by choosing the local observable
$\mathcal{O}(x^{\prime},t^{\prime})$ as some conserved density
$q_{n}(x^{\prime},t^{\prime})$ of the model

$\frac{\partial\langle
q_{n}(x^{\prime},t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{Eul}}}{\partial
s}=\int_{0}^{t}\mbox{d}\tau\,\langle
q_{n}(x^{\prime},t^{\prime})j_{i_{\ast}}(x,\tau)\rangle^{c,\mathrm{Eul}}_{\vartheta_{0}(s)}.$
(57)

The homogeneous GGE at every fluid cell $(x^{\prime},t^{\prime})$ is specified
by the Lagrange multipliers
$\\{\beta^{n}(x^{\prime},t^{\prime};s)\\}_{x^{\prime},n}$ or, equivalently, by
the average conserved densities $\langle
q_{n}(x^{\prime},t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{Eul}}=\langle
q_{n}(0,0)\rangle_{\vartheta(x^{\prime},t^{\prime};s)}$ in Eq. (14). Equation
(57) can therefore be considered as the “equation of motion” of the state
coordinates $\langle q_{n}(0,0)\rangle_{\vartheta(x^{\prime},t^{\prime};s)}$
for their trajectory, parametrized by $s$, in the manifold of inhomogeneous
GGEs. For convenience, we express these coordinates on an arbitrary time slice
$t^{\prime}$.

As a matter of fact, Eq. (57) relates the tangent vector of the trajectory,
the l.h.s., to a function of the coordinate itself, the r.h.s.. Since the
expression of the correlator on the r.h.s. is known, from Eq. (29) in Subsec.
2.3, the equation of motion is well defined and it fixes the flow of the
inhomogeneous GGE in Eq. (9) as a function of $s$. Indeed we now show that Eq.
(57) is equivalent to the flow equation (44). Exploiting the chain rule one
has

$\displaystyle\frac{\partial\langle
q_{n}(x^{\prime},t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{Eul}}}{\partial
s}{}$ $\displaystyle=\int\mbox{d}y\sum_{j}\frac{\delta\langle
q_{n}(x^{\prime},t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{Eul}}}{\delta\beta^{j}(y,t^{\prime};s)}\,\,\frac{\partial\beta^{j}(y,t^{\prime};s)}{\partial
s}$
$\displaystyle=-\sum_{j}(C_{\vartheta(x^{\prime},t^{\prime};s)})_{nj}\frac{\partial\beta^{j}(x^{\prime},t^{\prime};s)}{\partial
s}=-\left[\frac{\partial\beta(x^{\prime},t^{\prime};s)}{\partial
s}C_{\vartheta(x^{\prime},t^{\prime})}\right]_{n},$ (58)

where in the last step we used vector-matrix notation and the symmetry of the
correlation matrix $C_{\vartheta(x^{\prime},t^{\prime})}$, defined in Eq.
(25). Notice that in Eq. (58) we have further exploited the fluctuation
dissipation relation in Eq. (20)

$-\frac{\delta\langle
q_{i}(x^{\prime},t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{Eul}}}{\delta\beta^{j}(y,t^{\prime};s)}=\langle
q_{i}(x^{\prime},t^{\prime})q_{j}(y,t^{\prime})\rangle_{\vartheta_{0}(s)}^{\mathrm{c,Eul}}=\delta(x^{\prime}-y)(C_{\vartheta(x^{\prime},t^{\prime};s)})_{ij},$
(59)

where the second equality comes from the fact that for equal-time connected
correlation functions the propagator
$\Gamma_{(y,t^{\prime})\rightarrow(x^{\prime},t^{\prime})}(\lambda,\mu)=\delta(x^{\prime}-y)\delta(\lambda-\mu)$.
The latter equality simply expresses that at the Euler scale fluid cells on
the same time slice separated in space are uncorrelated, as shown in Ref.
[30]. The inverse matrix $C_{\vartheta(x^{\prime},t^{\prime};s)}^{-1}$ can be
defined by means of the functions $h_{\mathrm{dr}}^{j}$ in Eq. (37) as

$(C_{\vartheta(x^{\prime},t^{\prime};s)}^{-1})^{jn}=\int\mbox{d}\lambda\,\rho_{p}^{-1}(x^{\prime},t^{\prime},\lambda;s)f^{-1}(x^{\prime},t^{\prime},\lambda;s)h_{\mathrm{dr}}^{n}(x^{\prime},t^{\prime},\lambda;s)h_{\mathrm{dr}}^{j}(x^{\prime},t^{\prime},\lambda;s),$
(60)

Multiplying Eq. (58) by $C_{\vartheta(x^{\prime},t^{\prime};s)}^{-1}$ in Eq.
(60) and equating with the r.h.s of Eq. (57), where the connected correlator
is given by Eq. (29), one eventually obtains the flow equation for the
Lagrange parameter $\beta^{n}(x^{\prime},t^{\prime};s)$ in Eq. (44). The flow
equation for $\varepsilon(x^{\prime},t^{\prime},\lambda;s)$ (46), (47) follows
from Eq. (44) according to the steps reported in Eq. (45).

