#  Navier-Stokes equations for low-temperature one-dimensional fluids

Andrew Urichuk  Laboratoire de Physique Théorique et Modélisation, CNRS UMR
8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France  Stefano
Scopa  Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY
Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France  SISSA and INFN,
via Bonomea 265, 34136 Trieste, Italy  Jacopo De Nardis  Laboratoire de
Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université,
95302 Cergy-Pontoise Cedex, France

###### Abstract

We consider one-dimensional interacting quantum fluids, such as the Lieb-
Liniger gas. By computing the low-temperature limit of its (generalised)
hydrodynamics we show how in this limit the gas is well described by a
conventional viscous (Navier–Stokes) hydrodynamic for density, fluid velocity
and the local temperature, and the other generalised temperatures in the case
of integrable gases. The dynamic viscosity is proportional to temperature and
can be expressed in a universal form only in terms of the emergent Luttinger
Liquid parameter $K$ and its compressibility. We show that the heating factor
is finite even in the zero temperature limit, which implies that viscous
contribution remains relevant also at zero temperatures. Moreover, we find
that in the semi-classical limit of small couplings, kinematic viscosity
diverges, reconciling with previous observations of Kardar-Parisi-Zhang
fluctuations in mean-field quantum fluids.

††preprint: APS/123-QED

Introduction.—  Quantum many-body interacting systems pose an immense
technical challenge to modern-day physics due to their exponential complexity.
A successful approach in the past years has been in borrowing concepts and
ideas from classical hydrodynamic theory and applying them to quantum systems
Peng  _et al._ (2023); Abanov  _et al._ (2020); Glorioso and Liu (2018); Lucas
and Fong (2018); Burchards  _et al._ (2022); Sirker  _et al._ (2009). The main
idea behind these methods is the same in classical and quantum physics: the
exponentially large information determining the state of the system is reduced
to few thermodynamic functions, the hydrodynamic fields, that well
characterize the local equilibrium state. For quantum gases in one spacial
dimension, the theory of generalised hydrodynamics (GHD) Castro-Alvaredo  _et
al._ (2016); Bertini  _et al._ (2016) (see also e.g. Doyon (2020); Alba  _et
al._ (2021); Essler (2022) for reviews) has shown to be able to perfectly
capture the dynamics of integrable e.g. Doyon  _et al._ (2017, 2018); Caux
_et al._ (2019); Bertini  _et al._ (2019); Doyon and Yoshimura (2017);
Bastianello  _et al._ (2018); Bertini  _et al._ (2018); Alba  _et al._ (2019)
and near-integrable quantum gases Bastianello  _et al._ (2020a, 2021); Møller
_et al._ (2022); Cataldini  _et al._ (2022); Møller  _et al._ (2021); Malvania
_et al._ (2021), as well as spin chains e.g. Urichuk  _et al._ (2019); Piroli
_et al._ (2017); Bulchandani  _et al._ (2017, 2018); Collura  _et al._ (2018);
Dupont and Moore (2020); Gruber and Eisler (2019); Scopa  _et al._ (2022a),
Fermionic systems Mestyán  _et al._ (2019); Scopa  _et al._ (2021a, 2022b);
Nozawa and Tsunetsugu (2020, 2021); Bertini  _et al._ (2022) and classical
field theories Vecchio  _et al._ (2023); Koch  _et al._ (2022); Bonnemain  _et
al._ (2022); Doyon and Spohn (2017). At small values of temperatures and for
gapless or gapped systems, GHD recovers historically well established results
on the dynamics of low-temperature systems, in particular the celebrated
Luttinger liquid theory Haldane (1981a, b); Giamarchi (2003); Cazalilla
(2004); Imambekov and Glazman (2009a, b); Imambekov  _et al._ (2012) and the
semi-classical approaches Sachdev and Damle (1997); Damle and Sachdev (1998).
In particular, by re-quantising the fluctuations on top of a classical
background, evolving with GHD at zero temperature, one recovers a Luttinger
Liquid theory on top of an evolving hydrodynamic fluid, connecting this way
GHD with the most relevant field theory description of one dimensional quantum
systems Ruggiero  _et al._ (2020); Collura  _et al._ (2020); Scopa  _et al._
(2021b, 2022a); Ruggiero  _et al._ (2021); Scopa  _et al._ (2023).

At zero temperature (or entropy) GHD is effectively a set of equations
describing the dynamics of the Fermi points of the fermionised degrees of
freedom Doyon  _et al._ (2017). In particular, it was shown that, at the level
of Euler hydrodynamics, therefore neglecting any viscosity, the GHD evolution
of a single Fermi sea (with two Fermi points) is exactly equivalent to the
conventional hydrodynamics (CHD) evolution of density $\rho$ and momentum (or
fluid velocity $\eta$) as the two relevant hydrodynamic modes Doyon  _et al._
(2017); Schemmer  _et al._ (2019). However, while the latter creates
hydrodynamic shocks and fails to be meaningful after the time $t^{*}$ of the
creation of the first shock Bettelheim  _et al._ (2006a); Bettelheim and
Glazman (2012); Bettelheim  _et al._ (2006b); Simmons  _et al._ (2020a), Euler
GHD is free of shocks as at time $t^{*}$ two new modes, i.e. two new Fermi
points are created, resolving this way the shock into a simple contact
singularity Doyon  _et al._ (2017). However, while the latter is clearly valid
for strictly integrable systems, it is unclear what is instead the correct
hydrodynamics for quasi or even non-integrable gases at very low temperatures,
that can describe different experimental settings.

In this letter, we show that as Euler CHD is plagued by hydrodynamic shocks,
one, on the other hand, cannot neglect viscosity terms that are finite as soon
as interparticle interactions are non-zero. Viscous, or diffusive terms, have
been indeed incorporated into GHD for a few years De Nardis  _et al._ (2018);
Nardis  _et al._ (2019); Durnin  _et al._ (2021); Medenjak  _et al._ (2020);
Nardis and Doyon (2023), and they have been shown to be essential for the
thermalisation of quasi-integrable systems Bastianello  _et al._ (2020b) and
for well-describing spin dynamics in integrable spin chains De Nardis  _et
al._ (2019, 2020a, 2021, 2020b); Nardis  _et al._ (2022a); Bulchandani  _et
al._ (2021). However, since integrable systems are typically ballistic, they
usually account for small perturbative effects on top of the ballistic
current. Here we shall show that the picture change drastically at very low
temperature: there indeed they enter the CHD as a dynamic viscosity
$\mu(\rho,\beta)$ they fully regularize its shocks, making viscous CHD a
perfectly valid hydrodynamics for low-temperature gapless systems. By taking a
low-temperature limit of GHD we here determine a simple and universal
expression for the dynamic viscosity which only depends on the density $\rho$
and on the Luttinger liquid parameter $K(\rho)$ for a given interaction
strength. Therefore, we claim that our result is universal for any one-
dimensional interacting system at low temperatures.

GHD and CHD.—  We start by deriving CHD by taking the low-temperature limit of
GHD, and we consider the Lieb-Liniger model Lieb and Liniger (1963) as a
reference, although our derivation is fully generic. The Lieb-Liniger model
for $N$ contact-interacting bosonic particles in an external potential $V(x)$
is given by the Hamiltonian

$\hat{H}=\frac{1}{2}\sum_{i=1}^{N}\partial_{x_{i}}^{2}+2c\sum_{i<j=1}^{N}\delta(x_{i}-x_{j})+\sum_{i=1}^{N}V(x_{i})$
(1)

and it represents a paradigmatic model for one-dimensional interacting systems
and cold atomic gases, see e.g. Ref. Cazalilla  _et al._ (2011); Imambekov and
Glazman (2008); Ristivojevic (2023). In the repulsive regime $c>0$, its
eigenstates are labelled by (fermionic) quasiparticles with bare energy
$\varepsilon(\theta)=\theta^{2}/2$ and momentum $k(\theta)=\theta$, with the
parameter $\theta\in[-\infty,\infty]$. In the thermodynamic limit, the state
is specified by a filling function $n(\theta)$ fixed by the temperature and
chemical potential, where $\theta$ are the rapidities or the quasi-momenta of
the particles. Within the framework of GHD, one assumes that at each position
$x,t$ there exists a fluid cell where the gas is locally thermodynamic, and
where we can introduce a local filling function $n(\theta;x,t)$. The time
evolution of the latter then reads, given also the external force
$\mathfrak{f}=-\partial_{x}V$, as

$\partial_{t}n(\theta;x,t)+v^{\rm
eff}(\theta;x,t)\partial_{x}n(\theta;x,t)+\mathfrak{f}\partial_{\theta}n(\theta;x,t)=0$
(2)

where $v^{\rm eff}(\theta;x,t)=(\partial_{\theta}\varepsilon)^{\rm
dr}/(\partial_{\theta}k)^{\rm dr}$ is the dressed velocity of the
quasiparticles on top of the background fixed by the filling $n(\theta;x,t)$.
The latter can be found by knowing the explicit dressing operation, which
reads as $f^{\rm dr}=(1-\varphi n)^{-1}\cdot f$ with the scattering matrix
$\varphi(\theta)=c/(\pi(c^{2}+\theta^{2}))$ acting as a convolution operator
$\varphi\cdot f=\int\varphi(\theta-\alpha)f(\alpha)d\alpha$.

At low temperatures, the filling function becomes very close to a sharp Fermi
sea, such that $\delta
n(\theta;x,t)\to-\sum_{\sigma}\sigma\delta(\theta-\theta(x,t)^{\sigma})$ with
$\theta^{\sigma}$ the two Fermi edges, indexed by $\sigma=\pm 1$, such that
eq. (2) becomes an equation only for the two Fermi edges
$\theta^{\sigma}(x,t)$, namely at the only points where the delta functions
are not zero. It is useful to always choose a co-moving reference frame,
namely we describe the fluid as being at each position $x$ in a Fermi sea with
Fermi edges $-q,q$ and Fermi velocity $v_{F}=\pi\rho/K$, with $K$ the
Luttinger liquid parameter Giamarchi (2003), obtained by a fluid velocity
boost (i.e. employing the Galilean invariance of the fluid)
$\eta=(\theta^{+}+\theta^{-})/2$, i.e. $q=\theta^{+}-\eta$,
$-q=\theta^{-}-\eta$. In this notation, we obtain,

$\sum_{\sigma}\delta(\theta-\theta^{\sigma})[\partial_{t}\theta^{\sigma}+(\eta+\sigma
v_{F}(\rho))\partial_{x}\theta^{\sigma}-\mathfrak{f}]=0,$ (3)

with the Fermi edge $q(x,t)$ fixed by the local density $\rho(x,t)$. The
latter can be shown, see Supplementary Material (SM) SM , to be fully
equivalent to CHD for density and fluid velocity readings as

$\displaystyle\partial_{t}\rho$ $\displaystyle+$
$\displaystyle\partial_{x}(\eta\rho)=0$ $\displaystyle\partial_{t}\eta$
$\displaystyle+$
$\displaystyle\eta\partial_{x}\eta+\rho^{-1}{\partial_{x}\mathcal{P}_{s}^{(0)}(\rho)}{}=\mathfrak{f},$
(4)

with $\mathcal{P}_{s}^{(0)}(\rho)$ is the static pressure of the gas at zero
temperature, fixed by the equation of state of the system. In the specific
example of the Lieb-Liniger gas, the static pressure at zero temperature is
given by the integral of the dressed energy
$\mathcal{P}^{(0)}_{s}(\rho)=-\int_{-q}^{q}d\theta\varepsilon^{\rm
dr}(\theta)/2\pi$ and where $q$ is obtained by inverting the relation
$\rho=\int_{-q}^{q}1^{\rm dr}(\theta)d\theta/2\pi$. Finally, we should also
mention that we have used the momentum field $p$ and we have introduced the
fluid velocity $\eta=p/\rho$.

The equation for the fluid velocity $\eta$ in eq. ( Navier-Stokes equations
for low-temperature one-dimensional fluids) takes the form of the celebrated
Burgers equation, which is known to display hydrodynamic shocks Simmons  _et
al._ (2020b). In Euler GHD such shocks are resolved by adding to eqs. (3) two
extra Fermi edges, i.e. by introducing $2n$ hydrodynamic fields
$\theta^{\sigma}$ with $\sigma=1,\ldots,2n$ that describe $n$ split Fermi seas
Doyon  _et al._ (2017). In the phase-space $(\theta,x)$ this translates into
the existence of a continuous one-dimensional contour $\Gamma(s)$ that
separates the region where $n(\theta,x)=1$ from the region where
$n(\theta,x)=0$.

Clearly, this picture is based on the underlying integrability of the model,
i.e. in the fact that all the rapidities $\theta$ are conserved quantities.
For non-integrable generic systems, it is clear that this phenomenon cannot be
the one that regularises the shocks. As shocks in the Burgers equation are
regularised by a finite viscosity, we shall then see in the coming section how
to indeed introduce viscosity in CHD.

Viscous CHD.—  We now wish to include diffusive or viscosity effects. In order
to do so, we shall consider local thermodynamic states with finite but small
temperatures, as there are no viscosity effects at strictly zero temperatures.
We consider therefore local canonical equilibrium states with given density
$\rho$ (or chemical potential $\mu$), fluid velocity $\eta$ and temperature
$T=1/\beta$. In the LL model, this corresponds to filling functions of the
form $n(\theta;x,t)=(1+e^{\beta\varepsilon(\theta;x,t)})^{-1}$ given in terms
of its pseudo-energy

$\varepsilon=\epsilon_{0}(\theta-\eta(x,t))-(q(x,t))^{2}-\frac{\varphi\cdot\log(1+e^{-\beta\varepsilon})}{\beta(x,t)},$
(5)

with bare energy $\epsilon_{0}(\theta)=\theta^{2}/2$. We then move to
introduce diffusion terms into CHD by taking the high $\beta$ limit of the
diffusive GHD, which is known from refs. De Nardis  _et al._ (2018); Nardis
_et al._ (2019) and it is given by adding to the r.h.s. of eq. (2) the
diffusive part

$R\cdot\partial_{x}(R^{-1}\cdot\tilde{D}\cdot\partial_{x}n(\alpha)),$ (6)

with the kernels $R_{\theta,\theta^{\prime}}$ and
$\tilde{D}_{\theta,\theta^{\prime}}$ presented in SM . As diffusive GHD
contains $\partial_{x}^{2}$ terms, a description only in terms of the Fermi
edges becomes impossible, as we cannot simply use that the derivative of the
filling is a delta function at the edges, as this would produce
$\delta^{\prime}(\theta-\theta^{\sigma})=\partial\delta(\theta-\theta^{\sigma})$
whose meaning outside integration is unclear.

The only way to clarify their role is to plug them within the hydrodynamic
equation for the only two modes which are relevant here, i.e. density and
momentum (fluid velocity). This already clarifies the essential difference of
viscous CHD from Euler GHD: while the latter can be simply obtained as an
equation for the evolution of the Fermi edges (3) it can be trivially extended
to any number of them during time evolution. Viscous CHD instead requires
fixing the number of Fermi edges from the start, and, as we shall see, does
not require producing new Fermi edges, as hydrodynamic shocks are regularised
by viscosity. By means of a long but straightforward calculation deferred to
the SM SM , we then arrive to

$\displaystyle\partial_{t}\rho$ $\displaystyle+$
$\displaystyle\partial_{x}(\eta\rho)=0,$ $\displaystyle\partial_{t}\eta$
$\displaystyle+$
$\displaystyle\eta\partial_{x}\eta+\frac{\partial_{x}\mathcal{P}_{s}(\rho,T)}{\rho}=\frac{\partial_{x}(\mu(\rho,T)\partial_{x}\eta)}{\rho}+\mathfrak{f},$
(7)

with the dynamic viscosity at the leading order is linear in temperature, as
already remarked previously Maki and Zhang (2023), and it reads in terms of
the slope in the density dependence of the emergent Luttinger liquid parameter
$K$ as

$\mu(\rho,T)=\rho\frac{T}{4\pi}K(\partial_{\rho}\log K)^{2}+O(T^{3}),$ (8)

which can also be rewritten in terms of the derivative of the Fermi velocity
as $\mu(\rho,T)=T{\left(1-K\partial_{\rho}v_{F}/\pi\right)^{2}}/{4v_{F}}$.
This result can be understood similarly as in the kinetic picture of diffusion
in integrable models proposed in Gopalakrishnan  _et al._ (2018): the
excitations close to the Fermi edges move with Fermi velocity
$v_{F}(\rho)=\pi\rho/K(\rho)$ but as there are local density fluctuations
$\rho\to\rho+\delta\rho$ within the fluid, the Luttinger parameter (i.e. the
Fermi velocity) also fluctuates $K\to K+\partial_{\rho}K\delta\rho$, giving a
diffusive spreading to the trajectories. The dependence of the Luttinger
parameter on density (and therefore on the position in an inhomogeneous fluid)
is what makes inhomogeneous Luttinger liquids for interacting systems non-
conformal invariant Bastianello  _et al._ (2020c) and it is a direct effect of
non-trivial interactions. In free Fermi gases, the dependence of the Luttinger
parameter on the density trivialises, $K\to 1$, and viscosity vanishes, as
expected for non-interacting particles. Indeed, in the limit of large
coupling, the density profile of LL model is equivalent to that of a free
fermionic model, the so-called Tonks gas. The first corrections in $1/c$ of
the viscosity reads as,

$\mu(\rho,T)=\frac{4\rho}{\pi}\frac{T}{c^{2}}+O(c^{-3}),$ (9)

which manifests the actual dependence with respect to the reduced temperature
$t=T/c^{2}$ at small $t$. The pressure is given by taking the first correction
to static pressure in $T^{2}$,
$\mathcal{P}^{(0)}_{s}(\rho)\to\mathcal{P}_{s}(\rho,T)$ with

$\mathcal{P}_{s}(\rho,T)=\mathcal{P}^{(0)}_{s}(\rho)+T^{2}\rho\tilde{\chi}_{e}/2,$
(10)

with $\tilde{\chi}_{e}={K}/({3\rho^{2}})$. Eq. ( Navier-Stokes equations for
low-temperature one-dimensional fluids) gives the evolution of density and
momentum of the fluid, which is one-to-one with the chemical potential $q$ and
the boost $\eta$. Given that the system is at a finite temperature, and this
also represents a hydrodynamic variable, an extra equation is needed: the one
for the evolution of the energy density $e$. We define energy density at rest
as $e=E/\rho-\eta^{2}/2$, where $E$ is the total energy of the fluid, which in
the LL model is computed via $E=\int d\theta n(\theta)(\theta^{2}/2+V(x))^{\rm
dr}$. By proceeding in an analogous manner as to derive the eqs. ( Navier-
Stokes equations for low-temperature one-dimensional fluids), we obtain

$\partial_{t}e+\eta\partial_{x}e+\frac{\mathcal{P}_{s}(\rho,T)}{\rho}\partial_{x}\eta=\frac{\mu(\rho,T)}{\rho}(\partial_{x}\eta)^{2}.$
(11)

where, as expected, the kinematic viscosity $\nu=\mu/\rho$ now enters the
equation. We can convert this equation into an equation for the evolution of
the temperature field $T(x,t)=1/\beta(x,t)$ using the definition of the energy
susceptibility at fixed density $\delta e/\delta
T\Big{|}_{\rho}=\tilde{\chi}_{e}T$. Given the expression at zero temperature
for generic interacting systems, we then obtain

$\displaystyle\partial_{t}T$ $\displaystyle+$
$\displaystyle\eta\partial_{x}T=-\mathcal{P}^{T}_{s}T\partial_{x}\eta+\frac{\mu(\rho,T)}{T\rho\tilde{\chi}_{e}}(\partial_{x}\eta)^{2}$
(12)

with
$\mathcal{P}^{T}_{s}={\pi\left(1/\rho+{\partial_{\rho}v_{F}}/v_{F}\right)/(3v_{F}\tilde{\chi}_{e})}$.
Notice that one may also expect a term accounting for thermal conduction
$\partial_{x}(k(\rho,T)\partial_{x}T)$ on the rhs of (12), but given that
thermal conductivity $k(\rho,T)\sim T^{2}$ we neglect this term as of the
order of temperature to 2nd power. Eqs. ( Navier-Stokes equations for low-
temperature one-dimensional fluids) and (12) take exactly the same form as the
standard Navier-Stokes equation for a fluid, i.e. continuity equation,
conservation of momentum and conservation of energy, and it is quite
remarkable that we can derive it in an exact, non-perturbative way.

