# Periodically driven Rydberg chains with staggered detuning

Bhaskar Mukherjee1, Arnab Sen2, K. Sengupta2 1 Department of Physics,
University College London, Gower Street, London WC1E 6BT, UK.
2School of Physical Sciences, Indian Association for the Cultivation of
Science, Jadavpur, Kolkata 700032, India

###### Abstract

We study the stroboscopic dynamics of a periodically driven finite Rydberg
chain with staggered ($\Delta$) and time-dependent uniform ($\lambda(t)$)
detuning terms using exact diagonalization (ED). We show that at intermediate
drive frequencies ($\omega_{D}$), the presence of a finite $\Delta$ results in
violation of the eigenstate thermalization hypothesis (ETH) via clustering of
Floquet eigenstates. Such clustering is lost at special commensurate drive
frequencies for which $\hbar\omega_{d}=n\Delta$ ($n\in Z$) leading to
restoration of ergodicity. The violation of ETH in these driven finite-sized
chains is also evident from the dynamical freezing displayed by the density-
density correlation function at specific $\omega_{D}$. Such a correlator
exhibits stable oscillations with perfect revivals when driven close to the
freezing frequencies for initial all spin-down ($|0\rangle$) or Neel
($|{\mathbb{Z}}_{2}\rangle$, with up-spins on even sites) states. The
amplitudes of these oscillations vanish at the freezing frequencies and
reduces upon increasing $\Delta$; their frequencies, however, remains pinned
to $\Delta/\hbar$ in the large $\Delta$ limit. In contrast, for the
$|{\bar{\mathbb{Z}}_{2}}\rangle$ (time-reversed partner of
$|{\mathbb{Z}}_{2}\rangle$) initial state, we find complete absence of such
oscillations leading to freezing for a range of $\omega_{D}$; this range
increases with $\Delta$. We also study the properties of quantum many-body
scars in the Floquet spectrum of the model as a function of $\Delta$ and show
the existence of novel mid-spectrum scars at large $\Delta$. We supplement our
numerical results with those from an analytic Floquet Hamiltonian computed
using Floquet perturbation theory (FPT) and also provide a semi-analytic
computation of the quantum scar states within a forward scattering
approximation (FSA). This allows us to provide qualitative analytical
explanations of the above-mentioned numerical results.

##  I Introduction

The physics of non-equilibrium quantum systems has been studied extensively in
recent years rev1 ; rev2 ; rev3 ; rev4 ; rev5 ; rev6 ; rev7 ; rev8 . The
theoretical studies in the field have been boosted by possibility of
experimental realization using ultracold atom platforms rev9 ; exp1 ; exp2 ;
exp3 ; exp4 . Out of the several possible drive protocols that could be used
to take a quantum system out of equilibrium, periodic drives have attracted a
lot of recent attention rev6 ; rev7 ; rev8 . For such protocols characterized
by a time period $T$, the unitary evolution operator $U(t,0)$ governs the
dynamics of the system. At times $t=nT$, (where $n\in Z$) $U$ can be written
in terms of the Floquet Hamiltonian $H_{F}$: $U(nT,0)=\exp[-iH_{F}nT/\hbar]$,
where $\hbar$ is Planck’s constant. Thus the stroboscopic dynamics of the
system is completely governed by its Floquet Hamiltonian rev8 .

The interest in such drive protocols stems from several interesting physical
phenomena which occurs in periodically driven systems. These include drive-
induced non-trivial topology topo1 ; topo2 ; topo3 , dynamical localization
dloc1 ; dloc2 ; dloc3 ; dloc4 , dynamical phase transitions dtran1 ; dtran2 ;
dtran3 , realization of time crystals tc1 ; tc2 ; tc3 , dynamical freezing df1
; df2 ; df3 ; df4 ; df5 ; ethv2  and the possibility of tuning ergodicity
property of the driven system ethv1 . More recently, some of these properties
have also been investigated in the context of quasiperiodically qpd1 ; qpd2 ;
rand1  and aperiodically rand1  driven systems.

A central paradigm for understanding the behavior of non-integrable many-body
systems described by local Hamiltonians that are driven out of equilibrium
comes from the eigenstate thermalization hypothesis (ETH) eth1 ; eth2 ; eth3 ;
eth4 . ETH postulates that the reduced density matrix of any eigenstate with
finite energy density is thermal in the thermodynamic limit, the corresponding
temperature being determined by its energy density. ETH has also proved
successful in explaining properties of non-integrable systems in the presence
of a periodic drive; here eigenstates of the Floquet Hamiltonian play a
similar role eth4 .

The violation of ETH in many-body systems can occur via several routes mbl1 ;
mblrev1 ; mblrev2 ; mblrev3 ; mblrev4 ; ethv0 ; ethv01 ; ethv02 ; ethv03 ;
ethv04 ; ethv05a ; ethv05b ; ethv06 ; ethv07 ; ethv1 ; ethv2 ; ethv08 ; ethv09
; hsf1 ; hsf2 ; hsf3 ; hsf4 . For example, ETH does not hold for integrable
models where the presence of an extensive number of additional conserved
quantities makes the system non-ergodic. Moreover, a loss of ergodicity and
hence violation of ETH may also happen due to strong disorder when a system
enters a many-body localized (MBL) phase mbl1 ; mblrev1 ; mblrev2 ; mblrev3 ;
mblrev4 . Such violation may also occur due to Hilbert space fragmentation
where the system Hilbert space organizes into dynamically disconnected sectors
hsf1 ; hsf2 ; hsf3 ; hsf4 .

Another weaker version of ETH violation has also been found in Rydberg atom
systems ethv0 ; ethv01 ; ethv02 ; ethv03 ; ethv05b ; ethv1 ; ethv2 .
Experiments on such systems showed coherent long-lived oscillation of density
of Rydberg excitations during evolution starting from a Neel state with one
Rydberg excitation at every even
($|{\mathbb{Z}}_{2}\rangle=|..1_{2j},0_{2j+1},1_{2j+2},....\rangle$) or odd
($|{\bar{\mathbb{Z}}_{2}}\rangle=|...1_{2j-1},0_{2j},1_{2j+1},....\rangle$)
site; in contrast, ETH predicted thermalization occurred when the dynamics
started from the Rydberg vacuum state ($|0\rangle=|..0_{j},0_{j+1},\rangle$)
exp4 . The reason for such ETH violating oscillations were traced to the
existence of a special class of eigenstates called quantum scars ethv0 ;
ethv01 ; ethv02 ; ethv03 ; ethv04 ; ethv05a ; ethv05b ; ethv06 ; ethv07 ;
ethv1 ; ethv2 ; ethv08 ; ethv09 . These are many-body eigenstates with finite
energy density but anomalously low entanglement entropy. Consequently,
dynamics starting from an initial state which has large overlap with them do
not show thermalization. The role of such states in periodically driven
Rydberg chain has also been analyzed ethv1 ; ethv2 ; it was shown that their
presence may lead to the possibility of drive-frequency induced tuning
ergodicity property of these chains ethv1 , presence of subthermal steady
states, and the phenomenon of dynamical freezing ethv2 . These properties do
not have any analogue for scar-induced dynamics following a quench.