Once the flow equation is established, proving Eq. (43) for the SCGF is
simple. By applying a derivative with respect to $s$ to $G(s,x,t)$ in Eq.
(32), with $\rho_{0}$ as given by Eq. (9), one obtains

$\displaystyle\frac{\partial G(s,x,t)}{\partial s}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\lim_{z\rightarrow\infty}\langle
j_{i_{\ast}}(zx,z\tau)\rangle^{(s)}_{\mathrm{inh},z}=\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\,\langle
j_{i_{\ast}}(x,\tau)\rangle^{\mathrm{Eul}}_{\vartheta_{0}(s)}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\,\langle
j_{i_{\ast}}(0,0)\rangle_{\vartheta(x,\tau;s)},$ (61)

where in the first equality we have used the definition in Eq. (52) for the
average over the $s$-tilted ensemble, while Eq. (56) has been exploited in the
second and third equality. Integrating in $s$, and exploiting the fact that
$G(s=0,x,t)=0$, from the definition in Eq. (32) of the SCGF, one can see that
Eq. (61) reduces to Eq. (43).

Regarding the numerical evaluation of Eqs. (43) and (47), we remark that they
are expressed in terms of quantities available from the TBA for $s=0$, i.e.,
in the unbiased case, such as $v^{\mathrm{eff}}(x,\delta,\lambda;0)$,
$h_{i_{\ast}}^{\mathrm{dr}}(x,\delta,\lambda;0)$ and
$V^{j_{i_{\ast}}}(x,\tau;0)$. These functions can be efficiently evolved in
time using the solution of the GHD equation in Eq. (17) via the characteristic
curves in Eqs. (18) and (19), as discussed in Ref. [29]. The most difficult
element to numerically compute, which, instead, is not directly provided by
the TBA, is the indirect propagator
$\Delta_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}$. For the calculation of
the latter, however, an efficient numerical scheme has been developed in Ref.
[37] within the iFluid open-source code [100]. Accordingly, the only step
which has not yet been pursued in the literature, so far, is the numerical
solution of the flow equation in Eq. (47). This passage can be achieved by
writing Eq. (47) as an integral equation (by formally integrating in $s$ the
right and the left hand side). The latter can be in turn solved by iteration
starting from the knowledge of the right hand side of Eq. (47) for $s=0$,
where all the TBA functions and the propagator
$\Delta_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}$ are known as just
explained. The resulting expression for the pseudo-energy
$\varepsilon(x^{\prime},t^{\prime},\lambda;s)$ as a function of $s$ can then
be plugged into Eq. (43), together with the expression in Eq. (15) for the
average current $\langle j_{i}(0,0)\rangle_{\vartheta(x,\tau;s^{\prime})}$, to
eventually compute $G(s,x,t)$. We plan to carry on this numerical analysis in
a future work.

In concluding this Subsection, we emphasize that the present derivation of
$G(s,x,t)$ is strongly based on the result of Ref. [30], recalled in Subsec.
2.3, for two-point functions in inhomogeneous and dynamical GGEs. Accordingly,
Equations (43), (44) and (47) are restricted to integrable models, differently
from the results in Eqs. (39) and (42) for homogeneous GGEs. The calculation
of the scaled cumulant generating for inhomogeneous and dynamical states in
generic, not necessarily integrable, models is an unexplored and challenging
problem which goes beyond the analysis of the present manuscript.

###  4.3 Analysis of the cumulants

We report here the first three cumulants, which can be obtained from the
series expansion in Eqs. (32) and (33) of the general expression given by Eqs.
(43) and (47). The first cumulants provide important information about the
shape of the probability distribution of $J_{i_{\ast}}=\Delta q_{i_{\ast}}/t$
and they are the easiest to access experimentally. Moreover, the cumulants can
be computed numerically, e.g., with Monte-Carlo simulations in classical
systems such as the hard-rod gas, and therefore they can be used to test the
predictions of the Euler-scale formula in Eqs. (43) and (47) with simulations
of the microscopic dynamics. Higher cumulants can be in principle derived as
well, even if the derivation becomes combinatorially more cumbersome as the
order increases.

