# Phases and dynamics of ultracold bosons in a tilted optical lattice

K. Sengupta  School of Physical Sciences, Indian Association for the
Cultivation of Science, 2A and 2B Raja S. C. Mullick Road, Jadavpur 700032,
India.

###### Abstract

We present a brief overview of the phases and dynamics of ultracold bosons in
an optical lattice in the presence of a tilt. We begin with a brief summary of
the possible experimental setup for generating the tilt. This is followed by a
discussion of the effective low-energy model for these systems and its
equilibrium phases. We also chart the relation of this model to the recently
studied system of ultracold Rydberg atoms. Next, we discuss the non-
equilibrium dynamics of this model for quench, ramp and periodic protocols
with emphasis on the periodic drive which can be understood in terms of an
analytic, albeit perturbative, Floquet Hamiltonian derived using Floquet
perturbation theory (FPT). Finally, taking cue from the Floquet Hamiltonian of
the periodically driven tilted boson chain, we discuss a spin model which
exhibits Hilbert space fragmentation and exact dynamical freezing for wide
range of initial states.

##  I Introduction

The physics of ultracold bosonic atoms in an optical lattice has attracted
tremendous attention in recent years bref1 ; bref1a ; bref2 ; bref3 ; brefrev
. This enthusiasm stemmed from the fact that such a boson system acts as one
of the simplest emulator of the Bose-Hubbard model bref5 . Thus the study of
the low-energy physics of these bosons allows one to access the superfluid-
insulator quantum phase transition for ultracold bosons which is well-known to
be present in the phase diagram of a clean Bose-Hubbard model fisher1 ;
nandini1 ; tvr1 ; free1 ; nicolas1 ; tref1 .

The Mott phase of the bosons in this system, at integer filling, constitutes a
localized state with no broken symmetry. It was long realized that additional
broken translational or other discrete symmetries may lead to interesting
strongly-interacting phases of matter within the Mott phase. To this end,
several theoretical suggestions have been put forth. These include study of
bosons with nearest-neighbor repulsive interaction leading to the possibility
of different competing density-wave ground states at half-filling due to
broken translational symmetry. The precise nature of these states depends on
the geometry of the underlying lattice leading to a projective symmetry group
(PSG) based classification of the possible Mott phases balents1 ; balents2 ;
yb1 . In addition, systems with multiple species and/or spinor bosons have
also been studied; the low-energy physics of their Mott phases are mostly
controlled by effective models arising out of an order-by-disorder mechanism.
The effective Hamiltonians obtained for such bosons may be described in terms
of interacting spins which may represents either real spin or species degrees
of freedom. These Hamiltonians lead to several spin (or species) ordered
bosonic ground states girvin1 ; demler1 ; kush1 ; kush2 ; arun1 ; galit1 .
However, experimental realization of such strongly correlated states of
bosonic systems have not been yet achieved.

Instead, such a symmetry broken Mott phase was experimentally realized through
a slightly unexpected route. The key idea was to generate an ”electric field”
for these neutral bosons bref1 ; bref2 ; bref3 . As shall be detailed later,
such a field may be generated experimentally by shifting the center of the
trap which confines the bosons leading to a linear potential term in their
Hamiltonian. Alternatively, it can also be realized by application of a
linearly spatially varying Zeeman magnetic field to spinor bosons. The
presence of such a field leads to the realization of bosonic Mott ground
states with broken $Z_{2}$ symmetry subir2 ; bref2 . In addition, it has
recently been realized that the model used subir2  to describe such systems
can also describe the physics of Rydberg chains gr1 ; gr2 ; gr3 ; gr4 . The
study of non-equilibrium dynamics of these Rydberg atoms has shown that they
may host several anomalous features scarlitrev ; scarlit1 ; scarlit2 ;
scarlit3 ; scarlit4 ; scarlit5 ; ryddyn1 ; ryddyn2 ; ryddyn3  including scar-
induced dynamics scarlitrev ; scarlit1 ; scarlit2 ; scarlit3 ; scarlit4 ;
scarlit5 ; scarothers1 ; scarothers2 ; scarothers3 ; scarothers4  and
possibility of drive induced tuning of ergodicity properties ryddyn1 ; ryddyn2
; ryddyn3 . The main aim of this chapter is to provide a summary of some of
the theoretical and experimental aspects of this rapidly developing field.

The rest of this chapter is organized as follows. In Sec. II, we provide a
brief discussion of the experimental platforms that lead to the realization of
such tilted bosons. This is followed by Sec. III where we describe a low-
energy model which describes the ground states phases of these system and
discuss its relation with models describing a chain of ultracold Rydberg
atoms. This is followed by Sec. IV where we discuss non-equilibrium dynamics
of the model. Next in Sec. V we discuss possible related models with
interesting properties which may be realized using such boson platforms.
Finally, we conclude in Sec. VI.

##  II Experimental Platforms

In this section, we shall describe the essential ingredients of three
experimental setups. The first two involves generating a tilt for ultracold
bosons in a optical lattice while the third involves Rydberg atoms in one-
dimensional (1D) lattice.

The first experiment on tilted bosons in an optical lattice was carried out in
Ref. bref1, . In this experiment, spinor ${}^{87}{\rm Rb}$ atoms in their
angular momenta $F=2$ and $m_{F}=2$ state (where $F$ is the total and $m_{F}$
is the azimuthal angular momenta) were cooled in a trap. The trap was chosen
to be a cigar shaped magnetic trap with radial and axial frequencies $\nu_{\rm
radial}=240\,{\rm Hz}$ and $\nu_{\rm axial}=24\,{\rm Hz}$. The trap included
$2\times 10^{5}$ bosonic atoms. After the condensate was the formed, the
radial trapping frequency was relaxed to $240\,{\rm Hz}$ over a time period of
$500$ms. This led to a spherically symmetric condensate.

To generate an optical lattice, six orthogonal lasers with wavelength
$\lambda=852$nm were applied. This resulted in a potential of

$\displaystyle V(x,y,z)$ $\displaystyle=$ $\displaystyle
V_{0}(\sin^{2}kx+\sin^{2}ky+\sin^{2}kz)$ (1)

where $k=2\pi/\lambda$ and $\lambda$ is the wavelength of light used for the
lasers. For such optical lattices, all energies are typically measured in
terms of the recoil energy given by $E_{r}=\hbar^{2}k^{2}/(2m)$; this is the
basic energy scale in the problem which can be created out of the wavelength
$\lambda$ of the laser and mass $m$ of the atoms. It can be shown that in such
a lattice, the atoms see an approximately harmonic potential with strength
$V^{\prime}\simeq\sqrt{V_{0}E_{r}}$ leading to a trapping frequency
$\nu_{r}\simeq V^{\prime}/h\simeq 30$kHz. In the experimental setup of Ref.
bref1, , the strength of the trapping potential could be up to
$V^{\prime}=22E_{r}$. These values of parameters were sufficient to obtain a
Mott insulating state of bosons with one boson per site of the optical
lattice.

In addition, to apply the tilt, the center of the trap confining the bosons
was shifted. This led to a shifted harmonic potential. The bosons thus sees a
linear gradient since $V_{\rm shifted}=K_{0}(x-x_{0})^{2}/2\simeq
V_{x_{0}=0}-cx$, where $c=K_{0}x_{0}$, $x_{0}$ is the shift of the trap
center, and we have ignored an irrelevant constant term. Thus the shift is
analogous to having an electric field for the neutral bosons with
$eE=K_{0}x_{0}$; the magnitude of the field can be controlled by controlling
the shift. It was found in Ref. bref1,  that in the Mott phase, the presence
of such a shift leads to resonant energy absorption at special values of
electric field $eE$ (or shift $x_{0}$) which satisfies $eE=nU$ where $U$ is
the on-site interaction between the bosons and $n$ is an integer.

The measurement which confirmed this resonant absorption in the experiments of
Ref. bref1,  involved several steps. First, the condensate was subjected to
optical lattice potential whose amplitude was slowly ramped up (over a period
of $80$ms) to the final value. This value is chosen ($V^{\prime}=22E_{r}$)
such that the system would be in its Mott state with one boson per site.
Second, the system is allowed to equilibrate in this potential for a period of
$20$ms. During this time, the tilt of a fixed magnitude is applied to the
system. Third the optical lattice potential is reduced to $V^{\prime}=9E_{r}$
(for which the bosons are in a superfluid state) within a short time interval
of $3$ms. Finally, the trap and lattice is turned off, and the real space
imaging of the flying out bosons are carried out. It is to be noted that after
turning off the trap and the lattice, the bosons undergo a free flight. Thus
their position distribution during the flight provides information about their
momentum distribution (or initial velocity distribution) inside the trap just
before it was turned off. Consequently, in the superfluid phase, the position
distribution of the bosons would have a large central peak signifying the
presence of a large number of bosons in the state $\vec{k}=0$ inside the trap.
The experiment in Ref. bref1,  further noted that if the system absorbed
energy when the tilt is applied, it is not going to equilibrate to the
superfluid ground state when the lattice potential is reduced. Instead, it
will be in an excited state where the boson wavefunction have larger weight in
the $\vec{k}\neq 0$ states in the trap. This will broaden their position
distribution leading to a broader central peak during imaging. A large width
of the central peak of the image therefore is a signature of large energy
absorption due to the perturbation; such a large width is observed around
$eE=nU$.

The measurement technique of Ref. bref1,  did not allow for a direct
measurement of the number distribution of the bosons within the lattice in the
Mott phase. This is clearly desirable if one wanted to distinguish between
several competing ground states in the Mott regime. The later experiments,
performed in Ref. bref2, ; bref3, , made significant progress in this
direction. In these experiments, which also used ${}^{87}{\rm Rb}$ atoms,
lasers with wavelength of $\lambda=680$nm were used to generate the optical
lattice. The trap used to confine the bosons was also optical. The maximum
allowed lattice depth achieved in these experiments were $V^{\prime}=45E_{r}$
which brought the bosons close to their Mott state in the atomic limit. The
lattice obtained was a 2D lattice; however, the experiment had separate
control over the lattice depth in the $x$ (along the chain) and the $y$
(between the chains) direction. The inter-chain lattice depth could be ramped
to very high to achieve an effectively 1D optical lattice (almost disconnected
chains).

The tilt generated in these experiments were carried out via application of
Zeeman magnetic field which varied linearly in space. Such a Zeeman field
leads to a potential term $H_{1}=-g\mu_{B}B_{0}\sum_{j}j\hat{n}_{j}$ where
$\mu_{B}$ is the Bohr magneton, $g$ is the gyromagnetic ratio, $B_{0}$ is the
amplitude of the field on the first site ($j=1$) and $\hat{n}_{j}$ is the
number operator for the bosons. Such a field therefore creates an effective
electric field for the bosons with $eE=g\mu_{B}B_{0}$. Note that the intensity
of this field can be controlled by tuning the magnetic field which is
experimentally much more convenient than shifting the trap center.

To measure the density distribution of the bosons inside a trap, the
experiments in Ref. bref2, ; bref3,  used an ingenious fluorescence imaging
technique. In this technique, the depth of the lattice potential is suddenly
increased just before the measurement so that the bosons within the lattice
freeze for a long time scale. Then fluorescent light is applied to the bosons;
the frequency of this light is chosen in such a way that any boson pair on a
lattice site can scatter via light assisted collision and move out of the
trap. Thus such a fluorescent light leaves behind an empty lattice site if
there were, initially, an even number of bosons on that site; in contrast, if
there are odd number of bosons, at least one boson remains on the site. Thus
an imaging of the bosons after the fluorescent light is applied provides
information about their parity of occupation in the Mott state. If a boson
remains on a lattice site, it may scatter several photons leading to bright
spots. Thus it provides a direct distinction between Mott states with even and
odd occupations (for example between $0-2-0-2...$ and $1-1-1-1...$
occupations) of bosons; the sites with even occupations lead to dark spots
while that with odd occupation appear bright. This proves to be very useful
while measuring boson occupation in the presence of a tilt.