More recently, there have been several studies on possible role of confinement
in condensed matter systems. The concept of confinement in 1D quantum
electrodynamics is well-known; it was shown that such systems are
characterized by a parameter $\theta$ which is related to the background
electric field. For $\theta=\pi$, the system is deconfined and charge
excitations can propagate; for other values of $\theta$, these excitations are
confined qedcon1 ; qedcon2 . Such ideas have also been applied to the
simulation of lattice gauge theories (LGTs) using ultracold atom platforms
simu1 ; simu2 ; simu3 ; simu4 ; deb1 ; deb2 , Ising like chains conf1 , and
Rydberg chains with staggered detuning described by the Hamiltonian ssg ; fss
; ethv0 ; conf2 ; conf3

$\displaystyle H_{\rm ryd}$ $\displaystyle=$
$\displaystyle-w\sum_{j}\tilde{\sigma}_{j}^{x}-\frac{1}{2}\sum_{j}(\lambda+\Delta(-1)^{j})\sigma_{j}^{z}$
(1)

Here $\Delta>0$ and $\lambda$ denote the amplitude of staggered and uniform
detuning respectively, $w>0$ is the coupling strength between ground and
excited states of the Rydberg atoms, $\sigma_{j}^{z}=2\hat{n}_{j}-1$ is
related to the number operator $\hat{n}_{j}$ for a Rydberg excitations on site
$j$. The Hamiltonian given by Eq. 1 is to be supplemented by a constraint that
two adjacent sites can not both have Rydberg excitations. Thus the model has a
Rydberg blockade radius of one lattice site. This constraint is implemented
using the projected spin operators
$\tilde{\sigma}_{j}^{x}=P_{j-1}\sigma_{j}^{x}P_{j+1}$, where
$P_{j-1}=(1-\sigma_{j}^{z})/2$. The dynamics of such Rydberg chains was
studied in context of quench protocols in Ref. conf2, ; conf3, . It was
observed that the presence of the confinement terms leads to slow dynamics of
the Rydberg atoms. However, periodic drive protocols for such chains have not
been studied so far.

In this work, we study a periodically driven Rydberg chain with a staggered
detuning term. Here we consider the uniform detuning parameter $\lambda$ to be
a periodic function of time given by

$\displaystyle\lambda(t)$ $\displaystyle=$ $\displaystyle\lambda_{0}\quad
t\leq T/2$ (2) $\displaystyle=$ $\displaystyle-\lambda_{0}\quad t>T/2$

where $T=2\pi/\omega_{D}$, $\omega_{D}$ is the drive frequency and we focus in
the regime of large drive amplitude:$\lambda_{0}\gg\Delta,w$. In the rest of
this work we focus on the dynamics of the density-density correlation function
${\rm O}_{j2}=\langle\hat{n}_{j}\hat{n}_{j+2}\rangle$ of these driven Rydberg
atoms and the half-chain entanglement entropy $S_{L/2}$ of their Floquet
eigenstates. We use ED to study the properties of $S_{L/2}$ and ${\rm O}_{j2}$
numerically. We supplement our numerical finding by comparing them with those
obtained from an analytic, albeit perturbative, Floquet Hamiltonian derived
using Floquet perturbation theory (FPT) and also present a calculation of some
Floquet eigenstates using forward scattering approximation (FSA) applied to
the analytic Floquet Hamiltonian.

The main results that we obtain from such a study are as follows. First, we
show, via study of $S_{L/2}$, the presence of clustering of Floquet
eigenstates in these systems for strong staggered detuning and at intermediate
frequencies leading to violation of ETH. These clusters can be classified into
two categories. The first type, dubbed as primary clusters, is a result of
near-integrable nature of the system for large staggered detuning. The second
type of clusters occurs within each primary cluster at a much smaller
quasienergy scale. We show that these secondary clusters result from an
emergent conservation law found earlier in non-driven staggered Rydberg chains
ethv05b . Second, we also show that the secondary clustering is destroyed,
leading to ergodicity restoration within each primary cluster, at commensurate
drive frequencies $\hbar\omega_{D}=n\Delta$, where $n$ is an integer; this
allows drive-induced tuning of ergodicity property of the Floquet eigenstates
within each primary cluster. Third, we show that for almost all intermediate
drive frequencies, ${\rm O}_{22}$ does not reach its ETH predicted steady
state value; this phenomenon is seen for $|{\mathbb{Z}}_{2}\rangle$,
$|0\rangle$ and the $|{\bar{\mathbb{Z}}_{2}}\rangle$ initial states. For
dynamics starting from $|{\mathbb{Z}}_{2}\rangle$ and $|0\rangle$ initial
states, we find specific drive frequencies for which the system is dynamically
frozen; ${\rm O}_{22}$ remains pinned to its initial values at these
frequencies. Around these freezing frequencies, ${\rm O}_{22}$ displays
oscillatory behavior around its initial values with perfect revivals; the
amplitudes of these oscillations vanish as the freezing frequency is
approached and reduces as $\Delta$ is increased. In contrast, the frequency of
these oscillation are pinned to $\Delta/\hbar$ for a wide range of $\Delta$ in
the intermediate and large $\Delta$ regime. We provide a semi-analytic
explanation of this phenomenon. Fourth, in contrast to the above cases, for
the $|{\bar{\mathbb{Z}}_{2}}\rangle$ initial state, $O_{22}$ remains frozen
over a range of frequencies leading to a dynamically frozen phase. We provide
a qualitative explanation for this phenomenon; our analysis shows that for odd
$j$, $O_{j2}$ will have a frozen phase for dynamics starting from
$|{\mathbb{Z}}_{2}\rangle$. This dichotomy is a direct consequence of the
$Z_{2}$ (between even and odd sites) symmetry breaking due to the presence of
the staggered detuning. Fifth, we study the fate of the quantum many-body
scars in the Floquet eigenspectrum as a function of the staggered detuning.
For small $\Delta$, these scars have almost equal overlap with both the Neel
states; in contrast, we find that scars with $E_{F}<0$ ($E_{F}>0$) have
stronger overlap with $|{\mathbb{Z}}_{2}\rangle$
($|{\bar{\mathbb{Z}}_{2}}\rangle$). For large enough $\Delta$, the mid
spectrum scars ($E_{F}\simeq 0$) do not show significant overlap with either
of the Neel states; instead we show that these represent a separate type of
scar states exhibiting high overlap either with number states with broken
$Z_{4}$ symmetry ($|{\mathbb{Z}}_{4}\rangle$) or with states having single
Rydberg excitation ($|1\rangle$). These scar states results in long-time
coherent oscillations for dynamics starting from $|{\mathbb{Z}}_{4}\rangle$ or
$|1\rangle$ initial states and are therefore qualitatively different from
their counterparts in the PXP model ethv01 . We also present a computation of
the scars at small and intermediate $\Delta$ using a FSA formalism turpap1 .

The plan of the rest of the paper is as follows. In Sec. II, we obtain the
analytic Floquet Hamiltonian using FPT. This is followed by Sec. III where we
study the structure of the Floquet eigenspectrum for strong staggered detuning
and discuss the resultant clustering. Next, in Sec. IV, we analyze the
dynamical freezing of $O_{22}$ and its dynamics near the freezing points. This
is followed by Sec. V where we discuss the change in properties of the quantum
many-body scars in the Floquet eigenspectra as $\Delta$ is tuned from small to
large values. Finally, we summarize our main results and conclude in Sec. VI.
Some details of the calculation are presented in the appendix.

##  II Derivation of the Floquet Hamiltonian

In this section, we derive an analytic Floquet Hamiltonian for the driven
Rydberg atoms described by Eq. 1 in the presence of the drive protocol given
by Eq. 2. To this end, we shall focus on the limit $\lambda_{0}\gg\Delta,w$;
however, no restriction is placed on the relative magnitude of $\Delta$ and
$w$. For the rest of this section, we shall set $\hbar=1$.

In what follows, we shall use FPT to obtain the Floquet Hamiltonian rev9 ;
ethv2 . Since $\lambda_{0}\gg\Delta,w$, we treat the uniform detuning term
exactly and rewrite Eq. 1 as $H=H_{0}(t)+H_{1}+H_{2}$ where

$\displaystyle H_{0}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\lambda(t)\sum_{j}\sigma_{j}^{z},$ $\displaystyle
H_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{j}\Delta(-1)^{j}\sigma_{j}^{z},\quad
H_{2}=-w\sum_{j}\tilde{\sigma}_{j}^{x}.$ (3)

We shall treat $H_{1}$ and $H_{2}$ perturbatively. This leads to the evolution
operator

$\displaystyle U_{0}(t,0)$ $\displaystyle=$ $\displaystyle
T_{t}\left[e^{-i\int_{0}^{t}H_{0}(t^{\prime})dt^{\prime}}\right]$ (4)
$\displaystyle=$ $\displaystyle e^{i\lambda_{0}t\sum_{j}\sigma_{j}^{z}/2}\quad
t\leq T/2$ $\displaystyle=$ $\displaystyle
e^{i\lambda_{0}(T-t)\sum_{j}\sigma_{j}^{z}/2}\quad t\geq T/2,$

We note that $U_{0}(T,0)=I$ which leads to $H_{F}^{(0)}=0$. Also, for
subsequent computation, we define the states $|m\rangle$ as eigenstates of
$H_{0}$ that have $m$ up-spins and $L-m$ down-spins. Such states need to obey
the constraint imposed by the model; thus $m_{\rm max}=L/2$ is the maximum
number of up-spins.