The first cumulant is trivial from Eq. (43) and it is just the time integral
of the GHD expression of the current in Eq. (15)

$\frac{\partial G(s,x,t)}{\partial
s}\Big{|}_{s=0}=c_{1}=\frac{1}{t}\,\int_{0}^{t}\mbox{d}\tau\int\frac{\mbox{d}\lambda}{2\pi}(E^{\prime})^{\mathrm{dr}}(x,\tau\lambda)\vartheta(\lambda,x,\tau)h_{i_{\ast}}(\lambda).$
(62)

For higher-order cumulants we need the useful relation, with
$X^{\mathrm{dr}}(x^{\prime},t^{\prime},\lambda;s)$ a generic dressed quantity
(we drop for brevity all the independent variables),

$\partial_{s}X^{\mathrm{dr}}=\frac{\partial\varepsilon}{\partial
s}\,f\,X^{\mathrm{dr}}-\left(\frac{\partial\varepsilon}{\partial
s}\,f\,X^{\mathrm{dr}}\right)^{\mathrm{dr}},$ (63)

which directly follows upon differentiating with respect to $s$ Eq. (6).
$\partial\varepsilon/\partial s$ is specified by the flow equation (46) and
(47). The second cumulant is related to the Gaussianity of the distribution
close to the mean value and, from Eqs. (43) and (63), its expression reads

$c_{2}=-\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\int\frac{\mbox{d}\lambda}{2\pi}\,v^{\mathrm{eff}}(x,\tau,\lambda)\rho_{p}(x,\tau,\lambda)\frac{\partial\varepsilon}{\partial
s}\Big{|}_{s=0}f(x,\tau,\lambda)h_{i_{\ast}}^{\mathrm{dr}}(x,\tau,\lambda)h^{\mathrm{dr}}_{i_{\ast}}(x,\tau,\lambda),$
(64)

which, using (46) turns out to be

$\displaystyle c_{2}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\int_{0}^{t}\mbox{d}\tau^{\prime}\,\langle
j_{i_{\ast}}(x,\tau)j_{i_{\ast}}(x,\tau^{\prime})\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}},$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\int_{0}^{t}\mbox{d}\tau^{\prime}\\!\int\\!\mbox{d}\lambda\,[\Gamma_{(x,\tau^{\prime})\rightarrow(x,\tau)}V^{j_{i_{\ast}}}(x,\tau^{\prime})](\lambda)\,\rho_{p}(x,\tau,\lambda)f(x,\tau,\lambda)V^{j_{i_{\ast}}}(x,\tau,\lambda),$
(65)

that is in agreement with the expression for the connected current-current
two-point correlation function one obtains from Eq. (29), therefore providing
an important consistency check of Eqs. (43) and (46). The expression of the
third cumulant $c_{3}$ is related to the asymmetry or skewness of the
distribution. Its expression was not known before and it is given by

$\displaystyle c_{3}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\int_{0}^{t}\mbox{d}\tau^{\prime}\int_{0}^{t}\mbox{d}\tau^{\prime}\,\langle
j_{i_{\ast}}(x,\tau)j_{i_{\ast}}(x,\tau^{\prime})j_{i_{\ast}}(x,\tau^{\prime\prime})\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}}$
$\displaystyle=-\frac{1}{t}\int_{0}^{t}\mbox{d}\tau\int\mbox{d}\lambda\,v^{\mathrm{eff}}(x,\tau,\lambda)\rho_{p}(x,\tau,\lambda)f(x,\tau,\lambda)\times$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\,\,\,\times\left[2\,\frac{\partial\varepsilon}{\partial
s}\Big{|}_{s=0}\left(\frac{\partial\varepsilon}{\partial
s}\Big{|}_{s=0}fh^{\mathrm{dr}}_{i_{\ast}}\right)^{\mathrm{dr}}-h^{\mathrm{dr}}_{i_{\ast}}\left(\frac{\partial^{2}\varepsilon}{\partial
s^{2}}\Big{|}_{s=0}+\left(\frac{\partial\varepsilon}{\partial
s}\right)^{2}\Big{|}_{s=0}\tilde{f}\right)\right],$ (66)

with

$\tilde{f}=\frac{\mbox{d}f}{\mbox{d}\varepsilon}\frac{1}{f}+f.$ (67)

We notice that both Eq. (64) for $c_{2}$ and Eq. (66) for $c_{3}$ are written
in terms of $\partial\varepsilon/\partial s$, which is given by the flow
equation (46) and (47). Once the latter is numerically solved, also the
cumulants can be therefore evaluated numerically. We note that in the cases
where the initial state $\vartheta_{0}$ is invariant under the rescaling
$(x,t)\rightarrow(ax,at)$, with $a>0$ an arbitrary constant, and therefore
Eqs. (48)-(51) apply, the cumulants $c_{k}(\xi)$ in Eq. (33) of the time
integrated current are scaling function of $\xi$, i.e., the connected
correlation functions $\langle[\Delta
q_{i_{\ast}}(x,t)]^{k}\rangle_{\vartheta_{0}}^{c,\mathrm{Eul}}$ become scaling
functions of $\xi$ once they are rescaled by $t$. We have numerically checked,
by simulating the hard-rod gas dynamics from an initial step inhomogeneity in
the inverse temperature $\beta^{2}(x,0)$ as in Eqs. (11) and (12) and in Fig.
1, that the first three cumulants $c_{k}(\xi)$, with $k\leq 3$, of the
particle flow ($h_{0}(\lambda)=1$) are functions of $\xi$. In this case, we
have observed a linear growth as a function of $t$ of the connected
correlation functions, $\langle[\Delta q_{0}(\xi
t,t)]^{k}\rangle^{c}_{\mathrm{inh},z}$ with $\xi$ fixed and $k\leq 3$, which
implies that the cumulants are finite (the same scaling behavior is expected
also for higher cumulants with $k>3$) and the large deviation principle
applies, as commented after Eq. (34). In the homogeneous and stationary limit
of the results in Eqs. (43), (44) and (47), the same scaling behavior as a
function of $t$ for the cumulants was already observed in Ref. [71]. The
fineteness of the cumulants in integrable models implies that there is no
divergence in the derivatives of $G(s,x,t)$ w.r.t. $s$ and therefore no
dynamical phase transition in the time-integrated current $J_{i_{\ast}}$
statistics, see, e.g., Refs. [101, 102, 103]. The understanding of the precise
reason behind the absence of divergences in the $s$-derivatives of $G(s,x,t)$
in integrable systems is still under investigation and it will not be
addressed in this manuscript.

