# Localization of matter waves in lattice systems with moving disorder

Chenyue Guo  Wilczek Quantum Center and Key Laboratory of Artificial
Structures and Quantum Control, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240, China  Zi Cai<EMAIL_ADDRESS>Wilczek
Quantum Center and Key Laboratory of Artificial Structures and Quantum
Control, School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Shanghai Research Center for Quantum Sciences,
Shanghai 201315, China

###### Abstract

We study the localization phenomena in a one-dimensional lattice system with a
uniformly moving disordered potential. At a low moving velocity, we find a
sliding localized phase in which the initially localized matter wave
adiabatically follows the moving potential without diffusion, thus resulting
in an initial state memory in the many-body dynamics. Such an intriguing
localized phase distinguishes itself from the standard Anderson localization
in two aspects: it is not robust against interaction, but persists in the
presence of slowly varying perturbations. Such a sliding localized phase can
be understood as a consequence of interference between the wavepacket paths
under moving quasi-periodic potentials with various periods that are
incommensurate with the lattice constant. The experimental realization and
detection were also discussed.

Introduction – The absence of the diffusion of waves in a disordered medium
(dubbed Anderson localization) originates from the interference between
various scattering paths, thus it is ubiquitous in wave physicsAnderson
(1958). Such a phenomenon is expected to be robust against weak interactions,
and has reattracted enormous interest recently in the context of many-body
localization (MBL)D.M.Basko et al. (2006); Oganesyan and Huse (2007); Žnidarič
et al. (2008); Pal and Huse (2010). Even though disorders naturally exist in
solid-state settings, they can also be artificially introduced into
intrinsically clean systems (e.g.,ultracold atom or trapped ion) in a
controllable matterClément et al. (2006); Bermudez et al. (2010). Owing to its
unique features such as the perfect isolation and high degree of parameter
tunabilityBloch et al. (2008), the cold-atom system represents a new
perspective for studying localizationFallani et al. (2007); Billy et al.
(2008); Roati et al. (2008); Schreiber et al. (2015). For instance, it allows
the exploration of localization physics in a far-from-equilibrium system
within a strong driving regime inaccessible in conventional solid-state
settingsBordia et al. (2017); Vershinina et al. (2017); Liu et al. (2018), and
thus is beyond the scope of the linear response theory derived by MottMott
(1968).

In cold atom experiments, a quantum many-body system can be driven out of
equilibrium by periodically or stochastically modulating the system
parameters. Recently, an intriguing driving protocol other than a regular
(periodic) or completely irregular (stochastic) driving protocols was
proposedXu and Wu (2018). Therein, neither a spatial translational symmetry,
nor a temporal translational symmetry (TTS) are present for the driving
potential, which, instead, exhibits nontrivial intertwined space-time
symmetries that cannot be decomposed into a direct product of spatial and
temporal symmetries. Owing to the absence of discrete TTS, such a driving
protocol significantly differs from the periodic driving, thus the widely
employed Floquet description no longer apply, nor it resembles the stochastic
or quasi-periodic onesDumitrescu et al. (2018); Ren et al. (2020); Long et al.
(2021); Zhao et al. (2021) due to its strong correlation along a particular
space-time direction.

Such an intriguing driving protocol avails new possibilities exploring non-
equilibrium physics beyond the scope of periodically driven systemsEckardt
(2017); Harper et al. (2020); Oka and Kitamura (2019). As a example, in this
study, we explore the localization physics in a driven-disordered system with
a sliding space-time translational symmetry($\mathbf{SSTS}$). The proposed
system is a one-dimensional (1D) quantum model subjected to a time-dependent
inhomogeneous potential as a superposition of a dynamic disordered potential
moving at a constant velocity and a static periodic one (see Fig.1). As would
be demonstrated, such a moving disordered potential with tunable velocity can
be readily achieved in the cold atomic systems, even though it is unrealistic
in solid-state settings. Contrary to the periodically-driven disordered
systems which are generally delocalized by low-frequency perturbationsMartinez
and Molina (2006); Ponte et al. (2015); Lazarides et al. (2015); Abanin et al.
(2016); Vershinina et al. (2017), in the proposed model, a sliding localized
phase ($\mathbf{SLP}$) was observed at a sufficiently low velocity of the
moving potential. Even though the center of mass (COM) of the matter wave
adiabatically follows the moving potential, it does not cause diffusion. Such
an SLP persists in the non-interacting many-body systems, which exhibit an
initial state “memory” effect. However, dissimilar to the MBL systems, SLP
will be delocalized even by weak interactions. The system is delocalized at a
high moving velocity. Moreover, instead of diffusion, we find a ballistic
transport in the single-particle dynamics, and an algebraic decay of the
initial state memory at the many-body level.

Figure 1: (Color online) Schematic of the static periodic and moving
disordered potentials in our model.

