# Effective thermalization of a many-body dynamically localized Bose gas

Vincent Vuatelet  Université de Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de
Physique des Lasers, Atomes et Molécules, F-59000 Lille, France  Adam Rançon
Université de Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de Physique des
Lasers, Atomes et Molécules, F-59000 Lille, France

###### Abstract

Dynamical localization is the analog of Anderson localization in momentum
space, where the system’s energy saturates and the single-particle wave-
functions are exponentially localized in momentum space. In the presence of
interactions, in the context of a periodically kicked Bose gas, it has been
argued that dynamical localization persists. Focusing on the Tonks (strongly
interacting) regime, we show that the many-body dynamically localized phase is
effectively thermal, a clear deviation from the breaking of ergodicity
observed in standard many-body localized systems. We relate the effective
temperature to the driving parameters, and thus quantitatively describe the
loss of coherence at large distances in this phase. Contrary to the non-
interacting case, the momentum distribution decays as a power-law at large
momenta, characterized by an effectively thermal Tan’s contact. This is a rare
example where driving and many-body (dynamical) localization lead to an
effectively ergodic state.

##  I Introduction

Anderson localization of classical and quantum waves is a universal phenomenon
induced by disorder Anderson (1958); Evers and Mirlin (2008). Whether or not
it survives in the presence of interactions has been under intense scrutiny in
recent years, both theoretically and experimentally Nandkishore and Huse
(2015); Abanin  _et al._ (2019). It is now well understood that while
interactions tend to destroy localization, a strong enough disorder will give
rise to Many-Body Localization (MBL), at least in low dimensions. MBL can be
understood in terms of an effective integrability due to the existence of an
extensive number of local integrals of motions, breaking ergodicity and
preventing thermalization Serbyn  _et al._ (2013); Huse  _et al._ (2014). The
same mechanism prevents driven MBL systems from absorbing an infinite amount
of energy and thus prevents runaway heating Ponte  _et al._ (2015a, b).

Dynamical localization is the quantum chaos analog of Anderson localization,
but taking place in momentum space Fishman  _et al._ (1982). In the
paradigmatic quantum kicked rotor (QKR), periodic kicks give rise to a
ballistic propagation in momentum space, while the (pseudo) random phase
accumulated during the free propagation in between kicks by each momentum
state plays the role of disorder, resulting in destructive quantum
interferences and dynamical localization. Experimental realizations of the
atomic QKR have allowed for detailed investigations of the Anderson physics:
observation of Anderson transition Chabé  _et al._ (2008), characterization of
its critical properties Lemarié  _et al._ (2010); Lopez  _et al._ (2012),
localization at the upper critical dimension Manai  _et al._ (2015), the
effects symmetries on weak localization Hainaut  _et al._ (2018a), and
classical-to-quantum transition at early times Hainaut  _et al._ (2018b).

Whether interactions destroy dynamical localization is a fundamental question
that challenges our understanding of driven interacting quantum systems. This
has been studied for various toy-models Adachi  _et al._ (1988); Wen-Lei and
Quan-Lin (2009); Keser  _et al._ (2016); Rozenbaum and Galitski (2017);
Notarnicola  _et al._ (2018, 2020), as well as for the kicked Lieb-Liniger
model, a realistic model for cold atoms experiments Bloch  _et al._ (2008). At
the mean-field level, it has been argued both on theoretical and numerical
grounds that interactions destroy dynamical localization, which is replaced by
a subdiffusion in momentum space Shepelyansky (1993); Pikovsky and
Shepelyansky (2008); Flach  _et al._ (2009); Gligoricá  _et al._ (2011);
Cherroret  _et al._ (2014); Lellouch  _et al._ (2020). However, it is well
known that mean-field theory breaks down in one dimension Cazalilla  _et al._
(2011), questioning these predictions. Beyond mean-field, an early study for
two bosons hinted that interactions may also destroy dynamical localization
Qin  _et al._ (2017), but the validity of these results has been recently
questioned Chicireanu and Rançon (2021). Finally, Rylands et al. have argued
that dynamical localization persists in the presence of interactions, leading
to a Many-Body Dynamically Localized (MBDL) phase Rylands  _et al._ (2020).
The MBDL phase can be described by a steady-state density matrix
$\hat{\rho}_{ss}$, which in general should belong to a generalized Gibbs
ensemble Lazarides  _et al._ (2014); Vidmar and Rigol (2016). However, this
regime and its density matrix have yet to be characterized.

In this article, we study the MBDL phase of the kicked Lieb-Liniger gas in the
infinite interaction (Tonks) regime. Our main result is that the steady-state
of the system is very well described by the density matrix of a thermal gas,
seemingly in contradiction with the fact that the system is integrable and has
an extensive number of conserved charges (the occupation of the Floquet
eigenstates). We stress that this effective thermalization takes place while
the system is still periodically driven. This is a rare instance where driving
and Many-Body (Dynamical) Localization give rise to an effectively ergodic
state. We relate this temperature to the system’s parameters (kicks strength
and period). This allows us to quantitatively characterize two experimentally
relevant observables: the momentum distribution that does not decay
exponentially, as in the non-interacting limit, but as a power-law, which is
to be expected for interacting quantum systems Olshanii and Dunjko (2003); Tan
(2008); and the coherence function, which decays exponentially, demonstrating
the absence of phase coherence.

The article is organized as follows. Sec. II, we present the model and the
physical observables studied. In Sec. III, we summarize our numerical results
for the momentum distribution and the coherence function in the localized
regime, while in Sec. IV we interpret those results in terms of an effective
thermalization of the system, and we give arguments to explain such a
thermalization in Sec. V. The discussion of our results appears in Sec. VI.

##  II Model

We consider $N$ interacting bosons of mass $m$, the dynamics of which is
described by the periodic Hamiltonian

$\hat{H}(t)=\sum_{i}\left(\frac{\hat{p}_{i}^{2}}{2}+K\cos(\hat{x}_{i})\sum_{n}\delta(t-n)\right)+g\sum_{i<j}\delta(\hat{x}_{i}-\hat{x}_{j}).$
(1)

The one-body term corresponds to the QKR Hamiltonian $\hat{H}_{QKR}(t)$, while
the other describes the contact interaction (we also define
$\hat{H}_{TG}=\hat{H}|_{K=0}$). Here and in the following, time is in units of
the period $\tau$ of the kicks and length in unit of the inverse of the kick-
potential wavenumber $k_{K}$. Momenta are normalized such that
$[\hat{x}_{i},\hat{p}_{j}]=i{\mathchar 22\relax\mkern-9.0muk}\delta_{ij}$,
with ${\mathchar 22\relax\mkern-9.0muk}=\hbar k_{K}^{2}\tau/m$ the effective
Planck constant. The system is of size $L=2\pi$, and we assume periodic
boundary conditions, implying that momenta are quantized in unit of
${\mathchar 22\relax\mkern-9.0muk}$ (we will use units such that the Boltzmann
constant $k_{B}=1$).