Next, we consider the first order correction to the Floquet Hamiltonian given,
within FPT, by

$\displaystyle U_{1}(T,0)$ $\displaystyle=$
$\displaystyle-i\int_{0}^{T}dt_{1}U_{0}^{\dagger}(t_{1},0)(H_{1}+H_{2})U_{0}(t_{1},0)$
(5)

To evaluate $U_{1}$, we note that $H_{1}$ commutes with $U_{0}$. Moreover, the
contribution from $H_{2}$ has already been obtained in Refs. ethv1, ; ethv2, .
From these we find,

$\displaystyle\langle m|U_{1}(T,0)|n\rangle$ $\displaystyle=$ $\displaystyle
iT\sum_{m}\left(\sum_{j}\frac{\Delta}{2}(-1)^{j}\langle
m|\sigma_{j}^{z}|m\rangle|m\rangle\langle m|\delta_{mn}+\sum_{s_{j}=\pm
1}\frac{4iw}{\lambda_{0}T}\sin\frac{\lambda_{0}T}{4}e^{i\lambda_{0}Ts_{j}}|m\rangle\langle
m+s_{j}|\delta_{n,m+s_{j}}\right)$ (6)

Using Eq. 6, one obtains the first order Floquet Hamiltonian, given by
$H_{F}^{(1)}=iU_{1}(T,0)/T$ to be

$\displaystyle
H_{F}^{(1)}=H_{1}-w_{r}\sum_{j}\left(\cos\gamma\tilde{\sigma}_{x}+\sin\gamma\tilde{\sigma}_{y}\right),\,\,w_{r}=\frac{w\sin\gamma}{\gamma}$
(7)

where $\gamma=\lambda_{0}T/4$. We note that the second term in $H_{F}^{(1)}$,
obtained in Refs. ethv1, ; ethv2, , vanishes at $\gamma=n\pi$ (or
$\lambda_{0}=2n\omega_{D}$) for non-zero integer $n$; thus at these points, we
expect the effect of the staggered detuning term ($H_{1}$) to be particularly
strong.

It turns out that there is no second order contribution to $H_{F}$ since

$\displaystyle U_{2}(T,0)$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}dt_{1}U_{0}^{\dagger}(t_{1},0)(H_{1}+H_{2})U_{0}(t_{1},0)$
(8)
$\displaystyle\times\int_{0}^{t_{1}}dt_{2}U_{0}^{\dagger}(t_{2},0)(H_{1}+H_{2})U_{)}(t_{2},0)$
$\displaystyle=$ $\displaystyle U_{1}(T,0)^{2}/2$

Eq. 8 can be obtained from a straightforward computation; however, it can also
be checked as follows. First we note that the terms $\sim H_{1}$ in the
expression of $U_{2}$ commutes with $U_{0}$. This indicates that the term
originating from $H_{1}^{2}$ in $U_{2}(T,0)$ must trivially satisfy Eqn. 8.
Second, we recall that it is known ethv1 ; ethv2  that the term $\sim
H_{2}^{2}$ does not yield any second order contribution to $H_{F}$. This is
due to the presence of a symmetry noted in Ref. ethv1, . For $\Delta=0$, it
was shown that $\\{Q,H_{F}\\}=0$, where $Q=\prod_{j}\sigma_{j}^{z}$;
consequently, there can be no terms with product of even number of
$\tilde{\sigma}_{j}^{\pm}$ operators in $H_{F}$. Moreover, a straightforward
computation shows that the contribution from all terms which involve both
$H_{1}$ and $H_{2}$ also vanish due to commutation of $H_{1}$ with $U_{0}$.
Thus we have $H_{F}^{(2)}=0$.

Finally, we compute the third order term in the Floquet Hamiltonian. To this
end, we consider the third order contribution to $U$, given by

$\displaystyle U_{3}(T,0)$ $\displaystyle=$ $\displaystyle
i\int_{0}^{T}dt_{2}U_{0}^{\dagger}(t_{1},0)(H_{1}+H_{2})U_{0}(t_{1},0)$ (9)
$\displaystyle\times\int_{0}^{t_{1}}dt_{2}U_{0}^{\dagger}(t_{2},0)(H_{1}+H_{2})U_{0}(t_{2},0)$
$\displaystyle\times\int_{0}^{t_{2}}dt_{2}U_{0}^{\dagger}(t_{3},0)(H_{1}+H_{2})U_{0}(t_{3},0).$

To obtain the third order Floquet Hamiltonian, we first note that the
commutation of $H_{1}$ with $U_{0}$ ensures that there is no third order
contribution to $H_{F}$ from the term with three $H_{1}$ operators. The first
non-trivial contribution to the third order $H_{F}$ comes from the term in Eq.
9 which has two $H_{1}$ and one $H_{2}$ operators. The contribution of this
term to $U_{3}(T,0)$ can be computed as charted in details in Ref. ethv2, .
The result is

$\displaystyle U_{3a}(T,0)$ $\displaystyle=$
$\displaystyle\frac{i\Delta_{0}^{2}w}{4}\sum_{j_{1},j_{2},j_{3}}\sum_{s=\pm}(-1)^{j_{1}+j_{2}}(2c_{1s}\sigma_{j_{1}}^{z}\sigma_{j_{2}}^{z}\tilde{\sigma}_{j_{3}}^{s}$
(10)
$\displaystyle+c_{2s}\sigma_{j_{1}}^{z}\tilde{\sigma}_{j_{3}}^{s}\sigma_{j_{2}}^{z})$
$\displaystyle c_{1\pm}$ $\displaystyle=$ $\displaystyle\pm T^{3}\frac{-i\pm
2\gamma(1\pm 2i\gamma)-ie^{\pm
2i\gamma}\left(2\gamma^{2}-1\right)}{32\gamma^{3}}$ $\displaystyle c_{2\pm}$
$\displaystyle=$ $\displaystyle\pm T^{3}\frac{i\mp 2\gamma-ie^{\pm
2i\gamma}\left(2\gamma^{2}+1\right)}{16\gamma^{3}}$

The structure of $U_{3a}$ shows that it is necessary to categorize the terms
into two groups. The first constitutes the case where at least one of the
$j_{i}$s are different from the other two. For these, $U_{3a}(T,0)$ either
vanishes or can be written as

$\displaystyle U_{3a}(T,0)$ $\displaystyle=$
$\displaystyle\frac{i\Delta_{0}^{2}w}{4}\sum_{j_{1}\neq j_{2}\neq
j_{3}}\sum_{s=\pm}(-1)^{j_{1}+j_{2}}(2c_{1s}+c_{2s})$ (11)
$\displaystyle\times\sigma_{j_{1}}^{z}\sigma_{j_{2}}^{z}\tilde{\sigma}_{j_{3}}^{s}=\frac{U_{1}(T,0)^{3}}{3!}$

where we have used the fact that
$2c_{1s}+c_{2s}=iT^{2}(\exp[i\lambda_{0}Ts]-1)/(\lambda_{0}s)$ for $s=\pm 1$.
Thus such terms do not contribute to $H_{F3}$. However, when
$j_{1}=j_{2}=j_{3}$, we find that the last term in Eqn. 10 acquires a negative
sign due to the anticommutation of $\sigma_{j}^{z}$ and
$\tilde{\sigma}_{j}^{s}$. This leads to

$\displaystyle U_{3a}(T,0)$ $\displaystyle=$
$\displaystyle\frac{i\Delta_{0}^{2}w}{4}\sum_{j}\sum_{s=\pm}(2c_{1s}-c_{2s})\tilde{\sigma}_{j}^{s}$
(12)

Comparing this with $U_{1}(T,0)^{3}/3!$, we find that there is a non-trivial
contribution to the Floquet Hamiltonian given by

$\displaystyle H_{F1}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{w\Delta_{0}^{2}}{2T}\sum_{j}(c_{2+}\tilde{\sigma}_{j}^{+}+{\rm
h.c.})$ (13)

This term leads to a frequency dependent renormalization of the coefficient of
$w$ in the first order Floquet Hamiltonian.