Model – First, we considered a time-dependent single-particle Hamiltonian,
which is defined in the 1D continuum space as follows:

$\hat{H}(x,t)=\frac{\hat{P}^{2}}{2m}+V_{s}(x)+\tilde{V}(x,t)$ (1)

where $V_{s}(x)=V_{s}(x+a)$ is a static periodic potential that is produced by
the optical lattice with lattice constant $a$. $\tilde{V}(x,t)$ represents the
time-dependent disordered potential that breaks the spatial translational
symmetry at any given time. To drive the system out of equilibrium, the
disordered potential is suddenly pushed forward at a constant velocity
$\mathbf{v}$ indicating $\tilde{V}(x,t)=\tilde{V}(x-vt,0)$. The explicit form
of $\tilde{V}(x,t)$ would be formulated subsequently. $\hat{H}(x,t)$ preserves
the SSTS even though it breaks the temporal and spacial translational symmetry
separately:

$\hat{H}(x,t)=\hat{H}(x+a,t+a/v).$ (2)

Provided the static periodic potential is sufficiently deep, we adopted the
single-band approximation and derived the tight-binding Hamiltonian in the 1D
lattice as:

$\hat{H}(t)=\sum_{i}[-J(C_{i}^{\dagger}C_{i+1}+h.c)+V_{i}(t)\hat{n}_{i}]$ (3)

where $C_{i}$ ($C_{i}^{\dagger}$) is the annihilation (creation) operator of
the spinless fermion on site $i$ and $\hat{n}_{i}=C_{i}^{\dagger}C_{i}$. $J$
is the nearest-neighboring (NN) single-particle hopping amplitude and
$V_{i}(t)=\tilde{V}(x=ai,t)$ is the moving disordered potential at time t.
Assuming $V_{i}(t=0)$ was randomly sampled from a uniform random distribution
with $V_{i}(t=0)\in[-\Delta,\Delta]$. To recover the continuum function
$\tilde{V}(x,t=0)$ from a set of given points $\\{V_{i}(t=0)\\}$, we performed
cubic spline interpolation to smoothly connect these points on the different
sites, and derived the continuum function $\tilde{V}(x,t=0)$. The results did
not significantly depend on specific interpolation methods Sup . Immediately
after determining $\tilde{V}(x,t=0)$, $V_{i}(t)$ at any given time can be
obtained using the identities $\tilde{V}(x,t)=\tilde{V}(x-vt,0)$ and
$V_{i}(t)=\tilde{V}(x=ai,t)$. Despite the similarity in their SSTS, we should
emphasize that the proposed potential is significantly different from that in
the space-time crystalXu and Wu (2018); Gao and Niu (2021), since our
potential has only one lattice vector as shown in Eq.(2), thus it is not a
crystal in the 1+1 D space-time.

Figure 2: (Color online) (a) Density distributions of the wavepacket at
different time slices during the single-particle dynamics in the SLP
($v=0.01Ja$) and delocalized phase ($v=Ja$). The inset magnifies the lower
panel. Evolution of the (b) COM and (c) width of the wavepacket under
different moving velocities. Here and hereafter, $tv$ rather than $t$ would be
employed to characterize the evolution time, thus allowing the comparison of
the dynamics with different $v$ on the same footing. (d)Comparison between
$M(t)$ in SLP ($v=0.01Ja$) and delocalized phase ($v=Ja$) during the non-
interacting many-body dynamics beginning from the same ($|1010\cdots\rangle$)
initial state. The inset is the envelope of the oscillations in Fig.2 (d) in a
log-log plot. (e)Dynamics of M(t) in the non-interacting systems with
different system sizes $L$ for the SLP ($v=0.01Ja$ for the upper panel) and
delocalized phase ($v^{\prime}=Ja$ for the lower panel). The inset is a finite
size scaling of the amplitudes of the persistent periodic oscillations ($M$)
in the SLP ($v=0.01Ja$). (f) Dynamics of M(t) in the small systems with
different NN interaction strengths $U$. The inset is the envelope of the
oscillations in Fig.2 (f) in a log-log plot. The system sizes are chosen as
$L=1000$ for (a)-(c), $L=240$ for (d) and $L=16$ for (f). The discrete time
step $\Delta t=0.001J^{-1}$ for all the cases except $v=0.01Ja$ where $\Delta
t=0.01J^{-1}$ and $\Delta=4J$ for (a)-(f). A random disorder realization was
selected for (a)-(c) and the ensemble averages over 300 disorder realizations
were performed for (d)-(f).

Single-particle dynamics – We first consider the single-particle situation,
where the initial state was chosen as the ground state of Hamiltonian (3) with
$t=0$. Starting from such a spatially localized wavepacket, we study the time
evolution of the wavefunction $|\psi(t)\rangle$ by directly solving the time-
dependent Schrodinger Equation with different moving velocities. The density
distributions $\rho_{i}(t)=\langle\psi(t)|\hat{n}_{i}|\psi(t)\rangle$ during
the time evolution were plotted in Fig.2 (a), which revealed that the COM of
the wavepacket $X(t)=\sum_{i}i\rho_{i}(t)$ could not catch up with the moving
potential at a relatively high velocity ($X(t)<X(0)+vt$ as shown in Fig.2 b).
Consequently, the wavepacket experienced a sequence of rapidly varying
disordered potential, which consistently perturbed the wavepacket
stochastically as noise, and finally delocalized it. This delocalization can
be characterized by the width of the wavepacket
$w(t)=\sqrt{\sum_{i}\rho_{i}(t)[i-X(t)]^{2}}$. Fig.2 (c) shows that $w(t)$
roughly grows linearly after a long time, indicating a ballistic transport
($w(t)\sim t$) rather than diffusion ($w(t)\sim t^{\frac{1}{2}}$) reported
previously in noisy disordered systemsGopalakrishnan et al. (2017); Ren et al.
(2020). This discrepancy exists because the SSTS (Eq.2) resulted in a strong
space-time correlation for the disorder-induced “noise”, thus stopping it from
being “white” noise. Such a ballistic transport resembles the dynamics of the
disorder-free case, indicating that a fast-moving disordered potential does
not qualitatively change the long-time dynamics of lattice systems, it only
renormalizes the hopping amplitude $J$.