In the free case $(g=0)$, we recover the physics of the QKR, where any single-
particle wave-function is localized in momentum space at long time (larger
than the localization time) and decays exponentially in momentum space, with
the same “localization length” $p_{loc}$ (for larger $K/{\mathchar
22\relax\mkern-9.0muk}$, one finds $p_{loc}\propto K^{2}/{\mathchar
22\relax\mkern-9.0muk}$ Shepelyansky (1986); Lemarié  _et al._ (2009)) 111To
be precise, we define $p_{loc}$ of a single-particle state $|\psi\rangle$ as
$p_{loc}^{2}=\langle\psi|\hat{p}^{2}|\psi\rangle-\langle\psi|\hat{p}|\psi\rangle^{2}$,
which is proportional to the momentum scale on which the dynamically localized
states decay exponentially.. In particular, the total energy of the system
saturates to a constant value at long time.

Here, we focus on the Tonks regime, $g\to\infty$, allowing us to write the
exact time-dependent wave-function $\Psi_{B}(\\{x\\};t)$ of the system using
the Bose-Fermi mapping Girardeau (1960); Lenard (1964); Buljan  _et al._
(2008); Jukić  _et al._ (2008); Pezer  _et al._ (2009),

$\Psi_{B}(\\{x\\};t)=\prod_{i<j}{\rm sign}(x_{i}-x_{j})\Psi_{F}(\\{x\\};t),$
(2)

where $\Psi_{F}(\\{x\\};t)=\frac{1}{\sqrt{N!}}\det[\psi_{i}(x_{j},t)]$ is the
free fermions wave-function constructed from the $N$ single-particle orbitals
$\psi_{i}(x,t)$, which evolve according to the QKR Hamiltonian, $i{\mathchar
22\relax\mkern-9.0muk}\partial_{t}|\psi_{i}(t)\rangle=\hat{H}_{QKR}(t)|\psi_{i}(t)\rangle$.
We assume that the system starts in its ground state, i.e. the fermionic wave-
function describes a Fermi sea with Fermi momentum $p_{F}\propto N$ and ground
state energy $E_{0}$.

For a Tonks gas, all bosonic local observables (such as the energy or the
density) are given by those of free fermions. Therefore, since the dynamics of
the single-particle orbitals $\psi_{i}(x,t)$ is that of the non-interacting
QKR, we directly infer that they all dynamically localize at long time. The
energy (of both fermions and bosons) will thus saturate to a finite value
$E_{f}\simeq E_{0}+N\frac{p_{loc}^{2}}{2}$ for time larger than the
localization time, which is interpreted as MBDL Rylands  _et al._ (2020), see
Fig. 1. Since the fermions orbitals reach a steady-state in the MBDL phase, we
expect the system to be described by a steady-state density matrix
$\hat{\rho}_{ss}$, belonging a priori to the generalized Gibbs ensemble
Lazarides  _et al._ (2014), see discussion in Sec. V. Here we focus on the
properties of the system in this MDBL steady state, and thus do not write time
dependence of observables.

Non-local observables such as the steady-state one-body density matrix (OBDM)

$\begin{split}\rho(x,y)=N\int dx_{2}\ldots
dx_{N}&\Psi_{B}^{*}(x,x_{2},\ldots,x_{N})\times\\\
&\Psi_{B}(y,x_{2},\ldots,x_{N}),\end{split}$ (3)

and its Fourier transform, the momentum distribution

$n_{k}=\frac{1}{L}\int dxdy\,e^{ik(x-y)}\rho(x,y),$ (4)

are significantly different with those of free fermions. Since dynamical
localization is a non-local phenomenon, we therefore expect these observables
to significantly differ from that of free particles 222This is to be
contrasted with the case of a Tonks gas loaded in a random potential, where
localization in real space of the fermions is naturally preserved for the
bosons Radić  _et al._ (2010); Seiringer and Warzel (2016).. We therefore
focus on those observables in the steady-state (time much larger than the
localization time) in the following.

The time-evolution of each single-particle orbital is performed numerically by
discretizing space and using Fast Fourier Transform to alternate between real
space for the kicks and momentum space for the free propagation. The
observables are computed using the method of Refs. Rigol and Muramatsu (2005a,
b).

Figure 1: Time evolution of the total energy of the system $E_{f}(t)$ for
$N=61$ particles for $K=20,30$ and $40$ (${\mathchar
22\relax\mkern-9.0muk}=6$). Figure 2: Steady-state momentum distribution for
$N=51$ particles at ${\mathchar 22\relax\mkern-9.0muk}=6$ for $K=20$, $30$ and
$40$ in log-log scale (the different $n_{k}$ have been shifted for better
visibility in the main panel). The dashed line shows the asymptotic behavior
$n_{k}\simeq\mathcal{C}_{ss}/k^{4}$ at large momenta, with $\mathcal{C}_{ss}$
computed using the effectively thermal density matrix (see text).The inset
shows the same quantities in linear scale. Figure 3: Steady-state coherence
function $g_{1}(r)$ for $N=101$ particles at ${\mathchar
22\relax\mkern-9.0muk}=6$ for $K=20$, $30$ and $40$. Insets (a): Same data, in
semi-log scale, emphasizing the exponential decay in the MBDL, compared to the
$1/\sqrt{r}$ decay of the initial condition (dashed curve); (b): Occupation of
the zero-momentum state $n_{k=0}$. It grows as $\sqrt{N}$ in the ground state
(dotted line), but saturate to a finite value in the MBDL regime.

##  III MBDL momentum distribution and coherence

The groundstate of the Tonks gas is characterized by quasi-long-range order,
$n_{k}\propto 1/\sqrt{k}$ at small momenta and $n_{k=0}\propto\sqrt{N}$, where
the sublinear scaling implies the absence of true long-range order Vaidya and
Tracy (1979). Fig. 2 shows the momentum distribution in the ground state and
in the localized regime for $N=51$ bosons, ${\mathchar
22\relax\mkern-9.0muk}=6$ and various values of $K$, in log-log scale. The
divergence at small momenta of the momentum distribution is rounded (see
inset), while we observe a power-law decay at large momenta,
$n_{k}\simeq\mathcal{C}/k^{4}$. This behavior is a universal feature of
interacting quantum systems, where $\mathcal{C}$ is the so-called Tan’s
contact Olshanii and Dunjko (2003); Tan (2008). We conclude that while the
interactions do not destroy dynamical localization, in the sense that the
system does not heat up to infinite temperature, they do significantly alter
the exponential localization in momentum distribution of the bosons.