The next contribution to the Floquet Hamiltonian comes from the terms in Eqn.
9 with two $H_{2}$ and one $H_{1}$ operators. The calculation of this term,
$U_{3b}(T,0)$, follows a similar route as that followed in Ref. ethv2,  and
leads to

$\displaystyle U_{3b}(T,0)$ $\displaystyle=$
$\displaystyle\frac{i\Delta_{0}w^{2}}{2}\sum_{j_{1},j_{2},j_{3}}\sum_{s_{1},s_{2}=\pm}(-1)^{j_{3}}(2d_{1s_{1}s_{2}}\tilde{\sigma}_{j_{1}}^{s_{1}}\tilde{\sigma}_{j_{2}}^{s_{2}}\sigma_{j_{3}}^{z}+d_{2s_{1}s_{2}}\tilde{\sigma}_{j_{1}}^{s_{1}}\sigma_{j_{3}}^{z}\tilde{\sigma}_{j_{2}}^{s_{2}})$
(14) $\displaystyle d_{1++}$ $\displaystyle=$ $\displaystyle
T^{3}\frac{-(i/4+\gamma)+ie^{2i\gamma}(i+2\gamma)+e^{12i\gamma}(3i/4+2\gamma)}{32\gamma^{3}},$
$\displaystyle d_{1+-}$ $\displaystyle=$ $\displaystyle
T^{3}\frac{\gamma\left(2-i\gamma\right)-\gamma
e^{2i\gamma}-i\left(1-e^{2i\gamma}\right)/2}{16\gamma^{3}}$ $\displaystyle
d_{2++}$ $\displaystyle=$ $\displaystyle
T^{3}\frac{i/4-\gamma+3e^{4i\gamma}/4+e^{2i\gamma}(-i+2\gamma)}{16\gamma^{3}},\quad
d_{2+-}=\frac{T^{3}}{8\gamma^{3}}\left(\frac{1}{2}\sin 2\gamma-\gamma\cos
2\gamma\right)$ (15)

and $d_{a\pm\pm}=d_{a\mp\mp}^{\ast}$ for $a=1,2$. The argument leading to the
Floquet Hamiltonian is similar to the one discussed earlier and only the on-
site terms in $U_{3b}$ contribute to $H_{F}^{(3)}$. The final result is

$\displaystyle H_{F2}^{(3)}$ $\displaystyle=$
$\displaystyle-\frac{w^{2}\Delta_{0}}{T}\sum_{j}(-1)^{j}(d_{2+-}\tilde{\sigma}_{j}^{+}\sigma_{j}^{z}\tilde{\sigma}_{j}^{-}+{\rm
h.c.})$ (16)

Note that the action of $H_{F2}^{(3)}$ on any eigenstate of $\sigma_{j}^{z}$
is same as that of $-\sigma_{j}^{z}$. Thus this term can be considered as a
correction of the staggered detuning term in the first order Floquet
Hamiltonian. From Eq. 14, we find that for $\tan 2\gamma>2\gamma$, this
correction is negative and leads to a reduction in the magnitude of the on-
site detuning.

The final term in the Floquet Hamiltonian comes from the term in Eq. 9 which
has three $H_{2}$ operators. This term has already been computed in Ref.
ethv2, ; for completeness, we write the contribution of this term to $H_{F}$
here. This is given by

$\displaystyle H_{F3}^{(3)}$ $\displaystyle=$
$\displaystyle\sum_{j}A_{0}[(\tilde{\sigma}_{j-1}^{+}\tilde{\sigma}_{j+1}^{+}+\tilde{\sigma}_{j+1}^{+}\tilde{\sigma}_{j-1}^{+})\tilde{\sigma}_{j}^{-}-6\sigma_{j}^{+}]+{\rm
h.c}$ $\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle[e^{6i\gamma}+3e^{2i\gamma}(1+4i\gamma)+2(1-3e^{12i\gamma})]$
(17) $\displaystyle\times\frac{w^{3}T^{2}e^{-12i\gamma}}{192i\gamma^{3}}$

This term contains a three-spin term as well as a higher-order correction to
the first order Floquet term $\sim w$.

This completes our derivation of the Floquet Hamiltonian for the driven
Rydberg chain with staggered detuning to third order in Floquet perturbation
theory. In subsequent sections, we shall use this along with numerical results
to analyze the properties of the driven chain.

##  III Strong staggered detuning

In this and the next two sections, we present the main results of our study
obtained by exact numerics and compare them to that obtained from the analytic
Floquet Hamiltonian derived in Sec. II. In the present section, we shall
restrict ourselves to the strong staggered detuning limit ($\Delta\geq w$),
while other parameter regimes will be explored in Secs. IV and V.

The exact numerical results that we present are obtained using ED on finite
sized chain as follows. We first numerically diagonalize
$H[\pm\lambda_{0}]=H_{0}[\pm\lambda_{0}]+H_{1}+H_{2}$ (Eq. 3) to obtain their
energy eigenvalues $\epsilon_{a}^{\pm}$ and eigenvectors
$|\xi_{a}^{\pm}\rangle$. The evolution operator for the square pulse protocol
(Eq. 2) can then be written as

$\displaystyle U(T,0)$ $\displaystyle=$ $\displaystyle
e^{-iH_{F}[-\lambda_{0}]T/(2\hbar)}e^{-iH_{F}[\lambda_{0}]T/(2\hbar)}$ (18)
$\displaystyle=$
$\displaystyle\sum_{a,b}c_{ab}^{-+}e^{-i(\epsilon_{a}^{-}+\epsilon_{b}^{+})T/(2\hbar)}|\xi_{a}^{-}\rangle\langle\xi_{b}^{+}|$

where $c_{ab}^{-+}=\langle\xi^{-}_{a}|\xi_{b}^{+}\rangle$. We then numerically
diagonalize the unitary matrix $U$ to obtain its eigenvalues
$\lambda_{n}=\exp[-iE_{n}^{F}T/\hbar]$ and eigenvectors $|\chi_{n}\rangle$.
The exact Floquet Hamiltonian of the system can then be obtained as

$\displaystyle H_{F}$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}^{F}|\chi_{n}\rangle\langle\chi_{n}|$ (19)

In what follows, we shall use Eq. 19 for numerical computation of the Floquet
Hamiltonian. Most the numerical results in the rest of this section, unless
explicitly mentioned otherwise, correspond to $\lambda=15$ and $w=1$.

###  III.1 Clustering of Floquet Eigenstates

In this subsection, we discuss the structure of the Hilbert space spanned by
Floquet eigenstates of the driven staggered Rydberg chain. To this end, we
plot the half-chain entanglement entropy, $S_{L/2}$, of these eigenstates as a
function of their quasienergy $E_{F}$. The details of the numerical procedure
used for this computation is same as that charted in Ref. ethv1, . The first
step involves writing down the density matrix
$\rho_{n}=|\chi_{n}\rangle\langle\chi_{n}|$ corresponding to the $n^{\rm th}$
Floquet eigenstate. One then integrates out the contribution to this state
from Fock states residing on one half of the chain; such an operation has to
be carried out numerically without violating the constraint condition as
detailed in Ref. ethv1, . This leads to the reduced density matrix
$\rho_{n}^{\rm red}$. The von-Neumann entanglement entropy is then computed
from $\rho^{\rm red}_{n}$ following the standard prescription:
$S_{L/2}^{(n)}\equiv S_{L/2}=-{\rm Tr}[\rho_{n}^{\rm red}\ln\rho_{n}^{\rm
red}]$. For finite-sized periodically driven systems, the prediction of ETH
for $S_{L/2}$ is given by the Page formula: $S_{L/2}^{\rm Page}=\ln{\rm
HSD}-1/2$, where ${\rm HSD}$ denotes Hilbert space dimension of the half-chain
with open boundary condition, and ${\rm HSD}=377$ for $L=24$.