The situation is qualitatively different at a low velocity. Fig.2 (b) shows
that the COM adiabatically followed the moving potential ($X(t)=X(0)+vt$),
while its width $w(t)$ was bounded even after sufficiently a long time (see
Fig.2 c), indicating the absence of diffusion. The uniform motion of COM and
the absence of the diffusion of the wavepacket were the main features of SLP
at the single-particle level. The existence of the SLP indicates the validity
of the adiabatic theorem in our modelSup . The localization/delocaliztion of a
wavepacket does not only depend on the moving velocity, but also on the
initial state energy. For instance, if we begin from another eigenstate of
$\hat{H}(t=0)$ close to the band edge, the physics is similar as analyzed
above, while for an initial state in the band center, even an infinitesimal
velocity would be sufficient to delocalize the wavepacket. Consequently, for a
given velocity, a critical initial energy exists to separate the sliding
localized and delocalized states. The existence of such a critical initial
energy, is crucial for our subsequential discussion regarding the many-body
cases with and without interactions.

It is helpful to compare our results with the wavepacket dynamics under a
suddenly moving single trap. For instance, the dynamics of a bound state in
the presence of a uniformly moving attractive $\delta$ potential (with a
velocity $v$) can be solved exactlyGranot and Marchewka (2009), which showed
that the wavepacket will be split into three parts with different velocities:
one adiabatically follows the moving trap at a velocity $v$, while the other
two escape from the trap (with a velocity $0$ and $2v$ respectively), and
there is no dynamics phase transition. In our model, the two wavepackets
escaping from the original trap will be localized by the disordered potential
in the SLP, thus the wavepacket will adiabatically follow the moving
potential.

Figure 3: (Color online) (a) Comparison between the dynamics of the width of
wavepacket in the presence of a single moving quasi-periodic potential
$V_{k_{0}}(x,t)=\Delta\cos 2\pi k_{0}(x-vt)$ with and a superposition of two
quasi-periodic potentials $V^{\prime}_{k_{0}}(x,t)=\frac{\Delta}{2}[\cos 2\pi
k_{0}(x-vt)+\cos 2\pi k^{\prime}_{0}(x-vt)]$. The parameters are chosen as
$\Delta=4J$, $L=1000$, $v=0.01Ja$, $k_{0}=\frac{\sqrt{5}-1}{2}$ and
$k^{\prime}_{0}=\frac{\sqrt{6}-1}{2}$. (b)Sketch of the nearest-neighboring
tunneling of the wavepacket induced by a moving qPP during the period
$[0,t^{*}]$ with $t^{*}=\frac{a_{0}-a}{v}$.

Many-particle dynamics without interaction – One of the valuable features of
localization is the memory effect of the initial-state information, which will
be partially preserved during time evolution. For instance, if we begin from a
half-filled charge-density-wave(CDW) state $|1010\cdots\rangle$, the density
imbalance between two sublattices of the 1D lattice
$M(t)=\frac{1}{L}\sum_{i}(-1)^{i}\rho_{i}(t)$ in the initial state would be
memorized ($M(t\rightarrow\infty)\neq 0$) in localized phasesSchreiber et al.
(2015); however, in delocalized phase, it would be washed out after extended
period ($M(t\rightarrow\infty)\rightarrow 0$). Thus, to characterize the
feature of SLP from the perspective of many-body physics, we selected the
$|1010\cdots\rangle$ initial state and studied the time evolution under the
noninteracting Hamiltonian (3) with different $v$ by solving the equation of
motion of the equal-time single-particle Green’s function
$G_{ij}(t)=\langle\psi(t)|C_{i}^{\dagger}C_{j}|\psi(t)\rangle$, from which all
the physical quantities at any given time, including $M(t)$, were derived.

Without interaction, the dynamics of the many-body (fermionic) systems can be
understood as the collective behavior of different single-particle states,
each of which evolves independently. At a low velocity, the system
adiabatically followed the moving potential, so is the CDW state. A CDW state
moving at a constant velocity can be characterized by the persistent periodic
oscillation of $M(t)$ (Fig.2 d). Notably the oscillation amplitude is smaller
than that in the initial state ($M(0)=0.5$), which can be explained by the
aforementioned delocalized single-particle states that barely contributed to
the sliding CDW order. $M(t)$ in systems with different system sizes were
plotted in Fig.2 (e). Therein, a finite-size scaling (see the inset and Sup :)
shows that such periodic oscillations in SLP persisted in the thermodynamic
limit. Although such a sliding CDW state resembles the “moving solid” proposed
in the steady states of CDW systems driven by applied electric fieldsKoshelev
and Vinokur (1994); Marley et al. (1995); Balents and Fisher (1995), there is
significant difference: the CDW order in the “moving solid” is induced by
interactions, while that in our study originated from the initial state
memory. At a high velocity, the long-time dynamics of $M(t)$ noticeably
exhibited an algebraic decay $\sim t^{-\alpha}$ (accomplished by an
oscillation) with a non-universal exponent $\alpha$ depending on $v$.