The coherence of the Tonks gas in the MBDL regime can also be characterized by
the coherence function

$g_{1}(r)=\frac{1}{L}\int dR\,\rho(R-r/2,R+r/2).$ (5)

In its ground state, the gas has algebraic correlations, $g_{1}(r,t=0)\propto
1/\sqrt{r}$, corresponding to quasi-long-range order Cazalilla  _et al._
(2011). Fig. 3 shows that in the MBDL regime, the coherence function decays
exponentially fast at large distance, implying that the kicks have destroyed
the coherence of the quasi-condensate. This is in agreement with the fact that
$n_{k=0}$ does not scale with the number of particles (see inset (b) of Fig.
3).

##  IV Effective thermalization of MBDL

The absence of quasi-long-range coherence of the localized regime is similar
to that of a thermal Tonks gas Cazalilla  _et al._ (2011). We now show that
the system is very well described in the MDBL by the a thermal density matrix
$\hat{\rho}_{ss}\simeq\hat{\rho}_{th}$, where $\hat{\rho}_{th}$ is the thermal
density matrix of the Tonks gas

$\hat{\rho}_{th}\propto e^{-(\hat{H}_{TG}-\mu_{eff}\hat{N})/T_{eff}},$ (6)

with effective temperature $T_{eff}$ and effective chemical potential
$\mu_{eff}$ that depends on the system’s parameters and the number of
particles.

Thanks to the Bose-Fermi mapping, if there is indeed effective thermalization,
we expect the momentum distribution $n_{k}^{F}$ of the underlying free
fermions to be described by a Fermi-Dirac distribution, allowing us to extract
$T_{eff}$ and $\mu_{eff}$. We therefore begin by analyzing the thermal
properties of the free fermions, and then address the thermal-like properties
of the Tonks gas in the localized regime.

###  IV.1 Fermions

An example of the momentum distribution of the fermions in the localized
regime is shown in Fig. 4 (symbols). Contrary to the momentum distribution of
the bosons, it is rather noisy, as typical for disordered systems. To better
fit the momentum distribution of the fermions, it is convenient to introduce a
modified QKR Hamiltonian depending on a parameter $q$ 333The quantity $q$
plays the role of a conserved quasi-momentum in the single-particle QKR
without periodic boundary conditions.

$\hat{H}_{q}=\frac{(\hat{p}+q{\mathchar
22\relax\mkern-9.0muk})^{2}}{2}+K\cos(\hat{x})\sum_{n}\delta(t-n).$ (7)

Note that we never average the bosonic observables (we always consider the
physical value $q=0$), such as the OBDM or the momentum distribution. The
highly non-linear transformation relating the bosonic observables to that of
the fermions averages out the fluctuations. In App. A, we show below that the
temperature that can be estimated from $q=0$ is very well correlated with that
extracted from the average fermionic distribution.

In Fig. 4, in addition to the momentum distribution $n_{k}^{F}$ at $q=0$
discussed above, we also show the $n_{k}^{F}$ the momentum distribution
averaged over 150 random values of $q$ (full line). The smoothing effect of
the averaging procedure is very clear. On the same figure, we also show a
Fermi-Dirac distribution at an effective temperature $T_{eff}$ and effective
chemical potential $\mu_{eff}$ such that this thermal distribution explains
very well the data (dashed line).

Figure 4: Comparaison between raw and averaged distribution for the many-body
momentum distribution. In this case, $N=61$, $K=30$, ${\mathchar
22\relax\mkern-9.0muk}=6$.  Figure 5: Effective temperature
$T_{eff}/\varepsilon_{F}$ as a function of $p_{loc}/p_{F}$ for various
particle numbers. The collapse of the data shows the linear scaling for small
enough $p_{loc}/p_{F}$,
$T_{eff}/\varepsilon_{F}\simeq\frac{2\sqrt{3}}{\pi}p_{loc}/p_{F}$ (black
line). Inset: Momentum distribution of the fermions $n_{k}^{F}$ in the
localized regime (symbols), fitted by a Fermi-Dirac distribution with
temperature $T_{eff}$ and chemical potential $\mu_{eff}$, for $N=101$ and
${\mathchar 22\relax\mkern-9.0muk}=6$.

The effective temperature and chemical potential are obtained by imposing that

$\begin{split}&\sum_{k\in\mathbb{Z}}f_{FD}(k,T_{eff},\mu_{eff})=N,\\\
&\sum_{k\in\mathbb{Z}}\frac{{\mathchar
22\relax\mkern-9.0muk}^{2}k^{2}}{2}f_{FD}(k,T_{eff},\mu_{eff})=E_{f},\end{split}$
(8)

where $E_{f}$ is the energy obtained from the averaged momentum distribution
$n_{k}^{F}$, and $f_{FD}$ is the Fermi-Dirac distribution

$f_{FD}(k,T,\mu)=\frac{1}{e^{\frac{{\mathchar
22\relax\mkern-9.0muk}^{2}k^{2}-\mu}{2T}}+1}.$ (9)

We observe that the fit is very good, see the inset of Fig. 5, as long as
$p_{F}\gg p_{loc}$ (corresponding to small enough $K$), see also App. A for a
detailed analysis of the parameters regime where the thermal fit works. We
focus on this effectively thermal regime here. This corresponds to low
effective temperatures compared to the initial Fermi energy
$\varepsilon_{F}=p_{F}^{2}/2$, which allows us to find an explicit expression
of the effective temperature in terms of the two natural quantities $p_{loc}$
and $p_{F}$.

The initial condition of the system corresponds to the ground state, the
energy of which is

$E_{0}=\frac{N\varepsilon_{F}}{3},$ (10)

for a one-dimensional Fermi gas, with $\varepsilon_{F}=\frac{p_{F}^{2}}{2}$
the Fermi energy, which in our units read $\varepsilon_{F}=\frac{N^{2}}{8}$
($N\gg 1$). On the other hand, in the localized regime, the final energy reads

$E_{f}=E_{0}+N\frac{p_{loc}^{2}}{2}.$ (11)

Assuming that the system is thermal, the Sommerfeld expansion of the energy
gives

$E(T_{eff})\simeq\frac{N\varepsilon_{F}}{3}+\frac{N\pi^{2}}{12}\frac{T_{eff}^{2}}{\varepsilon_{F}}+\ldots,$
(12)

Equating $E_{f}=E(T_{eff})$, we obtain

$\frac{T_{eff}}{\varepsilon_{F}}=\frac{2\sqrt{3}}{\pi}\frac{p_{loc}}{p_{F}}.$
(13)

Note that the effective temperature is indeed small (compared to the Fermi
energy) for small $p_{loc}/{p_{F}}$, validating our initial assumption.