Figure 1: Evolution of the excess density $\delta\rho=\rho-\rho_{\infty}$ for
an initial bump obtained as the ground state density of (1) with interaction
$c=1$ and Gaussian potential $V(x)=-\mu-A\exp(-x^{2}/\sigma^{2})$, released at
$t>0$; here $\sigma=1$, $A$ and $\mu$ are set such that the background density
$\rho_{\infty}=1$ and maximum $\rho_{0}=1.2$. We compare the result by zero-
entropy GHD (solid thick lines) and diffusive GHD at different values of
inverse temperatures $\beta$ (see legend). The result of viscous CHD is shown
for large enough values of $\beta$ as it converges to a well-defined limit
(dashed line). Times are $t=0,3,6,12.5,18.5,25,32.5$ and increase from
leftmost to rightmost peak in the figure. Inset: evolution of the inverse
temperature and generalised temperatures starting from a thermal case at low
temperature in the integrable gas, here for simplicity the Lieb-Liniger gas at
large coupling (Tonks limit). The temperatures are obtained by fitting the
quasiparticle occupations $n(\theta;x,t)$ at a given position $x_{0}$ during
the front dynamic expansion. We clearly see the moment when the Fermi sea
split as the moment of temperatures inversion.

We first notice that the heating factor in eq. (12) is $\sim\mu/T$ namely it
is order zero in temperature. Therefore, even if the system is initially at
zero temperature, the rapid growth of the velocity gradient $\partial_{x}\eta$
heats the system, giving therefore finite viscosity $\mu$ to the dynamics of
$\eta$ and regularising its shocks. Moreover, given that the dynamic viscosity
in eq. (8) is expressed only in terms of the universal features of the low-
temperature effective field theory, namely the Luttinger liquid constant $K$
and its density-dependence (which is non-trivial only for interacting quantum
gases), we conjecture that its form is universal for generic one-dimensional
quantum fluids. The argument is simple: at low temperatures, all quantum
fluids become effectively integrable, as their description is in terms of
Luttinger liquid freely propagating bosonic modes. Their diffusion is
therefore expected to be the same as for integrable quasiparticles, where the
kinetic picture explained above applies. We also stress that there is no
fundamental difference in the nature of transport at small temperatures
between integrable and non-integrable systems, contrary to what is claimed in
previous literature Matveev and Pustilnik (2017).

Front dynamics in integrable gases.—  We now focus on the fate of an initial
density bump, for example in a system at a given temperature $T_{0}$ and
density $\rho_{0}$. Such a setting is paradigmatic to understanding the
response of a system to external perturbations and we here use it to establish
the main differences between viscous CHD and integrable diffusive GHD at low
temperature, see Fig. 1. When the system is fully integrable, for example in
the absence of forces $\mathfrak{f}$, one could expect that not only
temperature and chemical potential can characterise a local stationary state,
but a large number of Lagrange multipliers associated to higher charges, i.e.
a Generalised Gibbs Ensembles (GGE), deviating this way from the behaviour of
a generic non-integrable system. Namely, even if the gas is initially in a
thermal state, higher chemical potentials could be activated as time grows. In
particular, a first deviation from thermal is by considering the pseudo-energy
of eq. (5) and substituting

$\epsilon_{0}\to\frac{\theta^{2}}{2}+\gamma_{3}\frac{\theta}{3!}+\gamma_{4}\frac{\theta^{4}}{4!}$
(13)

with $\gamma_{n}$ the chemical potentials associated to the conserved charge
$\hat{Q}_{n}$, whose expectation value reads as $Q_{n}=\int d\theta
n(\theta)(\theta^{n})^{\rm dr}/n!$. Such a state is a valid stationary state
in the LL gas, due to its integrability. By extending the calculations done
for the case of temperature (12) we find, see SM , that the evolution of such
temperatures is expressed by quite lengthy partial differential equations. The
main relevant fact is that while $\partial_{t}T\sim O(T)$, we instead find
$\partial_{t}\gamma_{n}\sim O(T^{0})$, namely even by starting with a thermal
gas at low temperature, the integrable gas tends to generate finite
generalised temperatures with time evolution. Such temperature dynamics lead
to the splitting of the Fermi seas (occurring at a certain position $x_{0}$
when $\gamma_{4}(x_{0},t)\neq 0$ and $\beta(x_{0},t)\to-\infty$) and to a much
different shock regularisation dynamics compared to the CHD one, even if both
are regular hydrodynamics. Indeed, viscous CHD converges to a given profile,
with a finite shock width, as the temperature is lowered, and it never
develops a shock, even if naively the viscosity $\mu$ tends to zero. The
reason is the evolution of temperature (12), where, given that the heating
factor $\mu/T$ is finite in the zero temperature limit, even if temperature
initially is zero, the creation of a shock heats the system and therefore
increases the viscosity, self-regulating this way the shock. In Fig. 1, zero-
temperature GHD agrees with diffusive GHD at finite temperature, signalling
how the zero-temperature approximation is often able to capture out-of-
equilibrium fluids. On the other hand, observed deviations from viscous CHD
are attributed to the absence of higher conservation laws, as discussed above.

The small coupling limit and KPZ physics.—  As already discussed in the first
Lieb-Liniger paper Lieb and Liniger (1963), the limit of small coupling of the
LL gas does not simply recover free bosons. Indeed, when $c\to 0^{+}$, its
ground state becomes the one describing the so-called (semi-classical)
condensate solution of the Nonlinear Schrödinger equation (NLS) Ishikawa and
Takayama (1980). This is characterised by vanishing Fermi momentum
$q\sim\sqrt{c}$ but with diverging dressed functions $1^{\rm dr}\sim
1/\sqrt{c}$ in order to keep the density $\rho$ finite in the limit. The
relevant excitations become then the Bogoliubov excitations with spectrum
given by $\varepsilon_{k}\sim|k|$ and therefore with a degenerate group
velocity $v_{k}\sim{\rm sgn}(k)$. As $\partial_{\rho}v_{F}(\rho)\sim\lim_{q\to
0}\partial_{q}v_{q}$ the latter diverge, giving a divergent dynamic viscosity
at low coupling as

$\mu(\rho,T)\sim\frac{T}{\sqrt{c}},$ (14)

signalling the breakdown of the viscous CHD and the emergence of Kardar-
Parisi-Zhang physics Kardar  _et al._ (1986). The latter is well-known to
emerge in the stochastic Burgers equation, i.e. given a white
$\delta-$correlated noise $w$, a local perturbation of a hydrodynamic field
$\phi$ satisfying
$\partial_{t}\phi+\partial_{x}\left(v\phi+\kappa\phi^{2}+w\right)=0,$ moves
with finite velocity $v$ and spreads super-diffusively, as opposed to the
diffusive case of eq. ( Navier-Stokes equations for low-temperature one-
dimensional fluids) whenever $\mu$ is finite. Such a phenomenon is known to
appear in generic finite-components one-dimensional fluids, as described by
the non-linear fluctuating hydrodynamic (NLFH) theory Spohn (2014), which can
be successfully applied also to lattice (i.e. non-integrable) NLS Mendl and
Spohn (2015). Again, the argument is simple: whenever the Euler currents
contain non-linearities, one should expect that the introduction of a small
noise (which for example can describe the interaction with other non-
hydrodynamic modes) always leads to the KPZ universal fixed point. However,
this is not the case at finite coupling as, while it is true that eq. (
Navier-Stokes equations for low-temperature one-dimensional fluids) contains
the non-linearity $\eta^{2}$, there exists a continuous, thermally activated,
spectrum of modes around the Fermi points with velocities
$v_{F}\pm\delta(\theta)$ (with $\delta(\theta)$ a tiny function of $\theta$)
that are responsible for a finite diffusion constant in the system. It is only
in the small coupling limit that the velocities of all such excitations become
degenerate, therefore diminishing the number of effective hydrodynamic modes
to only the three macroscopic ones. In this limit, therefore, the theory of
NLFH applies and KPZ physics emerges, as signalled by the divergent dynamic
viscosity. We should stress that the existence of KPZ physics in the NLS at
low temperature was first established in Kulkarni and Lamacraft (2013) and we
here give a first analytical proof of its divergent diffusion constant.
Moreover, it is interesting to notice that the degeneracy of the hydrodynamic
modes leads to super-diffusive KPZ physics similarly also in the Heisenberg
spin chain (and any other integrable model with non-abelian symmetry)
Ljubotina  _et al._ (2019); Bulchandani  _et al._ (2021) at finite
temperatures. There the relevant degenerate excitations are not the ones
around the Fermi points but the so-called giant magnons De Nardis  _et al._
(2020c); Nardis  _et al._ (2022b), namely magnonic excitations with large spin
and vanishing velocities.

Conclusion.—  We have here shown that the standard Navier-Stokes equation for
the evolution of density, momentum and temperature can be derived naturally
from the low-temperature expansion of the GHD for the LL gas. We have found
universal expressions for the linear part of in temperature of the dynamic
viscosity, which we conjecture to apply to generic one-dimensional fluids. We
have shown that the viscosity is zero in the free fermions limit, as expected,
and that it diverges in the semi-classical limit of weakly interacting bosons
at small temperatures, which despite many numerical works, it was never
established from first principles. The divergent viscosity signals the
emergence of KPZ super-diffusive spreading Kulkarni and Lamacraft (2013), in
analogy to the one observed in integrable spin chains Wei  _et al._ (2022).

We have shown that the viscous terms regularise hydrodynamic instabilities in
one-dimensional gases, although the inclusion of generalised temperatures is
necessary in order to predict the full form of the shock front in the
integrable gas. Moreover, one should also expect that when the system is
strongly out of equilibrium and gradients of $\eta$ are strong, the system
strongly heats locally, invalidating, therefore, the zero temperature
approximation. Our findings therefore suggest that zero-entropy hydrodynamics
becomes harder to physically realise whenever interactions are present.

Differently from previous attempts to derive viscosity in quantum fluids by
perturbative corrections to Luttinger liquids, see for example Sirker  _et
al._ (2009); Matveev and Pustilnik (2017); Maki and Zhang (2023); DeGottardi
and Matveev (2020), we here derive non-perturbatively using the generalised
hydrodynamic of a specific model, the LL gas, and we extend our result to
generic systems, given the universality of its formulation. Clearly, a
different derivation only involving Luttinger liquid modes would also be
desirable in the future. Our result is ready to be checked by means of
numerical simulations Eberharter  _et al._ (2023); Vanderstraeten  _et al._
(2018); Van Damme  _et al._ (2021) and to apply to different quantum fluids as
such as chiral edge modes Wen (1990, 1992); Nardin and Carusotto (2023), to
open the way for a full fluctuating hydrodynamic theory Delacretaz and Mishra
(2023) of Luttinger liquid field theories, and to unveil Burgers-like
turbulent phases Gallone  _et al._ (2022) in low-dimensional quantum fluids.

Acknowledgements.—  We are thankful to B. Doyon, K. Kheruntsyan, I. Bouchoule,
and J. Dubail for insightful discussions. We are in debt to M. Panfil for
sharing his notes on dressed kernel identities. We thank G. Del Vecchio Del
Vecchio for the related collaboration. This work has been partially funded by
the ERC Starting Grant 101042293 (HEPIQ) (J.D.N., S.S and A.U.) and by the ERC
Consolidator Grant 771536 (NEMO) (S.S.). S.S. is thankful to LPTM (Cergy-
Paris) for the kind hospitality during the development of this work.

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Supplementary Material
Navier-Stokes equations for low-temperature one-dimensional fluids

###### Contents

  1. A TBA for Lieb Liniger
    1. A.1 Thermal energy and hydrodynamic fields
    2. A.2 Dressing relation
  2. B Zero temperature Euler GHD and conventional hydrodynamics (CHD)
    1. B.1 Euler CHD equations
    2. B.2 Equivalence of Euler GHD with CHD at $T=0$
  3. C Low-temperature viscous momentum equation from GHD
    1. C.1 Evaluation of derivative of the momentum diffusion kernel
  4. D Equation for the energy density
  5. E Dynamics of the temperature and inclusion of other conserved quantities
    1. E.1 Derivation of the evolution of charges, eq. (SE.69)
    2. E.2 Explicit form of (SE.61)
    3. E.3 Small temperature corrections to non-symmetric Fermi weights
    4. E.4 General viscosity expression eq. (SE.58)
  6. F Final result: low-temperature viscous CHD
  7. G Proofs and checks of various Identities
    1. G.1 Numerical check of viscosity
    2. G.2 Hydrodynamic fields at $T=0$ and their expressions
    3. G.3 Proof of the low-temperature expansion of eq. (SC.39)
    4. G.4 Specific heat at low temperatures
    5. G.5 Hydrodynamic field corrections at $O(T^{2})$
    6. G.6 Dressed functions identities at low temperature
    7. G.7 Diffusion at large coupling
    8. G.8 Commutativity of low-temperature limit and spatial derivative

## Outline

This supplementary material is arranged as follows:

In App. A the Lieb-Liniger model is used to introduce our local states by use
of the thermodynamic Bethe ansatz (TBA). This section includes a review of the
dressing relation, which appears throughout the various calculations. As a
warm-up $T=0$ generalized hydrodynamics (GHD) is considered in App. B, where
relations between the hydrodynamic fields and TBA quantities are used to
reproduce the known equivalence between $T=0$ conventional hydrodynamics (CHD)
and GHD. The diffusive GHD equations are introduced, which are then
specialized to the cases of momentum and energy. At low temperatures the
diffusive part is determined first for momentum, App. C, followed by energy in
App. D. Next, the question of additional Lagrange parameters is considered in
App. E. This consideration requires the derivation of additional evolution
equations for the corresponding conserved charge, which leads to an evolution
equation for the additional Lagrange parameters. At low temperature initial
states these additional parameters are found to grow as $O(T)$, unlike
temperature, which was found earlier to have corrections $O(1)$. These low $T$
diffusive corrections are considered in detail for both cases, and result in a
closed set of low $T$ diffusion equations referred to as the viscous GHD
equations. These equations are collected in their final form in App. F, where
the large $c$ limit is noted. This is followed up with a numerical comparison
between the predicted low $T$ diffusion and the finite temperature diffusion
kernel corrections to the momentum in App. G.1, where additional details about
the method can be found. Small temperature corrections to the hydrodynamic
fields are evaluated in detail and reported in Apps. G.2-G.5. Various
identities and useful relations for the dressed functions at small
temperatures are collected in App. G.6. This includes the zero temperature
behaviour of several hydrodynamic fields, a proof of an identity used to
derive the low $T$ diffusion, general small temperature corrections to
hydrodynamic fields, and a suite of dressed function identities. An
alternative calculation for large $c$ is also considered in App. G.7, which
takes advantage of the thermal energy being explicitly known and is found to
match the large $c$ limit previously obtained. Additionally, the commutation
of the small temperature limit and the spatial derivative are demonstrated in
the large $c$ case.