Many-particle dynamics with interaction – To examine the effect of interaction
on the SLP in our model, the NN interactions between fermions were introduced
as follows:

$\hat{H}_{I}(t)=\sum_{i}[-J(C_{i}^{\dagger}C_{i+1}+h.c)+V_{i}(t)\hat{n}_{i}+U\hat{n}_{i}\hat{n}_{i+1}]$
(4)

where U is the strength of the NN interaction. Employing the exact
diagonalization method, the time evolution of $M(t)$ was studied beginning
from the $|1010\cdots\rangle$ state under the Hamiltonian (4) for a relatively
small system ($L=16$) and compared with the non-interacting case. Fig.2 (f)
shows that even a weak interaction in the SLP ($v=0.01J$) can delocalize the
system and lead to an algebraic decay of $M(t)$. This is strikingly different
from the static disorder-induced localization, which is generally robust
against weak interaction. Such an interaction-induced delocalization can also
be explained by the aforementioned critical initial energy, above which the
single-particle states are delocalized. Generally, the interaction would
induce scatterings between the different single-particle states, thus mix the
localized and delocalized states. Therefore, those delocalized states above
the critical initial energy act as a bath that is coupled to the localized
states, and delocalize the whole system.

Discussion– Under Fourier transformation
$\tilde{V}(x,t)=\frac{1}{\sqrt{L}}\sum_{k}V_{k}e^{ik(x-vt)}$, the moving
disorder potential turns to a superposition of a set of moving periodic
potentials with different wavelengths, which are typically incommensurate with
the lattice constant $a$. Therefore, to understand the SLP, one should first
focus on the dynamics of a particle in the presence of a moving quasi-period
potential (qPP) with a period incommensurate with $a$. Regarding the static
case ($v=0$), it is known that a qPP with $\Delta>2J$ would also result in a
localization that is similar to the Anderson localization in the disordered
systemAubry and Andre (1980). Thus, their dynamical behaviors in the presence
of the moving potential could be expected to be similar. However, Fig.3 (a)
shows this is not the case. In the presence of a slowly moving qPP
$V_{k_{0}}(x,t)=\Delta\cos 2\pi k_{0}(x-vt)$ with $v=0.01J$, starting from the
initial state as the ground state of $V_{i}=V_{k_{0}}(ia,0)$, the variance of
the wavepacket w(t) is plotted in Fig.3 (a) (the solid black curve), which
grows linearly with time, indicating a ballistic transport behavior similar
with that of disorder-free case.

The delocalization of a particle under a slowly moving qPP can be understood
via the Landau-Zener tunneling. Considering a qPP with a period $a_{0}$
incommensurate but slightly larger than $a$, and further assuming that
initially the minima of the qPP and lattice potential coincide at site i, we
choose the initial state as the ground state at $t=0$, which spatially
localized around site i. Since $a_{0}$ is slightly larger than $a$, the
potential energies on site $i$ and $i\pm 1$ at $t=0$ are close with each other
but separated from those on other sites. We focus on site $i$ and $i-1$, whose
potential energies vary with the movement of qPP. As shown in Fig.3 b, at time
$t^{*}=\frac{a_{0}-a}{v}$, the energy minimum has been shifted from site $i$
to $i-1$ ($V_{i-1}(t^{*})=V_{i}(0)$), indicating an energy level crossing.
Considering the particle tunneling between site $i$ and $i-1$, this physics
resembles the Landau-Zener tunneling: for a slow moving velocity $v\ll J$, the
wavepacket will follow the energy minimum thus tunnel from site $i$ to site
$i-1$ at $t=t^{*}$. With further movement of qPP, the wavepacket will keep
tunneling, thus give rise to a ballistic transport with a renormalized
tunneling rate depending on $t^{*}$Sup .

This picture is helpful for us to understand the SLP in the moving disordered
potential, which can be considered as a superposition of moving qPPs with
different $a_{0}$. For a single qPP, a wavepacket tunneling from site $i$ to
$i-1$ carries a phase depending on the tunneling time $t^{*}$, which in turn
is determined by $a_{0}$ of the qPP. Therefore, in the presence of many qPPs
with random $a_{0}$, the wavepacktes carrying different phases will
interference with each other at site $i-1$, thus suppress the effective
tunneling and lead to localization. This point can be numerically verified by
comparing the dynamics of $w(t)$ with a single qPP and more than one qPP. As
shown in Fig.3 (a), in the presence of two qPPs with different periods, $w(t)$
significantly deviates from the linear growth, indicates that the effective
tunneling is strongly suppressed.