###  4.4 The non-interacting limit

In this Subsection we show how to specify the general result in Eqs. (43) and
(44) to non-interacting systems. In particular, we show that the flow equation
for the Lagrange multipliers simply reduces to a shift as a function of $s$,
as already noted in the homogeneous case in Refs. [71, 72]. This property is
characteristic of non-interacting systems and can be ascribed to the fact that
the Euler equations in Eq. (13) are in this case linear.

The fundamental simplification happening in non-interacting systems is that no
dressing is present since the scattering phase $\Theta(\lambda,\mu)$, or
equivalently $T(\lambda,\mu)$ in Eq. (3), vanishes identically. The flow
equation in Eq. (44) therefore simplifies drastically. Specifically, the
propagator $\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}(\lambda,\mu)$
in Eq. (28) for free models is uniquely given by the homogeneous contribution
(see also Appendix A)

$\Gamma_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}(\lambda,\mu)=\frac{1}{|v_{g}(\lambda)|}\delta\left(\frac{x^{\prime}-x}{v_{g}(\lambda)}-(t^{\prime}-\tau)\right)\delta(\lambda-\mu),$
(68)

where
$v^{\mathrm{eff}}(x,t,\lambda)=v_{g}(\lambda)=\mbox{d}E(\lambda)/\mbox{d}\lambda$
is the group velocity since there is no dressing. We observe that for the
calculation of $G(s,x,t)$ in Eq. (43), where the mean current $\langle
j_{i_{\ast}}(0,0)\rangle_{\vartheta(x,\tau;s^{\prime})}$ is evaluated on fluid
cells at the fixed space point $x$ and times $\tau\in(0,t)$ (see Fig. 1), one
has to specialize the propagator to equal space points $x^{\prime}=x$, i.e.,
$\Gamma_{(x,\tau)\rightarrow(x,t^{\prime})}(\lambda,\mu)$. Note that in
interacting systems the indirect propagator
$\Delta_{(x,\tau)\rightarrow(x^{\prime},t^{\prime})}(\lambda,\mu)$ vanishes as
a consequence of the fact that the quasi-particles trajectories do not depend
on the state (see also Appendix A). Inserting Eq. (68) into Eq. (44), with
$t^{\prime}-(x^{\prime}-x)/v_{g}(\lambda)<t$, one has

$\displaystyle\frac{\partial\beta^{n}(x^{\prime},t^{\prime};s)}{\partial s}$
$\displaystyle=-\int_{0}^{t}\mbox{d}\tau\int\mbox{d}\lambda\,v_{g}(\lambda)h_{i_{\ast}}(\lambda)h^{n}(\lambda)\,\delta\left(\frac{x^{\prime}-x}{v_{g}(\lambda)}-(t^{\prime}-\tau)\right)\frac{1}{|v_{g}(\lambda)|}$
$\displaystyle=-\int\mbox{d}\lambda\,\mbox{sgn}(v_{g}(\lambda))h_{i_{\ast}}(\lambda)h^{n}(\lambda)=-\mbox{sgn}(A)_{i_{\ast}}^{n},$
(69)

with the last step following from (36) without the dressing. For the same
reason $A_{i}^{j}$ is independent of $s$ and Eq. (69) can be trivially
integrated

$\beta^{n}(x^{\prime},t^{\prime};s)=\beta^{n}(x^{\prime},t^{\prime})-s\,\mbox{sgn}(A)_{i_{\ast}}^{n}.$
(70)