##  Appendix A TBA for Lieb Liniger

###  A.1 Thermal energy and hydrodynamic fields

In this section the Galilean invariance of the Fermi weight is explored,
specifically the consequences to be considered. This identity will allow us to
move between a rapidity coordinate system with symmetric, and anti-symmetric
Fermi points. For convenience these two setups are referred to as being the
symmetric and anti-symmetric bases, respectively. For us the low-temperature
Fermi weights are given by a thermal energy $\epsilon$, which is determined
through the TBA relation in the presence of a chemical potential $\mu(q)>0$.
For the case of symmetric Fermi points, $\theta=\pm q$, the thermal energy is
given by

$\displaystyle\epsilon(\theta)$ $\displaystyle=$
$\displaystyle\frac{\theta^{2}}{2}-\mu(q)-\frac{1}{\beta}\int
d\alpha\,\varphi(\theta-\alpha)\ln|1+\text{e}^{-\beta\epsilon(\alpha)}|,$
$\displaystyle\epsilon(\pm q)$ $\displaystyle=$ $\displaystyle 0.$ (SA.1)

A more general scenario can be obtained by considering a shift in the
rapidity. Thus the symmetric thermal energy can be shifted into a non-
symmetric scenario by noting that
$\theta\in\\{q,-q\\}=\\{\theta^{+},\theta^{-}\\}-\eta$ with
$\eta=\frac{\theta^{+}+\theta^{-}}{2}$. Furthermore, Galilean invariance of
the system guarantees that any two non-symmetric Fermi points $\theta^{+}$,
$\theta^{-}$ are given by

$\displaystyle\epsilon(\theta)$ $\displaystyle=$
$\displaystyle\frac{(\theta-\eta)^{2}}{2}-\mu(q)-\frac{1}{\beta}\int
d\alpha\,\varphi(\theta-\alpha)\ln|1+\text{e}^{-\beta\epsilon}|,$
$\displaystyle\epsilon(\theta^{\pm})$ $\displaystyle=$ $\displaystyle 0.$
(SA.2)

So the symmetric thermal energy in a local chemical potential $\mu(q)$ is
determined from either of the above integral equations and either are used to
define the Fermi weight as

$\displaystyle
n(\theta)=\left[1+\text{e}^{\beta\epsilon(\theta)}\right]^{-1}.$ (SA.3)

Note that a Galilean shift of the rapidity, $\theta\to\theta-\eta$, leaves the
Fermi weight invariant. At $\beta\to\infty$ with $\mu(q)>0$ the Fermi weight
becomes a step function and satisfies the identity

$\displaystyle\partial_{\theta}n(\theta)=-\sum_{\sigma=\pm}\sigma\delta(\theta-\theta^{\sigma})+O\left(\beta^{-2}\right).$
(SA.4)

From the Fermi weight hydrodynamic fields are introduced as

$\displaystyle\langle Q_{j}\rangle=\int
d\theta\,n(\theta){\left(\frac{\theta^{j}}{j!}\right)}^{\rm dr},$ (SA.5)

where an overall factor $1/2\pi$ is reintroduced in the final step and the
dressed functions ${f}^{\rm dr}$ will be defined below.

###  A.2 Dressing relation

From the the definition of the thermal energy it is possible to identify the
derivative of the thermal energy with the dressed energy derivative. For Lieb-
Liniger with symmetric Fermi points this explicitly reads as

$\displaystyle\partial_{\theta}\epsilon=\theta+\int
d\alpha\,\varphi(\theta-\alpha)n(\alpha)\partial_{\theta}\epsilon(\alpha)=\theta+\int
d\alpha\,\varphi(\theta-\alpha)n(\alpha){\theta}^{\rm
dr}(\alpha)=:{\theta}^{\rm dr}(\theta).$ (SA.6)

More generally this type of integral relation is referred to as a dressing
relation. This relation is used in conjunction with a driving function
$f(\theta)$ to define a dressed function ${f}^{\rm dr}(\theta)$ as

$\displaystyle{f}^{\rm dr}(\theta)=f(\theta)+\int
d\alpha\,\varphi(\theta-\alpha)n(\alpha){f}^{\rm dr}(\alpha).$ (SA.7)

With the note that a left and right dressing is exactly equivalent, which
implies

$\displaystyle\int d\theta{f}^{\rm dr}(\theta)n(\theta)g(\theta)=\int d\theta
f(\theta)n(\theta){g}^{\rm dr}(\theta).$ (SA.8)

It is also possible to dress multivariate functions like the scattering kernel

$\displaystyle{\varphi}^{\rm dr}(\theta,\mu)$ $\displaystyle=$
$\displaystyle\varphi(\theta-\mu)+\int
d\alpha\,\varphi(\theta-\alpha)n(\alpha){\varphi}^{\rm dr}(\alpha,\mu),$
(SA.9)

with a note that once again a left or right dressing are exactly equivalent.

There are also useful composite variables such as the effective velocity of
the quasi-particles

$\displaystyle v^{\rm eff}(\theta)=\frac{{\theta}^{\rm dr}(\theta)}{{1}^{\rm
dr}(\theta)},$ (SA.10)

which are characterized by these dressed functions.

Note, however, that unlike the thermal energy these dressed quantities are not
required to be invariant under Galilean boost. Instead the behaviour of the
dressed function under a Galilean transformation depends on the form of the
driving function $f$. For example, given Fermi points $\theta^{\pm}$ that are
then boosted so $\theta\to\theta-\frac{\theta^{+}+\theta^{-}}{2}=\theta-\eta$,
which results in symmetric Fermi points $\pm
q=\pm\left(\frac{\theta^{+}-\theta^{-}}{2}\right)$, then the dressed functions
behave as

$\displaystyle{f}^{\rm dr}(\theta-\eta)={f}^{\rm
dr}_{s}(\theta)={(f(\theta-\eta))}^{\rm dr}.$ (SA.11)

Consider as an explicit example $f=\theta$ then

$\displaystyle{\theta}^{\rm dr}(\theta-\eta)$ $\displaystyle=$
$\displaystyle{(\theta-\eta)}^{\rm dr}={\theta}^{\rm dr}(\theta)-\eta{1}^{\rm
dr}(\theta).$

Furthermore, the Fermi velocity can be determined by evaluating the effective
velocity at the symmetric Fermi point $v^{\rm eff}(q)=v_{F}$. Note that with
the Galilean invariance the general effective velocity satisfies

$\displaystyle v^{\rm eff}(\theta^{\pm})=\frac{{\theta}^{\rm
dr}(\theta^{\pm})}{{1}^{\rm dr}(\theta^{\pm})}=\frac{{\theta}^{\rm dr}(\pm
q)}{{1}^{\rm dr}(\pm q)}+\eta=\pm v_{F}+\eta.$ (SA.13)

##  Appendix B Zero temperature Euler GHD and conventional hydrodynamics (CHD)

Generalized hydrodynamics (GHD) determines the time evolution of conserved
charges $q_{i}$, which at the Euler scale involves evolution equations of the
form

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\partial_{t}\langle
Q_{i}\rangle+\partial_{x}\langle J_{i}\rangle=\partial_{t}\int
d\theta(q_{i}n{1}^{\rm dr})+\partial_{x}\int d\theta(q_{i}v^{\rm eff}n{1}^{\rm
dr}).$ (SB.14)

With the subscript ’$i$’ denoting the index of the conserved charge, the first
few of which are density, $q_{0}=1$, momentum $q_{1}=\theta$, and energy
$q_{2}=\theta^{2}/2$. What distinguishes GHD from conventional hydrodynamics
(CHD) is the presence of an infinite number of conserved charges, our goal
here is to demonstrate that at $T=0$ the Euler scale GHD reduces to CHD. This
is a well known result, however it should be noted that this result implies
the presence of shocks at non-zero temperatures. A similar procedure for
truncating the GHD equations at finite but low-temperature will then be
necessary for introducing a diffusive correction to regularize these shocks.

###  B.1 Euler CHD equations

From GHD the CHD equations can be understood as demonstrated as emerging from
a truncation of independent evolution equations, which occurs in the zero
temperature limit of GHD. At the Euler scale the first three modes for GHD
equations read

$\displaystyle\partial_{t}\rho+\partial_{x}p$ $\displaystyle=$
$\displaystyle\partial_{t}\int d\theta(n{1}^{\rm dr})+\partial_{x}\int
d\theta\,v^{\rm eff}(n{1}^{\rm dr})=0,$
$\displaystyle\partial_{t}p+\partial_{x}\mathcal{P}$ $\displaystyle=$
$\displaystyle\partial_{t}\int d\theta\theta(n{1}^{\rm dr})+\partial_{x}\int
d\theta\,\theta v^{\rm eff}(n{1}^{\rm dr})=0,$
$\displaystyle\partial_{t}E+\partial_{x}J_{E}$ $\displaystyle=$
$\displaystyle\partial_{t}\int d\theta\theta^{2}(n{1}^{\rm
dr})+\partial_{x}\int d\theta\,\theta^{2}v^{\rm eff}(n{1}^{\rm dr})=0.$
(SB.15)

It should be noted that when the, generally, infinite number of evolution
equations in GHD truncate to a finite number the result is some form of CHD.

A more conventional form of CHD can be written by noting $p=\eta\rho$ with
$\eta=(\theta^{+}+\theta^{-})/2$ being the fluid velocity so that the momentum
equation can be recast as

$\displaystyle\partial_{t}p+\partial_{x}\mathcal{P}=\rho\partial_{t}\eta+\rho\eta\partial_{x}\eta+\partial_{x}\mathcal{P}_{s}=0,$
(SB.16)

where $\mathcal{P}_{s}=\mathcal{P}-\eta^{2}\rho$ is the stationary pressure.
The energy equation is likewise written instead as an evolution equation in
terms of a stationary energy density

$e=\frac{E}{\rho}-\frac{\eta^{2}}{2}.$ (SB.17)

By use of their definition from TBA, the hydrodynamic fields for energy and
energy current satisfy, where $i=t,x$

$\displaystyle\partial_{i}E$ $\displaystyle=$
$\displaystyle\eta\rho\partial_{i}\eta+\frac{\partial_{i}q}{v_{F}}(\mathcal{P}+E-\eta^{2}\rho),$
$\displaystyle\partial_{i}J_{E}$ $\displaystyle=$
$\displaystyle(\mathcal{P}+E)\partial_{i}\eta+\eta
v_{F}^{2}\partial_{i}\rho-\eta^{2}\rho\partial_{i}\eta+\eta\partial_{i}E,$
(SB.18)

with the Fermi velocity $v_{F}=v^{\rm eff}(q)$. So the energy equation may be
rewritten at low temperatures as

$\displaystyle\frac{1}{\rho}\left(\partial_{t}E+\partial_{x}J_{E}\right)$
$\displaystyle=$
$\displaystyle\partial_{t}e+\frac{\mathcal{P}_{s}}{\rho}\partial_{x}\eta+\eta\partial_{x}e+\frac{e}{\rho}\left(\partial_{t}\rho+\eta\partial_{x}\rho+\rho\partial_{x}\eta\right)+\eta(\partial_{t}\eta+\eta\partial_{x}\eta+v_{F}^{2}\partial_{x}\rho).$

These considerations then allow us to identify the first three GHD modes with
the CHD equations:

$\displaystyle\partial_{t}\rho+\partial_{x}(\eta\rho)$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\partial_{t}\eta+\eta\partial_{x}\eta+\frac{\partial_{x}\mathcal{P}_{s}}{\rho}$
$\displaystyle\overset{\rm{Euler}}{=}$ $\displaystyle 0,$
$\displaystyle\partial_{t}e+\frac{\mathcal{P}_{s}}{\rho}\partial_{x}\eta+\eta\partial_{x}e$
$\displaystyle\overset{\rm{Euler}}{=}$ $\displaystyle 0.$ (SB.20)

To prove the equivalence of GHD and CHD it must be demonstrated that GHD
truncates so that these relations, or a subset thereof, are sufficient for
determining the dynamics.

###  B.2 Equivalence of Euler GHD with CHD at $T=0$

At $T=0$ explicit identities for the hydrodynamic fields are determined from
TBA. Again considering the first three modes the identities for the momentum
and density equations are

$\displaystyle\partial_{i}\rho$ $\displaystyle=$ $\displaystyle 2({1}^{\rm
dr}(\theta^{+}))^{2}(\partial_{i}q),$ $\displaystyle\partial_{i}p$
$\displaystyle=$ $\displaystyle 2({1}^{\rm
dr}(\theta^{+}))^{2}\left(v_{F}\partial_{i}\eta+\eta(\partial_{i}q)\right),$
$\displaystyle\partial_{i}\mathcal{P}$ $\displaystyle=$ $\displaystyle
2({1}^{\rm dr}(q))^{2}\left[2(\partial_{i}\eta)\eta
v_{F}+\left(\eta^{2}+v_{F}^{2}\right)\partial_{i}q\right].$ (SB.21)

Both of these equations imply conditions on the Fermi points
$\theta^{+},\theta^{-}$ and are more conveniently characterized by a symmetric
form $\theta^{\pm}=\eta\pm q$ in terms of the symmetric Fermi points $q$ and
the fluid velocity $\eta$. Specifically, they imply the Euler scale zero
entropy GHD equations for density and momentum

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{t}q+v_{F}\partial_{x}\eta+\eta\partial_{x}q,$
$\displaystyle 0$ $\displaystyle\overset{\rm{Euler}}{=}$
$\displaystyle\partial_{t}\eta+\eta\partial_{x}\eta+v_{F}\partial_{x}q.$
(SB.22)

recalling that the Fermi velocity $v_{F}$, is the effective velocity evaluated
at the the Fermi point. As a consequence of this result the Euler scale
evolution of the symmetric Fermi point $q=\frac{\theta^{+}-\theta^{-}}{2}$ and
the fluid velocity $\eta=\frac{\theta^{+}+\theta^{-}}{2}$ dictates the
dynamics of the model. Thus, at $T=0$ GHD truncates to a finite number of
equations establishing the CHD/ GHD equivalence. The precise meaning of this
truncation is that the additional equations become redundant. Consider the
energy evolution equation at the Euler scale, which from the identities

$\displaystyle\partial_{i}E$ $\displaystyle=$
$\displaystyle(\partial_{i}\eta)2\eta{\theta}^{\rm dr}_{s}(q){1}^{\rm
dr}+(\partial_{i}q)\left[{(\theta^{2}_{s})}^{\rm dr}(q)+\eta^{2}{1}^{\rm
dr}(q)\right]{1}^{\rm dr},$ $\displaystyle\partial_{i}J_{E}$ $\displaystyle=$
$\displaystyle\partial_{i}\eta\left([{(\theta^{2}_{s})}^{\rm
dr}(q)+\eta^{2}{1}^{\rm dr}(q)]{\theta}^{\rm dr}(q)+2\eta^{2}{\theta}^{\rm
dr}_{s}(q){1}^{\rm dr}(q)\right)+\partial_{i}q\left(\eta{1}^{\rm
dr}(q)[{(\theta^{2}_{s})}^{\rm dr}(q)+\eta^{2}{1}^{\rm
dr}(q)]+2\eta({\theta}^{\rm dr}_{s}(q))^{2}\right).$

imply that the energy evolution equation at $T=0$ is equivalent to an
’additional’ evolution equation for $\eta,q$. However, after some massaging
find

$\displaystyle\partial_{t}E+\partial_{x}J_{E}$
$\displaystyle\overset{\rm{Euler}}{=}$ $\displaystyle 0,$ $\displaystyle
4\eta{\theta}^{\rm dr}{1}^{\rm
dr}\left[\partial_{t}\eta+\eta\partial_{x}\eta+v_{F}\partial_{x}q\right]+2\left[{(\theta^{2}_{s})}^{\rm
dr}+\eta^{2}{1}^{\rm dr}\right]{1}^{\rm
dr}\left[\partial_{t}q+v_{F}\partial_{x}\eta+\eta\partial_{x}q\right]$
$\displaystyle\overset{\rm{Euler}}{=}$ $\displaystyle 0.$ (SB.24)

Comparing this with the momentum and density equations at $T=0$, see eq.
(SB.22), makes clear that the first term is zero, due to the momentum
equation, and the second also zero, due to the density equation. Consequently,
at Euler scale at $T=0$ the energy evolution equation is merely a restatement
of the density and momentum evolution equations. Whenever the temperature is
not strictly zero, the zero entropy GHD equations exhibit shocks, which in the
literature have been resolved by considering the post shock state as having
multiple Fermi seas. Mathematically, these shocks are not surprising.