Finally we briefly discuss some experiments on ultracold bosonic Rydberg atoms
in a 1D optical lattice gr1 ; gr2 ; gr3 ; gr4 . The atoms used for such
experiments are again ${}^{87}{\rm Rb}$. These atoms are confined in a 1D
optical lattice as discussed earlier. In these experiments, the atoms are
subjected to a Raman laser which induces a transition between the ground
($|g\rangle=|5S_{1/2};F=2;m_{F}=-2\rangle$) and the Rydberg excited
($|r\rangle=|70S_{1/2};F=1/2;m_{F}=-1/2\rangle$) state of the atoms via an
intermediate state ($|p\rangle=|6P_{3/2};F=3,m_{F}=3\rangle$). The experiment
used two lasers with wavelength $420$nm and $1024$nm (corresponding to single
photon Rabi frequencies of $\Omega_{B}=2\pi\times 36$ and
$\Omega_{R}=2\pi\times 60$ MHz respectively) so that there is a detuning
$\delta$ between the $|g\rangle$ and the $|p\rangle$ levels:
$\hbar\Omega_{B}=\delta+(E_{p}-E_{g})$. Similarly the detuning $\Delta$
between the $|r\rangle$ and the $|g\rangle$ levels are given by
$\Delta=\hbar(\Omega_{B}+\Omega_{R})-(E_{r}-E_{g})$. For $\Delta=0$ the atoms
would be in a equal linear superposition of $|g\rangle$ and $|r\rangle$
states. $\Delta$ can be tuned in experiments to have either positive or
negative values. The effective coupling between the atoms in the ground and
Rydberg state is given by $\Omega=\hbar^{2}\Omega_{B}\Omega_{R}/(2\delta)$.
Further experimental details regarding the setup can be found in Ref. gr1, .

The atoms experience a strong repulsive dipolar interactions ($V(r)\sim
1/|r|^{6}$) when excited in the Rydberg state. This interaction can be tuned
by controlling the relative position of the atoms in the lattice; in
particular, a regime can be reached which precludes two Rydberg excited atoms
within a certain length $R$. This phenomenon is called the dipole blockade and
$R$ is termed as the blockade radius. In experiments, it is possible to
control the dipolar interaction strength between the atoms leading to an
effective tuning of the blockade radius. Such a tuning of the blockade radius
may lead to translational symmetry broken phases as follows. In experiments
the blockade radius can be tuned to next neighbor; moreover, the parameters
are so adjusted that it is energetically favorable for each individual atom to
be in its Rydberg excited state. A competition between these two phenomenon
leads to a state where atoms in every alternate site can be in the state
$|r\rangle$ leading to a symmetry broken state. Similarly states with blockade
radius of two lattice sites can be achieved; these states, for large negative
$\Delta$, shall have a Rydberg excited in every three sites. In the next
section, we shall find that the physics of such a system can be understood in
terms of a model which has several common features with the tilted Bose
Hubbard model.

##  III Model and Phases

The low-energy effective model for the bosons in a tilted 1D optical lattice
has been derived in Ref. subir2, . An extension of this model has been studied
in Ref. fendley1, . In the first part of this section, we briefly sketch the
method of derivation of this model from the microscopics. In the next part, we
shall discuss the similarity of the model with a spin model appropriate for
describing the Rydberg atoms discussed in the previous section. The extension
of this model to higher dimension subir4 ; others1 ; bhaskar1a , its
application to tilted dipolar bosons others2 ; others3 , tilted spin chains
others4 , and other models with modified constraint bhaskar1b  has also been
worked out; however, in the rest of this work, we shall restrict ourselves to
the initial 1D model proposed in Ref. subir2, .

Figure 1: (a) Schematic representation of the parent Mott state with
$n_{0}=1$. (b) The state with a single dipole. (c) A state with two
neighboring dipoles which is equivalent to a single dipole of length two and
is not a part of the low energy subspace. (d) A state with maximal number of
dipoles which is also a part of the low-energy subspace.

In the presence of a tilt, the Hamiltonian of ultracold bosons in a 1D optical
lattice is given by

$\displaystyle H_{b}$ $\displaystyle=$ $\displaystyle-w\sum_{\langle
jj^{\prime}\rangle}(b_{j}^{\dagger}b_{j^{\prime}}+{\rm
h.c.})-\mu\sum_{j}\hat{n}_{j}+\frac{U}{2}\sum_{j}\hat{n}_{j}(\hat{n}_{j}-1)-{\mathcal{E}}\sum_{j}j\hat{n}_{j}$
(2)

where $b_{j}$ denotes the boson annihilation operator at site $j$,
$\hat{n}_{j}=b_{j}^{\dagger}b_{j}$ is the boson number operator, $w$ is their
nearest-neighbor hopping amplitude, $U$ is the amplitude of the on-site
interaction between the bosons, and $\mu$ denotes their chemical potential. In
Eq. 2, $\langle jj^{\prime}\rangle$ indicates that $j$ and $j^{\prime}$ are
nearest neighbors, and ${\mathcal{E}}$ denotes amplitude of the applied
electric field in units of energy. In absence of the field, the bosons are in
the in Mott state so that $\mu,U\gg w$; moreover in what follows, we shall
address the physics of the system for $U,{\mathcal{E}}\gg
w,|U-{\mathcal{E}}|$.

Before embarking on the description of the ground state of this model, it is
useful to think about the limit $U=0$. In this case the model represents
Wannier-Stark ladder for the bosons. The single-particle Schrodinger equation
for such non-interacting bosons is given by (with energies $\epsilon=E+\mu$)

$\displaystyle\epsilon\psi_{j}$ $\displaystyle=$
$\displaystyle-{\mathcal{E}}j\psi_{j}-w\left[\psi_{j-1}+\psi(j+1)\right]$ (3)

The solution to this problem is well-known wsref . The energy of the bosons
are given by $\epsilon_{m}={\mathcal{E}}m$ where $m$ denotes integers ranging
from $-\infty$ to $\infty$. Note that the energy is not bounded from below and
this arises from the fact that the potential is also not bounded on an
infinite lattice. The eigenfunctions of the bosons are given by

$\displaystyle\psi_{m}(j,t)=J_{j-m}(2w/{\mathcal{E}})\exp[-i{\mathcal{E}}mt/\hbar],$
(4)

where $J_{n}(x)$ denotes the Bessel function. Note that the wavefunction
returns to its original value at regular time intervals
$t_{0}=2\pi\hbar/\mathcal{E}$ which indicates Bloch oscillations. The bosons
are strongly localized to their respective site for $w\ll{\mathcal{E}}$. This
behavior can also be understood from the fact that $J_{j-m}(x)$ has
appreciable weight at $j=m$ for $x\ll 1$ leading to more than exponentially
localized wavefunctions subir2 . This behavior is in contrast to the classical
expectation where an electric field shall accelerate the bosons to the end of
the chain causing an electric breakdown. This feature shall be a key component
in constructing the effective theory in the limit ${\mathcal{E}}\gg w$. Such a
breakdown can happen in realistic quantum systems due to electric field
assisted tunneling to higher single particle bands; however for ultracold
bosons, such bands are well separated in energy from the lowest bands. This
leads to a tunneling time which is larger than the typical system lifetime and
allows us to ignore such electric breakdown. In the absence of such breakdown,
the system remains in the metastable parent state $|\psi(t=0)\rangle$. Thus
the strategy adapted in Ref. subir2,  for finding out the equilibrium phase of
the tilted bosons was to start from the metastable parent Mott state and find
out the manifold of states whose energies are close to the parent Mott state
in the regime ${\mathcal{E}},U\gg w,|{\mathcal{E}}-U|$.

To understand the nature of the effective theory of the interacting bosons, we
therefore start from the parent Mott state (the Mott ground state in the
absence of the electric field) with $n_{0}$ bosons on every lattice site (see
Fig. 1(a)) and ask how the electric field may destabilize such a bosonic
system. A possibility is that this can be achieved by addition of an extra
particle or hole in over the parent Mott state. The energy cost of adding a
particle to the Mott state is $E_{p}=U(n_{0}+1)-\mu$ while that for a hole is
$E_{h}=\mu-Un_{0}$. Note that $E_{p},E_{h}>0$ as long as the system is in the
Mott ground state. However, once added, the particle[hole] sees the electric
field and is thus described by an effective Hamiltonian which is identical to
Eq. 3 with $\mu=0$ and $w\to w(n_{0}+1)[wn_{0}]$. They are therefore localized
and can not reduce their energy via hopping (which would have been the case if
${\mathcal{E}}$ was absent). Thus such excitations are not effective in
destabilizing the Mott states. This situation is therefore different from that
of bosons without the tilt, where such additional particles or holes
destabilize the Mott state to bring about the superfluid-insulator transition
free1 ; nicolas1 .

The excited states which are part of the low-energy manifold around the parent
Mott state correspond to dipole excitations where a particle hops from a
lattice site to the next (Fig. 1(b)) subir2 . Such an excitation creates an
additional particle on the site $j+1$ and a hole at site $j$ and thus have an
energy cost

$\displaystyle E_{\rm dipole}$ $\displaystyle=$ $\displaystyle
E(n_{0}+1)+E(n_{0}-1)-2E(n_{0})-{\mathcal{E}}=U-{\mathcal{E}}$ (5)

A state with one or multiple dipoles thus become a part of the low energy
manifold for $U\simeq{\mathcal{E}}$. These states are schematically
represented in Fig. 1(d). These dipoles live on the link $\ell$ between two
adjacent lattice site $j$ and $j+1$. Identifying the parent Mott state as the
dipole vacuum, the operators describing the creation of such dipoles can be
written in terms of the boson operators as

$\displaystyle d_{\ell}^{\dagger}|{\rm vac}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n_{0}(n_{0}+1)}}b_{j}b_{j+1}^{\dagger}|{\rm
Mott}\rangle$ (6)

We note the following properties of the dipoles. First, there can be at most
one dipole on any given link. This is seen by noting that creating another
dipole on the same link cost an energy $E_{2}=4U-2{\mathcal{E}}$ (for
$n_{0}>1$). Thus a state with two dipoles on the same link is not a part of
the low energy manifold. Second, two dipoles on adjacent links cost an energy
$U-2{\mathcal{E}}$ (Fig. 1(c)) (this is equivalent to a length two dipoles
where a boson hops two sites as shown in Fig. 1(c)) and thus such states are
also not a part of this manifold. In fact it can be shown that all longer
length dipoles do not play a role subir2 . This leads to an effective dipole
Hamiltonian supplemented by the constraints

$\displaystyle H_{d}$ $\displaystyle=$
$\displaystyle-w^{\prime}\sum_{\ell}(d_{\ell}+d_{\ell}^{\dagger})+\lambda\sum_{\ell}\hat{n}_{\ell},\quad\hat{n}_{\ell}\leq
1,\quad\hat{n}_{\ell}\hat{n}_{\ell+1}=0\ $ (7)

where $\hat{n}_{\ell}=d_{\ell}^{\dagger}d_{\ell}$, $\lambda=U-{\mathcal{E}}$
and $w^{\prime}=w\sqrt{n_{0}(n_{0}+1)}$. We note that the model does not
conserve dipole numbers since spontaneous hopping of bosons leads to
formation/annihilation of dipoles. Moreover, the model is non-integrable due
to the presence of the second constraint. The number of states in the Hilbert
space of this Hamiltonian does not scale as $2^{L}$ (as that of hardcore
bosons or spins); it can be shown that they scale as $\varphi^{L}$ for large
$L$ where $\varphi=(1+\sqrt{5})/2$ is the golden ratio. It was shown that it
is possible to write down a recursion relation for the Hilbert space
dimension, ${\mathcal{N}}_{L}$, of the constrained dipole system with periodic
boundary condition:
${\mathcal{N}}_{L}={\mathcal{N}}_{L-1}+{\mathcal{N}}_{L-2}$. This relates
${\mathcal{N}}_{L}$ to Fibonacci numbers $F_{L}$ for integer $L$:
$N_{L}=F_{L}$ ryddyn1 .

It is easy to see that $H_{d}$ admits a representation in terms of spin-half
Pauli matrices due to the constraint $\hat{n}_{\ell}\leq 1$. Indeed the
mapping

$\displaystyle\sigma_{\ell}^{z}$ $\displaystyle=$ $\displaystyle
2\hat{n}_{\ell}-1,\quad\sigma_{\ell}^{x}=(d_{\ell}+d_{\ell}^{\dagger}),\quad\sigma_{\ell}^{y}=i(d_{\ell}-d_{\ell}^{\dagger})$
(8)

leads to the spin Hamiltonian pekker1 ; scarlit2  (up to an irrelevant
constant)

$\displaystyle H_{s}$ $\displaystyle=$
$\displaystyle\sum_{\ell}(-w^{\prime}\tilde{\sigma}_{\ell}^{x}+\lambda\sigma_{\ell}^{z}/2),\quad\tilde{\sigma}_{\ell}^{a}=P_{\ell-1}\sigma_{\ell}^{a}P_{\ell+1},\quad
P_{\ell}=(1-\sigma_{\ell}^{z})/2$ (9)

Here we have implemented the constraint $\hat{n}_{\ell}\hat{n}_{\ell+1}=0$ by
using a local projection operator $P_{\ell}$ and it is understood that $H_{\rm
spin}$ operates in the constrained Hilbert space where one can not have two up
spins on the neighboring link. The projection operator $P_{\ell}$ ensures that
this constraint is obeyed by $H_{s}$. We note here that the model, at
$\lambda=0$, has been dubbed as the $PXP$ model scarlitrev ; scarlit1 ;
scarlit2 ; scarlit3 ; scarlit4 ; scarlit5 .