Fig. 5 shows that indeed Eq. (13) works very well for $p_{loc}/{p_{F}}\ll 1$.
Note that while the effective temperature scales linearly with the particle
number, the relative thermal broadening of the Fermi distribution
$T_{eff}/\varepsilon_{F}$ vanishes as $N^{-1}$.

###  IV.2 Implications for the bosons

Assuming that the steady-state density matrix $\hat{\rho}_{ss}$ is thermal
allows us to quantitatively characterize the momentum distribution and the
coherence function of the Tonks gas in the localized regime, an a priori
formidable task without this insight.

At short distance, the coherence function of a Tonks gas is known to be non-
analytic due to the interactions,
$g_{1}(r)\sim\frac{\pi\mathcal{C}}{6L}|r|^{3}$. For a thermal Tonks gas of $N$
bosons at temperature $T$, the contact reads
$\mathcal{C}_{th}(T,N)=\frac{8NE(T,N)}{L^{2}{\mathchar
22\relax\mkern-9.0muk}^{2}}$ Vignolo and Minguzzi (2013). We therefore infer
that the contact in the MBDL regime $\mathcal{C}_{ss}$ should be given by
$\mathcal{C}_{ss}=\mathcal{C}_{th}(T_{eff},N)$. Fig. 2 shows that the power-
law decay is very well explained by $\mathcal{C}_{th}(T_{eff},N)/k^{4}$,
showed as dashed lines.

At long distances, the exponential decay of $g_{1}(r)$ of a Tonks gas at
finite temperature, $g_{1}(r)\propto e^{-2|r|/r_{c}}$, is also known Cazalilla
_et al._ (2011); Its  _et al._ (1991), and in the low-temperature limit we
expect $r_{c}=\frac{{\mathchar 22\relax\mkern-9.0muk}v_{F}}{T_{eff}}$, where
$v_{F}=\frac{{\mathchar 22\relax\mkern-9.0muk}N}{2}$ is the Fermi velocity in
our units. Therefore, due to the effective thermality of the MBDL phase, we
expect the correlation length $r_{c}$ to be independent of the particle number
and to be inversely proportional to $p_{loc}$. Combining the thermal scaling
and the expression in Eq. (13), we do expect the scaling
$r_{c}=\frac{\pi}{\sqrt{3}}\frac{{\mathchar 22\relax\mkern-9.0muk}}{p_{loc}}$,
which can be rewritten as

$r_{c}p_{F}=\frac{{\mathchar
22\relax\mkern-9.0muk}\pi}{\sqrt{3}}\frac{p_{F}}{p_{loc}}.$ (14)

Fig. 6 shows good agreement between Eq. (14) and the correlation length
extracted from the steady-state (see App. B for details). The inset shows that
Eq. (14) describes well the exponential decay of the coherence function.

Figure 6: Correlation length $r_{c}$ as a function of $p_{loc}$ for various
$N$. The collapse of the data shows that it is independent of the particle
number. The line corresponds to the scaling
$r_{c}=\frac{\pi}{\sqrt{3}}\frac{{\mathchar 22\relax\mkern-9.0muk}}{p_{loc}}$.
Inset: Coherence function $g_{1}(r)$ (blue line) and the expected exponential
decay with $r_{c}=\frac{{\mathchar
22\relax\mkern-9.0muk}\pi}{\sqrt{3}p_{loc}}$, for $N=101$, $K=40$, ${\mathchar
22\relax\mkern-9.0muk}=6$.

##  V Explanation of the effective thermalization

Let us now argue why the MBDL steady-state appears thermal. This is best
understood using the fermionic degrees of freedom, which are non-interacting
and evolve according to $\hat{H}_{QKR}$. Introducing the evolution operator
over one period $\hat{U}_{QKR}$ and its Floquet eigenstates
$\hat{U}_{QKR}|\phi_{\alpha}\rangle=e^{-i\omega_{\alpha}}|\phi_{\alpha}\rangle$,
it can be written in second quantization as
$\hat{U}_{QKR}=\exp\left(-i\sum_{\alpha}\omega_{\alpha}\hat{f}^{\dagger}_{\alpha}\hat{f}_{\alpha}\right)$.
Now, the occupation numbers of the Floquet eigenstates
$n_{\alpha}=\langle\hat{f}^{\dagger}_{\alpha}\hat{f}_{\alpha}\rangle$ are
obviously constants of motion, and since there is an extensive number of them,
the system is integrable. We therefore expect the steady-state to be described
by the (periodic) generalized Gibbs ensemble (GGE) Collura  _et al._ (2013);
Lazarides  _et al._ (2014), $\hat{\rho}_{ss}\simeq\hat{\rho}_{GGE}$, with

$\hat{\rho}_{GGE}\propto
e^{-\sum_{\alpha}\lambda_{\alpha}\hat{f}^{\dagger}_{\alpha}\hat{f}_{\alpha}},$
(15)

where the Lagrange multipliers
$\lambda_{\alpha}=\log((1-n_{\alpha})/n_{\alpha})$ are such that ${\rm
Tr}\left(\hat{\rho}_{GGE}\hat{f}^{\dagger}_{\alpha}\hat{f}_{\alpha}\right)=n_{\alpha}$
444The Periodic Gibbs ensemble allows for a time dependence of the steady-
state density matrix $\rho_{GGE}(t)$ with $0\leq t<1$, with period $1$. We
only consider here the case $t=1^{-}$ (just before the next kick). We have
checked that other choices do not change the qualitative picture.. It is a
priori surprising that this GGE density matrix, depending on an extensive
number of Lagrange multipliers, is well described by a thermal density matrix,
depending only on $T_{eff}$ and $\mu_{eff}$. To understand this, we write it
in terms of a non-local operator in momentum space

$\hat{\rho}_{GGE}\propto
e^{-\sum_{p,q}M_{p,q}\hat{f}^{\dagger}_{p}\hat{f}_{q}},$ (16)

with $M_{p,q}=\sum_{\alpha}\langle
p|\phi_{\alpha}\rangle\lambda_{\alpha}\langle\phi_{\alpha}|q\rangle$.
Therefore, for generic dynamics and initial states, one should expect a large
number of non-local conserved quantities and a clear departure from a thermal
state. However, in the present case, we note that the Floquet eigenstates are
exponentially localized in momentum space, over a scale of order
$p_{loc}$Fishman  _et al._ (1982), implying that: (i) only the states with
$|p_{\alpha}|\lesssim p_{F}+p_{loc}$ are occupied ($n_{\alpha}\simeq 1\,({\rm
resp.}\,0)$ for $|p_{\alpha}|\ll p_{F}$ (resp. $|p_{\alpha}|\gg p_{F}$)), with
$n_{\alpha}$ interpolating between $1$ and $0$ around $|p_{\alpha}|\simeq
p_{F}$ on a width of order $p_{loc}$; (ii) $M_{p,q}\simeq 0$ if $|p-q|\gg
p_{loc}$, meaning that it is almost diagonal, $M_{p,q}\simeq\delta_{p,q}h_{p}$
for some $h_{p}$. In practice, we find that $h_{p}\simeq
f_{p}\equiv(-\mu_{eff}+p^{2}/2)/T_{eff}$ to a good approximation as shown in
Fig. 7, justifying the effective thermalization
$\hat{\rho}_{ss}\simeq\hat{\rho}_{th}$. Note that this effective
thermalization depends crucially on point (i), as other initial conditions far
from the ground state, or a too large $p_{loc}$ implying that too many
eigenstate are populated, do not allow for a description of the steady-state
in terms of a thermal density matrix Vua .