##  Appendix C Low-temperature viscous momentum equation from GHD

GHD is capable of describing more than just the Euler scale dynamics, having
been extended to account for diffusive and more recently dispersive dynamics.
Here, the low T diffusive behaviour of GHD is considered, understanding the
diffusion as a low-temperature correction to the conserved charges such as the
density and momentum. The goal being to determine low-temperature viscous GHD
equations, which are found at leading order to characterize a type of CHD with
appropriate viscosity $\mu(\rho,\beta)$. From GHD the evolution equation of a
given conserved charge, whose local density is given by $q_{i}$, in the
presence of diffusion is given by

$\displaystyle\partial_{t}(q_{i}n{1}^{\rm dr})+\partial_{x}(q_{i}v^{\rm
eff}n{1}^{\rm dr})=\frac{q_{i}(\theta)}{2}\partial_{x}\left(\int d\alpha
D(\theta,\alpha)\partial_{x}(n{1}^{\rm dr})(\alpha)\right).$ (SC.25)

With the general form of the diffusion kernel $D(\theta,\alpha)$ being known
from GHD as

$\displaystyle D(\theta,\alpha)$ $\displaystyle=$ $\displaystyle\int d\mu
d\nu\,\theta(1-n\varphi)^{-1}(\theta,\mu){1}^{\rm
dr}(\mu)\tilde{D}(\mu,\nu)\frac{1}{{1}^{\rm
dr}(\nu)}(1-n\varphi)(\nu,\alpha),$ $\displaystyle\tilde{D}(\theta,\mu)$
$\displaystyle=$ $\displaystyle-2\pi\delta(\theta-\mu)\int
d\alpha\left[n(\alpha)(1-n(\alpha))({1}^{\rm dr}(\alpha))|v^{\rm
eff}(\theta)-v^{\rm eff}(\alpha)|\left(\frac{{\varphi}^{\rm
dr}(\theta,\alpha)}{{1}^{\rm dr}(\theta)}\right)^{2}\right]$ (SC.26)
$\displaystyle+2\pi n(\alpha)(1-n(\alpha))\left(\frac{{\varphi}^{\rm
dr}(\theta,\mu)}{{1}^{\rm dr}(\theta)}\right)^{2}{1}^{\rm dr}(\theta)|v^{\rm
eff}(\theta-v^{\rm eff}(\mu)|.$

The factor $2\pi$ appearing due to writing everything in terms of ${1}^{\rm
dr}$. So rewritten in terms of the hydrodynamic fields the GHD equations for
the density and momentum are

$\displaystyle\partial_{t}\rho+\partial_{x}(\eta\rho)$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\partial_{t}\eta+\frac{1}{2}\partial_{x}(\eta^{2})+\frac{\partial_{x}\mathcal{P}_{s}(q,T)}{\rho}$
$\displaystyle=$ $\displaystyle\frac{1}{\rho}D(p).$ (SC.27)

With $D(p)=\partial_{x}[\theta(1-n\varphi)^{-1}{1}^{\rm
dr}\tilde{D}\frac{1}{{1}^{\rm dr}}(1-n\varphi)]$ being the diffusive
contribution to the momentum from GHD. The low-temperature diffusive
corrections to the conserved charges can now be determined from the full
diffusion expression from GHD by expanding the $\tilde{D}$ operator. After
this expansion, a more convenient form for the full diffusion kernel is found
by using the identity

$\displaystyle\int d\alpha\,(1-n\varphi)(\theta,\alpha)\partial_{x}(n{1}^{\rm
dr})(\alpha)={1}^{\rm dr}(\theta)\partial_{x}n(\theta).$ (SC.28)

By specializing to a thermal state, like those described in App. A, the
diffusion kernel can be written as

$\displaystyle D(\theta,\alpha)$ $\displaystyle=$ $\displaystyle\int d\mu
d\nu\,(1-n\varphi)^{-1}(\theta,\mu){1}^{\rm
dr}(\mu)\tilde{D}(\mu,\nu)\frac{1}{{1}^{\rm
dr}(\nu)}(1-n\varphi)(\nu,\alpha),$ $\displaystyle\tilde{D}(\theta,\mu)$
$\displaystyle=$ $\displaystyle-2\pi\delta(\theta-\mu)\int
d\alpha\left[\frac{\partial_{\alpha}n}{\beta\partial_{\alpha}\varepsilon(\alpha)}{1}^{\rm
dr}(\alpha)|v^{\rm eff}(\theta)-v^{\rm eff}(\alpha)|\left(\frac{{\varphi}^{\rm
dr}(\theta,\alpha)}{{1}^{\rm dr}(\theta)}\right)^{2}\right]$ (SC.29)
$\displaystyle+2\pi\frac{\partial_{\theta}n}{\beta\partial_{\theta}\varepsilon(\theta)}\left(\frac{{\varphi}^{\rm
dr}(\theta,\mu)}{{1}^{\rm dr}(\theta)}\right)^{2}{1}^{\rm dr}(\theta)|v^{\rm
eff}(\theta)-v^{\rm eff}(\mu)|.$

Along with the
$\partial_{x}n=(\partial_{\theta}n)\frac{\partial_{x}\epsilon}{\partial_{\theta}\epsilon}$,
resulting in

$\displaystyle\int
d\alpha\tilde{D}(\theta,\alpha)\partial_{x}n(\alpha)=\frac{2\pi}{\beta^{2}}\int
d\alpha\left(\frac{{\varphi}^{\rm dr}(\theta,\alpha)}{{1}^{\rm
dr}(\theta)}\right)^{2}|v^{\rm eff}(\theta)-v^{\rm eff}(\alpha)|\left[{1}^{\rm
dr}(\theta)\partial_{x}(\beta\epsilon(\alpha))-{1}^{\rm
dr}(\alpha)\partial_{x}(\beta\epsilon(\theta))\right]\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\alpha}n}{\partial_{\alpha}\epsilon}.$

The form of the square bracketed term is simplified by noting that the spatial
derivatives of the thermal energy satisfy

$\displaystyle\frac{\partial_{x}(\beta\epsilon(\mu))}{\partial_{\mu}\epsilon}v^{\rm
eff}(\mu)-\frac{\partial_{x}(\beta\epsilon(\theta))}{\partial_{\theta}\epsilon}v^{\rm
eff}(\theta)=\beta\left(v^{\rm eff}(\theta)-v^{\rm
eff}(\mu)\right)\partial_{x}\eta+\left(\frac{\epsilon(\mu)}{{1}^{\rm
dr}(\mu)}-\frac{\epsilon(\theta)}{{1}^{\rm
dr}(\theta)}\right)\partial_{x}\beta,$ (SC.31)

which follows by of the spatial derivative identity (SG.145). Along with the
additional simplification $\frac{\partial_{x}\epsilon(\theta)}{{1}^{\rm
dr}(\theta)}=\frac{\partial_{x}\epsilon(\theta)}{\partial_{\theta}\epsilon}v^{\rm
eff}(\theta)$. As a consequence the momentum diffusion kernel can be written
in a symmetric form as

$\displaystyle\int d\theta d\mu\,v^{\rm eff}(\theta)({1}^{\rm
dr}(\theta))^{2}\tilde{D}(\theta,\mu)\partial_{x}n(\mu)=\frac{2\pi}{2\beta}\int
d\theta
d\mu\,\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}({1}^{\rm
dr}(\theta))({\varphi}^{\rm dr}(\theta,\mu))^{2}({1}^{\rm
dr}(\mu))\frac{\left(v^{\rm eff}(\theta)-v^{\rm eff}(\mu)\right)^{4}}{|v^{\rm
eff}(\theta)-v^{\rm eff}(\mu)|}\partial_{x}\eta,$
$\displaystyle+\frac{2\pi}{2\beta^{2}}\int d\theta
d\mu\,\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}({1}^{\rm
dr}(\theta){1}^{\rm dr}(\mu))({\varphi}^{\rm dr}(\theta,\mu))^{2}\frac{(v^{\rm
eff}(\theta)-v^{\rm eff}(\mu))^{3}}{|v^{\rm eff}(\theta)-v^{\rm
eff}(\mu)|}\left(\frac{\epsilon(\mu)}{{1}^{\rm
dr}(\mu)}-\frac{\epsilon(\theta)}{{1}^{\rm
dr}(\theta)}\right)\partial_{x}\beta,$ $\displaystyle=:$
$\displaystyle\int\frac{d\theta
d\mu}{2\beta}\,\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}\left[H(\theta,\mu)\partial_{x}\eta+h(\theta,\mu)\left(\frac{\epsilon(\mu)}{{1}^{\rm
dr}(\mu)}-\frac{\epsilon(\theta)}{{1}^{\rm
dr}(\theta)}\right)\frac{\partial_{x}\beta}{\beta}\right].$ (SC.32)

With what we refer to as the symmetric and antisymmetric functions
$H(\theta,\mu)$ and $h(\theta,\mu)$, respectively, defined as

$\displaystyle H(\theta,\mu)=2\pi({1}^{\rm dr}(\theta))({\varphi}^{\rm
dr}(\theta,\mu))^{2}({1}^{\rm dr}(\mu))\frac{\left(v^{\rm eff}(\theta)-v^{\rm
eff}(\mu)\right)^{4}}{|v^{\rm eff}(\theta)-v^{\rm eff}(\mu)|},$ (SC.33)
$\displaystyle h(\theta,\mu)=2\pi({1}^{\rm dr}(\theta))({\varphi}^{\rm
dr}(\theta,\mu))^{2}({1}^{\rm dr}(\mu))\frac{\left(v^{\rm eff}(\theta)-v^{\rm
eff}(\mu)\right)^{3}}{|v^{\rm eff}(\theta)-v^{\rm eff}(\mu)|}.$ (SC.34)

Note that both of these functions are invariant under Galilean boosts, in
addition to the symmetric/ antisymmetric behaviour under interchange of the
arguments $H(\theta,\mu)=H(\mu,\theta)$ and $h(\theta,\mu)=-h(\mu,\theta)$.
Note that the $\partial_{x}\beta$ vanishes, due to similar considerations when
considering the energy diffusion in sec. D.

Combined, these considerations allow the momentum diffusion contribution to be
written as

$\displaystyle\frac{1}{2}\partial_{x}\left(\int d\theta v^{\rm
eff}(\theta)({1}^{\rm
dr}(\theta))^{2}\left[\tilde{D}\partial_{x}n\right](\theta)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{4}\int d\theta
d\mu\,\partial_{x}\left[\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}H(\theta,\mu)\frac{\partial_{x}\eta}{\beta}\right].$
(SC.35)

If the integral is evaluated in the $1/\beta\to 0$ limit then it follows that

$\displaystyle\frac{1}{2}\partial_{x}\left(\int d\theta v^{\rm
eff}(\theta)({1}^{\rm
dr}(\theta))^{2}\left[\tilde{D}\partial_{x}n\right](\theta)\right)$
$\displaystyle=$
$\displaystyle\frac{8\pi}{\beta}\partial_{x}\left[v_{F}({\varphi}^{\rm
dr}(q,-q))^{2}(\partial_{x}\eta)\right],$ (SC.36)

and the symmetric function at the Fermi points reduces to

$\displaystyle H(q,-q)=H(-q,q)=2\pi({\varphi}^{\rm dr}(q,-q))^{2}({1}^{\rm
dr}(q))^{2}8v_{F}^{3}.$ (SC.37)

Equivalently, since the derivative and zero temperature limits commute, see
App. G.8, the derivative of the integrand can be taken prior to integration.
This procedure is carried out explicitly in App. C.1 and demonstrated as
equivalent.

###  C.1 Evaluation of derivative of the momentum diffusion kernel

Here the spatial derivative of the integrand is considered in detail and
proven to coincide with the reported visosity $\mu(\rho,\beta)$. To evaluate
the spatial derivative of the integrand there are two key identities. The
first being a result of the low-temperature behaviour of the Fermi sea and the
second taking into account the melting on the edges of the filling factor

$\displaystyle\partial_{\theta}n$ $\displaystyle\to$
$\displaystyle-\sum_{\sigma=\pm}\sigma\delta(\theta-\theta^{\sigma}),$ (SC.38)
$\displaystyle\beta(2n-1)\partial_{\theta}n$ $\displaystyle\to$
$\displaystyle-\sum_{\sigma=\pm}\frac{\sigma}{(\partial_{\theta}\epsilon)(\theta^{\sigma})}\delta^{\prime}(\theta-\theta^{\sigma}),$
(SC.39)

the second line being proven in App. G.3. These relations then yield a
particularly useful combined expression

$\displaystyle\partial_{x}\left(\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\right)=-\sum_{\sigma=\pm}\frac{\sigma}{(\partial_{\theta}\epsilon)(\theta^{\sigma})}\delta^{\prime}(\theta-\theta^{\sigma})\frac{\partial_{x}\epsilon}{\partial_{\theta}\epsilon}.$
(SC.40)

Recall that $q=\frac{\theta^{+}-\theta^{-}}{2}$, such that
$\theta^{\pm}=\frac{\theta^{+}+\theta^{-}}{2}\pm q$. So acting on each term
with the spatial derivative yields

$\displaystyle\partial_{x}\left(\int d\theta v^{\rm eff}(\theta)({1}^{\rm
dr}(\theta))^{2}\left[\tilde{D}\partial_{x}n\right](\theta)\right)=\frac{1}{2}\int
d\theta\int
d\mu\sum_{\sigma,\sigma^{\prime}}\frac{\sigma^{\prime}\delta(\mu-\theta^{\sigma^{\prime}})}{\partial_{\mu}\epsilon}\frac{\sigma\delta(\theta-\theta^{\sigma})}{\partial_{\theta}\epsilon}\partial_{x}\Big{(}\frac{(\partial_{x}\eta)H(\theta,\mu)}{\beta}\Big{)},$
$\displaystyle-\int\frac{d\theta\int
d\mu}{2\beta}\sum_{\sigma,\sigma^{\prime}}\frac{\sigma^{\prime}\delta(\mu-\theta^{\sigma^{\prime}})}{\partial_{\mu}\epsilon(\theta^{\sigma^{\prime}})}\frac{\sigma\delta(\theta-\theta^{\sigma})}{\partial_{\theta}\epsilon(\theta^{\sigma})}\left[\partial_{\theta}\left(\frac{\partial_{x}\epsilon(\theta)}{\partial_{\theta}\epsilon}H(\theta,\mu)\right)+\partial_{\mu}\left(\frac{\partial_{x}\epsilon(\mu)}{\partial_{\mu}\epsilon}H(\theta,\mu)\right)\right](\partial_{x}\eta).$
(SC.41)

The second term picking up an overall ‘$-$’ due to an integration by parts
after inserting the appropriate identities. The first term will be considered
in more detail later, but for now a lot of effort is needed to determine a
more reasonable form of the second term.

By combining many of the dressed function identities and noting that the
symmetric function $H(\theta,\mu)$ is invariant under interchange of
$\theta,\mu$ it is possible to write identities for the derivatives of the
symmetric function as

$\displaystyle(\partial_{\theta}+\partial_{\mu})H(\theta,\mu)\bigg{|}_{\theta,\mu=q,-q}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\partial_{q}H(\theta,\mu)\bigg{|}_{\theta,\mu=q,-q}$
$\displaystyle=$ $\displaystyle H(q,-q)\left(6{\varphi}^{\rm
dr}(q,q)-4{\varphi}^{\rm dr}(q,-q)\right),$
$\displaystyle(\partial_{\theta}-\partial_{\mu})H(\theta,\mu)\bigg{|}_{\theta,\mu=q,-q}$
$\displaystyle=$ $\displaystyle\frac{H(q,-q)}{v_{F}}(3-4v_{F}{\varphi}^{\rm
dr}(q,-q)-2v_{F}{\varphi}^{\rm dr}(q,q)).$ (SC.42)

By combining these derivative identities for the symmetric function and
velocity the low-temperature diffusion can be evaluated to be

$\displaystyle\partial_{x}$ $\displaystyle\left(\int d\theta v^{\rm
eff}(\theta)({1}^{\rm
dr}(\theta))^{2}[\tilde{D}\partial_{x}n](\theta)\right)=\frac{2H(q,-q)}{(\partial_{\theta}\epsilon(\theta^{+}))^{2}}\partial_{x}\left(\frac{\partial_{x}\eta}{\beta}\right)+\frac{2(\partial_{x}\eta)(\partial_{x}q)}{\beta(\partial_{\theta}\epsilon(\theta^{+}))^{2}}\left(\partial_{q}+(\partial_{\theta}-\partial_{\mu})-\frac{2\partial_{\theta}v^{\rm
eff}(\theta)}{v^{\rm
eff}(\theta)}\right)H(\theta,\mu)\bigg{|}_{\theta,\mu=q,-q},$ (SC.43)
$\displaystyle=\frac{\partial_{x}^{2}\eta}{\beta(\partial_{\theta}\epsilon(\theta^{+}))^{2}}H(q,-q)+\frac{(\partial_{x}\eta)(\partial_{x}q)}{\beta
v_{F}(\partial_{\theta}\epsilon(\theta^{+}))^{2}}H(q,-q)\left[1+4v_{F}{\varphi}^{\rm
dr}(q,q)-4v_{F}{\varphi}^{\rm dr}(q,-q)\right].$

This formula can be rewritten more succinctly as

$\displaystyle\partial_{x}\left(\int d\theta v^{\rm eff}(\theta)({1}^{\rm
dr}(\theta))^{2}\left[\tilde{D}\partial_{x}n\right](\theta)\right)$
$\displaystyle=$ $\displaystyle
16\pi\partial_{x}\left[\frac{v_{F}}{\beta}({\varphi}^{\rm
dr}(q,-q))^{2}(\partial_{x}\eta)\right].$ (SC.44)

With the viscosity $\mu(q,\beta)$ identified as

$\displaystyle\mu(q,\beta)=\frac{8\pi}{\beta}v_{F}({\varphi}^{\rm
dr}(q,-q))^{2}.$ (SC.45)

##  Appendix D Equation for the energy density

At the Euler scale, the low-temperature energy evolution equation was found to
be a restatement of the density and momentum relations. By adding diffusion
this is no longer the case, due to the presence of the temperature field in
the density and momentum equations. Since the diffusion is an $O(T)$ effect,
there is no avoiding the temperature field. A standard calculation akin to
that done in the Euler case, but now retaining the diffusive effects, then
implies

$\displaystyle\frac{1}{\rho}\left(\partial_{t}E+\partial_{x}J_{E}\right)$
$\displaystyle=$
$\displaystyle\partial_{t}e+\frac{P_{s}}{\rho}\partial_{x}\eta+\eta\partial_{x}e+0+\frac{\eta}{\rho}D(p)=\frac{1}{\rho}D(E).$
(SD.46)

To close our diffusive equations at low-temperature there are several steps.
First, the diffusive term is evaluated, as it was for the momentum. With the
low-temperature energy diffusion characterized, the energy field can be recast
in terms of a temperature field, which closes the set of equations. The
density, momentum, and temperature evolution equations are then brought
together and collectively understood as constituting low-temperature viscous
GHD.