Figure 1: Left: Plot of $S_{L/2}$ as a function of $\phi=E_{F}T/\hbar$ for
$\omega_{D}=100w/\hbar$. The red dotted line is the ETH predicted Page value
of $S_{L/2}$. Right panel: Plot of $\langle Z_{\pi}\rangle$ for Floquet
eigenstates as a function of $\phi$ for same $\omega_{D}$. For both plots
$L=24$, $\Delta/w=1.2$, and $\lambda/w=15$. All energies are scaled in units
of $w$ and frequencies in units of $w/\hbar$. The color scheme indicates
higher density of states for warmer colors. See text for details.

The result of such a computation is shown in Figs. 1 and 2 for $\Delta=1.2w$.
From Fig. 1, we find that the Hilbert space is ergodic at high drive
frequencies for which $w_{r}\simeq w$. In contrast, as seen from Fig. 2, it
fragments into several clusters at intermediate $\omega_{D}$. We dub these
clusters as primary clusters. Their origin can be straightforwardly understood
as follows.

We note that the leading terms in the Floquet Hamiltonian given by $H_{F1}$.
For $\lambda T/\hbar\leq 1$ ($\omega_{D}=100w/\hbar$), when $w_{r}\sim w$, the
amplitude of the staggered and the PXP terms in $H_{F1}$ are comparable and
their eigenstates span the entire Hilbert space as expected for a non-
integrable model. This behavior is shown in the left panel of Fig. 1 and is
further highlighted by plotting the expectation value of

$\displaystyle Z_{\pi}=\frac{1}{2}\sum_{j}(-1)^{j}\sigma_{j}^{z}$ (20)

for these states, as shown in the right panel of Fig. 1. We find that the
quasienergy difference between the states with different $\langle
Z_{\pi}\rangle$ are $O(w)$; hence $w_{r}\sim w$ can hybridize states with
different $\langle Z_{\pi}\rangle$ leading to ergodicity.

In contrast, for $\omega_{D}\sim 3.5w/\hbar$ and $3.65w/\hbar$, $w_{r}\ll
w,\Delta$. In this limit, the model is near-integrable and the eigenstates
cluster into groups as shown in the top panels of Fig. 2. Each of these group
have states with definite values $\langle Z_{\pi}\rangle$ and dimensionless
quasienergy

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{E_{F}T}{\hbar}\simeq-\frac{\Delta T}{\hbar}\langle
Z_{\pi}\rangle$ (21)

as shown in the bottom panels panel of Fig. 2. It is crucial to note here that
$\phi$ is $2\pi$ periodic; thus Floquet eigenstates with quasienergy outside
the range $-\pi\leq\phi\leq\pi$ have to be folded back, using this
periodicity, to the first Floquet-Brillouin zone. The primary clustering is
therefore a consequence of near-integrability of $H_{F}$ for small
$w_{r}/\Delta$.

The eigenstates within the central primary cluster seen in the top panel of
Fig. 2 has $\langle Z_{\pi}\rangle=\pm 3n$ while the other clusters have
$\langle Z_{\pi}\rangle=\pm 3n+1$ and $\pm 3n+2$ where $n$ is an integer. The
number of such primary clusters can be deduced from the fact that for the
parameter regime of these plots $\Delta/(\hbar\omega_{D})\sim 1/3$.
Importantly, $\Delta$ is still incommensurate to the drive frequency by a
small amount (the exact commensuration occurs at $\Delta=\omega_{D}/3\sim
1.2w$) so that states within the same cluster are non-degenerate. This
incommensuration can be parameterized by a dimensionless ratio given by

$\displaystyle x=\frac{\hbar\omega_{D}}{\Delta}-3.$ (22)

For $\Delta=1.2$, $x\simeq-0.0834$ for $\omega_{D}=3.5w/\hbar$ and $0.04167$
for $\omega_{D}=3.65w/\hbar$. The primary clusters, for $w_{r}\ll\Delta$, are
separated with quasienergies $\phi_{1}$ while the states within each cluster
are separated by smaller quasienergy scale $\phi_{2}$, where

$\displaystyle\phi_{1}$ $\displaystyle=$ $\displaystyle\delta E_{1}T/\hbar\sim
2\pi/(3+x)$ $\displaystyle\phi_{2}$ $\displaystyle=$ $\displaystyle\delta
E_{2}T/\hbar\sim|x|2\pi/(3+x).$ (23)

In the parameter regime of our numerics, where $w_{r}\ll\delta E_{1}$, the
primary clustering is not destroyed by $w_{r}$ for finite sized chains with
$L\leq 24$.

Figure 2: Top left panel: Plot of $S_{L/2}$ as a function of Floquet
quasienergies $\phi=E_{F}T/\hbar$ for $\omega_{D}=3.5w/\hbar$. Top right
panel: A similar plot for $\omega_{D}=3.65w/\hbar$. Bottom left panel: Plot of
$\langle Z_{\pi}\rangle$ for Floquet eigenstates as a function of $\phi$ for
$\omega_{D}=3.5w/\hbar$. Bottom right panel: Similar plot as the bottom left
panel but for $\omega_{D}=3.65w/\hbar$. For all plots $L=24$, $\Delta=1.2w$,
and $\lambda=15w$. All energies are scaled in units of $w$ and frequencies in
units of $w/\hbar$. The color scheme is same as in Fig. 1 and the red dotted
lines in the top panel indicate ETH predicted Page value of $S_{L/2}$. See
text for details.

Figure 3: Left panel: Plot of $S_{L/2}$ as a function of Floquet quasienergies
$\phi=E_{F}T/\hbar$ within the central primary cluster for
$\omega_{D}=3.5w/\hbar$. Top right panel: A similar plot for
$\omega_{D}=3.65w/\hbar$. For all plots $L=24$, $\Delta=1.2w$, and
$\lambda=15w$. All energies are scaled in units of $w$ and frequencies in
units of $w/\hbar$. The color scheme is same as in Fig. 1 and the red dotted
lines in the top panel indicate ETH predicted Page value of $S_{L/2}$. See
text for details.  Figure 4: Plot of $\langle Y\rangle$ for the Floquet
eigenstates as a function of $\phi=E_{F}T/\hbar$ for $\hbar\omega_{D}=3.65w$
(top panel), $3.58w$ (middle panel) and $3.5w$ (bottom panel). For all plots
$L=24$, $\lambda/w=15$, and $\Delta/w=1.2$. All energies are scaled in units
of $w$ and frequencies in units of $w/\hbar$. The color schme is same as in
Fig. 1 and the black dashed lines are guide to the eye. See text for details.

A closer inspection of the primary clusters reveal that each of them, with
definite values of $Z_{\pi}$, exhibit further clustering. The left[right]
panel of Fig. 3 shows such clustering for $\omega_{D}=3.5[3.65]w/\hbar$. We
dub this phenomenon as secondary clustering. The origin of such secondary
clustering can be understood via construction of an effective Floquet
Hamiltonian in the regime $w_{r}\sim\delta E_{2}\ll\delta E_{1}$. The analysis
is same as what has been carried out in Ref. ethv05b,  and have been detailed
in the appendix. Such a Hamiltonian, given by Eqs. 45 and 46, shows that
$w_{r}$ can have significant non-zero matrix elements only between states
within each primary cluster. Moreover, as shown in Ref. ethv05b, , the
presence of a finite $w_{r}$ does not lead to ergodic structure within each
primary cluster. Instead, a perturbative analysis, similar to that carried out
in Ref. ethv05b,  and detailed in the appendix, indicates that the secondary
clustering occurs due to an emergent conserved quantity (Eq. 46)

$\displaystyle Y=-\sum_{j}(-1)^{j}P_{j-1}\sigma_{j}^{z}P_{j+1}$ (24)

within each $Z_{\pi}$ sector. These secondary clusters correspond to groups of
states with same values of $\langle Y\rangle$. This can be clearly seen from
Fig. 4 where $\langle Y\rangle$ is plotted for eigenstates in each of the
primary cluster. We find a definite value of $\langle Y\rangle$ for all
Floquet eigenstates both at $\hbar\omega_{D}=3.65w$ and $3.5w$ as can be seen
from the top and bottom panels of Fig. 4 respectively. We note that the
emergence of $Y$ relies on the perturbative parameter $w_{r}/\delta E_{2}\leq
1$. Such a perturbation theory will break down for at the exact commensuration
point where $x$, and hence $\delta E_{2}$, vanishes. This leads to loss of
secondary clustering. At these commensuration points, the Floquet eigenstates
can not be associated with a definite $\langle Y\rangle$ value as can be seen
from the middle panel of Fig. 4. We shall discuss this case in Sec. III.2.