Experimental realization and detection–  The proposed model can be
experimentally simulated by loading atoms into a quasi-1D optical lattice, in
which the disordered and quasi-periodic potentials can be introduced in
controllable ways: the former was realized by by implementing a speckle
potential generated by a laser beam passing through a diffusion plateClément
et al. (2006); Billy et al. (2008), while the latter could be introduced by
imposing an additional optical lattice whose period is approximately
incommensurate with that of the original oneRoati et al. (2008). In both
cases, the moving potentials could be realized by shifting either the
diffusion plate or the phase of the additional lasers at a constant velocity.
For an optical lattice with $a=512nm$ and $J\simeq 400Hz$, one can estimate
that the realistic moving velocity corresponds to the typical parameter regime
discussed above is in the order of a magnitude of $10\mu m/s$. Regarding the
detections, the density imbalance $M(t)$ can be be directly or indirectly
measured via the superlattice band-mapping techniqueTrotzky et al. (2012), or
heterodyne detection method respectivelyLandig et al. (2016); Hruby et al.
(2018)

Conclusion and outlook –  In summary, the dynamics of quantum systems with
moving disordered potentials were studied, and a sliding localized phase,
which availed a new perspective for studying of driven-disordered systems, was
clarified. This result does not only apply to matter waves in quantum systems,
but also to a wide range of classical wave phenomena in disordered media, such
as lightWiersma et al. (1997), soundWeaver (1990) and microwavesDalichaouch et
al. (1991). Future developments will include generalization/extension of our
findings to higher-dimensional systems, in which the motion of disordered
potentials along a certain dimension might produce intriguing anisotropic
localized phases, in which the matter wave is delocalized along the driving
direction, but still localized along other directions that are perpendicular
to it. A more general question is whether there exists an effective time-
independent description analog to the Floquet Hamiltonian for this type of
driven systems with intertwined space-time symmetry. If there exists any, what
are the “local” integrals of motions that account for the initial state memory
in the sliding localized phase? Finally, even though we focused on
localization physics, the proposed driving protocol can be studied in a
broader context, especially in the interacting quantum systems where the
interplay between the spontaneous symmetry breaking and the moving disorder
might give rise to nontrivial phenomena, e.g., a generalization of the Imry-Ma
argumentsImry and Ma (1975) for the robustness of symmetry breaking to such a
moving disorder.

Acknowledgments.—This work is supported by the National Key Research and
Development Program of China (Grant No. 2020YFA0309000), NSFC of China (Grant
No.12174251), Natural Science Foundation of Shanghai (Grant No.22ZR142830),
Shanghai Municipal Science and Technology Major Project (Grant
No.2019SHZDZX01).

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Supplementary material for “Matter waves in lattice systems with moving
disorder”

###### Contents

  1. I Details of the numerical simulations
  2. II Other driving protocols
  3. III Initial state dependence: the existence of a critical initial energy
  4. IV Wavepacket dynamics in the presence of moving quasi-periodic potential

##  I Details of the numerical simulations

In this section, we check the dependence of the sliding localized phase on
different disorder realizations, system sizes, and spline interpolations.

###  I.1 Disorder realizations

In the maintext, the single-particle dynamics is calculated under a given set
of randomly chosen disorder realization $\\{V_{i}(t=0)\\}$, it is thus
important to check the our results do not crucially depend on the choices of
the disorder realizations. To this end, we calculate the dynamics of the
center of mass (COM) and width of the wavepackets for different disorder
realizations $\\{V_{i}(t=0)\\}$ starting from the initial state as the ground
state of the corresponding Hamiltonian. Without losing generality, we choose
those disorder realizations whose ground state locates close to the center of
the 1D lattice to avoid the boundary effect during the evolution as possible
as we can. As shown in Fig.4 (a), the long-time dynamics of the wavepackets
with different disorder realizations are qualitatively identical: at a low
velocity ($v=0.01Ja$), the COMs of the wavepackets under different disordered
potential adiabatically follow the moving potential: $X(t)=X(0)+vt$, while in
all these cases, their widths keep oscillating within a finite regime, which
indicates an absence of diffusion. Therefore, the existence of the sliding
localized phases doesn’t depend on the choices of the disorder realization.

Figure 4: (Color online) The time evolution of COM (upper panel) and width
(lower panel) of the wave packets under three different disorder realizations
with system size (a) L=1000 and (b) L=4000, $v=0.01Ja$ and $\Delta=4J$.

Figure 5: (Color online) (a) The dynamics of M(t) for a non-interacting many-
body system starting from the $|1010\cdots\rangle$ with parameters $L=240$,
$\Delta=4J$, $v=0.01Ja$. (b) The equal-time single particle Green’s function
at two different time slices ($t_{1}$ and $t_{2}$), which respectively
corresponds to a wave peak and node of $M(t)$ as shown in (a). Ensemble
average over 300 disorder realizations are performed in (a) and (b).

Figure 6: (Color online)(a) The envelop of M(t) for different system sizes in
the many-body dynamics of SLP. (b) Finite-size scaling of the saturated
amplitude of the oscillation M. $\Delta=4J$ and $v=0.01Ja$ for (a) and (b).
300 disordered realizations are performed for each system size.

Figure 7: (Color online)(a) A comparison between the cubic spline
interpolation and the linear interpolation from a same set of random
$\\{V_{i}(t=0)\\}$ (b) The dynamics of COM (upper panel) and width (lower
panel) of the wavepacket under the moving potential $\tilde{V}(x,t)$ derived
from the linear interpolation with parameters $L=1000$, $\Delta=4J$,
$v=0.01Ja$.