The diffusive correction to the energy expression is then computed here, where
for convenience of notation we introduce $v_{E}=\frac{{(\theta_{s}^{2})}^{\rm
dr}}{2({1}^{\rm dr})}$ with the note that

$\displaystyle D(E)$ $\displaystyle=$
$\displaystyle\partial_{x}\int\frac{d\theta d\mu}{2}\frac{{(\theta^{2})}^{\rm
dr}(\theta)}{2}({1}^{\rm
dr}(\theta))\tilde{D}(\theta,\mu)\partial_{x}n(\mu)=\partial_{x}\int\frac{d\theta
d\mu}{2}\left(v_{E}(\theta)+\eta v^{\rm
eff}(\theta)+\frac{\eta^{2}}{2}\right)({1}^{\rm
dr}(\theta))^{2}\tilde{D}(\theta,\mu)\partial_{x}n(\mu),$ (SD.47)
$\displaystyle=$ $\displaystyle
D_{v_{E}}+\mu(\rho,\beta)(\partial_{x}\eta)^{2}+\eta D(p)+\eta^{2}D(\rho).$

Only the first term $D_{v_{E}}$ is new, the viscosity in the second term has
been calculated already, see (SC.45), the third term is exactly the diffusive
correction to momentum, and the third term the diffusive correction to
density, which vanishes. Recall now the definition of the anti-symmetric
function $h(\theta,\mu)$

$\displaystyle h(\theta,\mu)=2\pi\frac{({1}^{\rm dr}(\theta){1}^{\rm
dr}(\mu))({\varphi}^{\rm dr}(\theta,\mu))^{2}}{\beta}\frac{(v^{\rm
eff}(\theta)-v^{\rm eff}(\mu))^{3}}{|v^{\rm eff}(\theta)-v^{\rm eff}(\mu)|},$
(SD.48)

which is helpful for simplifying the calculation. With the antisymetric
function and noting that $v_{E}(q)=v_{E}(-q)$ the diffusive contribution
$D_{v_{E}}$, after applying the spatial derivative, is given by

$\displaystyle D_{v_{E}}$ $\displaystyle=$ $\displaystyle\int\frac{d\theta
d\mu}{4\beta}\left(\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}\partial_{x}(h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))\partial_{x}\eta)+\right.$
(SD.49)
$\displaystyle+\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}(\partial_{x}q)(\partial_{x}\eta)\left(\partial_{\theta}\frac{h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))}{v^{\rm
eff}(\theta)}+\partial_{\mu}\frac{h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))}{v^{\rm
eff}(\mu)}\right),$
$\displaystyle\left.+\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}(\partial_{x}\eta)^{2}\left[\partial_{\theta}[h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))]+\partial_{\mu}[h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu)]\right]\right).$

Note that we have dropped the terms proportional to $\partial_{x}\beta$, since
these terms contain $O((v_{E}(\theta)-v_{E}(\mu))^{2})$ and so will always
vanish when $\theta,\mu\to q,-q$. The other terms can be determined as

$\displaystyle\int\frac{d\theta
d\mu}{4\beta}\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}\left(\partial_{\theta}\frac{h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))}{v^{\rm
eff}(\theta)}+\partial_{\mu}\frac{h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))}{v^{\rm
eff}(\mu)}\right)=0.$ (SD.50)

Along with the second term contributing

$\displaystyle\int\frac{d\theta
d\mu}{4\beta}\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}(\partial_{x}\eta)^{2}\left[(\partial_{\theta}+\partial_{\mu})[h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu)]\right]=\frac{16\pi
v_{F}}{\beta}({\varphi}^{\rm dr}(q,-q))^{2}(\partial_{x}\eta)^{2}.$

Likewise, the only surviving part of the first term is the spatial derivative
of $v_{E}$. By use of dressed function identities it then follows that the
first term can be written as

$\displaystyle\int\frac{d\theta
d\mu}{4\beta}\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\partial_{\mu}\epsilon}\partial_{x}(h(\theta,\mu)(v_{E}(\theta)-v_{E}(\mu))\partial_{x}\eta)=-\frac{16\pi
v_{F}}{\beta}({\varphi}^{\rm dr}(q,-q))^{2}(\partial_{x}\eta)^{2}.$ (SD.52)

Taken together Eqs. (D) and (SD.52) imply the diffusive contribution from
$D_{v_{E}}$ vanishes

$\displaystyle D_{v_{E}}=\partial_{x}\int d\theta d\mu\,v_{E}(\theta)({1}^{\rm
dr}(\theta))^{2}\tilde{D}(\theta,\mu)\partial_{x}n(\mu)=0.$ (SD.53)

Consequently, the total energy diffusion is given by

$\displaystyle D(E)$ $\displaystyle=$ $\displaystyle
D_{v_{E}}+(\partial_{x}\eta)^{2}\left(\frac{8\pi}{\beta}({T}^{\rm
dr}(q,-q))^{2}v^{\rm eff}(q)\right)+\eta D(p)+\eta^{2}D_{\rho},$ (SD.54)
$\displaystyle=$
$\displaystyle(\partial_{x}\eta)^{2}\left(\frac{8\pi}{\beta}({\varphi}^{\rm
dr}(q,-q))^{2}v^{\rm eff}(q)\right)+\eta D(p).$

Finally, this is combined with the energy evolution equation, which after some
simplification reads

$\displaystyle\partial_{t}e+\eta\partial_{x}e+\frac{\mathcal{P}_{s}}{\rho}\partial_{x}\eta$
$\displaystyle=$ $\displaystyle\frac{\mu}{\rho}(\partial_{x}\eta)^{2}$ (SD.55)

##  Appendix E Dynamics of the temperature and inclusion of other conserved
quantities

Our goal here is to compute the evolution of the temperature and also to
consider the impact of additional Lagrange parameters (temperatures) that are
allowed to evolve at low temperatures in the integrable gas. This amounts to
redefining the thermal energy and so obtain a more general local filling
function. The simplest example is the case of two additional conserved
quantities, $Q_{3}$ and $Q_{4}$

$\displaystyle\beta\epsilon=\beta\left(\frac{(\theta-\eta)^{2}}{2}-\mu+\gamma_{3}\frac{(\theta-\eta)^{3}}{3!}+\gamma_{4}\frac{(\theta-\eta)^{4}}{4!}\right)+\int
d\alpha\,\varphi(\theta-\alpha)\ln\left|1+\text{e}^{-\beta\epsilon}\right|.$
(SE.56)

To begin only those temperatures that lead to an even Fermi weight will be
considered, which will then be generalized to the anti-symmetric case.

Consider the general hydrodynamic charges corresponding to these generalized
temperatures, which are explicitly given by

$\displaystyle\frac{\langle Q_{2}\rangle}{\rho}=e-\frac{\eta^{2}}{2}\,,$
$\displaystyle\rho\langle e\rangle=\int d\theta\frac{(\theta^{2})^{\rm
dr}}{2}n(\theta)\,,$ $\displaystyle\frac{\langle Q_{3}\rangle}{\rho}=\langle
q_{3}\rangle-\eta\langle q_{2}\rangle-\frac{\eta^{3}}{3!}\,,$
$\displaystyle\rho\langle q_{3}\rangle=\int d\theta\frac{(\theta^{3})^{\rm
dr}}{3!}n(\theta)\,,$ $\displaystyle\frac{\langle Q_{4}\rangle}{\rho}=\langle
q_{4}\rangle-{\eta}\langle q_{3}\rangle-\frac{\eta^{2}}{2}\langle
q_{2}\rangle-\frac{\eta^{4}}{4!}\,,$ $\displaystyle\rho\langle
q_{4}\rangle=\int d\theta\frac{(\theta^{4})^{\rm dr}}{4!}n(\theta)\,.$ (SE.57)

Likewise, the corresponding diffusive terms are characterized by generalized
viscosities $\mu_{n}$, given by the low-temperature diffusion for an arbitrary
conserved charge. At leading order in temperature the general form is given by

$\displaystyle\mu_{n}$ $\displaystyle=$
$\displaystyle\frac{8\pi}{\beta}\frac{{q}^{\rm dr}_{n}(q)}{{1}^{\rm
dr}(q)}({\varphi}^{\rm dr}(q,-q))^{2}=\frac{8\pi
v_{n}(q)}{\beta}({\varphi}^{\rm dr}(q,-q))^{2}\sim O(T),$ (SE.58)

where $v_{n}={q}^{\rm dr}_{n}/{1}^{\rm dr}$ and noting that the only
nontrivial viscosity terms are odd. The set of hydrodynamic equations for
symmetric Fermi weights with an additional charge is given by

$\displaystyle\partial_{t}\rho+\partial_{x}(\eta\rho)$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\partial_{t}\eta+\frac{1}{2}\partial_{x}(\eta^{2})+\frac{\partial_{x}\mathcal{P}_{s}(\rho,T)}{\rho}$
$\displaystyle=$
$\displaystyle\frac{1}{\rho}\partial_{x}(\mu(\rho,T)\partial_{x}\eta)\,,$
$\displaystyle\partial_{t}\langle q_{n}\rangle+\eta\partial_{x}\langle
q_{n}\rangle+\frac{\mathcal{P}_{n}}{\rho}\partial_{x}\eta$ $\displaystyle=$
$\displaystyle\frac{\mu_{n-1}}{\rho}(\partial_{x}\eta)^{2},$ (SE.59)

where the generalised static pressure given by $\mathcal{P}_{n}=\int
d\theta\,{\theta}^{\rm dr}n(\theta)q_{n}$, not to be confused with the
stationary pressure $\mathcal{P}_{s}$, and the generalized viscosity $\mu_{n}$
is defined in eq. (SE.58). We now move to the evolution of the temperatures,

$\displaystyle\frac{\delta\langle q_{n}\rangle}{\delta
T}(\partial_{t}T+\eta\partial_{x}T)+\sum_{i=3}^{4}\frac{\delta\langle
q_{n}\rangle}{\delta\gamma_{i}}(\partial_{t}\gamma_{i}+\eta\partial_{x}\gamma_{i})=P^{T}_{n}\rho\partial_{x}\eta+\frac{\mu_{n}}{\rho}(\partial_{x}\eta)^{2},$
(SE.60)

along with the identification

$\displaystyle P^{T}_{n}=\Big{[}\partial_{\rho}\langle
q_{n}\rangle-\frac{\mathcal{P}_{n-1}}{\rho^{2}}\Big{]}-\Big{[}\partial_{\rho}\langle
q_{n}\rangle-\frac{\mathcal{P}_{n-1}}{\rho^{2}}\Big{]}_{T=0}.$ (SE.61)

The general correction $P_{n}^{T}$ is given in eq. (E.2). This expression can
then determines the evolution of the temperature, eq. (SE.60) in a thermal
state, using $\delta e/\delta T=T\tilde{\chi}_{e}$,

$\displaystyle(\partial_{t}T+\eta\partial_{x}T)=\frac{TP_{2}^{T}}{\tilde{\chi}_{e}}\rho\partial_{x}\eta+\frac{\mu(\rho,T)}{\rho
T\tilde{\chi}^{T}_{e}}(\partial_{x}\eta)^{2}$ (SE.62)

Let us now consider the case where also different temperatures can be
activated. These additional Lagrange parameters satisfy the general general
relations at low temperatures of

$\delta\langle q_{n}\rangle/\delta T\sim T\quad,\delta\langle
q_{n}\rangle/\delta\gamma_{i}\sim T^{2},$ (SE.63)

which follows from the general expressions for the low temperature charges
from eq. (SE.78). Recombining this with the expression for the evolution of
the Lagrange parameters, for both $q_{2}$ and $q_{4}$, with
$\chi^{(n)}_{\gamma_{j}}=\frac{1}{T^{2}}\frac{\partial\langle
q_{n}\rangle}{\partial\gamma}$, and
$\chi^{(n)}_{T}=\frac{1}{T}\frac{\partial\langle q_{n}\rangle}{\partial T}$
find the evolution equations

$\displaystyle
T\chi_{T}^{(2)}(\partial_{t}T+\eta\partial_{x}T)+\sum_{i=3}^{4}T^{2}\chi_{\gamma_{j}}^{(2)}(\partial_{t}\gamma_{i}+\eta\partial_{x}\gamma_{i})=P_{2}^{T}\rho\partial_{x}\eta+\frac{\mu_{1}}{\rho}(\partial_{x}\eta)^{2},$
$\displaystyle
T\chi_{T}^{(4)}(\partial_{t}T+\eta\partial_{x}T)+\sum_{i=3}^{4}T^{2}\chi_{\gamma_{j}}^{(4)}(\partial_{t}\gamma_{i}+\eta\partial_{x}\gamma_{i})=P_{4}^{T}\rho\partial_{x}\eta+\frac{\mu_{3}}{\rho}(\partial_{x}\eta)^{2}.$
(SE.64)

Simple algebra then implies the following evolution equation for the
additional Lagrange parameters

$\displaystyle\sum_{i=3}^{4}T\left(\frac{\chi_{\gamma_{j}}^{(4)}}{\chi_{T}^{(4)}}-\frac{\chi_{\gamma_{j}}^{(2)}}{\chi_{T}^{(2)}}\right)(\partial_{t}\gamma_{i}+\eta\partial_{x}\gamma_{i})=\frac{1}{T}\left(\frac{P_{4}^{T}}{{\chi_{T}^{(4)}}}-\frac{P_{2}^{T}}{{\chi_{T}^{(2)}}}\right)\rho\partial_{x}\eta+\left(\frac{\mu_{3}}{T\chi_{T}^{(4)}\rho}-\frac{\mu_{1}}{T\chi_{T}^{(2)}\rho}\right)(\partial_{x}\eta)^{2}.$
(SE.65)

This appears to be $O(T^{-1})$, however, by inserting the leading order terms
for $\mu_{1}$, $\mu_{3}$, and $\chi_{T}^{(n)}$ then the diffusive part
vanishes

$\displaystyle\left(\frac{\mu_{3}}{T\chi_{T}^{(4)}\rho}-\frac{\mu_{1}}{T\chi_{T}^{(2)}\rho}\right)=O(T^{2}).$
(SE.66)

With the note that the $O(T^{2})$ is due to higher order corrections to the
viscosity and susceptibilities. In the case with only a single additional
Lagrange parameter, remove $\gamma_{3}$, the evolution equation for
$\gamma_{4}$ is then written as

$\displaystyle\partial_{t}\gamma_{4}+\eta\partial_{x}\gamma_{4}=\frac{1}{T^{2}}\left(\frac{P^{T}_{4}\chi^{(2)}_{T}-P_{2}^{T}\chi^{(4)}_{T}}{\chi_{T}^{(4)}\chi_{\gamma_{4}}^{(2)}-\chi_{\gamma_{4}}^{(4)}\chi_{T}^{(2)}}\right)\rho\partial_{x}\eta\sim
O(T^{0}),$ (SE.67)

this can be reinserted to determine the temperature evolution as

$\displaystyle\partial_{t}T+\eta\partial_{x}T$ $\displaystyle=$
$\displaystyle\frac{1}{T}\frac{P^{T}_{2}\rho\partial_{x}\eta}{\chi_{T}^{(2)}}+\frac{\mu_{1}(\partial_{x}\eta)^{2}}{T\chi_{T}^{(2)}}+\frac{1}{T}\frac{\chi^{(2)}_{\gamma_{4}}}{\chi_{T}^{(2)}}\left(\frac{P_{2}^{T}\chi^{(4)}_{T}-P^{T}_{4}\chi^{(2)}_{T}}{\chi_{T}^{(4)}\chi_{\gamma_{4}}^{(2)}-\chi_{\gamma_{4}}^{(4)}\chi_{T}^{(2)}}\right)\rho\partial_{x}\eta.$

Note that inserting $P_{n}^{T}\sim O(T^{2})$ then in the absence of diffusion
($\mu_{j}\to 0$) the growth for both $\gamma_{3}$ and $\gamma_{4}$ are O(T),
whereas $T$ changes as $O(T^{0})$.

A similar analysis is now carried out for the case of a non-symmetric Fermi
weight. Here, although the hydrodynamic evolution equations are more
complicated, the general expressions are

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{t}\rho+\partial_{x}(\rho\eta),$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\partial_{t}\langle
q_{k}\rangle+\eta\partial_{x}\langle q_{k}\rangle-\frac{\langle
q_{k-1}\rangle}{\rho}\partial_{x}\mathcal{P}_{1}+\frac{1}{\rho}\partial_{x}\mathcal{P}_{k}+\frac{\mathcal{P}_{k-1}}{\rho}\partial_{x}\eta,$
(SE.69)

such that the evolution of the Lagrange parameters are given in general by

$\displaystyle
T(\partial_{t}T+\eta\partial_{x}T)\chi^{(n)}_{T}+\sum_{j=3}(\partial_{t}\gamma_{j}+\eta\partial_{x}\gamma_{j})T^{2}\chi_{\gamma_{j}}^{(n)}$
$\displaystyle=$ $\displaystyle\frac{\langle
q_{n-1}\rangle}{\rho}\partial_{x}\mathcal{P}_{1}-\frac{\partial_{x}\mathcal{P}_{n}}{\rho}+P_{n-1}^{T}\rho\partial_{x}\eta,$
(SE.70)

where, as before, $P^{T}_{n-1}=\partial_{\rho}\langle
q_{n}\rangle-\frac{\mathcal{P}_{n-1}}{\rho^{2}}$ It should be noted that
$\langle q_{1}\rangle=0$ and $n\geq 2$, furthermore, naively there appears to
be $O(T^{0})$ terms present on the RHS that would lead to $O(T^{-1})$
corrections overall. The $T=0$ identities imply, however, that $\langle
q_{n-1}\rangle\partial_{x}\mathcal{P}_{1}/\rho\sim
v_{n}\partial_{x}q+O(T^{2})$ and $\partial_{x}\mathcal{P}_{n}/\rho\sim
v_{n}\partial_{x}q+O(T^{2})$, so this apparent leading term vanishes. The
corrections stemming from $O(T^{2})$ corrections are combined into the
shorthand variable $G_{n}\sim O(T^{2})$ defined as

$\displaystyle G_{n}$ $\displaystyle=$ $\displaystyle
P_{n-1}^{T}\rho\partial_{x}\eta+\frac{\langle
q_{n-1}\rangle}{\rho}\partial_{x}\mathcal{P}_{1}-\frac{\partial_{x}\mathcal{P}_{n}}{\rho}\sim
O(T^{2}).$ (SE.71)

With these consideration the impact of the third and fourth additional
Lagrange parameters $\gamma_{3}$ and $\gamma_{4}$ can be considered. Introduce
$\widehat{\mathcal{D}}=\partial_{t}+\eta\partial_{x}$ and so determine that
the evolution equations read

$\displaystyle\widehat{\mathcal{D}}T$ $\displaystyle=$
$\displaystyle\frac{1}{T}\frac{G_{2}\chi_{T}^{(3)}-G_{3}\chi_{T}^{(2)}}{\chi_{\gamma_{3}}^{(3)}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)}}+\left(\frac{\chi_{\gamma_{4}}^{(3)}\chi_{\gamma_{3}}^{(2)}-\chi_{\gamma_{4}}^{(2)}\chi_{\gamma_{3}}^{(3)}}{\chi_{\gamma_{3}}^{(3)}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)}}\right)T\widehat{\mathcal{D}}\gamma_{4},$
$\displaystyle\widehat{\mathcal{D}}\gamma_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\frac{G_{3}\chi_{T}^{(2)}-G_{2}\chi_{T}^{(3)}}{\chi^{(3)}_{\gamma_{3}}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)}}-\frac{\chi_{\gamma_{4}}^{(3)}\chi_{T}^{(2)}-\chi_{\gamma_{4}}^{(2)}\chi_{T}^{(3)}}{\chi^{(3)}_{\gamma_{3}}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)}}\widehat{\mathcal{D}}\gamma_{4},$
$\displaystyle\widehat{\mathcal{D}}\gamma_{4}$ $\displaystyle=$
$\displaystyle\frac{G_{4}(\chi^{(3)}_{\gamma_{3}}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)})-(\chi^{(4)}_{T}-\chi_{\gamma_{3}}^{(4)})\chi_{T}^{(3)}G_{2}+(\chi_{T}^{(4)}-\chi_{\gamma_{3}}^{(4)})\chi^{(2)}_{T}G_{3}}{T^{2}\chi_{\gamma_{4}}^{(4)}(\chi^{(3)}_{\gamma_{3}}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)})}$
(SE.72)
$\displaystyle+\frac{\chi_{T}^{(4)}(\chi^{(3)}_{\gamma_{4}}\chi_{\gamma_{3}}^{(2)}-\chi_{\gamma_{4}}^{(2)}\chi_{\gamma_{4}}^{(3)})-\chi_{\gamma_{3}}^{(4)}(\chi_{\gamma_{4}}^{(3)}\chi_{T}^{(2)}-\chi_{\gamma_{4}}^{(2)}\chi_{T}^{(3)})}{\chi_{\gamma_{4}}^{(4)}(\chi^{(3)}_{\gamma_{3}}\chi_{T}^{(2)}-\chi_{\gamma_{3}}^{(2)}\chi_{T}^{(3)})}.$

With the note that this implies that $T$, $\gamma_{3}$, and $\gamma_{4}$
evolve with corrections of order $O(T)$, $O(T^{0})$, and $O(T^{0})$
respectively.