###  III.2 Ergodicity restoration

In this section, we discuss the behavior of the driven chain when the
amplitude of the staggered detuning is commensurate with the drive frequency.
To this end, we note that the Floquet Hamiltonian derived using FPT in Sec. II
can be written as

$\displaystyle H_{F}$ $\displaystyle=$
$\displaystyle-\sum_{j}\Delta_{r}Z_{\pi}+H^{\prime}_{1}$ $\displaystyle
H^{\prime}_{1}$ $\displaystyle=$ $\displaystyle-
w_{r}\sum_{j}\tilde{\sigma}_{j}^{x}+H_{F3}^{(2)}+H_{F}^{(3)}+..$
$\displaystyle\Delta_{r}$ $\displaystyle=$
$\displaystyle\Delta\left[1-\frac{w^{2}T^{2}}{8\gamma^{3}}\left(\frac{1}{2}\sin
2\gamma-\gamma\cos 2\gamma\right)+...\right]$ (25)

where the ellipsis indicate higher order contribution to the Floquet
Hamiltonian and $\Delta_{r}$ denote the renormalized amplitude of the
staggered term. We note that for $\hbar\omega_{D}/\Delta_{r}=n$ where $n\in
Z$, one can write

$\displaystyle U(T,0)$ $\displaystyle=$ $\displaystyle
e^{-iH_{F}T/\hbar}=e^{-i(H^{\prime}_{1}T+(2\pi/n)Z_{\pi})/\hbar}$ (26)

Thus the Floquet eigenstates can be organized into clusters of states with
values of $\langle Z_{\pi}\rangle=\pm 3n$, $\pm 3n+1$ and $\pm 3n+2$. This is
seen in the top left panel of Fig. 5. We also note that each of the states in
a given primary cluster shall have identical quasienergy at $w_{r}=0$ since
$x$, and hence $\delta E_{2}$, vanishes. Thus in absence of $w_{r}$, they form
a flat band and the Hilbert space comprises of three such flat bands with
$\delta E_{1}=0$ and $\pm\hbar\omega_{D}/3$ in this limit. The number of such
clusters depends on $n$; this is exhibited in Fig. 6 where the structure of
the states with $n=2$ and $n=4$ primary clusters are shown.

Figure 5: Left panel: Plot of $S_{L/2}$ as a function of $\phi=E_{F}T/\hbar$
for $\omega_{D}=3.58w/\hbar$. Central panel: A similar plot showing states
within the primary cluster around $E_{F}=0$. Right Panel: Plot of $\langle
Z_{\pi}\rangle$ as a function of $\phi$. For all plots $\Delta/w=1.2$, and
$\lambda/w=15$. All energies are scaled in units of $w$ and frequencies in
units of $w/\hbar$. The color scheme is same as in Fig. 1 and the red dotted
lines in the left and central panels indicate ETH predicted Page value of
$S_{L/2}$. See text for details.  Figure 6: Top Panel: Plot of $S_{L/2}$ as a
function of Floquet quasienergies $\phi=E_{F}T/\hbar$ for $\Delta/w=1.8$
showing two primary clusters. Bottom panel: A similar plot for $\Delta/w=0.9$
showing four primary clusters. For both plots $\hbar\omega_{D}/w=3.6w$, and
$\lambda/w=15$. All energies are scaled in units of $w$ and frequencies in
units of $w/\hbar$. The color scheme is same as in Fig. 1 and the red dotted
lines indicate ETH predicted Page value of $S_{L/2}$. The black dashed lines
are guide to the eye. See text for details.

In the presence of a non-zero $w_{r}$, within each cluster, the effective
Floquet Hamiltonian is given by $H^{\prime}_{1}$. For small $w_{r}$, the
primary clustering still exists for finite $L$, since $w_{r}\ll\Delta$.
However, within each primary cluster, $H_{F}^{\rm eff}\simeq H^{\prime}_{1}$
since $\delta E_{2}=0$. Thus the effective Floquet Hamiltonian for each
primary cluster resembles a renormalized PXP Hamiltonian along with higher
order terms. The states within each primary cluster are therefore expected to
have a ergodic structure with no secondary clustering. Moreover, as seen from
the bottom panel of Fig. 5, they possess a continuous range of $\langle
Z_{\pi}\rangle$ values for any finite $w_{r}$. This property stems from strong
hybridization between states with definite integer values of $\langle
Z_{\pi}\rangle$ in the presence of non-zero $w_{r}$. We note that the
perturbative analysis of derived in Ref. ethv05b,  leading to the emergent
invariant $Y$ responsible for the secondary clustering becomes invalid here
since the perturbative parameter $w_{r}/\delta E_{2}\to\infty$ at the
commensuration point. This expectation is validated from exact numerics as
seen from the top right panel of Fig. 5 where the ergodic structure for states
within the central primary cluster is shown. We note that each of the clusters
contain both anomalous states with low $S_{L/2}$ along with thermal states
with high $S_{L/2}$. A similar behavior is expected for the $s=\pm 1$ primary
clusters.

Figure 7: Plot of the steady state value of $O_{22}\equiv O_{22}^{\rm DE}$ as
a function $\omega_{D}$ for $\Delta/w=1.2$ showing restoration of ergodicity
at special frequencies. Such restoration is characterized by sharp dips in the
$O_{22}^{\rm DE}$ value. The inset shows the dynamics of $O_{22}$ starting
from a $|{\mathbb{Z}}_{2}\rangle$ state as a function of $n$ for commensurate
($\hbar\omega_{D}/w=3.58$) and slightly incommensurate
($\hbar\omega_{D}/w=3.65$ and $\hbar\omega_{D}/w=3.5$) drive frequencies. The
coherent oscillations of $O_{22}$ (shown above for $\hbar\omega_{d}/w=3.5$ and
$3.65$) seen at incommensurate frequencies are replaced by a slower approach
to the steady state at exactly commensurate drive frequency
($\hbar\omega_{D}/w=3.58$). All energies (frequencies) are scaled in units of
$w$($w/\hbar$). See text for details.

The consequence of such ergodicity restoration within a primary cluster is
also reflected in the steady state value of $O_{22}$, $O_{22}^{\rm DE}$, as
shown in Fig. 7. The steady state value of $O_{22}$, in terms of the Floquet
spectrum, is given by

$\displaystyle O_{22}^{\rm DE}$ $\displaystyle=$
$\displaystyle\sum_{n}|c_{n}^{\rm init}|^{2}\langle
n|\hat{n}_{2}\hat{n}_{4}|n\rangle$ (27)

where $c_{n}^{\rm init}=\langle n|\psi_{0}\rangle$ and
$|\psi_{0}\rangle\equiv|{\bar{\mathbb{Z}}_{2}}\rangle,\,|{\mathbb{Z}}_{2}\rangle,\,{\rm
or}\,|0\rangle$ is the overlap of the Floquet eigenstates $|n\rangle$ with the
initial state. We find that $O_{22}^{\rm DE}$ exhibits a sharp dip at the
commensuration point when the dynamics starts from $|{\mathbb{Z}}_{2}\rangle$;
in contrast it exhibits a smaller peak for $|{\bar{\mathbb{Z}}_{2}}\rangle$
initial state. In both cases, its value approaches close to that predicted by
ETH. This is clearly a consequence of restoration of ergodicity within a
primary cluster. As one varies the frequency away from the commensuration
point, $O_{22}^{\rm DE}$ deviates sharply from its ETH predicted value
approximately around $w_{r}\simeq\delta E_{2}$. The sharpness of such
deviation is a consequence of small $w_{r}\simeq 0.045$ at $\lambda=15w$ and
$\hbar\omega_{D}=3.58w$. This can be controlled by varying $\lambda$ (for a
fixed $\omega_{D}$) as seen in Fig. 8 displaying a significantly wider ergodic
regime as a function of the drive frequency at $\lambda=5w$ and $\lambda=10w$.