Next, we note that addition of longer-range density-density interaction to
$H_{d}$, leads to the Hamiltonian fendley1

$\displaystyle H^{\prime}_{d}$ $\displaystyle=$ $\displaystyle
H_{d}+V\sum_{\ell}\hat{n}_{\ell}\hat{n}_{\ell+2}$ (10)

We shall not discuss the details of the phases or the dynamics of this model
here but refer the readers to Refs. fendley1, ; subir2a, ; subir3, ; ghosh1,
for details. The model displays a rich phase diagram and supports non-Ising
quantum phase transition. Moreover, this also serves the low-energy effective
model for the Rydberg atoms discussed in the last section in certain limit; we
shall detail this point towards the end of this section.

The ground phase diagram of $H_{d}$ can be understood in a straightforward
manner. For $\lambda\gg w^{\prime}$($U\gg{\mathcal{E}}+w^{\prime}$), the
dipole excitations are energetically costly and the ground state is the dipole
vacuum $|{\rm vac}\rangle$ (the parent Mott state $|{\rm Mott}\rangle$). In
contrast, for $|\lambda|\gg 0$ with $\lambda<0$ (${\mathcal{E}}\gg
U+w^{\prime}$), the ground state is clearly a state with maximal number of
dipoles. However, due to the constraint, there are two such maximal dipole
states. The first consists of dipoles are formed on the even links of the 1D
lattice while the second where they are formed on the odd links. These states,
for a chain of length $2L$ whose links are labeled from $0$ to $2L-1$, are
(see Fig. 1(d))

$\displaystyle|{\mathbb{Z}}_{2}\rangle$ $\displaystyle=$ $\displaystyle
d_{0}^{\dagger}d_{2}^{\dagger}...d_{2L-2}^{\dagger}|{\rm
vac}\rangle,\quad|{\bar{\mathbb{Z}}_{2}}\rangle=d_{1}^{\dagger}d_{3}^{\dagger}...d_{2L-1}^{\dagger}|{\rm
vac}\rangle,$ (11)

The ground state chooses one of the two states and hence breaks $Z_{2}$
symmetry. This implies that it must be separated from the dipole vacuum ground
state by a transition; this quantum phase transition belongs to the Ising
universality class and occurs at subir2

$\displaystyle{\mathcal{E}}_{c}=U+1.31\sqrt{n_{0}(n_{0}+1)}w$ (12)

Such a transition can be understood to be the result of competition between
the dipole number fluctuation arising from the first term in Eq. 7 and the
effect of the electric field in the second term which makes such fluctuation
energetically costly. A rough estimate of ${\mathcal{E}}_{c}$ can also be
obtained from a variational wavefunction approach. The ordered state which
breaks the $Z_{2}$ symmetry is characterized by an Ising order parameter. In
the dipole language this order parameter can be written as

$\displaystyle O$ $\displaystyle=$
$\displaystyle\frac{1}{2L}\sum_{\ell=0}^{2L-1}(-1)^{\ell}\hat{n}_{\ell}$ (13)

which is the dipole density at $k=\pi$ signifying the broken translational
symmetry of the ground state.

The presence of such a translational symmetry broken state was directly
verified in the experiment carried out in Ref. bref2,  using the fluorescence
imaging technique discussed earlier. The experiment started with a Mott state
having $n_{0}=1$ bosons per lattice site and generated a tilt using a linearly
varying Zeeman field. Thus the parent Mott state correspond to an odd number
of bosons per site and provided bright patterns in an imaging measurement. In
contrast, the maximal dipole state, which constitutes an even number of states
per site would provide a dark image. It was found that an increase of the
strength of the magnetic field providing the tilt indeed led to a dark
pattern; moreover, one could coherently interpolate between such bright and
dark imaging patterns by tuning the strength of the magnetic field. This
constituted the realization of the symmetry broken Mott state for ultracold
bosons in an optical lattice.

Before ending this section, we briefly comment on the Rydberg atom experiments
and the relation of the Hamiltonian of such Rydberg atoms with the model
developed here. The low-energy effective Hamiltonian of these atoms describes
the physics of the system for times which is smaller compared to typical decay
scales of Rydberg excited atoms. The Hamiltonian involves two states
$|g\rangle$ and $|r\rangle$. Defining $\tau_{\alpha}$ ($\alpha=x,y,z$) to be
standard Pauli matrices in the space of these two states, one can write

$\displaystyle H_{\rm Ryd}$ $\displaystyle=$
$\displaystyle-\sum_{j}(\Omega\tau_{j}^{x}+\Delta\tau_{j}^{z})+\sum_{jj^{\prime}}V_{jj^{\prime}}\hat{n}^{\prime}_{j}\hat{n}^{\prime}_{j^{\prime}}$
(14)

where $\Delta$ and $\Omega$ has been defined in Sec. II, $V_{jj^{\prime}}$ is
the dipolar interaction between two Rydberg excited atoms. Here
$\hat{n}^{\prime}_{j}=(1+\tau_{j}^{z})/2$ denotes the number operator for
Rydberg excitations. We note that for $V_{j,j+1}\gg\Omega,|\Delta|$ and
$V_{j,j+n}\ll\Omega,|\Delta|$ for $n>2$, Eq. 14 reduces to Eq. 10 with
$\Omega\to-w\sqrt{n_{0}(n_{0}+1)}$, $\Delta\to\lambda$ and $V_{j,j+2}\to V$.
Moreover if $V\ll\Omega,|\Delta|$, Eq. 14 reduces to Eq. 7 (and hence to Eq.
9). It is therefore expected that the ground state phase diagram of Rydberg
atoms would also reflect translational symmetry broken states for $\Delta<0$
and $|\Delta|\gg\Omega$.

In experiments carried out in Ref. gr1, , $\Delta$ could be tuned to large
negative values to achieve the translational symmetry broken states. Such
states had a period $2$ when $V_{j,j+1}\gg\Omega,|\Delta|$ and
$V_{j,j+n}\ll\Omega,|\Delta|$ for $n>1$, in accordance with the prediction of
the dipole model. In addition, it was possible to tune the strength of the
Rydberg interaction such that $V_{j,j+1},V_{j,j+2}\gg\Omega,|\Delta|$ and
$V_{j,j+n}\ll\Omega,|\Delta|$ for $n>2$. In this case, the ground state broke
$Z_{3}$ symmetry and led to a period $3$ density-wave in accordance with that
found in Ref. fendley1,  by analyzing the Hamiltonian given in Eq. 10.

##  IV Non-equilibrium dynamics

In this section, we shall discuss the non-equilibrium dynamics of the tilted
bosons in the presence of an optical lattice. The quench and the ramp dynamics
of the bosons shall be discussed in Sec. IV.1 while the periodic dynamics will
be addressed in Sec. IV.2.

###  IV.1 Quench and ramp protocols

It is well-known that ultracold atoms provide a perfect platform for studying
non-equilibrium dynamics of closed quantum systems rmpks ; dziar1 ; adutta1 ;
brefrev ; bookchap1 ; bookchap2 . One of the simplest protocol for such
dynamics is the quench, where a parameter in the Hamiltonian of the system is
changed suddenly. Consequently, the state of the system does not have time to
react. The old ground state of the system (or any initial state the system
might be in when the quench is performed) evolves according to the new
Hamiltonian. Since the old state is no longer an eigenstate of the new
Hamiltonian, it displays non-trivial dynamics. For non-integrable systems,
such dynamics is expected to be lead to a thermal steady state at long times.
This follows from the eigenstate thermalization hypothesis (ETH) ethref1 ;
ethref2 ; ethref3 ; ethrev  which predicts eventual thermalization of a
typical many-body state under unitary dynamics. At short times, the system is
expected to display transient oscillations.

For the tilted boson system, such oscillations were studied in Ref. ks1, . The
bosons were initially assumed to a state of dipole vacuum $|0\rangle$. At
$t=0$, the electric field ${\mathcal{E}}$ is changed from its initial value
${\mathcal{E}}_{i}<U$ to its final value ${\mathcal{E}}_{f}$. The state of the
system at time $t$ can then be described by

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}c_{n}e^{-iE_{n}t/\hbar}|n\rangle,\quad
H({\mathcal{E}}_{f})|n\rangle=E_{n}|n\rangle,\quad c_{n}=\langle n|0\rangle$
(15)

Under such evolution, the dipole order parameter given by Eq. 13 oscillates as

$\displaystyle O(t)$ $\displaystyle=$
$\displaystyle\sum_{m,n}c_{m}c_{n}O_{mn}\cos\omega_{mn}t,\quad O_{mn}=\langle
m|O|n\rangle,\quad\omega_{mn}=(E_{m}-E_{n})/\hbar$ (16)

where we have used the fact that $c_{m}$ could be chosen to be real. A
numerical evaluation of $O(t)$, carried out in Ref. ks1,  (left panel of Fig.
2), showed that the transient oscillations have maximal amplitude when
${\mathcal{E}}_{f}=U$ near the critical point; they are tiny deep inside both
the maximal dipole and dipole vacuum phases as shown in the left panel of Fig.
2. Thus transition can act as a qualitative marker for change in state of the
system.

Figure 2: Left panel: Plot of $O(t)$ as a function of time for several
representative values of ${\mathcal{E}}_{f}/w$. For all plots $U/w=40$ and
initial state is the dipole vacuum ground state $|0\rangle$. Right panel: Plot
of ${\bar{O}}\equiv\langle O\rangle_{t}$ as a function of
${\mathcal{E}}_{f}/w$ showing a peak near the critical point. In the figs $N$
represents the chain length in units of lattice spacing. This figure is
adapted from Ref. ks1, .

To understand why the amplitude peaks around $U={\mathcal{E}}_{f}$, we focus
on the time averaged value of these oscillations

$\displaystyle\bar{O}$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}O(t)=\sum_{m}c_{m}^{2}O_{mm}.$
(17)

The amplitude of these oscillations is maximal when the product of overlap
$c_{m}$ and order parameter expectation $O_{mm}$ is large. When
${\mathcal{E}}_{f}$ is deep inside the dipole vacuum, $c_{m}\sim 1$ when $m$
correspond to the old ground state. However, $O_{mm}\to 0$ for this state
leading to a small $\bar{O}$. In contrast, when ${\mathcal{E}}_{f}$
corresponds to deep inside the maximal dipole ground state, $O_{mm}$ is large
for the new ground state; however, $c_{m}\sim 0$ for this ground state
leading, once again, to a small ${\bar{O}}$. In between, near the critical
point, ${\bar{O}}$ could be large since both $c_{m}$ and $O_{mm}$ can be non-
zero for several $m$. This leads to a peak of ${\bar{O}}$ near the critical
point as shown in the right panel of Fig. 2.

More recently, the quench dynamics of Rydberg atoms has been studied
experimentally starting from the $|{\mathbb{Z}}_{2}\rangle$ state gr1 . It was
found that the evolution of such a state, following a quench of the parameter
$\Delta\to 0$, displays long-lived coherent oscillations. Since the system is
non-integrable, ETH predicts that such dynamics will lead to an eventual
thermal steady state; however, such a steady state was not observed in
experiments for dynamics starting from $|{\mathbb{Z}}_{2}\rangle$. In
contrast, dynamics starting from the all spin-down state ($|0\rangle$ or the
dipole vacuum state) showed expected, ETH predicted, thermalization. This
phenomenon therefore constituted a weak (initial-state dependent) violation of
ETH in such finite-sized chains.