###  E.1 Derivation of the evolution of charges, eq. (SE.69)

To prove the general form of the evolution equations the idea is to apply the
relevant shifts to the conserved charges, so that they are being considered in
the symmetric basis. These shifts are expanded and so the evolution equation
for each charge field can be written in terms of a sum over evolution equation
of conserved charges in the symmetric basis. These sums are rearranged and
noted to vanish term-wise, which yields the general form of the evolution
equation whenever the Fermi weight is symmetric. The diffusive case is then
considered, which requires the addition of an additional sum that is treated
analogously to the Euler case. Finally, the symmetric condition for the Fermi
weight is relaxed and the fully general expression is obtained.

In general the hydrodynamic evolution equations can be written with the
notation $\rho\langle q_{n}\rangle=\int
d\theta\frac{\theta^{n}}{n!}n(\theta)({1}^{\rm dr})$ and $\rho\langle
j_{n}\rangle=\mathcal{P}_{n}=\int d\theta\,\frac{\theta^{n}}{n!}({\theta}^{\rm
dr})n(\theta)$. The density and $\eta$ fields are identified as

$\displaystyle\rho=\int d\theta\,{1}^{\rm dr}n,$ $\displaystyle\rho\eta=\int
d\theta\,{\theta}^{\rm dr}n\theta,$ (SE.73)

this choice of $\eta$ fixes $\langle q_{1}\rangle=0$. These expression are
used to rewrite the continuity equation as

$\displaystyle\partial_{t}\left(\int d\theta(\theta+\eta)^{n}n{1}^{\rm
dr}\right)$ $\displaystyle+$ $\displaystyle\partial_{x}\left(\int
d\theta(\theta+\eta)^{n}n{(\theta+\eta)}^{\rm
dr}\right)=\sum_{k=0}^{n}\frac{n^{\underline{k}}}{n!}\left(\partial_{t}\left(\rho\eta^{n-k}\langle
q_{k}\rangle\right)+\partial_{x}\left[\rho\eta^{n-k+1}\langle
q_{k}\rangle+\rho\eta^{n-k}\langle j_{k}\rangle\right]\right),$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\frac{n^{\underline{k}}}{n!}\left(\eta^{n-k}\partial_{t}(\rho\langle
q_{k}\rangle)+(n-k)\langle
q_{k}\rangle\eta^{n-k-1}\rho\partial_{t}\eta\right.+\eta^{n-k}\partial_{x}(\rho\eta\langle
q_{k}\rangle)$ $\displaystyle+(n-k)\rho\langle
q_{k}\rangle\eta^{n-k}\partial_{x}\eta+\left.\eta^{n-k}\partial_{x}(\rho\langle
j_{k}\rangle)+(n-k)\rho\langle
j_{k}\rangle\eta^{n-k-1}\partial_{x}\eta\right),$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\left(\partial_{t}\rho+\partial_{x}(\eta\rho)\right)+\frac{n^{\underline{n+1}}}{n!\eta}\left(\langle
q_{n}\rangle\rho(\partial_{t}\eta+\eta\partial_{x}\eta)+\rho\langle
j_{n}\rangle\partial_{x}\eta\right)$
$\displaystyle+\sum_{k=1}^{n}\frac{n^{\underline{k}}}{n!}\eta^{n-k}\left(\partial_{t}(\rho\langle
q_{k}\rangle)+\partial_{x}(\rho\eta\langle
q_{k}\rangle)+\partial_{x}(\rho\langle j_{k}\rangle)+\langle
q_{k-1}\rangle\rho(\partial_{t}\eta+\eta\partial_{x}\eta)+\rho\langle
j_{k-1}\rangle\partial_{x}\eta\right)$

with the dropping power $n^{\underline{k}}=n(n-1)\dots(n-k+1)$, such that
$n^{\underline{n+1}}=0$. These terms must vanish term by term and so yields
the conservation equations for the general evolution equations, written with
$\mathcal{P}_{n}=\rho\langle j_{n}\rangle$, that for the Euler case read

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{t}\rho+\partial_{x}(\rho\eta),$ $\displaystyle 0$
$\displaystyle=$
$\displaystyle\partial_{t}\eta+\eta\partial_{x}\eta+\frac{\partial_{x}\mathcal{P}_{1}}{\rho},$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\partial_{t}\langle
q_{k}\rangle+\eta\partial_{x}\langle q_{k}\rangle-\frac{\langle
q_{k-1}\rangle}{\rho}\partial_{x}\mathcal{P}_{1}+\frac{1}{\rho}\partial_{x}\mathcal{P}_{k}+\frac{\mathcal{P}_{k-1}}{\rho}\partial_{x}\eta.$
(SE.75)

When the Fermi weight is symmetric all odd, $n=2k-1$, charges $\langle
q_{2k-1}\rangle=0$ and all even, $n=2k$, currents $\langle j_{2k}\rangle=0$
this reduces the general formula to the hydrodynamic equations for symmetric
Fermi weights.

Now the diffusive corrections can be inserted and treated in a closely
analogous fashion. Insert the diffusive correction

$\displaystyle\partial_{t}\left(\int d\theta(\theta+\eta)^{n}n{1}^{\rm
dr}\right)+\partial_{x}\left(\int
d\theta(\theta+\eta)^{n}n{(\theta+\eta)}^{\rm dr}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{x}\int d\theta
d\alpha\,\frac{(\theta+\eta)^{n}}{n!}D(\theta,\alpha)\partial_{x}(n{1}^{\rm
dr})(\alpha)$ (SE.76) $\displaystyle=$ $\displaystyle\partial_{x}\int
d\theta\,\frac{(\theta+\eta)^{n}}{n!}\hat{\mu}(\theta)\partial_{x}\eta,$

where the $\int
d\theta\frac{\theta^{n}}{n!}\hat{\mu}(\theta)\partial_{x}\eta=\mu_{n}\partial_{x}\eta$
is chosen in analogy with the low temperature result, although the precise
form does not affect the proof. From this notation we identify
$\partial_{x}(\mu_{n}\partial_{x}\eta)$ as the diffusion in the symmetric
basis for the $n$th conserved charge

$\displaystyle\left(\sum_{k=0}\frac{n^{\underline{k}}}{n!}\partial_{x}(\mu_{k}\eta^{n-k}\partial_{x}\eta)\right)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\frac{n^{\underline{k}}}{n!}\eta^{n-k}\left(\mu_{k-1}(\partial_{x}\eta)^{2}+\partial_{x}(\mu_{k}\partial_{x}\eta)\right).$
(SE.77)

With the diffusive matched to the term of the Euler evolution equation with
matching prefactor $n^{\underline{k}}$.

###  E.2 Explicit form of (SE.61)

These additional Lagrange parameters depend on the specific form under
consideration, however certain general expressions are still possible. For a
single Fermi sea the even hydrodynamic charges are given by

$\displaystyle\rho\langle q_{n}\rangle(\rho,T)$ $\displaystyle=$
$\displaystyle\rho\langle q_{n}\rangle(\rho,0)+\frac{\pi^{2}T^{2}{1}^{\rm
dr}}{3(\epsilon^{\prime}(q,\gamma_{4}))^{2}}{\left(\frac{\theta_{s}^{n-1}}{(n-1)!}\right)}^{\rm
dr}.$ (SE.78)

Likewise, at constant density and low-temperature the hydrodynamic currents

$\displaystyle(\rho\langle j_{n}\rangle)(\rho,T)=\int d\theta\,vq_{n}{1}^{\rm
dr}n=\int d\theta\,\theta{q}^{\rm dr}_{n}n.$ (SE.79)

These currents are related to pressure by the simple relation
$\mathcal{P}_{n}=\rho\langle j_{n}\rangle$. The general identity at low
temperature and a single Fermi sea for constant density $\rho$ the corrections
are given by

$\displaystyle\rho\langle j_{n}\rangle(\rho,T)$ $\displaystyle=$
$\displaystyle\rho\langle
j_{n}\rangle(\rho,0)+\frac{\pi^{2}T^{2}}{3(\epsilon^{\prime}(q,\gamma_{4}))^{2}}\left({\theta}^{\rm
dr}_{s}(q){\left(\frac{\theta_{s}^{n-1}}{(n-1)!}\right)}^{\rm
dr}+({\theta}^{\rm dr}{1}^{\rm
dr})v_{n}\left[\frac{1}{v_{F}}+2(V(q)-U(q))+f(\gamma_{4})\right]\right).$

For the generalized pressure energy relation note that

$\displaystyle\partial_{\rho}\langle q_{n}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\rho}\int
d\theta\,{\left(\frac{\theta^{n}}{n!}\right)}^{\rm dr}({1}^{\rm
dr})\partial_{\rho}n(\theta)-\frac{\langle q_{n}\rangle}{\rho},$
$\displaystyle\frac{\mathcal{P}_{n-1}}{\rho^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\rho^{2}}\int d\theta\,{\theta}^{\rm
dr}n(\theta){\left(\frac{\theta^{n-1}}{(n-1)!}\right)}^{\rm
dr}=-\frac{1}{\rho^{2}}\int
d\theta\,\left[{\left(\frac{\theta^{n}}{n!}\right)}^{\rm
dr}n(\theta)\partial_{\theta}{\theta}^{\rm
dr}+{\left(\frac{\theta^{n}}{n!}\right)}^{\rm
dr}(\partial_{\theta}n){\theta}^{\rm dr}\right],$ (SE.81) $\displaystyle=$
$\displaystyle-\frac{\langle q_{n}\rangle}{\rho}-\frac{\int
d\theta\,{\left(\frac{\theta^{n}}{n!}\right)}^{\rm
dr}(\partial_{\theta}n){\theta}^{\rm dr}}{\rho^{2}}.$

By subtracting these two formula the useful identity for computing the
difference of the charge and pressure is determined as

$\displaystyle\Big{[}\partial_{\rho}q_{n}-\frac{\mathcal{P}_{n}}{\rho^{2}}\Big{]}$
$\displaystyle=$ $\displaystyle\frac{\int d\theta\,\frac{{(\theta^{n})}^{\rm
dr}}{n!}\partial_{\theta}n\left({\theta}^{\rm dr}+{1}^{\rm
dr}\rho\frac{\partial_{\rho}(\beta\epsilon)}{\partial_{\theta}(\beta\epsilon)}\right)}{\rho^{2}}=\left[\partial_{\rho}\langle
q_{n}\rangle-\frac{\mathcal{P}_{n-1}}{\rho^{2}}\right]_{T=0}+P^{T}_{n}.$
(SE.82)

The identity $4{\varphi}^{\rm
dr}(q,-q)v_{F}=(1-2K(\rho)\partial_{\rho}v_{F}(\rho))$ is applied and
generalized effective velocities $v_{0}=1$, $v_{1}=v_{F}$, and
$v_{n}={(\theta^{n})}^{\rm dr}/(n!({1}^{\rm dr}))$. From these identifications
the scenario of a single Fermi sea with arbitrarily many Lagrange parameters
can be written out by noting $\epsilon^{\prime}={\theta}^{\rm
dr}\left(1+\sum_{j=3}\gamma_{j}\frac{v_{j-1}}{v_{F}}\right)$. With these
considerations the $O(T^{2})$ corrections when there are two and then an
arbitrary number of Lagrange parameters are respectively given by

$\displaystyle P^{T}_{n}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}T^{2}}{3v_{F}\rho}\left(\frac{\partial_{\rho}v_{n-1}}{v_{F}}-\frac{(v_{n-1}+v_{F}v_{n-2})}{\rho
v_{F}}-\frac{2v_{n-1}\partial_{\rho}v_{F}}{v_{F}^{2}}-\frac{2v_{n-1}}{v_{F}\rho}K(\rho)\partial_{\rho}v_{F}\right),$
$\displaystyle P_{n}^{T}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}T^{2}}{3v_{F}\rho}\left[\frac{\left(\frac{\partial_{\rho}v_{n-1}}{v_{F}}-\frac{(v_{n-1}+v_{F}v_{n-2})}{\rho
v_{F}}-\frac{2v_{n-1}\partial_{\rho}v_{F}}{v_{F}^{2}}-\frac{2v_{n-1}}{v_{F}\rho}K(\rho)\partial_{\rho}v_{F}\right)}{\left(1+\sum_{j=3}\gamma_{j}\frac{v_{j-1}}{v_{F}}\right)^{2}}-2\frac{v_{n-1}\sum_{j=3}\gamma_{j}\left[\frac{\partial_{\rho}v_{j-1}}{v_{F}}-\frac{v_{j-1}\partial_{\rho}v_{F}}{v_{F}^{2}}\right]}{\left(1+\sum_{j=3}\gamma_{j}\frac{v_{j-1}}{v_{F}}\right)^{3}}\right].$

###  E.3 Small temperature corrections to non-symmetric Fermi weights

For the case of symmetric Fermi weights the pressure and charge can be
straightforwardly expanded at low temperature and constant density in the case
of symmetric Fermi points as

$\displaystyle\mathcal{P}_{n}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{n}(\rho,0)+\frac{\pi^{2}T^{2}\rho}{6(\epsilon^{\prime}(q))^{2}}\left(v_{n-1}+\frac{v_{n}}{v_{F}}2K(\rho)\partial_{\rho}v_{F}\right),$
$\displaystyle\rho\langle q_{n}\rangle(\rho,T)$ $\displaystyle=$
$\displaystyle\rho\langle
q_{n}\rangle(\rho,0)+\frac{\pi^{2}T^{2}\rho}{6(\epsilon^{\prime}(q))^{2}}\frac{v_{n-1}}{v_{F}}.$
(SE.84)

The case of non-symmetric Fermi points is more complicated, however a similar
picture holds. Specifically, whenever there is a single Fermi sea the state
energy $\beta\epsilon$, for $k=x,t$ reads

$\displaystyle\partial_{k}(\beta\epsilon)=-(\partial_{k}\eta)(\partial_{\theta}\beta\epsilon)-\frac{(\partial_{\theta}\epsilon)(q)}{{1}^{\rm
dr}(q)}(\partial_{k}q){1}^{\rm
dr}(\theta)+\sum_{j}\partial_{k}\gamma_{j}\frac{\partial(\beta\epsilon)}{\partial\gamma_{j}}.$
(SE.85)

Non-symmetric Fermi weights occur whenever odd Lagrange parameters are non-
zero and the low temperature approximation is slightly modified to

$\displaystyle\int_{-\infty}^{\infty}d\theta\,n(\theta)f(\theta)$
$\displaystyle=$
$\displaystyle\int_{q_{L}}^{q_{R}}d\theta\,f(\theta)+\frac{\pi^{2}T^{2}}{6}\left[\frac{f^{\prime}(q_{R})-\left(\frac{\epsilon^{\prime\prime}(q_{R})}{\epsilon^{\prime}(q_{R})}-U(q_{R})\right)f(q_{R})}{(\epsilon^{\prime}(q_{R}))^{2}}-\frac{f^{\prime}(q_{L})-\left(\frac{\epsilon^{\prime\prime}(q_{L})}{\epsilon^{\prime}(q_{L})}-U(q_{L})\right)f(q_{L})}{(\epsilon^{\prime}(q_{L}))^{2}}\right],$
(SE.86) $\displaystyle=$
$\displaystyle\int_{q_{L}}^{q_{R}}d\theta\,f(\theta)+\Lambda_{f}(q_{R})-\Lambda_{f}(q_{L}).$

Based on the definition of the density and $\eta$ fields, then at low
temperatures the left right Fermi points are considered as $q_{L/R}(T)=\pm
q+\delta_{L/R}(T)$. At zero temperature the Fermi functions
$\delta_{L/R}(T)\overset{T\to 0}{=}0$ and so

$\displaystyle\Lambda_{\rho}(\theta)$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}T^{2}({1}^{\rm
dr}(\theta))}{6(\epsilon^{\prime}(\theta))^{2}}\left(2V(\theta)({1}^{\rm
dr}(\theta))+\left[\frac{\epsilon^{\prime\prime}(\theta)}{\epsilon^{\prime}(\theta)}-U(\theta)\right]({1}^{\rm
dr}(\theta))\right)$ $\displaystyle\rho$ $\displaystyle=$ $\displaystyle\int
d\theta\,n{1}^{\rm dr}\approx\int^{q_{R}}_{q_{L}}d\theta\,{1}^{\rm
dr}-\Lambda_{\rho}(q_{R})+\Lambda_{\rho}(q_{L})\approx\int^{q_{R}}_{q_{L}}d\theta\,{1}^{\rm
dr}-\Lambda_{\rho}(q)+\Lambda_{\rho}(-q)+O(T^{2}\delta(T)),$ $\displaystyle=$
$\displaystyle\rho_{0}+{1}^{\rm
dr}(q)(\delta_{R}+\delta_{L})-\Lambda_{\rho}(q)+\Lambda_{\rho}(-q)$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int d\theta\,n{\theta}^{\rm
dr}-\rho\eta\sim\int^{q_{R}}_{q_{L}}d\theta\,{\theta}^{\rm
dr}+O(T^{2}\delta(T))$ (SE.87) $\displaystyle=$
$\displaystyle\frac{q_{R}+q_{L}}{2}2({1}^{\rm dr}(q^{\prime})){\theta}^{\rm
dr}(q^{\prime})+{\theta}^{\rm dr}(q)(\delta_{R}+\delta_{L}).$

Taken together these expressions confirm that the corrections to the Fermi
points are $O(T^{2})$ in the low temperature limit.