###  I.2 System size dependence and localization length

The single-particle simulation in the main text is performed on a 1D lattice
with finite system site(L=1000), which also limits the maximum simulation time
($t_{max}\sim La/v$), after which the wave packet will be bounced by the
boundary of the 1D lattice under the open boundary condition. For a 1D system
with periodic boundary condition, in principle, the COM of a wavepacket
located on a ring is ill defined. To avoid the boundary/finitesize effect, we
choose those initial states localized close to the middle of the 1D lattice
($X(t=0)\simeq L/2$), and the typical simulation time $T<t_{max}/2$.

It is known that sometimes the localization phenomena may be sensitive to the
system size, especially for the weak disordered cases where the localization
length is long enough to be compatible to the system size. To make sure that
the system size ($L=1000$) and simulation time chosen in the main text are
long enough to capture the physics of the infinite long-time dynamics of a
system in the thermodynamic limit, here we study a larger system ($L=4000$),
which also allows us to simulate the dynamics with a longer time. The dynamics
COM and the width of the wave packets in a system with $L=4000$, $v=0.01Ja$,
but different disorder realizations are shown in Fig.4 (b), from which we can
see that there is no qualitatively difference between the results of $L=4000$
and $L=1000$ shown in Fig.4 (a).

In the single particle dynamics, it is difficult to derive the localization
length due to the intrinsic non-equilibrium nature of our model, which is
different from the conventional Anderson localization or periodically driven
systems where the localization length can be derived from the eigenstates of
the single-particle (Floquet) Hamiltonian. In our model, there is no effective
Hamiltonian description, thus the eigenstate analysis is invalid. However, as
shown in the main text, the many-body dynamics of the sliding localized phase
starting from the $|1010\cdots\rangle$ state will approach to an asymptotic
periodically oscillating dynamics, which allows us to study the equal-time
single particle correlation function
$G(r,t)=\langle\psi(t)|\frac{1}{L}\sum_{i}C_{i}^{\dagger}C_{i+r}|\psi(t)\rangle$
in this asymptotic regime. We calculate $G(r,t)$ at two different time slices
($t_{1}$ and $t_{2}$ as shown in Fig.5 a), which respectively correspond to a
wave peak and a wave node in the asymptotic oscillation regime. As shown in
Fig.5 (b), both $G(r,t_{1})$ and $G(r,t_{2})$ rapidly (exponentially) decay in
distance. By making an analogue to the Anderson localization, such a
exponential decay of the single-particle Green’s function enable us to
estimate the localization length (a few lattice constants), which is
significantly smaller than the typical system sizes in our simulation
(hundreds of lattice constant).

Figure 8: (Color online)(a) The dynamics of the COM ($X(t)$ in the upper
pannel) and width ($w(t)$ in the lower pannel) of the wavepacket in the
presence of disordered potentials with (a)a periodical kicked driving and (b)
a periodically modulating amplitude with the parameters $L=1000$, $\Delta=4J$,
$v=0.01Ja$. Neither of them exhibit the sliding localized phase at low $v$.

###  I.3 Finite size scaling of the many-body dynamics

At half-filling case, the SLP is characterized by a persistent oscillation of
the CDW order parameter
$M(t)=\frac{1}{L}\sum_{i}(-1)^{i}\langle\Psi(t)|\hat{n}_{i}|\Psi(t)\rangle$ if
we start from the initial state $|1010\cdots 10\rangle$. As for the stability
of the SLP, it is important to distinguish the persistent oscillation from an
extremely slow decay, and one needs to show that the oscillation amplitude
doesn’t decay to zero in the thermodynamic limit. To this end, we extended the
simulation time of the many-body dynamics, and provided some details of the
finite size scaling in this section. We plotted the envelop of the
oscillations of $M(t)$ for different system sizes up to a sufficiently long
time in In Fig.6 (a), from which we can find that $M_{en}(t)$ barely decays in
time. We define the saturated amplitude M as the average of the $M_{en}(t)$
over the period $240a/v<t<480a/v$, and study its dependence with system size.
Fig.6 (b) shows that M decays linearly with $1/L$, and in the thermodynamic
limit, M extrapolates to a finite value, which indicates that the persistent
oscillation do survive in thermodynamics limit in the SLP.

###  I.4 Interpolation dependence

To recover the continuum potential function $\tilde{V}(x,t=0)$ from a set of
discrete points $\\{V_{i}(t=0)\\}$, in our simulation we used the cubic spline
interpolation to assure the smoothness of the function $\tilde{V}(x,t=0)$.
Here, we will show that our results doesn’t crucially depends on such a
specific choice of interpolation. To this end, we choose a different
interpolation method (linear interpolation as shown in Fig.7 a) to derive
$\tilde{V}(x,t=0)$ from the same set of $\\{V_{i}(t=0)\\}$, and calculate the
dynamics of the wavepacket under such a non-smooth potential. As shown in
Fig.7 (b), in spite of the roughness of the potential derived by linear
interpolation, at a low moving velocity, the dynamics of the wavepacket is
qualitatively the same as that in the cubic spline interpolation, thus the
existence of the sliding localized phase doesn’t depend on the interpolation
method. However, we should emphasize that the interpolated potential should be
spatially continuous, otherwise it is impossible to define a adiabatic
process. For instance, for a steplike potential $\tilde{V}(x)=V_{i}$, if
$(i-\frac{1}{2})a<x<(i+\frac{1}{2})a$, there is no sliding localized phase no
matter how slow the potential moves. Such a steplike moving potential is equal
to the periodically kicked driving as we will analyzed in the subsequential
section.