In order to further simplify Eq. (69) we briefly recall some basic identities
about the Lagrange mulipliers $\\{\beta^{n}\\}$. As already commented after
Eq. (57), the GGE state at every fluid cell $(x^{\prime},t^{\prime})$ can be
equivalently described in terms of the Lagrange parameters
$\\{\beta^{n}(x^{\prime},t^{\prime})\\}_{x^{\prime},n}$ or with the densities
$\langle q_{n}(0,0)\rangle_{\vartheta(x^{\prime},t^{\prime})}$. As a matter of
fact, it simple to see from Eq. (13), using the definition of the matrix $C$
in Eq. (25), that the multipliers satisfy an hydrodynamic equation similar to
the one the densities fulfill, as shown in details in Refs. [9, 10],

$\partial_{t}\beta^{n}(x^{\prime},t^{\prime})+\sum_{j}\partial_{x^{\prime}}\beta^{j}(x^{\prime},t^{\prime})A_{j}^{n}(x^{\prime},t^{\prime})=0.$
(71)

Since $A_{i}^{j}$ in the absence of dressing is independent of space and time,
Eq. (71) is linear. Accordingly, one can introduce the normal mode function
$\mathcal{N}(x^{\prime},t^{\prime},\lambda)$ via a simple linear combination
of the $\\{\beta^{n}(x^{\prime},t^{\prime})\\}_{x^{\prime},n}$, i.e.,

$\mathcal{N}(x^{\prime},t^{\prime},\lambda)=\sum_{n}\beta^{n}(x^{\prime},t^{\prime})\,h_{n}(\lambda),$
(72)

where $h_{n}$ are the bare single-particle eigenvalues, see Eq. (4). Plugging
Eq. (72) into Eq. (71) and exploiting the eigenvalue equation for $A_{i}^{j}$
in Eq. (38) one has

$\partial_{t^{\prime}}\mathcal{N}(x^{\prime},t^{\prime},\lambda)+v_{g}(\lambda)\,\partial_{x^{\prime}}\mathcal{N}(x^{\prime},t^{\prime},\lambda)=0.$
(73)

Note that Eq. (73) generalizes to the interacting case upon replacing
$v_{g}(k)\rightarrow v^{\mathrm{eff}}(x^{\prime},t^{\prime},\lambda)$. As a
matter of fact, one can notice that Eq. (73) has the very same structure as
Eq. (17). Indeed, the function $\vartheta(x^{\prime},t^{\prime},\lambda)$ is
the normal mode function associated to the conserved densities $\langle
q_{n}(0,0)\rangle_{\vartheta(x^{\prime},t^{\prime},\lambda)}$, as already
commented after Eq. (17), in the same way as $\mathcal{N}$ is the mode
function of the Lagrange parameters $\\{\beta^{n}\\}$. The fundamental point,
specific of free systems, is that $\mathcal{N}$ is linearly related to the
$\\{\beta^{n}\\}$, according to Eq. (72). In the case of interactions, as a
consequence of the nonlinearity of Eq. (17), the simple relation in Eq. (73)
does not hold, see Refs. [9, 10, 76] for a complete discussion. Inserting Eq.
(72) into Eq. (70) and remembering Eq. (38) one has

$\mathcal{N}(x^{\prime},t^{\prime},\lambda;s)=\mathcal{N}(x^{\prime},t^{\prime},\lambda)-s\,\mbox{sgn}(v_{g}(\lambda))\,h_{i_{\ast}}(\lambda).$
(74)

Let us now specialize the discussion to the partitioning protocol initial
inhomogeneous state in Eqs. (11) and (12) with an initial thermal
inhomogeneity in $\beta^{2}(x^{\prime},0)$ and all the other Lagrange
parameters initially set to zero (see, for example, Fig. 1). For non-
interacting models this protocol has been addressed, e.g., in Refs. [104, 105,
106, 107, 108, 109, 62, 65, 63, 73]. In this case, the solution of Eq. (73) is
readily found to be

$\mathcal{N}(\xi^{\prime},\lambda)=\mathcal{N}_{l}(\lambda)\Theta(v_{g}(\lambda)-\xi^{\prime})+\mathcal{N}_{r}(\lambda)\Theta(\xi^{\prime}-v_{g}(\lambda)),$
(75)

with $\xi^{\prime}=x^{\prime}/t^{\prime}$, while $\mathcal{N}_{l,r}(\lambda)$
are fixed by the initial conditions for the left and right half of the system.
Then, from Eq. (72) computed at time $t^{\prime}=0$, one has

$\mathcal{N}(x^{\prime},0,\lambda)=\beta^{2}(x^{\prime},0)\,h_{2}(\lambda)=\beta_{r}\,h_{2}(\lambda)\Theta(x^{\prime})+\beta_{l}\,h_{2}(\lambda)\,\Theta(-x^{\prime}),$
(76)

where we identified

$\mathcal{N}_{r}(\lambda)=\beta_{r}\,h_{2}(\lambda),\qquad\mathcal{N}_{l}(\lambda)=\beta_{l}\,h_{2}(\lambda).$
(77)

Note that $i_{\ast}=2$ in Eq. (74) and $h_{2}(\lambda)=E(\lambda)$ since we
are now considering energy transport. Inserting Eqs. (75) and (77) into Eq.
(74) and diving both sides by $h_{2}(\lambda)$ one eventually has