###  E.4 General viscosity expression eq. (SE.58)

Based on the work done so far diffusion for general conserved charges, denoted
$q_{n}=\theta^{n}/n!$, can be computed. This is carried out by noting the
general form of the diffusion kernel

$\displaystyle D(q_{n})$ $\displaystyle=$
$\displaystyle\partial_{x}\int\frac{d\theta d\mu}{2}\frac{{(\theta^{n})}^{\rm
dr}(\theta)}{n!}({1}^{\rm
dr}(\theta))\tilde{D}(\theta,\mu)\partial_{x}n(\mu).$ (SE.88)

The kernel simplifies in the thermal case or more generally in the case for
$n(q)=n(-q)$ find

$\displaystyle\frac{\partial_{x}(\beta\epsilon(\mu))}{\partial_{\mu}\epsilon}v^{\rm
eff}(\mu)-\frac{\partial_{x}(\beta\epsilon(\theta))}{\partial_{\theta}\epsilon}v^{\rm
eff}(\theta)=\beta\left(v^{\rm eff}(\theta)-v^{\rm
eff}(\mu)\right)\partial_{x}\eta+\left(\frac{\epsilon(\mu)}{{1}^{\rm
dr}(\mu)}-\frac{\epsilon(\theta)}{{1}^{\rm
dr}(\theta)}\right)\partial_{x}\beta,$ (SE.89)

When there is a single Fermi sea the state energy $\beta\epsilon$ a general
differential, for $k=x,t$ reads

$\displaystyle\partial_{x}(\beta\epsilon)=-(\partial_{x}\eta)(\partial_{\theta}\beta\epsilon)-\frac{(\partial_{\theta}\beta\epsilon)(q)}{{1}^{\rm
dr}(q)}(\partial_{x}q){1}^{\rm
dr}(\theta)+\sum_{j}\partial_{x}\gamma_{j}\frac{\partial(\beta\epsilon)}{\partial\gamma_{j}}.$
(SE.90)

this expression can be conveniently rewritten with the anti-symmetric function

$\displaystyle h(\theta,\mu)=2\pi({1}^{\rm dr}(\theta){1}^{\rm
dr}(\mu))({\varphi}^{\rm dr}(\theta,\mu))^{2}\frac{(v^{\rm eff}(\theta)-v^{\rm
eff}(\mu))^{3}}{|v^{\rm eff}(\theta)-v^{\rm eff}(\mu)|}.$ (SE.91)

So that at low-temperatures the diffusion for an arbitrary conserved charge
has a general form given by

$\displaystyle D(q_{n})$ $\displaystyle=$
$\displaystyle\partial_{x}\int\frac{d\theta
d\mu}{4}\left(\frac{{(\theta^{n})}^{\rm dr}(\theta)}{n!({1}^{\rm
dr}(\theta))}-\frac{{(\theta^{n})}^{\rm dr}(\mu)}{n!({1}^{\rm
dr}(\mu))}\right)\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\mu}n}{\beta\partial_{\mu}\epsilon}h(\theta,\mu)\partial_{x}\eta\sim
O(T).$ (SE.92)

As in the case of the energy diffusion the conserved charge $q_{n}$ can be
boosted into the co-moving frame by $\theta\to\theta+\eta$. Such a calculation
involves expansions of
${\left[(\theta+\eta)^{n}-(\alpha+\eta)^{n}\right]}^{\rm dr}/n!$, which are
then simplified by noting ${(\theta^{n})}^{\rm
dr}(q)=(-1)^{n}{(\theta^{n})}^{\rm dr}(-q)$. None of this will impact the
leading temperature order correction, which is always $O(T)$.

In addition to these general considerations there is a correction to the
spatial derivative of the thermal energy, which reads

$\displaystyle\frac{\partial_{x}\epsilon}{\partial_{\theta}\epsilon}=-\partial_{x}\eta+\sum_{j}\frac{\partial_{x}\gamma_{j}}{\partial_{\theta}\epsilon(\theta)}\frac{\partial\epsilon}{\partial\gamma_{j}}\mp\frac{v^{\rm
eff}(\pm q)}{v^{\rm
eff}(\theta)}\partial_{x}q=-\partial_{x}\eta\mp\frac{v^{\rm eff}(\pm
q)}{v^{\rm eff}(\theta)}\partial_{x}q+\sum_{j}\frac{v_{j}(\theta)}{v^{\rm
eff}(\theta)}\partial_{x}\gamma_{j}$ (SE.93)

with the additional caveat that
$\frac{\partial\epsilon}{\partial\gamma_{j}}={(\theta^{j})}^{\rm dr}/j!$. With
$v_{j}=\frac{{(\theta^{j})}^{\rm dr}}{j!\,{1}^{\rm dr}(\theta)}$ and the sum
of velocities given by $\frac{\partial_{\theta}\epsilon}{{1}^{\rm dr}}$ as
well as $v_{1}=v_{F}$. A consequence of this term is that the diffusion
constant

$\displaystyle D^{\alpha}(Q_{n})$ $\displaystyle=$
$\displaystyle\partial_{x}\int\frac{d\theta d\alpha}{2}{q}^{\rm
dr}_{n}(\theta)\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\alpha}n}{\beta\partial_{\alpha}\epsilon}{1}^{\rm
dr}(\alpha)({\varphi}^{\rm
dr}(\theta,\alpha))^{2}|v_{F}(\theta)-v_{F}(\alpha)|\left[-\frac{\partial_{x}(\beta\epsilon(\theta))}{\partial_{\theta}\epsilon}v^{\rm
eff}(\theta)+\frac{\partial_{x}(\beta\epsilon(\alpha))}{\partial_{\alpha}\epsilon}v^{\rm
eff}(\alpha)\right].$

Insert the ratio for the derivative to find that in the general case

$\displaystyle D^{\alpha}(Q_{n})$ $\displaystyle=$
$\displaystyle\partial_{x}\int\frac{d\theta
d\alpha}{4}\left(v_{n}(\theta)-v_{n}(\alpha)\right)\frac{\partial_{\theta}n}{\partial_{\theta}\epsilon}\frac{\partial_{\alpha}n}{\beta\partial_{\alpha}\epsilon}[{1}^{\rm
dr}(\alpha){1}^{\rm dr}(\theta)]|v_{F}(\theta)-v_{F}(\alpha)|({\varphi}^{\rm
dr}(\theta,\alpha))^{2}$ $\displaystyle\left[(v^{\rm eff}(\theta)-v^{\rm
eff}(\alpha))\partial_{x}\eta-\sum_{j=3}(v_{j}(\theta)-v_{j}(\alpha))\partial_{x}\gamma_{j}\right]$

For even dressed charges it is straightforward to see that
$v_{n}(q)-v_{n}(-q)=0$, whereas $v_{n}(q)-v_{n}(-q)=2v_{n}(q)$ for odd values
of $n$. An interchange $q\leftrightarrow-q$ leaves the expression unchanged
and so the diffusion is non-zero only for odd $n$ and given by

$\displaystyle D_{n}$ $\displaystyle=$
$\displaystyle\partial_{x}\left[\frac{4({\varphi}^{\rm
dr}(q,-q))^{2}v_{n}(q)v_{F}(q)}{\beta v^{\rm eff}(q)v^{\rm
eff}(-q)}\left(v_{F}(q)\partial_{x}\eta+\sum_{j=3,5,\dots}\left(v_{j}(q)\gamma_{j+1}\partial_{x}\eta-
v_{j}(q)\partial_{x}\gamma_{j}\right)\right)\right].$ (SE.96)

Note that the when the additional Lagrange parameters $\gamma_{j}=0$ for all
$j=3,4\dots$ the diffusion reduces to the thermal case.

##  Appendix F Final result: low-temperature viscous CHD

By restoring the conventional hydrodynamic fields with a factor $1/2\pi$ such
as $\rho=\frac{1}{2\pi}\int d\theta\,n{1}^{\rm dr}$, and $p=\frac{1}{2\pi}\int
d\theta\,{\theta}^{\rm dr}n$, (analogously for pressure, energy, and energy),
together with

$K=(1^{\rm
dr})^{2}(q)/(2\pi)=\pi\rho/v_{F},\quad\quad\partial_{\rho}v_{F}=\frac{1-4\varphi^{\rm
dr}(q,-q)v_{F}}{{K}/\pi},$ (SF.97)

with the identity $v_{F}({1}^{\rm dr})^{2}(q)/\pi=\frac{{\theta}^{\rm
dr}(q)}{{1}^{\rm dr}(q)}({1}^{\rm dr})^{2}(q)/\pi={\theta}^{\rm dr}(q){1}^{\rm
dr}(q)/\pi=\rho$, we finally find

$\displaystyle\partial_{t}$ $\displaystyle\rho+\partial_{x}(\eta\rho)=0,$
$\displaystyle\partial_{t}$
$\displaystyle\eta+\frac{1}{2}\partial_{x}(\eta^{2})+\frac{\partial_{x}\mathcal{P}_{s}(\rho,T)}{\rho}=\frac{1}{\rho}\partial_{x}(\mu(\rho,T)\partial_{x}\eta),$
$\displaystyle(\partial_{t}$ $\displaystyle
T+\eta\partial_{x}T)=-T\frac{\pi\left(\frac{1}{\rho}+\frac{\partial_{\rho}v_{F}}{v_{F}}\right)}{3v_{F}\tilde{\chi}_{e}}\partial_{x}\eta+\frac{\mu(\rho,T)}{\rho
T\tilde{\chi}^{T}_{e}}(\partial_{x}\eta)^{2},$ (SF.98)

together with

$\displaystyle\tilde{\chi}_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{{\rho}}\frac{\pi}{3v_{F}},$ $\displaystyle\mu(\rho,T)$
$\displaystyle=$
$\displaystyle\frac{T}{4}\frac{\left(1-\frac{2K}{2\pi}\partial_{\rho}v_{F}\right)^{2}}{v_{F}}=4T{\varphi}^{\rm
dr}(q,-q)^{2}v_{F},$ $\displaystyle\partial_{x}\mathcal{P}_{s}(\rho,T)$
$\displaystyle=$ $\displaystyle\Big{(}v_{F}^{2}-\frac{\pi
T^{2}}{6}\frac{1-4v_{F}\varphi^{\rm dr}(q,-q)}{\rho
v_{F}}\Big{)}\partial_{x}\rho+\frac{\pi T}{3v_{F}}\partial_{x}T.$ (SF.99)

##  Appendix G Proofs and checks of various Identities

###  G.1 Numerical check of viscosity

Figure S1: Comparison between the low $T$ viscosity
$\mu(\rho,T)\partial_{x}\eta$ appearing in the rhs of ( Navier-Stokes
equations for low-temperature one-dimensional fluids) (solid lines) and the
finite $T$ diffusion $D(p)$ (dotted lines), see eq. (SC.26) for its
expression, where they are found to be an extremely close match. These
calculations were performed at $\beta=20$ at various values of $c$.

To verify that our low-temperature result for the viscosity

$\displaystyle\mu(\rho,T)$ $\displaystyle=$ $\displaystyle\frac{\pi
T}{2}\frac{(1-2K\partial_{\rho}v_{F})^{2}}{v_{F}}=8\pi T{\varphi}^{\rm
dr}(q,-q)^{2}v_{F}$ (SG.100)

is consistent with low-temperature GHD a numerical evaluation of the full
diffusion kernel, see eq. (SC.26), is carried out. The computed diffusion
kernel is then specialized to the momentum component and compared to our final
result for the low-temperature viscosity.

Our protocol is as follows: two Gaussian fields are introduced, the potential
$\mu(x)=1+\mu_{0}\text{e}^{-\nu x^{2}}$ and shifts in the rapidity given by
$\eta(x)=\eta_{0}\text{e}^{-\sigma x^{2}}$. These two fields are used to
determine the filling function by use of the thermal energy

$\displaystyle\beta\epsilon(\theta,x)=\beta\frac{(\theta-\eta(x))^{2}}{2}-\mu(x)-\int
d\alpha\,\varphi(\theta,\alpha)\ln\left|1+\text{e}^{-\beta\epsilon(\alpha,x)}\right|.$
(SG.101)

Once the filling function is determined the full diffusion kernel is
considered and applied to the bare charge to determine the general diffusion.
The filling function is also used to determine the hydrodynamic density $\rho$
and the Fermi points $\pm q$, which are then used to determine the Fermi
velocity $v_{F}=v^{\rm eff}(q)$ and dressed kernel ${\varphi}^{\rm dr}(q,-q)$
thus determining our proposed viscosity. Comparing both methods leads to
identical results, see fig. S1.

###  G.2 Hydrodynamic fields at $T=0$ and their expressions

To bridge the gap between generalized and conventional hydrodynamics it’s
useful to consider the low-temperature values of the hydrodynamic fields.
These identities can be evaluated in a fairly straightforward fashion by
integrating by parts and thereafter applying the relevant dressed function
identities. The hydrodynamic variables are collected below for convenient
reference

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\int d\theta\,{1}^{\rm
dr}n=2{\theta}^{\rm dr}(q){1}^{\rm dr}(q),$ $\displaystyle p$ $\displaystyle=$
$\displaystyle\int d\theta\,v{1}^{\rm
dr}n=\frac{1}{2}\left[{(\theta^{2})}^{\rm dr}(\theta^{+})-{(\theta^{2})}^{\rm
dr}(\theta^{-})\right]{1}^{\rm dr}(q)=\eta\rho,$ $\displaystyle\mathcal{P}$
$\displaystyle=$ $\displaystyle\int d\theta\,{\theta}^{\rm dr}n\theta=\int
d\theta\,{\theta}^{\rm
dr}_{s}n\theta_{s}+\eta^{2}\rho=\mathcal{P}_{s}+\eta^{2}\rho,$ $\displaystyle
E$ $\displaystyle=$ $\displaystyle\int d\theta\,\theta^{2}{1}^{\rm
dr}n=\frac{1}{6}\left[{(\theta^{3})}^{\rm dr}(\theta^{+})-{(\theta^{3})}^{\rm
dr}(\theta^{-})\right]{1}^{\rm dr}(q),$ $\displaystyle E+\mathcal{P}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[[{(\theta^{2})}^{\rm
dr}{\theta}^{\rm dr}](\theta^{+})-[{(\theta^{2})}^{\rm dr}{\theta}^{\rm
dr}](\theta^{-})\right],$ $\displaystyle E+\mathcal{P}-\eta^{2}\rho$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[{(\theta^{2})}^{\rm
dr}(\theta^{+})+{(\theta^{2})}^{\rm dr}(\theta^{-})\right]{\theta}^{\rm
dr}(q).$ (SG.102)

To extract the hydrodynamic equations from the truncated GHD equations both
the space and time derivatives are considered. These relations are exact at
zero temperature, however, their subleading temperature correction is
$O(\beta^{-2})$. At lowest order in temperature the space/ time derivatives of
the hydrodynamic fields can be evaluated explicitly in terms of either
hydrodynamic fields or equivalently TBA quantities respectively. These
derivative identities, with $i=x,t$, are collected below

$\displaystyle\partial_{i}\rho$ $\displaystyle=$ $\displaystyle 2({1}^{\rm
dr}(\theta^{+}))^{2}(\partial_{i}q),$ $\displaystyle\partial_{i}p$
$\displaystyle=$ $\displaystyle\rho\partial_{i}\eta+\eta\partial_{i}\rho,$
$\displaystyle=$ $\displaystyle 2({1}^{\rm
dr}(\theta^{+}))^{2}\left(v_{F}\partial_{i}\eta+\eta(\partial_{i}q)\right),$
$\displaystyle\partial_{i}\mathcal{P}$ $\displaystyle=$
$\displaystyle\partial_{i}(\rho\eta^{2})+v_{F}^{2}\partial_{i}\rho,$
$\displaystyle=$ $\displaystyle 2({1}^{\rm
dr}(q))^{2}\left[2(\partial_{i}\eta)\eta
v_{F}+\left(\eta^{2}+v_{F}^{2}\right)\partial_{i}q\right],$
$\displaystyle\partial_{i}E$ $\displaystyle=$
$\displaystyle\eta\rho\partial_{i}\eta+\frac{\partial_{i}q}{v_{F}}(\mathcal{P}+E-\eta^{2}\rho),$
$\displaystyle=$ $\displaystyle(\partial_{i}\eta)2\eta{\theta}^{\rm
dr}_{s}(q){1}^{\rm dr}+(\partial_{i}q)\left[{(\theta^{2}_{s})}^{\rm
dr}(q)+\eta^{2}{1}^{\rm dr}(q)\right]{1}^{\rm dr},$
$\displaystyle\partial_{i}J_{E}$ $\displaystyle=$
$\displaystyle(\mathcal{P}+E)\partial_{i}\eta+\eta
v\rho\partial_{i}q-\eta^{2}\rho\partial_{i}\eta+\eta\partial_{i}E,$
$\displaystyle=$ $\displaystyle\partial_{i}\eta\left([{(\theta^{2}_{s})}^{\rm
dr}(q)+\eta^{2}{1}^{\rm dr}(q)]{\theta}^{\rm dr}(q)+2\eta^{2}{\theta}^{\rm
dr}_{s}(q){1}^{\rm dr}(q)\right)+\partial_{i}q\left(\eta{1}^{\rm
dr}(q)[{(\theta^{2}_{s})}^{\rm dr}(q)+\eta^{2}{1}^{\rm
dr}(q)]+2\eta({\theta}^{\rm dr}_{s}(q))^{2}\right).$