The theoretical explanation of this phenomenon followed soon scarlitrev ;
scarlit1 ; scarlit2 ; scarlit3 ; scarlit4 ; scarlit5 . The details of this has
been summarized in Ref. scarlitrev,  and references therein. It was found at
$\Delta=0$, or in the $PXP$ limit, the eigenspectrum of $H_{s}$ (or
equivalently $H_{\rm Ryd}$ for large $V_{i,i+1}$) supports a special class of
eigenstates called quantum scars scarlitrev ; scarlit1 ; scarlit2 ; scarlit3 ;
scarlit4 ; scarlit5 ; scarothers1 ; scarothers2 ; scarothers3 ; scarothers4 .
These states have finite energy density but anomalously low half-chain
entanglement entropy scarlitrev . Being anomalous, they have very little
overlap with the thermal band of mid-spectrum eigenstates. Consequently, they
form an almost closed subspace. It was also found that they have a strong
overlap with the $|{\mathbb{Z}}_{2}\rangle$ state; thus dynamics starting from
the $|{\mathbb{Z}}_{2}\rangle$ state is almost confined within the closed
subspace formed by the scars leading to coherent long-lived oscillations. This
does not happen for dynamics starting from the $|0\rangle$ state since it has
very little overlap with the scar states. Further details of this phenomenon
and properties of quantum scars is summarized in Ref. scarlitrev, .

Figure 3: Plots showing behavior of (a) residual energy $Q$ as a function of
$v=\tau^{-1}$ (b) $Q/(Lv)$ as a function of $v$ (c) overlap $F$ as a function
of $L^{2}v$ and (d) $F/(L\sqrt{v})$ as a function of $L^{2}v$ for quench from
the dipole vacuum or paramagnetic ground state(PM) to the quantum critical
point (QCP). In the plot, all quantities are dimensionless, $U/w=40$, and the
lattice spacing $a$ has been set to unity. For ramp from the PM to the maximal
dipole or antiferromagnetic (AFM) state, the behavior of $Q$ and $F$ are shown
as a function of $v$ in panels (e) and (f) respectively. This figure is
adapted from Ref. pekker1, .

Next, we discuss the ramp dynamics of such a system addressed in Ref. pekker1,
. In this case, one ramps the electric field via a linear protocol from its
initial value ${\mathcal{E}}_{i}$ at $t=0$ to a final value
${\mathcal{E}}_{f}$ at $t=\tau$ with a rate $\tau^{-1}$:
${\mathcal{E}}(t)={\mathcal{E}}_{i}+({\mathcal{E}}_{f}-{\mathcal{E}}_{i})t/\tau$.
The values of ${\mathcal{E}}_{f}$ and ${\mathcal{E}}_{i}$ are so chosen so
that the system starts from the dipole vacuum state (ground state for
${\mathcal{E}}={\mathcal{E}}_{i}<U$) and reaches the critical point at
${\mathcal{E}}={\mathcal{E}}_{f}={\mathcal{E}}_{c}$. It is well known that
such a passage to the critical point leads to excitation production; the
density of these excitations, for slow ramp rates, scale with $\tau$ according
to the Kibble-Zurek (KZ) scaling law $n_{\rm ex}\sim\tau^{-\nu d/(z\nu+1)}$,
were $\nu$ and $z$ are the correlation length and dynamical critical exponents
and $d$ is the spatial dimension of the system rmpks ; dziar1 ; adutta1 ;
bookchap1 ; bookchap2 ; kibble1 ; zureck1 ; anatoli1 ; ks2 ; ks3 ; anatoli2 ;
sondhi1 . However such scaling laws are strictly appropriate when the system
size is large. For finite-sized systems such as the Rydberg chain, the system
size $L$ provides a length scale which restrict the applicability of the
scaling laws.

Figure 4: Plots of (a) dipole density $n_{d}$ as a function of the ramp
amplitude $U-{\mathcal{E}}_{f}$ for several $v=\tau^{-1}$, (b) excitation
density $D$ as a function of $v$ when ${\mathcal{E}}_{f}={\mathcal{E}}_{c}$,
and (c) $C_{ii+\ell}$ as a function of $\ell$ and $v$. For all plots $U/w=40$
and the lattice spacing $a$ has been set to unity. This figure is adapted from
Ref. pekker1, .

To understand why the presence of a finite-system size changes things, let us
consider that the length scale $L$ leads to a energy scale $\epsilon_{0}(L)$.
When the ramp rate $\tau^{-1}\leq\epsilon_{0}(L)/\hbar$, the system does not
see the critical point; instead the dynamics is similar to a two-level system
(the two states correspond to the instantaneous ground and first excited
states near the transition where excitations are formed) with avoided level
crossing, where the minimum gap is ${\rm O}(\epsilon_{0}(L))$. In this case,
the excitation density scales as $n_{\rm ex}\sim\tau^{-2}$ and is independent
of the critical exponents; this is known as Landau-Zenner (LZ) scaling
anatoli3 . Also, when $\tau^{-1}$ is large, i.e. for fast drives, $n_{\rm ex}$
saturates and KZ scaling is not obeyed. In between, there is a finite window
of drive rates for which one finds KZ scaling law. This behavior can be summed
up by noting that for finite-sized system, the scaling of excitation density
is described by

$\displaystyle n_{\rm ex}$ $\displaystyle=$ $\displaystyle N^{d}\tau^{-\nu
d/(z\nu+1)}f(N^{1/\nu+z}\tau^{-1}),$ (18)

where $N=L/a$, $a$ is the lattice spacing, and the scaling function $f(x)$
satisfies $f(x)=x^{2-\nu d/(z\nu+1)}$ for $x\ll 1$ and $f(x)=1$ for $x\gg 1$.
Note that $n_{\rm ex}$ crosses over from LZ to KZ scaling regime with
decreasing $\tau$ around $x\simeq 1$ $\tau\sim N^{2-\nu d/(z\nu+1)}$.

The dynamics of dipole chain for such ramps have been numerically studied
using exact diagonalization (ED) and time dependent matrix product states
(tMPS) methods. For ED, the analysis involves finding the instantaneous
eigenvalues and eigenvectors of the driven chain at $t=t_{f}=\tau$;
$H[t=\tau]|n\rangle=\epsilon_{n}|n\rangle$. One can then expand the time
dependent state $|\psi(t)\rangle=\sum_{n}c_{n}(t)|n\rangle$, where
$c_{n}(t)=\langle n|\psi(t)\rangle$. The Schrodinger equation for the driven
system can thus be written as coupled differential equations for the
coefficients $c_{n}(t)$ given by

$\displaystyle i\frac{dc_{n}(t)}{dt}$ $\displaystyle=$
$\displaystyle\epsilon_{n}c_{n}(t)+\sum_{m}\lambda_{mn}(t)c_{m}(t),\quad
c_{n}(0)=\langle
n|\psi(0)\rangle,\quad\lambda_{mn}(t)=(E_{f}-E_{i})(t/\tau-1)\langle
m|\sum_{j}\hat{n}_{j}|n\rangle$ (19)

These equations need to be solved numerically to obtain $|\psi(t)\rangle$.
Having obtained $|\psi(t)\rangle$, one may compute several quantities such as
residual energy $Q$, log fidelity $F$ (which is same as the excitation
density), the dipole density $n_{d}$ and the defect density $D$ given by

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\langle\psi(\tau)|H(\tau)|\psi(\tau)\rangle-\epsilon_{0},\quad
F=\ln|\langle\psi(\tau)|0\rangle\rangle|,\quad
C_{\ell\ell^{\prime}}=\langle\psi(\tau)|S^{z}_{\ell}S^{z}_{\ell^{\prime}}|\psi(\tau)\rangle$
$\displaystyle n_{d}$ $\displaystyle=$
$\displaystyle\langle\psi(\tau)|\sum_{\ell}\hat{n}_{\ell}|\psi(\tau)\rangle,\quad
D=n_{d}-\langle 0|\sum_{\ell}\hat{n}_{\ell}|0\rangle$ (20)

where $|0\rangle$ and $\epsilon_{0}$ denotes the wavefunction and energy of
the final ground state at $t=\tau$ and $S_{\ell}^{z}=2\hat{n}_{\ell}-1$.

A plot of these quantities are shown in Fig. 3 and 4 for several
representative values of $L$. It was found that they exhibit KZ scaling (with
exponent $\nu d/(z\nu+1)=1/2$ for $d=z=\nu=1$) within a finite window of ramp
rate which depends on $L$. As $L$ increases the KZ regime holds for slower
ramp rates; the crossover between the LZ and the KZ regime is shown by the
sharp drops in the figures. Thus these ramped boson chains can provide
experimental platform for testing KZ scaling.

Before ending this section, we would like to note that the ramp dynamics for
$H^{\prime}_{d}$ (Eq. 10) has also been studied in details ghosh1 ; subir3 .
Ref. ghosh1,  found KZ scaling consistent with the presence of a $3$-state
Potts transition in these chains. However, work of Ref. subir3,  which
accessed larger chain lengths, has predicted the existence of non-Ising like
critical points with $z\neq 1$ (where $z$ is the dynamical critical exponent)
in such chains. Since $H^{\prime}_{d}$ with large $V$ is easily reproduced in
Rydberg chains, these chains can also act as platform for hosting such non-
Ising quantum critical points. We shall not discuss this issue further in this
article but refer the interested reader to Refs. fendley1, ; subir3, ; ghosh1,
. Furthermore, the dynamics of tilted bosons in the presence of a two-rate
protocol sau1  has also been discussed; it was found that such protocols may
aid suppression of excitation formation in these systems uma1 .

###  IV.2 Periodic protocols

In this section, we shall discuss the periodically driven tilted boson chains
ryddyn1 ; ryddyn2 ; ryddyn3 . For the rest of this article, we shall
explicitly use the spin representation and work with $H_{s}$ (Eq. 9). The
connection of the spins with the original dipoles is given by Eq. 8. The
methods used shall be briefly discussed in Sec. IV.2.1 while the main results
shall be presented in Sec. IV.2.2.

####  IV.2.1 Methods

The properties of a periodically driven system is best described in terms of
its Floquet Hamiltonian $H_{F}$ which is related to the unitary time evolution
operator $U$ by $U(T,0)=\exp[-iH_{F}T/\hbar]$, where $T=2\pi/\omega_{D}$ is
the time period of the drive, $\omega_{D}$ is the drive frequency, and the
wavefunction of the driven system at any time $t$ is related to the initial
wavefunction by $|\psi(t)\rangle=U(t,0)|\psi(0)\rangle$. Thus the stroboscopic
dynamics of the system at $t=nT$, where $n$ is an integer is completely
controlled by its Floquet Hamiltonian flref1 ; flref2 ; asenrev

The Floquet Hamiltonian of any periodically can be computed by comparing two
equivalent expressions for $U(T,0)$

$\displaystyle U(T,0)$ $\displaystyle=$ $\displaystyle
T_{t}\left\\{\exp\left[-i\int_{0}^{T}dtH(t)/\hbar\right]\right\\}=\exp\left[-iH_{F}T/\hbar\right]$
(21)

For an interacting many-body system, it is usually not possible to obtain
$H_{F}$ exactly. This has led to several approximation schemes for such
computations asenrev . In what follows, we shall mostly use one of these
schemes, namely, the floquet perturbation theory (FPT) dsen1 ; tb1 ; ghosh2 ;
asenrev , to obtain the Floquet Hamiltonian for the driven dipole chain. This
method involves a perturbation in drive amplitude; the term in $H(t)$ with the
largest amplitude is treated exactly, while the other terms are treated using
standard time-dependent perturbation theory asenrev . In contrast to the
Magnus expansion technique, the drive period $T$ need not be a small parameter
here; thus, the method allows us to access the intermediate drive-frequency
regime which can not be accessed using Magnus expansion.

To study the properties of the periodically driven chain, we choose to vary
the electric field according to the square pulse protocol such that

$\displaystyle\lambda(t)$ $\displaystyle=$
$\displaystyle-(+)\lambda_{0}/2\quad{\rm for}\,\,t\leq(>)T/2$ (22)

where $\lambda_{0}$ denotes the amplitude of the drive. In what follows, we
shall address the regime where $\lambda_{0}\gg w$. In this regime, the drive
term is the one which has the largest amplitude. Thus we write

$\displaystyle H_{s}(t)$ $\displaystyle=$ $\displaystyle
H_{s0}(t)+H_{s1},\quad H_{s0}(t)=\lambda(t)\sum_{j}\sigma_{j}^{z},\quad
H_{s1}=-w\sum_{j}\tilde{\sigma}_{j}^{x}$ (23)

and treat $H_{s0}(t)$ exactly. Noting that $H_{s0}$ is diagonal in the spin
basis, we consider a complete set of states in the constrained Hilbert space
which has $m$ up-spins and denotes these states as $|m\rangle$. Note that the
positions of these spins (as long as they are not nearest neighbors) do not
change their instantaneous energy under action of $H_{s0}$; thus each
$|m\rangle$ represents a degenerate manifold of states. Using these states as
basis states one finds

$\displaystyle U_{0}(t,0)$ $\displaystyle=$ $\displaystyle
e^{i\lambda_{0}t\sum_{j}\sigma_{j}^{z}/(2\hbar)}\quad{\rm for}\quad t\leq T/2$
(24) $\displaystyle=$ $\displaystyle
e^{i\lambda_{0}(T-t)\sum_{j}\sigma_{j}^{z}/(2\hbar)}\quad{\rm for}\quad t\geq
T/2$

so that $\langle m|U_{0}|n\rangle\sim\delta_{mn}$. We also note that for
$t=T$, $U_{0}(T,0)=I$ (where $I$ is the identity matrix); thus $H_{F}^{(0)}=0$
for this protocol. This is a consequence of the symmetric nature of the drive
(Eq. 22) which leads to a vanishing average of $\lambda(t)$ over a drive
cycle.