$\beta(\xi^{\prime},\lambda;s)=\beta(\xi^{\prime},\lambda)-s\,\mbox{sgn}(v_{g}(\lambda)),$
(78)

where we defined
$\beta(\xi^{\prime},\lambda)=\mathcal{N}(\xi^{\prime},\lambda)/h_{2}(\lambda)$.
We observe that for the partitioning protocol initial state
$\beta(\xi^{\prime},\lambda)$ depends on $x^{\prime}$ and $t^{\prime}$ through
the scaling variable $\xi^{\prime}=x^{\prime}/t^{\prime}$, and therefore the
Lagrange multipliers are invariant upon simultaneous rescaling of space and
time $\beta^{n}(ax^{\prime},at^{\prime})=\beta^{n}(x^{\prime},t^{\prime})$ for
every $n$, $x^{\prime}$ and $t^{\prime}$. Accordingly, Eq. (49) is satisfied,
as one can explicitly see from Eq. (78), and $G(s,x,t)=G(s,\xi)$, with
$\xi=x/t$ according to the discussion before Eq. (51). Upon using Eq. (78)
into Eq. (51) one obtains

$G(s,\xi)=\int_{0}^{s}\mbox{d}s^{\prime}\int_{0}^{1}\mbox{d}t^{\prime}\int\frac{\mbox{d}\lambda}{2\pi}\,E(\lambda)v_{g}(\lambda)\vartheta(\xi/t^{\prime},\lambda;s^{\prime}),$
(79)

with

$\vartheta(\xi/t^{\prime},\lambda;s)=\vartheta_{\beta_{r}(s)}(\lambda)\Theta(\xi/t^{\prime}-v_{g}(\lambda))+\vartheta_{\beta_{l}(s)}(\lambda)\Theta(v_{g}(\lambda)-\xi/t^{\prime}),$
(80)

for the filling function $\vartheta$ similarly to Eq. (75), and
$\beta_{r,l}(s)=\beta_{r,l}-s\,\mbox{sgn}(v_{g}(\lambda))$. For non-
interacting systems, accordingly, the biasing of the measure necessary to
compute the SCGF can be simply obtained by performing a linear shift of the
inverse temperatures $\beta_{l,r}$ characterizing the partitioning protocol
initial condition in Eq. (76). The latter statement in the case of the
homogeneous stationary state ensuing from the partitioning protocol initial
state, obtained upon setting $\xi=0$ in Eqs. (79) and (80), was already known
as “extended fluctuation relation”, first proposed in Ref. [93]. The extended
fluctuation relation has been shown to be valid in conformal field theory [53,
54] and in free-particle models [93, 62, 63]. In Ref. [72], in particular, the
validity of the extended fluctuation relation has been shown to be a
characteristic feature of non-interacting systems, where the statistical
properties of the time-integrated current are directly determined by the
fluctuations of the initial state, induced by the shift of the Lagrange
parameters $\beta_{l,r}$, since the quasi-particle trajectories do not depend
on the surrounding state where the quasi-particles propagate. Within this
perspective, the result in Eqs. (79) and (80) constitute a generalization as a
function of $\xi$ to inhomogeneous and dynamical GGEs in Eq. (9) of the
extended fluctuation relation, in agreement with the results of Ref. [73]
(cf., Eqs. (84) and (85) therein). In Ref. [73], indeed, Eqs. (79) and (80)
have been first derived using stationary phase methods for the partitioning
protocol initial state with a thermal inhomogeneity in $\beta^{2}(x,0)$, as in
Eq. (76). The derivation of Eq. (74) presented in this Subsection is, instead,
based on Euler-scale hydrodynamics and applies more generally to any initial
state $\rho_{0}$ having the form in Eq. (9). Only in Eqs. (75)-(80) the result
has been specialized to the partitioning protocol state with a step profile of
$\beta^{2}(x,0)$. More importantly, the Euler-scale hydrodynamics allows for
the exact treatment of interactions, as we have seen in Secs. 4.1 and 4.2. In
the case of interacting systems, however, quasi-particles trajectories, due to
the nonlinearity of the GHD equation in Eq. (17), are modified by the
dynamical inhomogeneous background within which they move. Accordingly,
fluctuations do not depend directly on the initial-state fluctuations, but
also on the whole motion of the quasi-particles between time $0$ and $t$
through the inhomogeneous fluid background, as expressed by the time integral
in Eqs. (46) and (47).