Figure 8: Plot of the steady state value of $O_{22}\equiv O_{22}^{\rm DE}$ as
a function $\omega_{D}$ for $\Delta/w=1.2$ for several $\lambda$ showing
increase in width of the ergodic region as a function of $\lambda$ around
$\hbar\omega_{D}/w=3.58$. All energies (frequencies) are scaled in units of
$w(w/\hbar)$. See text for details.

The stroboscopic evolution of $O_{22}$ at these ergodicity restoring points do
not display coherent oscillations as can be seen from the inset of Fig. 7;
instead it displays a slow approach to the thermal steady state. We note that
this is in contrast to the dynamics of $O_{22}$ starting from
$|{\mathbb{Z}}_{2}\rangle$ for a renormalized PXP Floquet Hamiltonian. This
difference can be understood as follows. Since $w_{r}/\Delta\ll 1$, the
dynamics of $O_{22}$ is dominated by terms which connects states within the
same primary cluster. The matrix elements between any two such states, which
differ by at least three flipped spins for $\hbar\omega_{D}/\Delta_{r}\simeq
3$, must involve at least ${\rm O}(w_{r}^{3})$ term; consequently, the
effective Hamiltonian for such dynamics is different from the renormalized
PXP. Thus the scar-induced oscillations are not seen. Such an effective ${\rm
O}(w_{r}^{3})$ term naturally leads to a thermalization time which is ${\rm
O}(w_{r}^{-3})$ and hence large.

##  IV Dynamical freezing

In this section, we present our results on the density-density correlation
function $O_{j2}=\langle\hat{n}_{j}\hat{n}_{j+2}\rangle$ with focus on
dynamical freezing of these correlations at specific drive frequencies and
their oscillations near these freezing frequencies. For all numerical results
in this section, we shall choose $j=2$. The discussion on the freezing
phenomenon is given in Sec. IV.1 while the oscillation of $O_{22}$ around the
freezing point is analyzed in Sec. IV.2.

###  IV.1 Freezing at special drive frequencies

Figure 9: Plot of the steady state value of $O_{22}\equiv O_{22}^{\rm DE}$ as
a function of $\omega_{D}$ starting from the $|0\rangle$ (top panel),
$|{\mathbb{Z}}_{2}\rangle$ (middle panel) and $|{\bar{\mathbb{Z}}_{2}}\rangle$
(bottom panel) states for several representative values of $\Delta/w$ as
shown. The black dashed lines indicate the ETH predicted value of $O_{22}^{\rm
DE}$. For all plots $\lambda/w=15$. All energies are scaled in units of $w$
and frequencies in units of $w/\hbar$. See text for details.

We begin with a study of the behavior of $O_{22}^{\rm DE}$ as a function of
$\Delta/w$ as shown in Fig. 9. The plot shows its behavior for the initial
Neel ( $|{\mathbb{Z}}_{2}\rangle$ and $|{\bar{\mathbb{Z}}_{2}}\rangle$ ) and
vacuum ($|0\rangle$) states. It indicates strong deviation from the ETH
predicted value $O_{22}^{\rm ETH}\simeq 0.11$ at intermediate drive
frequencies. For the $|0\rangle$ initial state, $O_{22}^{\rm DE}$ reaches its
ETH predicted value at high drive frequencies where $w_{r}\sim w\simeq\Delta$
as expected; in contrast, for the $|{\mathbb{Z}}_{2}\rangle$ initial state,
$O_{22}^{\rm DE}$ reaches a super-thermal value. The latter behavior can be
attributed to the presence of scar states in the Floquet spectra with large
overlap with the $|{\mathbb{Z}}_{2}\rangle$ states ethv1 ; ethv2 ; we shall
discuss these states in Sec. V. For either of these initial states,
$O_{22}^{\rm DE}$, in the high-frequency limit, is almost independent of
$\Delta$ as long as $w\geq\Delta$. In contrast, it depends strongly on
$\Delta$ for $\hbar\omega_{D}\gg w$ and can take either superthermal or
subthermal values for $|{\bar{\mathbb{Z}}_{2}}\rangle$ initial state; this
constitutes a stronger violation of ETH which can not be explained by presence
of scars alone.

Figure 10: Top left panel: Plot of $O_{22}^{(m)}=\langle
m|\hat{n}_{2}\hat{n}_{4}|m\rangle\equiv O_{22}$ corresponding to Floquet
eigenstates $|m\rangle$ as a function of their quasienergy
$\phi=E_{F}^{(m)}T/\hbar$ for $\Delta/w=0.1$. Bottom left panel: Overlap of
these eigenstates with $|{\mathbb{Z}}_{2}\rangle$ (red dots),
$|{\bar{\mathbb{Z}}_{2}}\rangle$ (blue dots), and $|0\rangle$ (black dots).
Right panels: Same as the corresponding left panels but for $\Delta/w=1.2$.
All left panel plots have same Y axes range as their right panel counterparts.
All plots correspond to $L=14$, $\lambda/w=15$ and $\hbar\omega_{D}/w=6.9$.
All energies are scaled in units of $w$ and frequencies in units of $w/\hbar$.
See text for details.

To see why this is the case, we note that $\Delta/w\ll 1$, the value of
$O_{22}^{\rm DE}$ is expected to be identical irrespective of whether the
starting state is $|{\mathbb{Z}}_{2}\rangle$ or
$|{\bar{\mathbb{Z}}_{2}}\rangle$. This situation is highlighted in the left
panels of Fig. 10. The top left panel shows the expectation value
$O_{22}^{(m)}=\langle m|\hat{n}_{2}\hat{n}_{4}|m\rangle$ corresponding to the
Floquet eigenstate $|m\rangle$ as a function of their dimensionless
quasienergy $\phi=E_{F}^{(m)}T/\hbar$. The bottom panel shows the overlap of
the eigenstates $|m\rangle$ with the initial states. For small $\Delta/w=0.2$,
we find that the largest value of $O_{22}^{(m)}$ occurs for eigenstates
$|m\rangle$ which lie close to the middle of the Floquet spectrum
($E_{F}\simeq 0$). Furthermore, near the middle of the Floquet spectrum, some
of these states have significant overlap with $|{\mathbb{Z}}_{2}\rangle$ while
others have large overlap with $|{\bar{\mathbb{Z}}_{2}}\rangle$ as can be seen
the left bottom panel of Fig. 10. From Eq. 27, we find that this ensures that
$O_{22}^{\rm DE}$ would be almost identical irrespective of whether the
dynamics starts from $|{\mathbb{Z}}_{2}\rangle$ or
$|{\bar{\mathbb{Z}}_{2}}\rangle$ initial states; in fact they have exactly
identical values at $\Delta=0$.

This situation drastically changes at large $\Delta$, where all the Floquet
eigenstates with large values of $O_{22}^{(m)}$ have quasienergy $E_{F}>0$ as
shown in the top right panel of Fig. 10. Moreover, these eigenstates only have
large overlap with $|{\mathbb{Z}}_{2}\rangle$. Consequently, the value of
$O_{22}^{\rm DE}$ do not change appreciably for dynamics starting from
$|{\mathbb{Z}}_{2}\rangle$ as $\Delta$ is increased; however, its value
reduces drastically if the initial state is $|{\bar{\mathbb{Z}}_{2}}\rangle$.
Thus with increasing $\Delta$, $O_{22}^{\rm DE}$ drops rapidly to subthermal
values and approaches zero for the $|{\bar{\mathbb{Z}}_{2}}\rangle$ initial
state; this leads to qualitatively different $\Delta$-dependence of
$O_{22}^{\rm DE}$.