The first order perturbative correction to the evolution operator $U$ can be
obtained using standard time dependent perturbation theory. This is given by

$\displaystyle U_{1}(T,0)$ $\displaystyle=$
$\displaystyle\frac{-i}{\hbar}\int_{0}^{T}dtU_{0}^{\dagger}(t,0)H_{s1}U_{0}(t,0)$
(25)

We note that $\tilde{\sigma}_{j}^{x}$ flips a spin on the $j^{\rm th}$ site,
$\langle m|H_{s1}|n\rangle\sim\delta_{m,n\pm 1}$. Denoting a state
$|m+(-)\uparrow_{j}\rangle$ to be the one with one additional (less) spin up
residing at the $j^{\rm th}$ site, we can therefore write ryddyn1 ; ryddyn2 ;
ryddyn3

$\displaystyle U_{1}(T,0)$ $\displaystyle=$
$\displaystyle\sum_{m}\sum_{j}\sum_{s_{j}=\pm
1}c^{(1)}_{s_{j}}|m\rangle\langle m+s_{j}|,\quad
c^{(1)}_{s}=\frac{4iw}{\lambda_{0}}e^{i\lambda_{0}Ts/(4\hbar)}\sin\lambda_{0}T/(4\hbar)$
(26)

where $s=\pm 1$. Noting that $\tilde{\sigma}^{\pm}|m\rangle=|m\pm 1$, and
using Eq. 21 we get, to first order in $w/\lambda_{0}$

$\displaystyle H_{F}^{(1)}$ $\displaystyle=$
$\displaystyle-w\frac{\sin\gamma}{\gamma}\sum_{j}(\cos\gamma\tilde{\sigma}_{j}^{x}+\sin\gamma\tilde{\sigma}_{j}^{y}),\quad\gamma=\lambda_{0}T/(4\hbar).$
(27)

We note that for $T\to 0$, $\gamma\to 0$, and in this limit
$H_{F}^{(1)}=-w\sum_{j}\tilde{\sigma}_{j}^{x}=H_{\rm PXP}$. This is also
consistent with the Magnus result for $H_{F}$ which demands that $H_{F}$ shall
be average Hamiltonian given by

$\displaystyle H_{F}^{\rm magnus}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}H(t)dt=H_{\rm PXP}$ (28)

Also, at this order, $H_{F}^{(1)}$ represents a PXP like Hamiltonian up to a
rotation and an overall renormalization by a factor $\sin\gamma/\gamma$; it
was noted in Ref. ryddyn1,  that this result constitutes a resummation of a
class of terms in the Magnus expansion.

The higher order terms in the Floquet Hamiltonian can also be computed. This
has been carried out systematically in Ref. ryddyn3,  and leads to
$H_{F}^{(2)}=0$ at second order. This null result for $H_{F}$ to second order
owes it existence to the fact that there exists an operator
$C=\prod_{j}\sigma_{j}^{z}$ which satisfies $CU(T,0)C=U^{-1}(T,0)$ leading to
$\\{H_{F},C\\}=0$. As shown in Refs. scarlitrev, ; ryddyn1, , such anti-
commutation only allows for odd orders in the Floquet Hamiltonian. At third
order, a straightforward but tedious calculation ryddyn3

$\displaystyle H_{F}^{(3)}$ $\displaystyle=$
$\displaystyle-\alpha_{0}\sum_{j}\left[\left(\tilde{\sigma}_{j+1}^{+}\sigma_{j-1}^{+}+\tilde{\sigma}_{j-1}^{+}\tilde{\sigma}_{j+1}^{+}\right)\tilde{\sigma}_{j}^{-}-6\tilde{\sigma}_{j}^{+}\right]+{\rm
h.c.},$ $\displaystyle\alpha_{0}$ $\displaystyle=$
$\displaystyle\left[e^{3i\lambda_{0}T/(2\hbar)}+3e^{i\lambda_{0}T/(2\hbar)}(1+i\lambda_{0}T/\hbar)+2(1-3e^{i\lambda_{0}T/\hbar})\right]\frac{\hbar
w^{3}e^{-i\lambda_{0}T/\hbar}}{3i\lambda_{0}^{3}T}.$ (29)

The first term in $H_{F}^{(3)}$ involves three spin on neighboring sites. We
note that the form of this term, i,e., the order in which the
$\tilde{\sigma}^{\pm}$ operators appear, is dictated by the presence of the
constraint; the order of appearance ensures that two neighboring sites can
never have two up-spins. The second term of the Floquet Hamiltonian provides a
shift to $H_{F}^{(1)}$ and renormalize its coefficients. In what follows, we
shall use this perturbative Floquet Hamiltonian to understand the numerical
results.

The numerical approach to this problem, carried out in Ref. ryddyn1, ;
ryddyn3,  used exact diagonalization(ED). The first step in this direction
involves numerical diagonalizaton of $H[\pm\lambda_{0}]$. We denote the
corresponding energy eigenvalues and eigenvectors by

$\displaystyle H[\pm\lambda_{0}]|p_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\epsilon_{p}^{\pm}|p_{\pm}\rangle$ (30)

In terms of this, the evolution operator can be written as

$\displaystyle U(T,0)$ $\displaystyle=$ $\displaystyle
e^{-iH[\lambda_{0}]T/(2\hbar)}e^{-iH[-\lambda_{0}]T/(2\hbar)}=\sum_{p,q}c_{pq}^{-+}e^{-i(\epsilon_{p}^{-}+\epsilon_{q}^{+})T/(2\hbar)}|p^{-}\rangle\langle
q^{+}|,\quad c_{pq}^{-+}=\langle p^{-}|q^{+}\rangle$ (31)

Thus one has a finite-dimensional matrix (for finite sized $L$) which can then
be diaonalized. Since $U$ is an unitary operator. its eigenvalues are
unimodular. Thus one can express it in term of its eigenspectra as

$\displaystyle U(T,0)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}e^{-i\epsilon_{\alpha}^{F}T/\hbar}|\alpha\rangle\langle\alpha|,\quad
H_{F}|\alpha\rangle=\epsilon_{\alpha}^{F}|\alpha\rangle$ (32)

where $|\alpha\rangle$ are the eigenfunctions of $U(T,0)$,
$\epsilon_{\alpha}^{F}$ are the eigenvalues of the corresponding Floquet
Hamiltonian, and the last equation follows from Eq. 21. This procedure thus
allows access to the exact Floquet quasienergies and eigenfunctions for
finite-sized systems. The stroboscopic evolution, at $t=nT$ ( where $n$ is an
integer), for any operator ${\mathcal{O}}$, can thus be computed as ryddyn1

$\displaystyle{\mathcal{O}}_{n}$ $\displaystyle=$
$\displaystyle\langle\psi(0)|(U^{\dagger}(T,0))^{n}{\mathcal{O}}U^{n}(T,0)|\psi(0)\rangle=\sum_{\alpha,\beta}c_{\beta}^{\ast}c_{\alpha}e^{-i(\epsilon_{\alpha}^{F}-\epsilon_{\beta}^{F})T/\hbar}\langle\beta|{\mathcal{O}}|\alpha\rangle$
(33)

where $|\psi(0)\rangle$ is the initial state and
$c_{\alpha}=\langle\psi(0)|\alpha\rangle$.

In what follows, we shall also be computing the half-chain entanglement
entropy of the Floquet eigenstates. The procedure for this is as follows.
First, corresponding to any eigenstate $|\psi_{n}\rangle$, we construct a
density matrix $\rho_{n}=|\psi_{n}\rangle\langle\psi_{n}|$ which is defined on
the full chain with periodic boundary condition. Next we divide the chain into
two equal halves, $A$ and $B$, with open boundary condition and trace out the
contribution of states residing in $B$. Thus each matrix element of the
reduced density matrix $\rho^{A}_{n}$ after tracing out $B$ can be written as
ryddyn1

$\displaystyle\langle\rho_{n}^{A}\rangle_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\sum_{\mu\in
B=1}^{{\mathcal{N}}_{B}}\langle\alpha;\mu|\rho_{n}|\beta;\mu\rangle$ (34)

where ${\mathcal{N}}_{B}$ denotes the Hilbert space dimension corresponding to
states residing in $B$ with open boundary condition and the states
$|\alpha\rangle$ and $|\beta\rangle$ have weights in region $A$ ryddyn1 .
While carrying out this procedure, one has to be careful in excluding states
where the right end of $A$ and the left end of $B$ both has spin-up (or
dipoles) since these states were not part of the Hilbert space of the full
chain owing to the constraint. The half-chain entanglement $S_{L/2}^{(n)}$ can
then be obtained numerically using

$\displaystyle S^{(n)}_{L/2}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{{\mathcal{N}}_{A}}q_{j}\ln q_{j}$ (35)

where $q_{j}$ denotes the eigenvalues of $\rho_{n}^{A}$ obtained via numerical
diagonalization. We shall use $S^{(n)}_{L/2}$ to distinguish between states
with volume- and area-law entanglement entropies in the rest of this article.

####  IV.2.2 Results

To study the physics of the driven dipole chain, we concentrate on half-chain
entanglement $S_{L/2}$ (Eq. 35) of the Floquet eigenstates and the density-
density correlator $\langle
O_{j2}\rangle=\langle\psi(nT)|\hat{n}_{j}\hat{n}_{j+2}|\psi(nT)\rangle$, where
$\hat{n}_{j}=(1+\sigma_{j}^{z})/2$. In what follows, we shall discuss the
property of the stroboscopic evolution of $\langle O_{22}\rangle$ (i.e.
choosing $j=2$) starting from either the $|{\mathbb{Z}}_{2}\rangle$
(antiferromagnetic state with up-spins on even sites) or the $|0\rangle$
(ferromagnetic all spin-down state) state. The evolution of $\langle
O_{jj+2}\rangle$ for other values of $j$ is identical as long as $j$ is chosen
to be even ryddyn1 ; ryddyn2 ; ryddyn3

Figure 5: Left Panel: Plot of $\langle O_{22}\rangle$ as a function of number
of drive cycles $n$ for (a) $\hbar\omega_{D}=100$ and (b)
$\hbar\omega_{D}=0.01$. The red lines indicates dynamics starting from the
state $|0\rangle$ while the black lines indicate $|{\mathbb{Z}}_{2}\rangle$
The ETH predicted thermal steady state value of $\langle O_{22}\rangle$ is
$\sim 0.11$ as shown in blue dotted line. Plots (c) and (d) shows the
corresponding half-chain entanglement $S_{L/2}$. Note the presence of athermal
scar states below the thermal band for $\hbar\omega_{D}=100$. Such states are
absent for $\hbar\omega_{D}=0.01$. The red circles correspond to states which
has high overlap with the initial $|{\mathbb{Z}}_{2}\rangle$ state. For all
plots $\lambda=15$, $w=\sqrt{2}$, $\hbar$ is set to unity, and $L=18$. Right
Panel: Plot of $O_{22}$ as a function of $n$ for (a) $\hbar\omega_{D}=8.25$,
(b) $\hbar\omega_{D}=8$, (c) $\hbar\omega_{D}=7.75$, and (d)
$\hbar\omega_{D}=7.5$. The plot for (a) and (d) shows coherent oscillation
while those for (b) and (c) exhibits rapid approach to ETH predicted
thermalization. All other parameters are same that in the left panel. This
figure is adapted from Ref. ryddyn1, .