##  5 Conclusion

In this manuscript we have addressed the problem of computing the Euler-
scaling limit of the scaled cumulant generating function $G(s,x,t)$, defined
in Eq. (32) and depicted in Fig. 1, of the time-integrated currents associated
to ballistically transported conserved densities in integrable models. The
SCGF is of paramount importance since it gives access via the large-deviation
principle in Eq. (31) to the full probability distribution $p(J_{i_{\ast}})$
of the rare-atypical fluctuation of the rescaled time-integrated current
$J_{i_{\ast}}=\Delta q_{i_{\ast}}(x,t)/t$ far from its mean value $\langle
J_{i_{\ast}}\rangle$. Moreover, its series expansion provides all the
cumulants of the time-integrated current according to Eq. (33). We have
extended the analysis of Refs. [71, 72], which we have reviewed in Subsec.
3.2, for the calculation of the SCGF in homogeneous and stationary GGEs in Eq.
(2) to the more complex and interesting case of inhomogeneous and dynamical
GGE states of the form in Eq. (9), where we have defined the Euler-scaling
limit of $G(s,x,t)$ in Eq. (32).

Our findings are detailed in Sec. 4. Specifically, in Subsec. 4.1 we have
stated our main results given by Eqs. (43), (46) and (47), whose proof is
provided in Subsec. 4.2. Equation (43) expresses the SCGF as an integral of
the mean value of the current, given by the generalized hydrodynamics formula
in Eq. (15). The crucial point is that the mean value is computed over an
$s$-tilted inhomogeneous measure, see Eqs. (52) and (53), according the
exponential of the time-integrated current. All quantities which depend on the
inhomogeneous fluid state acquire, as a consequence, an additional dependence
on the parameter $s$, coupled to the time integral of the current. This
$s$-dependence can be determined by solving the flow equation, cf., Eqs. (46)
and (47), which describes exactly how the initial measure is affected by the
tilting procedure in terms of a flow, parametrized by $s$, in the manifold of
inhomogeneous GGEs. The flow equation is based on the knowledge of the Euler-
scale two-point correlation functions, whose expressions have been first
derived in Ref. [30] and later numerically tested in Ref. [37] for the hard-
rod gas. In Subsec. 4.3 we have provided in Eqs. (62), (64)-(66) the
expressions of the first three cumulants of the time-integrated current. We
have further numerically checked for the hard-rod fluid, in an initial
inhomogeneous state with an inverse temperature $\beta^{2}(x,0)$ profile as in
Eqs. (11) and (12) and in Fig. 1, that the first three cumulants $c_{k}$ of
the particle current, with $k\leq 3$, are finite (higher cumulants are
similarly expected to be finite). The numerical check of the finiteness of the
cumulants is methodologically fundamental since it confirms, a posteriori,
that the time integrals in Eqs. (62), (64)-(66) for the cumulants are
convergent, that the series expansion in Eqs. (32) and (33) is well defined
and that therefore the large deviation principle in Eq. (31) applies. In
Subsec. 4.4 we have analyzed the non-interacting limit of the general formulas
in Eqs. (43) and (44). In this case, due to the linearity of the hydrodynamic
equations, the flow equation drastically simplifies and, in the case of the
partitioning protocol initial state in Eq. (11), it reduces to a shift of the
Lagrange parameters characterizing the initial state, see Eq. (78). We have
further checked the the results obtained in this way reproduce those obtained
in Ref. [73] via stationary-phase methods for non-interacting systems. In the
latter case, the fluctuations of the time-integrated current are directly
determined by fluctuations of the initial state since quasi-particles
propagate along straight line characteristics with a velocity which is
independent on the state. Due to the inhomogeneity and the interactions, on
the contrary, quasi-particles move along curved trajectories whose shape
depends on the surrounding dynamical fluid state. Consequently, the
fluctuations of the time-integrated current depend on the whole trajectories
of the quasi-particles, as expressed by Eqs. (46) and (47).

As a future perspective, we plan to numerically evaluate the general
expressions in Eqs. (43), (46) and (47) for specific classical and quantum
interacting integrable models, such as the Lieb-Liniger, the hard rods, and
the sinh Gordon. The comparison between the expressions in Eqs. (62),
(64)-(66) and the numerical simulations of the hard-rod model will be also
carried out. From the analytical side it would be very interesting to extend
our results in Sec. 4 beyond the Euler-scaling limit of infinite variation
lengths of the inhomogeneities. This can be achieved by accounting for
diffusive corrections to the hydrodynamic equation in Eq. (17), as done in
Refs. [86, 87, 88, 89, 43]. In the specific case of classical models, such as
the hard-rod gas where diffusion has been investigated in detail in Refs. [11,
14, 75], this should somehow connect to the macroscopic fluctuation theory,
see Refs. [110, 111, 112], which is also based on diffusive hydrodynamics. We
surely plan to carry out this study in the future.

## Acknowledgements

G.P. thanks A. Gambassi for careful reading of the manuscript and for
collaboration on a related project and M. Kormos for useful comments. The
authors thank F. S. Møller for fruitful discussions and collaboration on a
related project.

#### Funding information

G.P. thanks the A. Della Riccia Foundation (Florence, Italy)–INFN– for
financial support and King’s College (London, United Kingdom) for hospitality
in the period when this work has been carried on.