Figure 7: Comparison of $h_{p}$ (symbols) with $f_{p}$ for varying $K$, $N=51$
and ${\mathchar 22\relax\mkern-9.0muk}=6$.

The fact that $M_{p,q}$ is not exactly diagonal means that the steady state is
not perfectly described by a thermal density matrix. In particular, it implies
that the natural orbitals of the OBDM are not exactly plane-waves, but have
width $p_{loc}$ and that the two-dimensional Fourier transform of the OBDM,
$L^{-1}\int dxdy\,e^{ik(x+y)}\rho(x,y;t)$ decays exponentially as
$\exp(-|k|/p_{loc})$ instead of being $N\delta_{k,0}$, see App. C.

##  VI Conclusion

We have studied the steady-state of a kicked Tonks gas. While dynamical
localization is preserved by the interactions, in the sense that the system
does not heat up to infinite temperature, we have shown that the momentum
distribution of the bosons is not exponentially localized, as in the non-
interacting case. Instead, it decays as a power-law given by Tan’s contact, as
expected for an interacting quantum many-body system. We have also shown that
the steady-state is very well described by a thermal density matrix, with an
effective temperature that scales linearly with both the Fermi and
localization momenta. This steady-state is therefore both many-body
dynamically localized and well described by a small number of constant of
motions, corresponding to the particle number and the energy of the localized
state, even though the dynamic is integrable, with an extensive number of
conserved quantities. This is in contrast with standard MBL, where ergodicity
breaking corresponds to emergent integrability and the existence of an
extensive set of quasi-local integrals of motion Abanin  _et al._ (2019). MBDL
should be observable in state-of-the-art cold atoms experiments by measuring
the steady-state momentum distribution using for instance the methods of Refs.
van Amerongen  _et al._ (2008); Fabbri  _et al._ (2011). As long as the
initial temperature is smaller than the effective temperature, effective
thermalization should dominate 555Using
$T_{eff}\simeq\sqrt{\varepsilon_{loc}\varepsilon_{F}}$, with
$\varepsilon_{loc}$ the localization energy per particles, and the estimates
$\varepsilon_{F}\simeq E_{R}N^{2}$ and $\varepsilon_{loc}\simeq\alpha E_{R}$
with $E_{R}$ the recoil energy and $\alpha$ typically of order a few hundreds
depending on the kicks parameters, we estimate that $T_{eff}\simeq 10NT_{R}$
($T_{R}=E_{R}/k_{B}$ the recoil temperature, typically of order $10^{-7}K$ in
cold atoms experiments).. It can be tested by measuring the momentum
distribution of the underlying fermions Wilson  _et al._ (2020); Malvania  _et
al._ (2021), extracting the corresponding temperature, and comparing with the
bosons’ observables.

In the few body-limit, it has been shown that finite or infinite interactions
give a rather similar dynamical localization of the kicked Lieb-Liniger model
Chicireanu and Rançon (2021). An interesting question is whether this
effective thermalization persists beyond the Tonks regime and allows for a
quantitative description of the many-body dynamical localization at finite
interactions.

Finally, it is well-known that if the kicks strength is modulated, the (non-
interacting) QKR displays a delocalization transition similar to the Anderson
transition Shepelyansky (1987); Casati  _et al._ (1989), which has been
observed experimentally in the atomic QKR Chabé  _et al._ (2008); Lemarié  _et
al._ (2009). We therefore expect that modulating the kicks in the kicked Lieb-
Liniger model will induce a phase transition from the MBDL to a new phase
where the system can heat up to infinite temperature. Understanding the
properties of such a delocalized phase is under progress.

## Acknowledgments

We thank R. Chicireanu, J.-C. Garreau, D. Delande, N. Cherroret, C. Tian and
H. Buljan for insightful discussions. This work was supported by Agence
Nationale de la Recherche through Research Grants QRITiC I-SITE ULNE/
ANR-16-IDEX-0004 ULNE, the Labex CEMPI Grant No.ANR-11-LABX-0007-01, the
Programme Investissements d’Avenir ANR-11-IDEX-0002-02, reference
ANR-10-LABX-0037-NEXT and the Ministry of Higher Education and Research,
Hauts-de-France Council and European Regional Development Fund (ERDF) through
the Contrat de Projets État-Region (CPER Photonics for Society, P4S).