##  II Other driving protocols

###  II.1 Disordered potential with a periodically kicked driving

To the check the dependence of the SLP on the driving protocol of the
disordered potential. We first consider a driving protocol with periodic
kicks: instead of moving uniformly, within each driving circle, the disordered
potential keeps static for a period of $\frac{a}{v}$, and then is suddenly
pushed forward by a lattice constant. Mathematically, this disordered
potential can be expressed as

$V_{1}(x,t)=V(x-\theta(t),0)$ (5)

with $\theta(t)=na$ for $(n-1)a/v<t<na/v$. Notice that on average, the
disordered potential moving forward at a velocity $v$, however, the wavepacket
dynamics under such a step-like driving protocol is qualitatively different
from that under an uniformly moving disordered potential. As show in Fig.8
(a), at a low averaged velocity ($v=0.01Ja$), the COM of the wavepacket cannot
follow the potential (it moves towards the opposite direction), and its width
keep grows in time, indicating the absence of localization. The reason behind
this is that there is no adiabatic limit for such a periodically kicked
driving driving protocol no matter how small v is due to the sudden movement
of the potential, thus it is impossible for the wavepacket adiabatically
following the moving potential.

###  II.2 Disordered potential with a periodically modulated amplitude

The second case considered involved a model of disordered potential with
periodically modulated amplitude, in which $\tilde{V}(x,t)$ takes the form of

$V_{2}(x,t)=\cos vt\times\tilde{V}(x,0)$ (6)

with $\tilde{V}(x,0)$ being defined the same as in the maintext. However,
different from the moving disordered potential in the maintext, $V_{2}(x,t)$
preserves the discrete temporal translational symmetry:
$V_{2}(x,t)=V_{2}(x,t+2\pi/v)$. Fig.8 (b) shows that in the presence of such a
periodically driven disordered potential, there is no localizations: the width
of the wavepacket keep growing even in the presence of a low frequency
$v=0.01aJ$.

Figure 9: (Color online). (a) The dynamics of the COM of the wavepacket
starting from initial states with different eigen-energies for the initial
state. (b) $\Delta X=X(T)-X(0)$ as a function of the initial state eigenenergy
$E_{n}$ with parameters $Tv=200a$, $v=0.01Ja$ and $L=2000$ and two different
disorder realizations (upper and lower pannels).

##  III Initial state dependence: the existence of a critical initial energy

In this section, we will study the initial state dependence of the single-
particle dynamics, and show that the initial state energy plays an important
role in determining the long-time dynamics of the wave packet. We choose
different initial states as the single-particle eigenstates of the Hamiltonian
at $t=0$, all of which are spatially localized in our 1D system. We use the
energy $E_{n}$ to characterize different initial states, where $E_{n}$ is the
eigenenergy of the $n$th eigenstate. As shown in Fig.9 (a), the long-time
dynamics of the wavepacket crucially depends on $E_{i}$: at a fixed (low)
velocity ($v=0.01aJ$), for those initial states close to the band edges (e.g.
$E=-4.33J$), the long-time dynamics ($X(t)=X_{0}+vt$) are qualitatively
identical to those starting from the ground state ($E=-5.41J$) and we still
find a sliding localized phase but with a lower critical velocity, while for
those initial states close the band center (e.g. $E=-0.02J$), the COM of the
wavepacket does not follow the potential (it move towards the opposite
direction of the potential as shown in Fig.9 a), and even an infinitesimal
driving velocity can delocalize the system and induce a ballistic transport
behavior.

For a given velocity, there exists a critical initial energy $E_{c}$ to
distinguish those sliding localized states from delocalized states.
Numerically, the critical energy $E_{c}$ is determined by the onset of the
deviation of the adiabatic COM dynamics ($X(t)=X_{0}+vt$), which is always
accompanied by the divergence of the width of the wave packet. For a fixed
$v=0.01Ja$, we plot $\Delta X=X(T)-X_{0}$ with $vT=200a$ as a function of the
eigenenergy $E_{n}$ in Fig.9 (b), from which we can find $\Delta X=200a$ for
$E_{n}<E_{c}$, while for $E_{n}<E_{c}$, $\Delta X<200a$. Fig.9 (b) also
suggests that the critical initial energy seems depends on the disorder
realizations, but we are not sure whether such a dependence is a finite size
effect or not.

Figure 10: (Color online). (a) The dynamics of width of the wavepacket in the
presence of a slowly moving quasi-periodic potential
$V_{k_{0}}(x,t)=\Delta\cos 2\pi k_{0}(x-vt)$ with different moving velocities;
(b) the dependence of the renormalized nearest-neighboring tunneling amplitude
(the slope of $w(t)$) as a function of moving velocity of the quasi-periodic
potential. The parameters are chosen as $k_{0}=\frac{\sqrt{5}-1}{2}$,
$\Delta=4J$, and $L=1000$.