We first consider the dynamics starting from the $|{\mathbb{Z}}_{2}\rangle$
initial state. For this state, for $\lambda_{0}/(\hbar\omega_{D})\ll 1$, the
dynamics is similar to that of quench studied in Ref. scarlitrev, . In this
regime, we expect scar-induced coherent oscillations with a frequency which is
determined by the energy separation between the quantum scar eigenstates. In
the opposite limit, $w/(\hbar\omega_{D})\gg 1$ (where FPT and Magnus expansion
both fail) it is expected that the system shall heat up due to the drive and
reach the thermal steady state as predicted by ETH. This expectation is
verified from the behavior of $\langle O_{22}\rangle$ shown for high and low
frequencies in the plots shown at the top of the left panel in Fig. 5. The
bottom plots of the left panel of Fig. 5 shows the half-chain entanglement
entropies of the Floquet eiegnstates at these drive frequencies; the high
frequency eigenstates clearly show the presence of athermal scars separated
from the thermal band of states. No such athermal states are seen at low drive
frequency; in this regime, all the states fall within the thermal band. The
red dots indicates eigenstates with large overlap with the initial
$|{\mathbb{Z}}_{2}\rangle$ state. These are the states which primarily drive
the dynamics. At high frequency, these states are athermal and lie well
outside the thermal band. Thus the dynamics exhibit coherent oscillations
since it involves a small, ${\rm O}(L)$, subspace of the full Hilbert space.
In contrast, at low frequency, the eigenstates having large overlap with
$|{\mathbb{Z}}_{2}\rangle$ are part of the thermal band leading to
thermalization.

Figure 6: Left Panel: Plots of half-chain entanglement entropy $S_{L/2}$ for
(a) $\hbar\omega_{D}=8.25$ (b)$\hbar\omega_{D}=8$, (c) $\hbar\omega_{D}=7.75$,
and (d) $\hbar\omega_{D}=7.5$. The plots shows athermal scar states below the
thermal band with low $S_{L/2}$ at $\hbar\omega_{D}=8.25$ and
$\hbar\omega_{D}=7.5$ while no such states are found for $\hbar\omega_{D}=8$
and $7.75$. The red circles exhibits states which has high overlap with the
initial $|{\mathbb{Z}}_{2}\rangle$. For all plots, $\lambda=15$, $w=\sqrt{2}$
and $L=18$ and we have set $\hbar=1$. Right Panel: Phase diagram in the
$\lambda-\omega_{D}$ plane showing ergodic (shaded) and coherent (white)
regions with reentrant crossover between them. All other parameters are same
as those in the left panel. This figure is adapted from Ref. ryddyn1, .

Naively, one would expect a crossover between these two regimes at some
intermediate drive frequency. In contrast, as shown, in the right panel Fig.
5, the system exhibits non-monotonic re-entrant behavior at multiple
intermediate frequencies. This is most easily noted by comparing the four
plots in the right panel of Fig. 5. Clearly, ETH predicted thermalization is
restored around $\hbar\omega_{D}\simeq 7.75w$; however, scar induced
oscillations take over at a lower drive frequency $\hbar\omega_{D}\simeq
7.5w$. The corresponding half-chain entanglement entropies, shown in the left
panels of Fig. 6, indicate the presence of scars at $\hbar\omega_{D}=7.5w$ and
their absence at $\hbar\omega_{D}=7.75w$. Further studies, carried out in Ref.
ryddyn1, , demonstrates multiple crossovers between ETH predicted ergodic and
scar-induced coherent oscillatory behaviors at intermediate frequencies
leading to the phase diagram shown in the right panel of Fig. 6. The density
of the ergodic regimes increases with lower frequency and they completely
cover the phase diagram at low drive frequencies leading to ergodic behavior
in this regime. However, at intermediate frequencies, as shown in Ref.
ryddyn1, , it is possible to tune into and out of such ergodic regimes by
tuning the drive frequency. This leads to the possibility of drive-frequency
induced tuning of ergodicity in a driven non-integrable system.

This tunability of ergodicity can be qualitatively understood from the
analytical Floquet Hamiltonian (Eqs. 27 and 29) as follows. For
$\hbar\omega_{D}\gg\lambda_{0}$, $\sin\gamma\sim\gamma$ and $H_{F}^{(1)}\simeq
H_{\rm PXP}$. Moreover since $w/\lambda_{0}\ll 1$, $H_{F}^{(3)}$ is negligible
compared to $H_{F}^{(1)}$ in this limit. Thus the Floquet Hamiltonian is of
the PXP form and supports quantum scars. For an initial state
$|{\mathbb{Z}}_{2}\rangle$, the dynamics gets maximal contribution from the
scar subspace leading to coherent oscillation scarlitrev . In contrast around
$\gamma\simeq m\pi$ (where $m$ is a non-zero integer), $H_{F}^{(1)}\to 0$ and
$H_{F}^{(3)}$ dominates. The Floquet Hamiltonian is then not of the PXP form
anymore and does not support scar states with large overlap with
$|{\mathbb{Z}}_{2}\rangle$. Consequently, the dynamics becomes ergodic around
$\omega_{D}=\omega_{D}^{c}=\gamma_{0}/(2m\hbar)$; the width of the ergodic
regime depends on the relative magnitudes of $H_{F}^{(1)}$ and $H_{F}^{(3)}$
as we move away $\omega_{D}^{c}$. Of course, this estimate of $\omega_{D}^{c}$
does not take into account the renormalization of $H_{F}^{(1)}$ from the
higher order terms (such as the one from $H_{F}^{(3)})$ and is thus not exact.
However since these terms are typically small in the intermediate frequency
regime by a factor of at least $w^{3}/\lambda_{0}^{3}\ll 1$, this estimate
turns out to be qualitatively correct.

Figure 7: Left Panel: Plot of $\langle O_{22}\rangle\sim O_{22}$ as a function
of $n$ for (a) $\hbar\omega_{D}=100$ (b)$\hbar\omega_{D}=8.5$, (c)
$\hbar\omega_{D}=7.88$, and (d) $\hbar\omega_{D}=7.25$. For all plots,
$\lambda=15$, $w=\sqrt{2}$ and $L=26$ and we have set $\hbar=1$. Right Panel:
The steady state value of $\langle O_{22}\rangle\equiv O_{22}^{\rm DE}$ as a
function of $\omega_{D}$ for several $L$. For all plots $\lambda=15$,
$w=\sqrt{2}$ and $\hbar$ is set to unity. This figure is adapted from Ref.
ryddyn3, .

Next, we consider the stroboscopic dynamics of $\langle O_{22}\rangle$
starting from the $|0\rangle$ state. We note that for
$\hbar\omega_{D}/\lambda_{0}\gg 1$, the dynamics exhibits thermalization
consistent with the ETH prediction as can be seen from the left panel of Fig.
5. Thus in the quench limit, scars do not play a role in the dynamics since
they have negligible overlap with the $|0\rangle$ state. As the drive
frequency is lowered, however, the situation changes as can be seen from left
panel of Fig. 7. Around $\hbar\omega_{D}=8.5w$, we find the presence of long-
time coherent oscillations while at $\hbar\omega_{D}=7.88w$, $\langle
O_{22}\rangle$ remains completely frozen to its initial value. The latter
behavior constitutes dynamic freezing in an otherwise ergodic non-integrable
system. Finally, for $\hbar\omega_{D}=7.26w$, we find that $\langle
O_{22}\rangle$ reaches a steady state value which is lower than the ETH
predicted value (shown by the blue dotted line). This constitutes a
qualitatively different violation of ETH since coherent scar-induced
oscillatory dynamics (such as the one seen for $\hbar\omega_{D}=8.5w$) usually
leads to super-thermal steady state values in contrast to the subthermal value
found in the present case. The steady state behavior of $\langle
O_{22}\rangle$ starting from $|0\rangle$ initial state is shown in the right
panel of Fig. 7. We find that the steady state reaches its thermal value,
shown by the dotted line, at large $\omega_{D}$. As the frequency is lowered,
it reacher superthermal steady state value. Here the dynamics indicates
coherent long-time stroboscopic oscillation. Just below $\hbar\omega_{D}<8w$,
the steady state value of $\langle O_{22}\rangle$ drops and reaches zero at
$\hbar\omega_{D}\simeq 7.88w$. This constitutes an example of dynamical
freezing dynfr1 ; dynfr2 . This is followed by a wide range of frequency at
which the steady state value remains subthermal. We also note the presence of
a second freezing point at around $\hbar\omega_{D}\simeq 3.95w$. These steady
state values of $O_{22}$ seem to be independent of system size within the
range of $L$ accessible within ED.

Figure 8: Leftmost Panel: Plot of $S_{L/2}$ as a function of $E_{F}$ for
$\hbar\omega_{D}=8.5$. The green circles indicate states having large overlap
with $|{\bar{Z}}_{2}\rangle$ while the red circles indicates those having
large overlap with $|0\rangle$. For all plots, $\lambda=15$, $w=\sqrt{2}$ and
$L=26$. Left-center Panel: Plot of $\langle|O_{22}|\rangle\equiv O_{22}$ for
all Floquet eigenstates. The red and the green circles indicates contribution
from states with large overlap with $|0\rangle$ and $|{\mathbb{Z}}_{2}\rangle$
respectively. The violet dotted line indicates the ETH prediction for
$O_{22}$. Right-center and Rightmost Panels: Same as the leftmost and the
left-center panels respectively but for $\hbar\omega_{D}=7.26$. This figure is
adapted from Ref. ryddyn3, .

The oscillatory behavior of $O_{22}$ starting from $|0\rangle$ can be
understood by observing the nature of the Floquet eigenstates as shown in the
leftmost panel of Fig. 8. The plot shows the presence of a different set of
athermal states (indicated by red circles) which has large overlap with the
$|0\rangle$. These states are different from the scars which has large overlap
with $|{\mathbb{Z}}_{2}\rangle$ shown by green circles. Thus the Floquet
hamiltonian at these frequencies supports at least two separate set of scar
states; this phenomenon has no analog in scars of the PXP model. Indeed, as
the frequency is increased, the set of scar states which has large overlap
with $|0\rangle$ merge into the continuum of thermal states leaving behind
only athermal $|{\mathbb{Z}}_{2}\rangle$ scars. This brings out the central
role of higher order terms in the Floquet Hamiltonian for generation of such
scars. The presence of such scars lead to coherent oscillatory dynamics of
$O_{22}$. Its steady state value turns out to be superthermal which can be
seen from the left-center panel of Fig. 8. This is due to the fact that for
most scar eigenstates that have large overlap with $|0\rangle$ (shown as red
circles in the left-center panel of Fig. 8), $\langle O_{22}\rangle$ is larger
than the ETH predicted thermal value.

Similar scar states with large overlap with $|0\rangle$ is shown by red
circles in the right-center panel of Fig. 8 for $\hbar\omega_{D}=7.26w$. Here
in contrast to the case $\hbar\omega_{D}=8.5$, the initial state $|0\rangle$
turns out to have relatively large overlap ($>0.01$) with a few scars states
as shown in the figure and small overlap with a large number of thermal
states. The presence of these thermal states do not allow for long time
coherent oscillations. However, the steady state value of $\langle
O_{22}\rangle$ is controlled by the athermal scar states since they have
relatively large overlap with $|0\rangle$. It turns out that these athermal
scar states have a subthermal value of $\langle O_{22}\rangle$ (as can be seen
from the rightmost panel of Fig. 8). Thus $\langle O_{22}\rangle$ quickly
decays to a subthermal steady state value leading to violation of ETH. We note
that such a violation do not involve oscillatory dynamics and thus constitute
a separate route of scar-induced ETH violation in finite boson chains which
has no analogue in chains subjected to quench. ryddyn3 .

Finally, we discuss the phenomena of dynamic freezing at
$\hbar\omega_{D}\simeq 7.88$ for $w=\sqrt{2}$ and $\lambda=15$. Qualitatively,
the existence of such freezing is easy to understand by noting that the state
$|0\rangle$ is annihilated by all non-PXP higher order terms of the Floquet
Hamiltonian. This follows from the fact that such terms necessarily involves
$\tilde{\sigma}_{j}^{-}$ operator which annihilates the state $|0\rangle$.
Thus for drive period, where the coefficient of $H_{F}^{(1)}$ vanishes,
$U(T,0)\to I$ and one encounters freezing. This argument, in conjunction with
the first order perturbative Floquet Hamiltonian, suggest that the freezing
would occur at frequencies for which $w_{r}=0$:
$\hbar\omega_{D}=\lambda/(2n)$. This predicts a freezing frequency of
$\hbar\omega_{D}=7.5$ which differs quite a bit with the exact result.