## References

  * Anderson (1958) P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
  * Evers and Mirlin (2008) Ferdinand Evers and Alexander D. Mirlin, “Anderson transitions,” Rev. Mod. Phys. 80, 1355–1417 (2008).
  * Nandkishore and Huse (2015) Rahul Nandkishore and David A. Huse, “Many-body localization and thermalization in quantum statistical mechanics,” Annual Review of Condensed Matter Physics 6, 15–38 (2015).
  * Abanin  _et al._ (2019) Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (2019).
  * Serbyn  _et al._ (2013) Maksym Serbyn, Z. Papić, and Dmitry A. Abanin, “Local conservation laws and the structure of the many-body localized states,” Phys. Rev. Lett. 111, 127201 (2013).
  * Huse  _et al._ (2014) David A. Huse, Rahul Nandkishore, and Vadim Oganesyan, “Phenomenology of fully many-body-localized systems,” Phys. Rev. B 90, 174202 (2014).
  * Ponte  _et al._ (2015a) Pedro Ponte, Z. Papić, Fran çois Huveneers, and Dmitry A. Abanin, “Many-body localization in periodically driven systems,” Phys. Rev. Lett. 114, 140401 (2015a).
  * Ponte  _et al._ (2015b) Pedro Ponte, Anushya Chandran, Z. Papic, and Dmitry A. Abanin, “Periodically driven ergodic and many-body localized quantum systems,” Annals of Physics 353, 196–204 (2015b).
  * Fishman  _et al._ (1982) Shmuel Fishman, D. R. Grempel, and R. E. Prange, “Chaos, quantum recurrences, and anderson localization,” Phys. Rev. Lett. 49, 509–512 (1982).
  * Chabé  _et al._ (2008) Julien Chabé, Gabriel Lemarié, Benoit Grémaud, Dominique Delande, Pascal Szriftgiser, and Jean Claude Garreau, “Experimental observation of the anderson metal-insulator transition with atomic matter waves,” Phys. Rev. Lett. 101, 255702 (2008).
  * Lemarié  _et al._ (2010) Gabriel Lemarié, Hans Lignier, Dominique Delande, Pascal Szriftgiser, and Jean Claude Garreau, “Critical state of the anderson transition: Between a metal and an insulator,” Phys. Rev. Lett. 105, 090601 (2010).
  * Lopez  _et al._ (2012) Matthias Lopez, Jean-Fran çois Clément, Pascal Szriftgiser, Jean Claude Garreau, and Dominique Delande, “Experimental test of universality of the anderson transition,” Phys. Rev. Lett. 108, 095701 (2012).
  * Manai  _et al._ (2015) Isam Manai, Jean-François Clément, Radu Chicireanu, Clément Hainaut, Jean Claude Garreau, Pascal Szriftgiser, and Dominique Delande, “Experimental observation of two-dimensional anderson localization with the atomic kicked rotor,” Phys. Rev. Lett. 115, 240603 (2015).
  * Hainaut  _et al._ (2018a) Clément Hainaut, Isam Manai, Jean-François Clément, Jean Claude Garreau, Pascal Szriftgiser, Gabriel Lemarié, Nicolas Cherroret, Dominique Delande, and Radu Chicireanu, “Controlling symmetry and localization with an artificial gauge field in a disordered quantum system,” Nature Communications 9, 1382 (2018a).
  * Hainaut  _et al._ (2018b) Clément Hainaut, Ping Fang, Adam Rançon, Jean-Francois Clément, Pascal Szriftgiser, Jean-Claude Garreau, Chushun Tian, and Radu Chicireanu, “Experimental observation of a time-driven phase transition in quantum chaos,” Phys. Rev. Lett. 121, 134101 (2018b).
  * Adachi  _et al._ (1988) S. Adachi, M. Toda, and K. Ikeda, “Quantum-classical correspondence in many-dimensional quantum chaos,” Phys. Rev. Lett. 61, 659–661 (1988).
  * Wen-Lei and Quan-Lin (2009) Zhao Wen-Lei and Jie Quan-Lin, “Quantum to classical transition in a system of two coupled kicked rotors,” Communications in Theoretical Physics 51, 465–469 (2009).
  * Keser  _et al._ (2016) Aydin Cem Keser, Sriram Ganeshan, Gil Refael, and Victor Galitski, “Dynamical many-body localization in an integrable model,” Phys. Rev. B 94, 085120 (2016).
  * Rozenbaum and Galitski (2017) Efim B. Rozenbaum and Victor Galitski, “Dynamical localization of coupled relativistic kicked rotors,” Phys. Rev. B 95, 064303 (2017).
  * Notarnicola  _et al._ (2018) Simone Notarnicola, Fernando Iemini, Davide Rossini, Rosario Fazio, Alessandro Silva, and Angelo Russomanno, “From localization to anomalous diffusion in the dynamics of coupled kicked rotors,” Phys. Rev. E 97, 022202 (2018).
  * Notarnicola  _et al._ (2020) Simone Notarnicola, Alessandro Silva, Rosario Fazio, and Angelo Russomanno, “Slow heating in a quantum coupled kicked rotors system,” Journal of Statistical Mechanics: Theory and Experiment 2020, 024008 (2020).
  * Bloch  _et al._ (2008) Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
  * Shepelyansky (1993) D. L. Shepelyansky, “Delocalization of quantum chaos by weak nonlinearity,” Phys. Rev. Lett. 70, 1787–1790 (1993).
  * Pikovsky and Shepelyansky (2008) A. S. Pikovsky and D. L. Shepelyansky, “Destruction of anderson localization by a weak nonlinearity,” Phys. Rev. Lett. 100, 094101 (2008).
  * Flach  _et al._ (2009) S. Flach, D. O. Krimer, and Ch. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett. 102, 024101 (2009).
  * Gligoricá  _et al._ (2011) G. Gligoricá, J. D. Bodyfelt, and S. Flach, “Interactions destroy dynamical localization with strong and weak chaos,” EPL (Europhysics Letters) 96, 30004 (2011).
  * Cherroret  _et al._ (2014) Nicolas Cherroret, Benoit Vermersch, Jean Claude Garreau, and Dominique Delande, “How nonlinear interactions challenge the three-dimensional anderson transition,” Phys. Rev. Lett. 112, 170603 (2014).
  * Lellouch  _et al._ (2020) Samuel Lellouch, Adam Rançon, Stephan De Bièvre, Dominique Delande, and Jean Claude Garreau, “Dynamics of the mean-field-interacting quantum kicked rotor,” Phys. Rev. A 101, 043624 (2020).
  * Cazalilla  _et al._ (2011) M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, “One dimensional bosons: From condensed matter systems to ultracold gases,” Reviews of Modern Physics 83, 1405–1466 (2011).
  * Qin  _et al._ (2017) Pinquan Qin, Alexei Andreanov, Hee Chul Park, and Sergej Flach, “Interacting ultracold atomic kicked rotors: loss of dynamical localization,” Scientific Reports 7, 41139 (2017).
  * Chicireanu and Rançon (2021) Radu Chicireanu and Adam Rançon, “Dynamical localization of interacting bosons in the few-body limit,” Phys. Rev. A 103, 043314 (2021).
  * Rylands  _et al._ (2020) Colin Rylands, Efim B. Rozenbaum, Victor Galitski, and Robert Konik, “Many-body dynamical localization in a kicked lieb-liniger gas,” Phys. Rev. Lett. 124, 155302 (2020).