Figure 11: (Color online). (a)The excess energy pumped into the system after
the potential is pushed forward by one lattice constant $\delta E$ as a
function of the moving velocity $v$ in the presence of disordered or quasi-
periodic potentials. (b)The instantaneous eigenenergies during this period
$0<tv<a$. The parameters are chosen as $L=64$ and $\Delta=4J$ for (a) and (b).
The inset of (b) indicates the number of the energy crossing between the
instantaneous ground state and 1st excited state as a function of the system
size $L$.

##  IV Wavepacket dynamics in the presence of moving quasi-periodic potential

###  IV.1 Ballistic transport and effective tunneling amplitude

In the maintext, we show the wavepacket dynamics under a moving quasi-periodic
potential is qualitatively different from that under a moving disordered
potential, at a low velocity, the latter will be localized, while the former
is characterized by a ballistic transport, where the width of the wavepacket
grows linearly with time. The slope of this linear growth can be considered as
the velocity of the ballistic transport, which is proportional to the
effective single-particle tunneling amplitude between adjacent sites
($J^{\prime}$). As shown in Fig. 10 (a), the slopes depend on the moving
velocity of the potential, from which we can derive a moving velocity
dependence of the effective tunneling amplitude. Fig.10 (b) suggests that
$J^{\prime}$ depends on $v$ in an oscillatory way, whose envelop roughly grows
linearly with increasing $v$. In the maintext, we consider the situation where
the period of the quasi-periodic potential ($a_{0}$) is slightly larger than
the lattice constant ($a$). At a small moving velocity, the typical tunneling
time between the adjacent sites is $t^{*}\sim\frac{a_{0}-a}{v}$, whereas the
effective tunneling amplitude $J^{\prime}\sim\frac{1}{t^{*}}$, thus is
proportional to $v$. For a general $a_{0}$, the moving quasi-periodic
potential may induce longer-range tunneling since the 1st excited state in
this case may not locate at the adjacent sites of the groundstate wavepacket.

###  IV.2 Breakdown of adiabaticity in the presence of moving quasi-periodic
potential

In the maintext, we mentioned that the qualitatively different behaviors
between the systems with moving disordered and quasi-periodic potential is
related with the breakdown of the adiabaticity in the latter, here a detailed
analysis of this point is provided. In general, for an isolated system with
some external parameter being slowly driven from some initial value to the
final one with a ramp speed $v$, assuming there is no energy level crossing on
the way, the adiabatic theorem states that the excess entropy or energy
density ($\delta e$) pumped into the system in this process as a function of
$v$ will approach to zero as $\delta e\sim v^{2}$ in the limit $v\rightarrow
0$. This is due to the fact that $\delta e$ is an analytic function of $v$,
but insensitive to its sign ($\delta e(v)=\delta e(-v)$).

Specific to our system, the time varying parameter is the phase shift
$\phi(t)$ of moving potential, which grows linearly as $\phi(t)\sim vt$, where
the moving velocity $v$ is the ramp speed. For simplicity, we consider the
single-particle case. Initially, $\phi(t=0)=0$, and we choose the initial
state as the ground state of $\hat{H}(t=0)$. The final phase shift is chosen
to satisfy the potential is pushed forward by one lattice constant
$\phi(t=t_{f})=a$, thus $t_{f}=a/v$. Due to the sliding space-time
translational symmetry, the Hamiltonian at the final stage $\hat{H}(t=t_{f})$
is equivalent to initial one $\hat{H}(t=0)$ by shifting one lattice constant
($i\rightarrow i+1$). Under the periodical boundary condition, this
equivalence indicates that the energy spectrums of $\hat{H}(t=t_{f})$ and
$\hat{H}(t=0)$ are the identical with each other. As a consequence, the excess
energy pumped into the system in this process (the disordered/quasiperiodic
potential is pushed forward by one lattice constant) is defined as:

$\delta e=\langle\psi(t_{f})|\hat{H}(t=t_{f})|\psi(t_{f})\rangle-E_{0}$ (7)

where $E_{0}$ is the ground state energy for both $\hat{H}(t=t_{f})$ and
$\hat{H}(t=0)$, which is also the system energy at $t=0$.
$|\psi(t_{f})\rangle$ is the wavefunction at the final time $t_{f}$.

The excess energy $\delta e$ as a function of $v$ for both cases with
disordered and quasi-periodic potential are plotted in Fig.11 (a), which
indicates a qualitative difference between them: as for a moving disordered
potential, the excess energy grows with the moving velocity as $\delta e\sim
v^{2}$, which suggests that the adiabatical theorem still holds in this case.
On the contrary, for the moving quasi-periodic potential, $\delta e\rightarrow
const$ in the limit of $v\rightarrow 0$, indicating a breakdown of the
adiabatical theorem, which can explain that even an infinitesimal velocity can
delocalize the system. We expect that such a breakdown of adiabatical theorem
is due to the presence of a large amount of level crossings ($\infty L$ as
shown in the inset of Fig.11 b) during this process, which pumps the particle
to excited states via a sequence of Landau-Zener tunnelings.