###  G.3 Proof of the low-temperature expansion of eq. (SC.39)

The goal of this section is to prove the identity

$\displaystyle\beta(2n-1)\partial_{\theta}n\to-\sum_{\sigma}\frac{\sigma}{\partial_{\theta}\epsilon(\theta^{\sigma})}\delta^{\prime}(\theta-\theta^{\sigma}).$
(SG.103)

Consider the following integral with a smooth function $f(\theta)$. Expansion
of the arguments will allow us to apply well known integral approximation
methods to evaluate the result

$\displaystyle\int d\theta
f(\theta)\beta(2n(\theta)-1)\partial_{\theta}n=-\int
d\theta\,f(\theta)\beta^{2}\frac{(1-\text{e}^{\beta\epsilon(\theta)})\text{e}^{\beta\epsilon}\epsilon^{\prime}(\theta)}{(1+\text{e}^{\beta\epsilon(\theta)})^{3}},$
(SG.104)

It is key that there are Fermi points that can be identified, specifically
$\epsilon(\theta^{\pm})=0$, such that expanding around these points the
thermal energy may be rewritten as

$\displaystyle\beta\epsilon(\theta)$ $\displaystyle\sim$
$\displaystyle\beta\epsilon(\theta^{\pm})+\beta\epsilon^{\prime}(\theta^{\pm})(\theta-\theta^{\pm)}+\frac{\beta}{2}\epsilon^{\prime\prime}(\theta^{\pm})(\theta-\theta^{\pm})^{2}+O((\theta-\theta^{\pm})^{2}),$
$\displaystyle=$ $\displaystyle
0+s+\frac{\epsilon^{\prime\prime}(\theta^{\pm})}{2\beta(\epsilon^{\prime}(\theta^{\pm}))^{2}}s^{2}+O\left(\frac{1}{\beta^{4}}\right),$
$\displaystyle\epsilon^{\prime}(\theta)$ $\displaystyle\sim$
$\displaystyle\epsilon^{\prime}(\theta^{\pm})+\epsilon^{\prime\prime}(\theta^{\pm})(\theta-\theta^{\pm})+O((\theta-\theta^{\pm})^{2}).$
(SG.105)

Transformation of the integration measure to our new measure $s$ is then given
by

$\displaystyle s=\beta\epsilon^{\prime}(\theta^{\pm})(\theta-\theta^{\pm})$
$\displaystyle ds=\beta\epsilon^{\prime}(\theta^{\pm})d\theta.$ (SG.106)

So that the integral under consideration is now given, in terms of these
variables $s$. For concreteness consider the upper half of the integral, such
that $\theta^{+}$ is expanded around. Furthermore, introduce
$\Lambda=\epsilon^{\prime\prime}(\theta^{+})/(2(\epsilon^{\prime}(\theta^{+}))^{2})$
and note that $\theta^{+}>\theta^{-}$ then the integral can be approximated as

$\displaystyle-$
$\displaystyle\int_{(\theta^{+}+\theta^{-})/2}^{\infty}d\theta\,f(\theta)\beta^{2}\epsilon^{\prime}(\theta)\frac{(1-\text{e}^{\beta\epsilon(\theta)})\text{e}^{\beta\epsilon}}{(1+\text{e}^{\beta\epsilon(\theta)})^{3}}\sim-\int^{\infty}_{\beta\epsilon(\theta^{+})(\theta^{-}-\theta^{+})/2}ds\,f\left(\frac{s}{\beta\epsilon^{\prime}(\theta^{\pm})}+\theta^{+}\right)\partial_{s}\left(\frac{\text{e}^{s+s^{2}\frac{\Lambda}{\beta}}}{(1+\text{e}^{s+s^{2}\frac{\Lambda}{\beta}})^{2}}\right),$
(SG.107)
$\displaystyle\sim-\beta\int^{\infty}_{-\infty}ds\,\left(f\left(\theta^{+}\right)+\frac{s}{\beta\epsilon^{\prime}(\theta^{\pm})}f^{\prime}\left(\theta^{+}\right)\right)\partial_{s}\left(\frac{\text{e}^{s}}{(1+\text{e}^{s})^{2}}+\frac{\Lambda}{\beta}\frac{\text{e}^{s}(1-\text{e}^{s})s^{2}}{(1+\text{e}^{s})^{3}}+O\left(\frac{1}{\beta^{2}}\right)\right).$

Then to leading order as $\beta\to\infty$ the integral yields

$\displaystyle\int^{\infty}_{\frac{\theta^{+}+\theta^{-}}{2}}d\theta
f(\theta)\beta(2n(\theta)-1)\partial_{\theta}n$ $\displaystyle=$
$\displaystyle-f(\theta^{+})\int^{\infty}_{-\infty}ds\,\beta\partial_{s}\frac{\text{e}^{s}}{(1+\text{e}^{s})^{2}}-\frac{f^{\prime}(\theta^{+})}{\epsilon^{\prime}(\theta^{+})}\int^{\infty}_{-\infty}ds\,s\partial_{s}\frac{\text{e}^{s}}{(1+\text{e}^{s})^{2}}$
(SG.108)
$\displaystyle+f(\theta^{+})\Lambda\epsilon^{\prime}(\theta^{+})\int_{-\infty}^{\infty}ds\,\frac{\text{e}^{s}(1-\text{e}^{s})s^{2}}{(1+\text{e}^{s})^{3}},$
$\displaystyle=$ $\displaystyle
f^{\prime}(\theta^{+})\frac{1}{\epsilon^{\prime}(\theta^{+})}.$

The first term vanishes on integration, likewise the third term is also
vanishing after integration by parts leaving only the second term. The
procedure is identical with the left Fermi points, albeit with an additional
sign flip and so it follows that

$\displaystyle\int^{\infty}_{-\infty}d\theta
f(\theta)\beta(2n(\theta)-1)\partial_{\theta}n=\left(f^{\prime}(\theta^{+})\frac{1}{\epsilon^{\prime}(\theta^{+})}-f^{\prime}(\theta^{-})\frac{1}{\epsilon^{\prime}(\theta^{-})}\right)+O\left(\frac{1}{\beta}\right).$
(SG.109)

Thus identify the final result with the identity (SC.39)

$\displaystyle\beta(2n-1)\partial_{\theta}n=-\sum_{\sigma}\frac{\sigma}{\partial_{\theta}\epsilon(\theta^{\sigma})}\delta^{\prime}(\theta-\theta^{\sigma}).$
(SG.110)

###  G.4 Specific heat at low temperatures

The low-temperature expansion for the specific heat $\tilde{\chi}_{e}$ can be
computed by standard methods

$\tilde{\chi}_{e}=\frac{1}{\rho}\lim_{T\to
0}T^{-1}\left(\frac{dE(\rho,T)}{dT}\right)\bigg{|}_{\rho={\rm const}}.$
(SG.111)

This will be obtained by determining the low-temperature energy at constant
density. Such a quantity is obtained by expanding first around a constant
chemical potential, then introducing a changing chemical potential correction
that keeps $\rho$ constant. By use of the general formula for a conserved
charge (SG.133) at constant chemical potential the low-temperature energy is
given in terms of $U(q)={\varphi}^{\rm dr}(q,-q)+{\varphi}^{\rm dr}(q,q)$ and
$V(q)={\varphi}^{\rm dr}(q,q)-{\varphi}^{\rm dr}(q,-q)$, in the comoving
frame, as

$\displaystyle\int d\theta n(\theta)\left(\frac{\theta^{2}}{2}\right)^{\rm
dr}$
$\displaystyle=\int_{-q(0)}^{q(0)}{\left[\frac{\theta^{2}}{2}\right]}^{\rm
dr}_{T=0}+\frac{\pi^{2}T^{2}}{3(\epsilon^{\prime}(q))^{2}}\left(\frac{({\theta}^{\rm
dr})^{2}}{v_{F}}-2({1}^{\rm dr})V{\left[\frac{\theta^{2}}{2}\right]}^{\rm
dr}-2({1}^{\rm
dr})\left(\frac{1}{v_{F}}-2U\right){\left[\frac{\theta^{2}}{2}\right]}^{\rm
dr}\right)(q).$ (SG.112)

To keep density constant the zero temperature Fermi points $q(0)$ must be
changed, such that the chemical potential is allowed to vary. Consider a
constant density $\rho_{\rm ref}=\rho(q_{B})$ and a second low-temperature
density at constant chemical potential $\rho(q(T))$. The low-temperature
corrections to density are known, see App. G.5, to be

$\displaystyle\rho(q(T))=\rho(q(0))-T^{2}f(q)=\rho(q(0))-\frac{\pi^{2}T^{2}}{3{\theta}^{\rm
dr}v_{F}}\left(2V(q)({1}^{\rm
dr}(q))+\left[\frac{1}{v_{F}}-2U(q)\right]({1}^{\rm dr}(q))\right).$ (SG.113)

By imposing $\rho(q(T))=\rho(q_{B})$ the condition of constant density
requires that

$\displaystyle\rho(q(T))=\rho(q(0))-T^{2}f(q)=\rho(q(0))+2({1}^{\rm
dr}(q(0)))(q_{B}-q(0)).$ (SG.114)

Consequently the Fermi points must satisfy

$\displaystyle 2(q_{B}-q(0))=-\frac{\pi^{2}T^{2}}{3{\theta}^{\rm
dr}v_{F}}\left(2V(q)+\left[\frac{1}{v_{F}}-2U(q)\right]\right),$ (SG.115)

to keep the density constant. Now inserting $q(0)=q_{B}+T^{2}f/2$ the low-
temperature correction to energy with constant density is determined to be

$\displaystyle\rho e(\rho,T)$ $\displaystyle=\rho
e(\rho,0)+\frac{\pi^{2}T^{2}}{3{\theta}^{\rm
dr}v_{F}}\left(2V(q)+\left[\frac{1}{v_{F}}-2U(q)\right]\right){\left[\frac{\theta^{2}}{2}\right]}^{\rm
dr},$
$\displaystyle+\frac{\pi^{2}T^{2}}{3(\epsilon^{\prime}(q))^{2}}\left(\frac{({\theta}^{\rm
dr})^{2}}{v_{F}}-2({1}^{\rm dr})V{\left[\frac{\theta^{2}}{2}\right]}^{\rm
dr}-2({1}^{\rm
dr})\left(\frac{1}{v_{F}}-2U\right){\left[\frac{\theta^{2}}{2}\right]}^{\rm
dr}\right)(q),$ $\displaystyle=\rho e(\rho,0)+\frac{\pi^{2}T^{2}}{3v_{F}}.$
(SG.116)

An application of the temperature derivative on the above formula then implies
the sought-after expression for the specific heat

$\displaystyle\tilde{\chi}_{e}=\frac{1}{T}\,(\partial_{T}e)_{\rho={\rm
const}}=\frac{1}{T}\,\partial_{T}\left(\frac{E_{s}}{\rho}\right)_{\rho={\rm
const}}=\frac{2\pi^{2}}{3\rho v_{F}}.$ (SG.117)

###  G.5 Hydrodynamic field corrections at $O(T^{2})$

In general a Sommerfeld approximation can be applied to obtain a general low-
temperature expansion Bertini and Piroli (2018)

$\displaystyle\int_{-\infty}^{\infty}n(\theta)f(\theta)$ $\displaystyle=$
$\displaystyle\int_{-q}^{q}f(\theta)+\frac{\pi^{2}T^{2}}{6(\epsilon^{\prime}(q))^{2}}\left[f^{\prime}(q)-f^{\prime}(-q)-\left(\frac{\epsilon^{\prime\prime}(q)}{\epsilon^{\prime}(q)}-U(q)\right)(f(q)+f(-q))\right].$
(SG.118)

This identity is sufficient to determine the subleading temperature
corrections to the integrals, which are key to determining the hydrodynamic
fields. This expansion as written characterizes the low-temperature expansion
for constant chemical potential. To account for the low-temperature expansion
of the relevant charges it will be relevant to consider both the temperature
correction to the dressed function, occurring due to the dressing, and the
corrections in the integral noted above. The low-temperature corrections due
to dressing a function is considered first. It should also be noted that
whether the function being dressed is even or odd also impacts the expansion
and so these cases are considered separately.

The low-temperature corrections to a general dressed function ${g}^{\rm dr}$
is considered. The dressed functions are assumed to be either symmetric
${g}^{\rm dr}(q)={g}^{\rm dr}(-q)$ or anti-symmetric ${g}^{\rm
dr}(q)=-{g}^{\rm dr}(-q)$. Recall the definition for a dressed function

$\displaystyle{g}^{\rm
dr}(\theta)=g(\theta)+\int_{-\infty}^{\infty}\varphi(\theta-\mu)n(\mu){g}^{\rm
dr}(\mu).$ (SG.119)

For the purposes of keeping track of the small temperature limits the zero
temperature dressed function is also introduced

$\displaystyle{g}^{\rm
dr}_{0}(\theta)=g(\theta)+\int^{\infty}_{-\infty}\varphi(\theta-\alpha)n_{0}(\alpha){g}^{\rm
dr}_{0}(\alpha)=g(\theta)+\int^{q}_{-q}\varphi(\theta-\alpha){g}^{\rm
dr}_{0}(\alpha).$ (SG.120)

So the goal is to consider the low T expansion of the integral on the RHS of
the dressing relation. So identify the variable for use of the Sommerfeld
expansion of eq. (SG.118) as

$\displaystyle f(\theta,\alpha)$ $\displaystyle=$
$\displaystyle\varphi(\theta-\alpha){g}^{\rm dr}(\alpha).$

For even functions, $g(q)=g(-q)$, the low-temperature expansion is carried out
with

$\displaystyle f^{\prime}(\theta,q)-f^{\prime}(\theta,-q)$ $\displaystyle=$
$\displaystyle-[(\partial_{\theta}\varphi(\theta-q)-\partial_{\theta}\varphi(\theta+q))+(\varphi(\theta,q)+\varphi(\theta,-q))({\varphi}^{\rm
dr}(q,q)-{\varphi}^{\rm dr}(q,-q))]{g}^{\rm dr}(q)$
$\displaystyle+[\varphi(\theta-q)+\varphi(\theta+q)]{g}^{\rm dr}(q),$
$\displaystyle f(\theta,q)+f(\theta,-q)$ $\displaystyle=$
$\displaystyle(\varphi(\theta,q)+\varphi(\theta,-q)){g}^{\rm dr}(q).$ (SG.122)

From the above the small temperature correction reads

$\displaystyle({g}^{\rm dr}-{g}^{\rm dr}_{0})$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}T^{2}{g}^{\rm
dr}(q)}{6(\epsilon^{\prime}(q))^{2}}\left(-[{(\partial_{\theta}\varphi(\theta-q)-\partial_{\theta}\varphi(\theta+q))}^{\rm
dr}\frac{}{}\right.$ (SG.123) $\displaystyle\left.+({\varphi}^{\rm
dr}(\theta,q)+{\varphi}^{\rm dr}(\theta,-q))({\varphi}^{\rm
dr}(q,q)-{\varphi}^{\rm
dr}(q,-q))]-\left[\frac{1}{v_{F}}-2U(q)\right]({\varphi}^{\rm
dr}(\theta,q)+{\varphi}^{\rm dr}(\theta,-q))\right)$
$\displaystyle+\frac{\pi^{2}T^{2}{(\partial g)}^{\rm
dr}(q)}{6(\epsilon^{\prime}(q))^{2}}\left[{\varphi}^{\rm
dr}(\theta,q)+{\varphi}^{\rm dr}(\theta,-q)\right].$

Note that on the RHS all functions are evaluated at zero temperature, however
either the zero temperature or finite temperature functions can be used
without altering the $O(T^{2})$ order and so the subscript is suppressed.

Now consider the antisymmetric case $g(q)=g(-q)$. Now the identities for the
low-temperature expansion read

$\displaystyle f^{\prime}(\theta,q)-f^{\prime}(\theta,-q)$ $\displaystyle=$
$\displaystyle-(\varphi^{\prime}(\theta-q)+\varphi^{\prime}(\theta+q)){\theta}^{\rm
dr}(q)+(\partial_{\theta}{\theta}^{\rm
dr})(q)(\varphi(\theta-q)-\varphi(\theta+q)),$ $\displaystyle
f(\theta,q)+f(\theta,-q)$ $\displaystyle=$
$\displaystyle(\varphi(\theta,q)-\varphi(\theta,-q)){g}^{\rm dr}(q).$ (SG.124)

With the temperature corrections to ${g}^{\rm dr}$ given by

$\displaystyle({g}^{\rm dr}-{g}^{\rm dr}_{0})$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}T^{2}{g}^{\rm
dr}(q)}{6(\epsilon^{\prime}(q))^{2}}\left(-[{(\partial_{\theta}\varphi(\theta-q)+\partial_{\theta}\varphi(\theta+q))}^{\rm
dr}\frac{}{}\right.$ (SG.125) $\displaystyle\left.+({\varphi}^{\rm
dr}(\theta,q)-{\varphi}^{\rm dr}(\theta,-q))({\varphi}^{\rm
dr}(q,q)+{\varphi}^{\rm
dr}(q,-q))]-\left[\frac{1}{v_{F}}-2U(q)\right]({\varphi}^{\rm
dr}(\theta,q)-{\varphi}^{\rm dr}(\theta,-q))\right)$
$\displaystyle+\frac{\pi^{2}T^{2}{(\partial g)}^{\rm
dr}(q)}{6(\epsilon^{\prime}(q))^{2}}\left[{\varphi}^{\rm
dr}(\theta,q)-{\varphi}^{\rm dr}(\theta,-q)\right].$

These relations are then combined with