  * Lazarides  _et al._ (2014) Achilleas Lazarides, Arnab Das, and Roderich Moessner, “Periodic thermodynamics of isolated quantum systems,” Phys. Rev. Lett. 112, 150401 (2014).
  * Vidmar and Rigol (2016) Lev Vidmar and Marcos Rigol, “Generalized gibbs ensemble in integrable lattice models,” Journal of Statistical Mechanics: Theory and Experiment 2016, 064007 (2016).
  * Olshanii and Dunjko (2003) Maxim Olshanii and Vanja Dunjko, “Short-distance correlation properties of the lieb-liniger system and momentum distributions of trapped one-dimensional atomic gases,” Physical Review Letters 91, 090401– (2003).
  * Tan (2008) Shina Tan, “Large momentum part of a strongly correlated fermi gas,” Annals of Physics 323, 2971 – 2986 (2008).
  * Shepelyansky (1986) D. L. Shepelyansky, “Localization of quasienergy eigenfunctions in action space,” Phys. Rev. Lett. 56, 677–680 (1986).
  * Lemarié  _et al._ (2009) Gabriel Lemarié, Julien Chabé, Pascal Szriftgiser, Jean Claude Garreau, Benoit Grémaud, and Dominique Delande, “Observation of the anderson metal-insulator transition with atomic matter waves: Theory and experiment,” Phys. Rev. A 80, 043626 (2009).
  * Note (1) To be precise, we define $p_{loc}$ of a single-particle state $|\psi\rangle$ as $p_{loc}^{2}=\langle\psi|\hat{p}^{2}|\psi\rangle-\langle\psi|\hat{p}|\psi\rangle^{2}$, which is proportional to the momentum scale on which the dynamically localized states decay exponentially.
  * Girardeau (1960) M. Girardeau, “Relationship between systems of impenetrable bosons and fermions in one dimension,” Journal of Mathematical Physics 1, 516–523 (1960).
  * Lenard (1964) A. Lenard, “Momentum distribution in the ground state of the one‐dimensional system of impenetrable bosons,” Journal of Mathematical Physics 5, 930–943 (1964).
  * Buljan  _et al._ (2008) H. Buljan, R. Pezer, and T. Gasenzer, “Fermi-bose transformation for the time-dependent lieb-liniger gas,” Phys. Rev. Lett. 100, 080406 (2008).
  * Jukić  _et al._ (2008) D. Jukić, R. Pezer, T. Gasenzer, and H. Buljan, “Free expansion of a lieb-liniger gas: Asymptotic form of the wave functions,” Physical Review A 78, 053602– (2008).
  * Pezer  _et al._ (2009) R. Pezer, T. Gasenzer, and H. Buljan, “Single-particle density matrix for a time-dependent strongly interacting one-dimensional bose gas,” Physical Review A 80, 053616– (2009).
  * Note (2) This is to be contrasted with the case of a Tonks gas loaded in a random potential, where localization in real space of the fermions is naturally preserved for the bosons Radić  _et al._ (2010); Seiringer and Warzel (2016).
  * Rigol and Muramatsu (2005a) Marcos Rigol and Alejandro Muramatsu, “Fermionization in an expanding 1d gas of hard-core bosons,” Physical Review Letters 94, 240403– (2005a).
  * Rigol and Muramatsu (2005b) Marcos Rigol and Alejandro Muramatsu, “Ground-state properties of hard-core bosons confined on one-dimensional optical lattices,” Physical Review A 72, 013604– (2005b).
  * Vaidya and Tracy (1979) H. G. Vaidya and C. A. Tracy, “One-particle reduced density matrix of impenetrable bosons in one dimension at zero temperature,” Phys. Rev. Lett. 42, 3–6 (1979).
  * Note (3) The quantity $q$ plays the role of a conserved quasi-momentum in the single-particle QKR without periodic boundary conditions.
  * Vignolo and Minguzzi (2013) Patrizia Vignolo and Anna Minguzzi, “Universal contact for a tonks-girardeau gas at finite temperature,” Phys. Rev. Lett. 110, 020403 (2013).
  * Its  _et al._ (1991) A.R. Its, A.G. Izergin, and V.E. Korepin, “Space correlations in the one-dimensional impenetrable bose gas at finite temperature,” Physica D: Nonlinear Phenomena 53, 187 – 213 (1991).
  * Collura  _et al._ (2013) Mario Collura, Spyros Sotiriadis, and Pasquale Calabrese, “Equilibration of a tonks-girardeau gas following a trap release,” Phys. Rev. Lett. 110, 245301 (2013).
  * Note (4) The Periodic Gibbs ensemble allows for a time dependence of the steady-state density matrix $\rho_{GGE}(t)$ with $0\leq t<1$, with period $1$. We only consider here the case $t=1^{-}$ (just before the next kick). We have checked that other choices do not change the qualitative picture.
  * (54) V. Vuatelet and A. Rançon, to be published.
  * van Amerongen  _et al._ (2008) A. H. van Amerongen, J. J. P. van Es, P. Wicke, K. V. Kheruntsyan, and N. J. van Druten, “Yang-yang thermodynamics on an atom chip,” Phys. Rev. Lett. 100, 090402 (2008).
  * Fabbri  _et al._ (2011) N. Fabbri, D. Clément, L. Fallani, C. Fort, and M. Inguscio, “Momentum-resolved study of an array of one-dimensional strongly phase-fluctuating bose gases,” Phys. Rev. A 83, 031604 (2011).
  * Note (5) Using $T_{eff}\simeq\sqrt{\varepsilon_{loc}\varepsilon_{F}}$, with $\varepsilon_{loc}$ the localization energy per particles, and the estimates $\varepsilon_{F}\simeq E_{R}N^{2}$ and $\varepsilon_{loc}\simeq\alpha E_{R}$ with $E_{R}$ the recoil energy and $\alpha$ typically of order a few hundreds depending on the kicks parameters, we estimate that $T_{eff}\simeq 10NT_{R}$ ($T_{R}=E_{R}/k_{B}$ the recoil temperature, typically of order $10^{-7}K$ in cold atoms experiments).
  * Wilson  _et al._ (2020) Joshua M. Wilson, Neel Malvania, Yuan Le, Yicheng Zhang, Marcos Rigol, and David S. Weiss, “Observation of dynamical fermionization,” Science 367, 1461–1464 (2020).
  * Malvania  _et al._ (2021) Neel Malvania, Yicheng Zhang, Yuan Le, Jerome Dubail, Marcos Rigol, and David S. Weiss, “Generalized hydrodynamics in strongly interacting 1d bose gases,” Science 373, 1129–1133 (2021).
  * Shepelyansky (1987) D.L. Shepelyansky, “Localization of diffusive excitation in multi-level systems,” Physica D: Nonlinear Phenomena 28, 103 – 114 (1987).
  * Casati  _et al._ (1989) Giulio Casati, Italo Guarneri, and D. L. Shepelyansky, “Anderson transition in a one-dimensional system with three incommensurate frequencies,” Phys. Rev. Lett. 62, 345–348 (1989).
  * Radić  _et al._ (2010) J. Radić, V. Bačić, D. Jukić, M. Segev, and H. Buljan, “Anderson localization of a tonks-girardeau gas in potentials with controlled disorder,” Physical Review A 81, 063639– (2010).
  * Seiringer and Warzel (2016) Robert Seiringer and Simone Warzel, “Decay of correlations and absence of superfluidity in the disordered tonks–girardeau gas,” New Journal of Physics 18, 035002 (2016).