##  Appendix A Indirect propagator for two-point correlation functions

In this Appendix we report for completeness the necessary formulas for the
evaluation of the indirect propagator $\Delta_{(y,0)\rightarrow(x,t)}$ defined
in Subsec. 2.3, which enters in the calculation of $G(s,x,t)$ in Eqs. (43) and
(47).

The indirect propagator, as shown in Ref. [30], satisfies the integral
equation

$\displaystyle\left[\Delta_{(y,0)\rightarrow(x,t)}V^{\mathcal{O}^{\prime}}\right](x,t,\lambda)=$
$\displaystyle\>2\pi\mathcal{D}(\mathcal{U}(x,t,\lambda,0),\lambda)\>\bigg{(}\left[W_{(y,0)\rightarrow(x,t)}V^{\mathcal{O}^{\prime}}\right](\lambda)+$
$\displaystyle+\int_{x_{0}}^{x}\mbox{d}z\left(\rho_{s}(z,t)f(z,t)\left[\Delta_{(y,0)\rightarrow(z,t)}V^{\mathcal{O}^{\prime}}(z,t)\right]\right)^{*\mathrm{d}\mathrm{r}}(\lambda)\bigg{)},$
(81)

with $V^{\mathcal{O}^{\prime}}$ defined in Eq. (24) and the characteristic
curve in Eqs. (18) and (19). The function $\mathcal{D}$ is usually named
“effective acceleration”. The latter has been first defined in Ref. [26] to
account for weakly inhomogeneous terms in the Hamiltonian ruling the time
evolution of the system, e.g., due to the presence of a confining potential.
In the case of Eq. (81), in a complementary way, it expresses the
inhomogeneity of the initial state in Eq. (9), while the dynamics is assumed
to be dictated by an homogeneous and time-independent Hamiltonian (as
commented at the beginning of Subsec. 2.3). The expression of the effective
acceleration is given by [26]

$\mathcal{D}(x,\lambda)=\frac{\partial_{x}\vartheta(x,0,\lambda)}{2\pi\rho_{p}(x,0,\lambda)f(x,0,\lambda)}\;.$
(82)

The function $\mathrm{W}_{(y,0)\rightarrow(x,t)}$ is dubbed source term and it
is given by

$\displaystyle\left[\mathrm{W}_{(y,0)\rightarrow(x,t)}V^{\mathcal{O}^{\prime}}\right]$
$\displaystyle(\lambda)=-\Theta(\mathcal{U}(x,t,\lambda,0)-y)\left(\rho_{s}(y,0)f(y,0)V^{\mathcal{O}^{\prime}}\right)^{*\mathrm{dr}}(y,0,\lambda)\;$
$\displaystyle+\int_{x_{0}}^{x}\mbox{d}z\\!\\!\sum_{\gamma\in\lambda_{\star}(z,t,y,0)}\\!\\!\\!\\!\\!\frac{\rho_{s}(z,t,\gamma)\vartheta(y,0,\gamma)f(y,0,\gamma)}{\left|\partial_{\lambda}\mathcal{U}(z,t,\gamma,0)\right|}T^{\mathrm{dr}}(z,t,\lambda,\gamma)V^{\mathcal{O}^{\prime}}(\gamma),$

where the star-dressing operation is defined as
$h^{*\mathrm{dr}}(\lambda)=h^{\mathrm{dr}}(\lambda)-h(\lambda)$, $\Theta(x)$
is the Heaviside step function, the set $\lambda_{\star}(x,t,y,0)$ has been
defined after Eq. (29), and $x_{0}$, defined in Eq. (19), has to be fixed such
that $\vartheta(x,t^{\prime},\lambda)=\vartheta(x,0,\lambda)$ for $x<x_{0}$
and $t^{\prime}\in[0,t]$, see Refs. [29, 30]. In principle one should think
that $x_{0}=-\infty$, although for numerical calculations, see Ref. [37], one
fixes $x_{0}$ to a value sufficiently far from the point $y$ and any other
inhomogeneity. Note that the indirect propagator
$\Delta_{(y,0)\rightarrow(x,t)}$ vanishes identically in the two limiting
cases of an homogeneous initial state, i.e.,
$\vartheta(x,0,\lambda)=\vartheta(\lambda)$ and therefore $\mathcal{D}=0$
identically in Eq. (82), and for non-interacting systems, as already commented
in Subsec. 4.4 in Eq. (68). In the case of non-interacting models the
vanishing of $\Delta_{(y,0)\rightarrow(x,t)}$ in Eq. (81) follows from the
fact that
$h^{*\mathrm{dr}}(\lambda)=h^{\mathrm{dr}}(\lambda)-h(\lambda)=h(\lambda)-h(\lambda)=0$
for any arbitrary function $h(\lambda)$ since $T=0$ and there is no dressing.
In the two cases of homogeneous initial states and non-interacting systems
correlations are then solely given by the direct term in Eqs. (28) and (29).

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