This discrepancy can be understood by noting that the first order Floquet
Hamiltonian neglects the normalization of $w_{r}$ due to contribution of
higher order term such as the one from $H_{F}^{(3)}$ (Eq. 29). To see this, we
consider an exact analytical calculation of $H_{F}$ for $L=3$. Here the exact
analytic calculation is feasible since the Hilbert space consists of just two
states in the zero momentum sector; this reduces the problem to a driven two-
state problem ryddyn3 . These two states are
$|0\rangle=|\downarrow,\downarrow,\downarrow\rangle$ and
$|1\rangle=(|\downarrow,\uparrow,\downarrow\rangle+|\uparrow,\downarrow,\downarrow\rangle+|\downarrow,\downarrow,\uparrow\rangle)/\sqrt{3}$.
The Hamiltonian, the space of these two states, reads

$\displaystyle H_{2}(t)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&-\sqrt{3}w\\\
-\sqrt{3}w&\lambda(t)\end{array}\right)$ (38)

For the square pulse protocol given by Eq. 22 the Floquet Hamiltonian
$H^{\prime}_{F}$ corresponding to $H_{2}(t)$ can be exactly computed ryddyn3 .
A straightforward calculation shows that

$\displaystyle\langle 0|H^{\prime}_{F}T|1\rangle$ $\displaystyle=$
$\displaystyle\cos^{-1}\left[1-\frac{12w^{2}}{\Delta_{0}^{2}}\\{1-\cos(\Delta_{0}T/2)\\}\right],\quad\Delta_{0}=\sqrt{\lambda_{0}^{2}+12w^{2}}$
(39)

This matrix element, which needs to be finite for the driven system to evolve,
vanishes for $\Delta_{0}T=4m\pi$ (where $m$ is an integer). For $\lambda=15$
and $w=\sqrt{2}$, $\hbar\omega_{D}\simeq 7.86$ for $m=1$, and this provides a
near exact match to the observed freezing frequency. Moreover, it also
predicts the second freezing point at $\hbar\omega_{D}\simeq 3.93$ which
correspond to $m=2$. We note that for $\lambda\gg w$, this frequency will
reduce to that predicted by first order Floquet theory
($\hbar\omega_{D}=\lambda/2m$) as expected. The reason for the accuracy of the
result obtained with this simplistic calculation is that the higher order
multiple spin terms in $H_{F}$ do not contribute to the freezing phenomenon as
discussed earlier. We note that at the freezing point the state $|0\rangle$ is
disconnected from all other states in the Hilbert space of the system; this
constitutes an example of (weak) fragmentation of the Hilbert space of
$H_{F}$.

Before ending this section, we note that the periodic dynamics of Rydberg
atoms has been studied experimentally in Ref. gr4, . The experiment used a
continuous drive protocol using a damped cosine drive and starting from
$|Z_{2}\rangle$; it was found the system exhibits a robust subharmonic
response to the drive. It was noted that this response depended on the initial
state and its relation to quantum scars in the system was discussed. However,
the dynamics of the system starting from $|0\rangle$ state or that in the
presence of a square pulse protocol has not been studied in this work.

##  V Hilbert space fragmentation: A minimal model

It has been recently been pointed out that the presence of dynamical
constraints in a many-body system may lead to fragmentation of its Hilbert
space into several disconnected sectors. This phenomenon, termed as Hilbert
space fragmentation (HSF), provides yet another route to violation of ETH in
non-integrable quantum systems fraglitrev ; fraglit1 ; fraglit2 ; fraglit3 ;
fraglit4 ; fraglit5 ; fraglit6 ; fraglit7 ; fraglit8 ; bm1 . This phenomenon
naturally arises in constrained systems where the presence of additional
conservation laws provide the constraint fraglitrev ; fraglit1 ; fraglit2 ;
fraglit3 ; fraglit4 ; fraglit5 ; fraglit6 ; fraglit7 ; fraglit8 . The
resulting physics has close similarity with those of fractons fracrev1 ;
fracrev2 . The realization of such constrained systems using circuit models
whose Hilbert space is identical to that of a $S=1$ spin chain of length $L$
provides a neat example in this context fraglit2 . In this case, it was shown
that the presence of two simultaneous $U(1)$ conserved quantities, namely
$Q=\sum_{j}S_{j}^{z}$ (which is analogous to total charge in a fractonic
model) and $P=\sum_{j}jS_{j}^{z}$ (dipole moment of a fractonic model),
provides the necessary constraints for fragmentation. Typically, in most of
the models studied, such conservation comes from the commutation
$[Q,H]=[P,H]=0$ and remains valid for all states in the Hilbert space;
moreover, they lead to an exponentially large number of inert zero-energy
states in the Hilbert space. We note here such fragmentation may even separate
states with same symmetry into different, disconnected, segments within the
Hilbert space. Such fragmentation is dubbed as ”strong” if the Hilbert space
is fragmented in exponentially many separate sectors; in this case, the number
of states which lead to violation of ETH increases exponentially with system
size fraglitrev ; fraglit3 . If this condition is not satisfied, the
fragmentation is termed as weak; in this sense, scars constitutes an example
of weak fragmentation of Hilbert space. A recent experiment involving Fermi-
Hubbard model has observed non-ergodic behavior in a titled Fermi-Hubbard
system which can possibly be attributed to such fragmemntation fragexp . In
the rest of this section, we shall discuss such fragmentation in a model which
was inspired by the Floquet Hamiltonian of the tilted Bose-Hubbard model; the
details of HSF and its realization in other model can be found in Ref.
fraglitrev, .

###  V.1 Model and fragmentation

The tilted Bose-Hubbard model constitutes an example of a spin-half system in
a constrained Hilbert space. However, the model does not show HSF. The reason
for this becomes clear when one analyzes the connectivity of the states of the
model in the number basis in the PXP limit. It turns out, as noted in Ref.
bm1, , that most states in the Hilbert space are connected under the action of
$H_{\rm PXP}$ via the state $|0\rangle$. Thus a Hamiltonian which would
annihilate the state $|0\rangle$ may lead to HSF. Taking cue from the
structure of the Floquet Hamiltonian for the periodic driven tilted Bose-
Hubbard model, Ref. bm1,  pointed out that one such possible model involving
spin-half Pauli matrices on a 1D chain is given by

$\displaystyle H_{\rm fr}$ $\displaystyle=$ $\displaystyle
w\sum_{j=1}^{L}\tilde{(}\sigma_{j-1}^{+}\tilde{\sigma}_{j+1}^{+}\tilde{\sigma}_{j}^{-}+{\rm
h.c.}),$ (40)

where $w$ is an arbitrary energy scale which shall be set to unity for the
rest of this section and it is understood that $H_{\rm fr}$ acts on the
constrained Hilbert where two up-spins can not be neighbors. We note that
$H_{\rm fr}$ is identical to the three-spin term in the third order Floquet
Hamiltonian (Eq. 29) but with its coefficient set to unity. Such a term
becomes the largest term in $H_{F}$ around the point where renormalized
$H_{F}^{(1)}$ (Eq. 27) vanishes. We note that $H_{\rm fr}$ do not have any
simultaneous conserved quantities and thus differ from a class of earlier
studied models fraglitrev .

The simplest class of states which demonstrates such fragmentation corresponds
to blocks of length $\ell=3$ with one or two up-spins in a background of down-
spins bm1 . These states can be written as

$\displaystyle|X_{1,j}\rangle$ $\displaystyle=$
$\displaystyle|...1_{j-1}0_{j},1_{j-1}...\rangle,\quad|X_{2,j}\rangle=|...0_{j-1}1_{j},0_{j-1}...\rangle$
(41)

where we have denoted up- and down-spins by $1$ and $0$ respectively and
ellipsis indicates down spins on all other sites. It is easy to see that under
action of $H_{\rm fr}$, these blocks transform to each other: $H_{\rm
fr}|X_{1,j}\rangle=|X_{2,j}\rangle$ and $H_{\rm
fr}|X_{2,j}\rangle=|X_{1,j}\rangle$. They are connected to any other states in
the Hilbert state and thus constitute a fragment. Their linear combination

$\displaystyle|X_{\pm,j}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|X_{1,j}\rangle\pm|X_{2,j}\rangle)$ (42)

yields eigenstates of $H_{\rm fr}$ with eigenvalues $\pm 1$ (we have set
$w=1$). Moreover, if two such blocks are spaced with at least two down-spins
separating them, they act as non-interacting entities and the total energy of
the system becomes a sum of the energy of the individual blocks. This leads to
a class of eigenstates with zero and integer (in units of $w$) energies:
$E=(n_{+}-n_{-})$, where $n_{\pm}$ are the number of isolated
$|X_{\pm}\rangle$ blocks in the state. We note that these blocks are localized
and are dubbed as ”bubbles” in Ref. bm1, .

These bubbles develop dispersion when two individual bubbles are allowed to
interact by placing them next to one another. The simplest of these states can
be understood analytically, and they form a closed Hilbert space fragment
spanned by the states bm1

$\displaystyle|\psi_{1,k}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j}e^{ikj}T_{j}|...X_{2,\,j}X_{2,\,j+3}...\rangle\equiv\sum_{j}e^{ikj}T_{j}|...0_{j-1}1_{j}0_{j+1}0_{j+2}1_{j+3}0_{j+4}....\rangle$
$\displaystyle|\psi_{2,k}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j}e^{ikj}T_{j}|...X_{1,\,j}X_{2,\,j+3}...\rangle\equiv\sum_{j}e^{ikj}T_{j}|...1_{j-1}0_{j}1_{j+1}0_{j+2}1_{j+3}0_{j+4}....\rangle$
(43)

where $T_{j}$ denotes the translation operator, the lattice spacing has been
set to unity, and the ellipsis denotes all down spins. It was shown in Ref.
bm1,  that $|\psi_{1(2),k}\rangle$ form a closed subspace with

$\displaystyle H_{\rm fr}\left(\begin{array}[]{c}|\psi_{1,k}\rangle\\\
|\psi_{2,k}\rangle\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&(1+e^{ik})\\\
(1+e^{-ik})&0\end{array}\right)\left(\begin{array}[]{c}|\psi_{1,k}\rangle\\\
|\psi_{2,k}\rangle\end{array}\right)$ (50)

This leads to a pair of eigenstates in the momentum space given by

$\displaystyle E_{k}$ $\displaystyle=$ $\displaystyle\pm 2\cos k/2$ (51)

It was noted in Ref. bm1,  that Eq. refenmom provide an analytical explanation
for the presence another class of eigenstates with $E=\pm 2$ (for $k=0$),
$E=\pm 1$ (for $k=2\pi/3$), and $E=0$ (for $k=\pi$). Further for $L\geq 8$
such that $L=4n$ ($n\in Z$), $k=\pi/2$ leads to $E=\pm\sqrt{2}$; this provides
a natural explanation of such eigenstates with simple irrational eigenvalues
that was found in the spectrum of $H_{\rm fr}$.

Apart from such simple fragments., more complicated fragments with much longer
bubbles leading to flat bands were discussed in Ref. bm1, . We shall not
discuss them in details here. However, we would like to point out that the
model exhibits a phenomenon, dubbed as secondary fragmentation in Ref. bm1, ,
which was not found in earlier works. Such secondary fragmentation happens
when basis states within a primary fragments forms a further closed subspace
under action of $H_{\rm fr}$; these states are constructed out of specific
linear combination of a fixed number of basis states and they turn out to be
orthogonal to other states within the same primary fragment. The eigenvalues
corresponding to such eigenstates, for the model discussed in Ref. bm1, , are
integers, and their number increases with increasing $L$; importantly, their
existence can not be straightforwardly tied local classical conservation
conditions.

###  V.2 Adding a staggered field

In this section, we shall focus on the structure of the zero-energy
eigenstates of the model in the ”bubble” sector. To this end, as pointed out
in Ref. bm1, , it is useful to add a staggered magnetic field to the model
leading to

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{\rm fr}+H_{\Delta},\quad
H_{\Delta}=\frac{\Delta}{2}\sum_{j}(-1)^{j}\sigma_{j}^{z}$ (52)

We note that there exists two operators

$\displaystyle Q$ $\displaystyle=$ $\displaystyle\prod_{j}\sigma_{j}^{z},\quad
C=\prod_{j}(\sigma_{2j}^{z}+\sigma_{2j+1}^{z})$ (53)

We note that $[H,Q]=0$ so that the spectrum of $H$ is symmetric around $E$.
Moreover, we note that $H_{\Delta}$ anticommutes with $C$. This allows for
presence of zero energy modes of $H$. The details of these zero modes has been
worked out in Ref. bm1, .