##  Appendix A Validity of the thermal fit of the steady-state of the
fermionic momentum distributions

We focus here on giving details on the effective thermalization and its range
of validity with respect to the parameters of the system.

As hinted in the inset of Fig. 5, the thermal fit of the steady-state
fermionic momentum distributions works well for $p_{loc}/p_{F}\ll 1$, while in
the opposite limit, it does not work, implying that the system does not
effectively thermalize. This can be quantified by introducing the error

$\varepsilon=\frac{\lVert n^{F}_{k}-f_{FD}(k,T_{eff},\mu_{eff})\rVert}{\lVert
n^{F}_{k}\rVert}.$ (17)

The error as a function of $N$ and $K$ is shown in the top panel of Fig. 8,
and as a function of $p_{F}$ and $p_{loc}$ in the bottom panel. The low value
of $\varepsilon$, i.e. a good the thermal fit, corresponds to the blue area in
Fig. 8). In the following, we only consider parameters such that
$\varepsilon\lesssim 5\%$, where the effective thermalization takes place.

Figure 8: Top panel: $\varepsilon(N,K)$ at fixed ${\mathchar
22\relax\mkern-9.0muk}=6$. Dark red color corresponds to $\varepsilon\geq
0.05$. Bottom panel: same as top panel but in function of $p_{loc}$ and
$p_{F}$.

Fig. 9 shows that the localized energy in the steady-state is well described
by the Sommerfeld expansion Eq. (12) in terms of the fitted effective
temperature $T_{eff}$, as long as the temperature is small enough, i.e. when
the thermal fit works well (dark blue symbols). Fig. 10 shows that in the same
regime, the effective temperature is also well described by our prediction Eq.
(13).

Figure 9: Final energy $E_{f}$ as a function of the fitted effective
temperature $T_{eff}$ for ${\mathchar 22\relax\mkern-9.0muk}=6$ and various
particle number. In practice, we plot $\frac{E_{f}-E_{0}}{N\varepsilon_{F}}$
as a function of $T_{eff}^{2}/\varepsilon_{F}^{2}$. The black line corresponds
to the Sommerfeld expansion in Eq. (12). The color code corresponds to that of
Fig. 8. Figure 10: Effective temperature $T_{eff}/\varepsilon_{F}$ as a
function of $p_{loc}/p_{F}$ for ${\mathchar 22\relax\mkern-9.0muk}=6$ and
various particle number. In practice, we plot
$\frac{E_{f}-E_{0}}{N\varepsilon_{F}}$ as a function of
$T_{eff}^{2}/\varepsilon_{F}^{2}$. The black line corresponds to the
prediction Eq. (13). The color code corresponds to that of Fig. 8.

Finally, let us address the effects of the averaging over $q$. Fig. 11 is a
scatter plot of $T_{eff}^{(q=0)}$, the effective temperature extracted from
the fermionic energy for $q=0$, and $T_{eff}$, the effective temperature
obtained from the averaged momentum distribution, for various values of $N$,
${\mathchar 22\relax\mkern-9.0muk}$ and $K$. We see a very clear correlation
between the two. This shows that while averaging is convenient to analyze the
fermionic degrees of freedom, the effective temperature and chemical potential
obtained will describe very well the non-averaged observables of the bosons.

Figure 11: Comparaison of temperature extracted from $E_{f}$ and from the
averaged distribution (respectively $T_{eff}^{(q=0)}$ and $T_{eff}$). The
colored dots are the datas (green: ${\mathchar 22\relax\mkern-9.0muk}=6$, red
: ${\mathchar 22\relax\mkern-9.0muk}=7$, blue: ${\mathchar
22\relax\mkern-9.0muk}=8$) and the black line is a guide to the eye. The
collapse of the datas around the black line show that $T_{eff}^{(q=0)}$ and
$T_{eff}$ are mostly the same.

##  Appendix B Extraction of $r_{c}$

We observed that the coherence function of the Tonks gas $g_{1}(r)$ decays
exponentially in the localized regime as it was shown in Fig. 3. Assuming that
it decays as $g_{1}(r)\propto e^{-2|r|/r_{c}}$, we can estimate the
correlation length $r_{c}$ by

$r_{c}=\sqrt{2\frac{\sum_{r}r^{2}g_{1}(r)-(\sum_{r}rg_{1}(r))^{2}}{\sum_{r}g_{1}(r)}},$
(18)

which is well described by Eq. (14) as discussed in the main text, where we
have focus on the case ${\mathchar 22\relax\mkern-9.0muk}=6$. We show in Fig.
12 that for other ${\mathchar 22\relax\mkern-9.0muk}$, our prediction is in
good agreement with the data.

Figure 12: Correlation length for different ${\mathchar
22\relax\mkern-9.0muk}$ (7, 8 from up to down). The black line correspond on
every figure to the relation Eq. (14).

##  Appendix C Natural orbitals

The OBDM can be decomposed in natural orbitals $\phi^{\eta}(x)$, which can be
interpreted as the many-body version of the wavefunctions occupied by the
bosons, and which are the eigenfunctions of the OBDM,

$\int dy\rho(x,y)\phi^{\eta}(y)=\lambda_{\eta}\phi^{\eta}(x),$ (19)

with the $\lambda_{\eta}$the occupation of $\eta$-th natural orbital. Fig. 13
(top) shows the most occupied natural orbital in momentum space for $N=51$,
$K=20$, ${\mathchar 22\relax\mkern-9.0muk}=6$ in semi-log scale. We observe
that it decays exponentially over a scale $p_{loc}$, as can been verified by
plotting a localized wave-function of the non-interacting QKR (which decays
over the same scale).

Fig. 13 (top) shows the two-dimensional Fourier transform of the OBDM,

$\rho(k,k^{\prime})=\frac{1}{L}\int dxdye^{ikx-ik^{\prime}y}\rho(x,y),$ (20)

where $\rho(k,k)$ is the momentum distribution. We observe that contrary to a
thermal OBDM, it is non-zero for $k\neq k^{\prime}$ (as expected by invariance
by translation for the thermal gas). However, it decays exponentially over the
scale $p_{loc}$, as can be seen in Fig. 13 (bottom).

Figure 13: In both panel $N=51$, $K=20$, ${\mathchar
22\relax\mkern-9.0muk}=6$. Top panel: 2D graphic of the logarithm value of the
OBDM in momentum space $\rho(k,k^{\prime})$. Bottom panel: comparison between
the anti-diagonal of the OBDM in momentum space and the most-occupied natural
orbitals. We compare them to the $k=0$ wave-function for the same parameters.