Since $H_{\Delta}$ is diagonal in the number basis, it makes sense to work in
the diagonal basis in $|X_{1(2)}\rangle$ states. In the space of these states,
it is possible to represent $H$ in terms of pseudospin operators
$\tau_{j}^{\alpha}$ ($\alpha=x,y,z$) such that
$\tau_{j}^{x}|X_{1,j}\rangle=|X_{2,j}\rangle$. In terms of these pseudospin
operators, one can write $H$ as bm1

$\displaystyle H$ $\displaystyle=$
$\displaystyle\epsilon_{0}\sum_{j}\sum_{\alpha=x,y,z}\eta^{\alpha}\tilde{\tau}_{j}^{\alpha}+\frac{\Delta}{2}(n_{\rm
odd}-n_{\rm
even}),\quad\epsilon_{0}=\sqrt{1+9\Delta^{2}/4},\quad\eta=(\eta^{x},\eta^{y},\eta^{z})=(1,0,3\Delta/2)/\epsilon_{0}$
(54)

where $n_{\rm odd/even}$ are the number of bubbles centered on odd or even
sites. Here we have defined $\tilde{\tau}^{y,z}_{j}=(-1)^{j}\tau^{y,z}_{j}$
and $\tilde{\tau}^{x}_{j}=\tau_{j}^{x}$ for all $j$, where $j$ denotes the
center of the bubbles and the sum over $j$ indicates sum over number of such
bubbles. Thus we find $H$ reduces to a collection of non-interacting
pseudospins (with $s=1/2$) on every site. This indicates that the sector will
contribute states to the Hilbert space of $H$ which has area-law entanglement.
Moreover, these states indicates a novel features when it comes to out-of-
equilibrium dynamics of initial states belonging to the bubble sector.

To see this, let us consider a square pulse protocol for which
$\Delta(t)=-(+)\Delta_{0}$ for $t\leq(>)T/2$, where $T=2\pi/\omega_{D}$ is the
driving frequency. We shall assume that we start the dynamics from a state
which belongs to the bubble sector; HSF then ensures that the dynamics will be
controlled by states within the sector. Since $H$ constitutes non-interacting
pseudospins on every center site of a bubble, $U(T,0)$ corresponding to a
square-pulse drive protocol can be found exactly. The reason for this stems
from the fact that here one deals with a $s=1/2$ pseudospin on every such
site; this is similar to an analogous problem for Ising or other integrable
model where an analogous structure can be seen in momentum space bookchap1 . A
straightforward analysis yields for $n_{\rm odd}=n_{\rm even}$ bm1

$\displaystyle U(T,0)$ $\displaystyle=$
$\displaystyle\prod_{j=1..L}\left(\begin{array}[]{cc}p_{j}&q_{j}\\\
-q_{j}^{\ast}&p_{j}\end{array}\right),\quad
p_{j}=\frac{(3\Delta_{0}/2)^{2}+\cos(\epsilon_{0}T)}{\epsilon_{0}^{2}},\quad
q_{j}=\frac{(-1)^{j}(3\Delta/2)[1-\cos(\epsilon_{0}T)]-i\epsilon_{0}\sin(\epsilon_{0}T)}{\epsilon_{0}^{2}}$
(57)

This shows that $U(T,0)=I$ when $\omega_{D}=\omega_{f}=\epsilon_{0}/n$. At
these frequencies, the stroboscopic dynamics of the system, starting from any
of the states in the bubble sector, exhibits dynamical freezing dynfr1 ;
dynfr2 ; uma1 ; ryddyn3 . Such freezing clearly stems from fragmentation and
is a non-perturbative exact phenomenon. Moreover, its existence shows up in
the dynamics of simple initial states in the Fock space (such as
$|X_{1,j}\rangle$) making it experimentally accessible.

###  V.3 Connection to lattice gauge theories

Finally, we point out the connection of these systems to lattice gauge
theories. The possibility of simulating lattice gauge theories using optical
lattice systems has been a subject of recent interest lgt1 ; lgt1a ; lgt2 ;
lgt3 ; lgt4 ; iaed1 . The reason for this is partly the possibility of
realization of an experimental platform for study of confinement. It is well-
known from the seminal work of Ref. coleref,  that in 1D quantum
electrodynamics, charges (charge $\pm e$ for particles and anti-particles)
display the phenomenon of confinement in the presence of a background of
electric field $E_{b}$. This confinement stems from the fact that the energy
of these charges increases linearly with the distance between them. This is
characterized by a parameter $\theta$ which is proportional to the background
electric field $E_{b}$: $\theta=2\pi E_{b}/e$. It was shown for all
$\theta\neq\pi$, a pair of charges remain confined since their energy grows
linearly with distance when they are attempted to be separated. It was also
pointed out in Ref. coleref,  that this form of confinement holds in $d=1$;
for higher dimensions, the presence of transverse photons changes the scenario
and leads to deconfinement.

A variant of this phenomenon is expected to be found in possible realization
of U(1) lattice gauge theories, generically termed as quantum link models lgt1
; lgt1a ; lgt2 ; lgt3 ; lgt4 ; iaed1 , using optical lattice platforms.
Indeed, a recent work on the PXP model in the presence of an additional
staggered magnetic field of strength $\Delta$ showed that such a model can be
mapped to a $U(1)$ lattice gauge theory with $\Delta=J(\theta/\pi-1)$ iaed1 ,
where $J$ is a microscopic energy scale. Thus the absence of $\Delta$ which
corresponds to $\theta=\pi$ in the gauge theory language, corresponds to the
PXP model which shows deconfined behavior. In contrast, the presence of a
large $\Delta$ leads to confining behavior whose signature can be picked up in
quench dynamics of such system iaed1 . We note here that an active field of
research in this area involves understanding the role of gauge invariance in
the dynamical evolution of these systems both experimentally gaugeexpt1  and
theoretically gaugeth1 ; gaugeth2 .

To understand the mapping of $H_{\rm fr}$ to lattice gauge theory, we begin by
writing the spin variables in the language of Kogut-Susskind fermions kosuss1
. This provides a simple dictionary which relates the Rydberg spins
$\sigma_{j}^{\alpha}$ to the fermionic matter (denoted by fields $\psi$) and
gauge fields (denoted by field $\hat{E}$) which are the degrees of freedom in
the gauge theory. The gauge (electric) field and the Rydberg spins live on the
link $\ell$ of the dual 1D lattice ( i.e. sites of the original lattice) while
the fermionic matter fields live on the sites $j$ of the dual lattice. The
electric fields take value $\pm 1/2$ and are related to the Rydberg spins via
the relation bm1

$\displaystyle E_{\ell}$ $\displaystyle=$ $\displaystyle
S_{\ell}^{z}=\eta_{j}\sigma_{\ell}^{z}/2$ (58)

where $\eta_{j}=\pm 1$ refers to the site $j$ at the left of the link $\ell$
and is $1(-1)$ if that site is odd(even). The fermionic matter field
$\psi_{j}$ is related to the electric field by the Gauss’s law $G_{j}=0$ where
lgt1

$\displaystyle G_{j}$ $\displaystyle=$ $\displaystyle
E_{\ell}-E_{\ell+1}-\hat{n}_{j}-[1+(-1)^{j}]/2,\quad\hat{n}_{j}=\psi_{j}^{\dagger}\psi_{j}$
(59)

It turns out that the constraint of having no two up-spins as neighbors is
exactly implemented by this law; moreover, the number of gauge-invariant
states for a chain of length $L$ exactly equals the number of states within
the constrained Hilbert space of the PXP model. In this language, the PXP
model, supplemented by the staggered magnetic field term, can be written as
iaed1

$\displaystyle H_{\rm spin}$ $\displaystyle=$
$\displaystyle-\sum_{j}\tilde{\sigma}_{j}^{x}+\frac{\Delta}{2}\sum_{j}(-1)^{j}\sigma_{j}^{z}\equiv-\sum_{i}(\psi_{i}^{\dagger}U_{\ell}\psi_{i+1}+{\rm
h.c.})+m\sum_{i}\psi_{i}^{\dagger}\psi_{i}$ (60)

where $U_{\ell}=S_{\ell}^{+}$ leading to $[E_{\ell},U_{\ell}]=U_{\ell}$ and
$\ell$ joins the dual lattice sites $i$ and $i+1$. Thus the staggered spin
term acts as the mass of the fermion fields.

It turns out that $H_{\rm fr}$ can also be written in the gauge theory
language leading to a Hamiltonian bm1

$\displaystyle H_{\rm fr}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{L}\psi_{i}^{\dagger}U_{\ell}U_{\ell+1}U_{\ell+2}\psi_{i+3}\Gamma_{i+1}\Gamma_{i+2}+{\rm
h.c.}$ (61)

where $\ell$ is the link between the sites $i$ and $i+1$ on the dual lattice
and $\Gamma_{i}=\hat{n}_{i}$ for odd and $(1-\hat{n}_{i})$ for even $i$
respectively. It was shown in Ref. bm1,  that $[H_{\rm fr},G_{j}]=0$ which
ensures that $H_{\rm fr}$ obeys Gauss’s law. The key point about the gauge
theory representations of $H_{\rm fr}$ (Eq. 61) is that the it only conserves
local charge. There is no dipole moment conservation associated with this
model unlike most models of Hilbert space fragmentation (see Refs. fraglit1, ;
fraglit2, ; fraglit3, ; fraglit4, ) studied earlier.

Instead, such fractonic behavior may appear as emergent phenomenon in certain
sectors of theory bm1 . This was demonstrated in Ref. bm1,  in the bubble
sector. To see this let us consider a single bubble state. All the down spin
outside this bubble are annihilated by $H_{\rm fr}$. In the gauge theory
language, this means that within this sector, all the fermionic charges
outside the bubble are immobile. It was shown that the dynamics involve only
the bubble which, here, constitutes dipoles involving three lattice sites
(length three dipoles). Only these dipoles have significant dynamics under
action of $H_{\rm fr}$ and this leads to emergent fractonic physics. We note
that this emergence occurs in a fragment of the Hilbert space which is not
necessarily a low-energy sector. Other fragmented sectors may show similar
emergence, and this is discussed in details in Ref. bm1, .

##  VI Discussion

In this review, we have touched upon several aspects of the tilted boson chain
which can be realized in 1D optical lattices hosting trapped ultracold bosons.
It turns out that the physics of this system has close parallel to that of
Rydberg atoms in the sense that both systems are described by similar
effective Hamiltonians in their low-energy sector.

The ground state phase diagram of such bosonic system provides a route to
realizing translational symmetry broken Mott states. In addition, they also
host a quantum phase transition between Mott states with broken and unbroken
translational symmetries. For the tilted boson chain, this transition belongs
to the Ising universality class and can be understood in terms of a dipole
model of the bosons. A modification of this dipole model may lead to
realization of non-Ising critical points, as has been seen in Rydberg atom
arrays.

The quench dynamics of these systems provided the key clue to unraveling the
presence of quantum scars in their Hilbert space. Such states lead to a weak
violation of the eigenstate thermalization hypothesis in finite chain as has
been seen in recent experiments involving these systems. The ramp dynamics of
these systems near their critical point may provide a platform to test Kibble-
Zurek scaling law; this is particularly interesting for the Rydberg arrays
where the transition is non-Ising like.

The study of periodic dynamics of such systems shows the possibility of tuning
their ergodicity properties using the drive frequency. It was found that these
systems shows unconventional phenomenon such as the presence of sub-thermal
steady states, long term coherent oscillations, and dynamical freezing in the
presence of a periodic drive. These effects can be analyzed using the Floquet
Hamiltonian of these driven system; moreover, they occur, in contrast to their
quench counterparts, for both $|0\rangle$ and $|{\mathbb{Z}}_{2}\rangle$
initial states and constitute different routes to violation of ETH in finite-
sized chains.

The Floquet Hamiltonian of this periodically driven system provides a class of
terms which can act as a minimal model for $s=1/2$ spins exhibiting HSF. This
model, unlike some earlier models displaying HSF, do not have simple conserved
quantities which causes the fragmentation; instead, such conservation emerges
in specific sector of the model. Also, the model provides an example of
secondary fragmentation which does not follow from classical conservation
laws. As a consequence of HSF, the model exhibits exact dynamical freezing for
an infinite number of drive frequencies and for an exponentially large number
initial classical Fock states.

Several aspects of this tilted boson chains and/or Rydberg ladders remain to
be studied. These include the effect of staggered detuning term on the driven
system, the physics of multiple interacting Rydberg chains and manifestation
of the physics of these driven system in higher dimensions. These studied are
expected to add to the already large class of the physical phenomenon seen in
these systems.

Acknowledgement: KS thanks D. Banerjee, B. K. Clark, U. Divakaran, P. Fendley,
S. M. Girvin, R. Ghosh, S. Kar, M. Kolodrubetz, B. Mukherjee, S. Nandy, D.
Pekker, S. Powell, S. Sachdev, A. Sen, and D. Sen for collaborations on
several related projects.

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