In addition, the top and the middle panel plots of Fig. 9 indicate the
presence of dynamical freezing at special values the drive frequencies
$\omega_{D}\simeq\lambda/(2n_{0}\hbar)$ (where $n_{0}\in Z$) for which
$w_{r}\simeq 0$ . At these values, the initial states $|0\rangle$,
$|{\mathbb{Z}}_{2}\rangle$ and $|{\bar{\mathbb{Z}}_{2}}\rangle$ becomes
approximate eigenstates of the Floquet Hamiltonian leading to freezing. The
neighborhood of these freezing frequencies is marked by strong deviation from
the ETH predicted value of $O_{22}$ as shown in the top and middle panels of
Fig. 9; this deviation increases with increasing $\Delta$. In contrast, for
$|{\bar{\mathbb{Z}}_{2}}\rangle$ initial state and for $\Delta>w$, the
freezing extends over a range of frequencies, as seen in the bottom panel of
Fig. 9.

###  IV.2 Oscillations around freezing frequencies

Figure 11: Top panel: Plot of $O_{22}$ as a function of the number of drive
cycles $n$ starting from the $|0\rangle$ (red line),
$|{\mathbb{Z}}_{2}\rangle$ (black line). The bottom panel shows the plot of
fidelity ${\mathcal{F}}$ as a function of $n$ for $|{\mathbb{Z}}_{2}\rangle$
initial state. For all plots $L=14$, $\lambda/w=15$ and
$\hbar\omega_{D}/w=6.9$. All energies(frequencies) are scaled in units of $w$
($w/\hbar$). See text for details.

The stroboscopic dynamics of $O_{22}$ as a function of $n$ around the freezing
frequencies is shown in the top panel Fig. 11. We find that $O_{22}$ does not
thermalize; instead, it displays long time oscillation with a fixed frequency
$\omega_{\rm osc}\simeq\Delta/\hbar$. We have checked this numerically for
$n>1000$ cycles. The amplitude of these oscillation decrease as the freezing
point is approached and also as $\Delta$ is increased; it is a monotonically
decreasing function of $w_{r}/\Delta$. However, the period of the oscillations
remains constant for $\Delta/w\geq 1$, as shown in Fig. 12. A similar feature
is seen from the fidelity
${\mathcal{F}}=|\langle{\mathbb{Z}}_{2}|\psi(nT)\rangle|^{2}$, where
$|\psi(nT)\rangle$ denotes the many-body state at time $t=nT$, as shown in the
bottom panel of Fig. 11. In contrast, the oscillation amplitude is close to
zero for a wide range of frequencies around the freezing point when one starts
from the $|{\bar{\mathbb{Z}}}_{2}\rangle$ state.

Figure 12: Top panel: Plot of the frequency (top panel) and the amplitude
(bottom panel) of coherent oscillation of $O_{22}$ as a function of $\Delta$
for $\hbar\omega_{D}/w=6.9$ and $\lambda/w=15$. All energies(frequencies) are
scaled in units of $w$($w/\hbar$). See text for details.

The characteristics of these oscillations at large $\Delta/w_{r}$ can be
qualitatively understood as follows. Let us consider the evolution operator
when $\Delta_{r}\gg w_{r}$. Clearly, in this limit, one can write

$\displaystyle U(nT,0)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}e^{-in\epsilon_{\alpha}^{F}T/\hbar}|\alpha\rangle\langle\alpha|$
$\displaystyle\epsilon_{\alpha}$ $\displaystyle=$ $\displaystyle
E_{0}+\alpha\Delta+{\rm O}(w_{r}^{2}/\Delta^{2})$
$\displaystyle|\alpha\rangle$ $\displaystyle=$
$\displaystyle\sum_{p}c_{p;q}^{\alpha}|p;q\rangle$ (28)

where $E_{0}$ denotes the energy of the state $|0\rangle$ for $w_{r}=0$ and
any eigenstate $|\alpha=\alpha_{0}\rangle$ of the Floquet Hamiltonian $H_{F}$
is mapped to a state in the number basis $|p;q\rangle=|p_{0},q_{0}\rangle$ for
$w_{r}=0$. We note here that the states $|p_{0};q_{0}\rangle$ have $p_{0}$
($q_{0}$) up-spins distributed on even (odd) sites of the chain without
violation of the constraint; these are eigenstates of $H_{F}^{(1)}$ for
$w_{r}=0$ with quasienergy

$\displaystyle E=E_{0}+(q_{0}-p_{0})\Delta.$ (29)

Here we shall assume that $c_{p_{0}+\mu;q_{0}}^{\alpha_{0}}\sim{\rm
O}((w_{r}/\Delta)^{|\mu|})$ for $\mu=\pm 1,\pm 2...$; thus for small
$w_{r}/\Delta$, $|c_{p_{0}+\mu;q_{0}}^{\alpha_{0}}|\ll
c_{p_{0};q_{0}}^{\alpha_{0}}$ for any $\alpha_{0}$. A similar relation holds
for $c_{p_{0};q_{0}+\mu}^{\alpha_{0}}$.

Let us now consider the initial state $|0\rangle$. The wavefunction after $n$
drive cycles can be obtained as

$\displaystyle|\psi(nT)\rangle$ $\displaystyle=$ $\displaystyle
U(nT,0)|0\rangle=\sum_{\alpha}c_{0;0}^{\alpha\ast}e^{-i\epsilon_{\alpha}nT/\hbar}|\alpha\rangle$
(30) $\displaystyle=$
$\displaystyle\sum_{\alpha}\sum_{p,q}c_{0;0}^{\alpha\ast}c_{p;q}^{\alpha}e^{-i\epsilon_{\alpha}pT/\hbar}|p;q\rangle$

where $c_{0;0}^{\alpha}=\langle p_{0}=0;q_{0}=0|\alpha\rangle$. Since the
correlator $O_{22}$ can receive finite contribution for states with $p_{0}>2$
up-spins, we find that the leading contribution to $O_{22}$ is given by

$\displaystyle
O_{22}^{(0)}(nT)\simeq|c_{0;0}^{0\ast}c_{2;0}^{2}+c_{0;0}^{1\ast}c_{2;0}^{1}e^{i\Delta
nT/\hbar}|^{2}$ (31)

where we have retained only the leading order term in $w_{r}/\Delta$ and used
$\epsilon_{\alpha_{0}}\simeq E_{0}-(p_{0}-q_{0})\Delta$ (Eq. 28). This
indicates that the oscillation amplitudes will decay with increasing $\Delta$
as $(w_{r}/\Delta)^{4}$; however the frequency of these oscillation will be
pinned to $\Delta$ as long as $w_{r}\ll\Delta$. The change in the oscillation
frequency is expected to occur with reduction of $\Delta$ when ${\rm
O}((w_{r}/\Delta)^{2})$ terms in the expression of $\epsilon_{\alpha}$ becomes
significant. The validation of this argument can be found in Fig. 12. We note
that for $\Delta_{r}\gg w_{r}$, the presence of a large on-site term which is
diagonal in the Fock basis leads to long-time coherent oscillations; in this
case, we do not find the rapid thermalization expected by ETH.

The coherent oscillations discussed above are expected to have larger
amplitude provided we start from the state $|{\mathbb{Z}}_{2}\rangle$. To see
this, we note that in this case, the wavefunction after $n$ drive cycles is
given by

$\displaystyle|\psi^{\prime}(nT)\rangle$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\sum_{p,q}c_{L/2;0}^{\alpha\ast}c_{p,q}^{\alpha}e^{-i\epsilon_{\alpha}nT/\hbar}|p;q\rangle$
(32)

where $c_{L/2;0}^{\alpha}=\langle{\mathbb{Z}}_{2}|\alpha\rangle$. In this
case, the leading order contribution to $O_{22}$ reads

$\displaystyle
O_{22}^{({\mathbb{Z}}_{2})}(nT)\simeq||c_{L/2,0}^{L/2\ast}|^{2}+c_{L/2,0}^{L/2-1\ast}c_{L/2-1,1}^{L/2-1}e^{i\Delta
nT/\hbar}|^{2}$ (33)

which leads to ${\rm O}(w_{r}^{2}/\Delta^{2})$ oscillation amplitude.

In contrast, if one starts from the $|{\bar{\mathbb{Z}}_{2}}\rangle$ states,
the oscillation amplitude is expected to be vanishingly small. This is due to
the fact that $|{\bar{\mathbb{Z}}_{2}}\rangle$ has insignificant overlap with
any $|p_{0};q_{0}\rangle$ state which contributes to
$\langle\hat{n}_{2}\hat{n}_{4}\rangle$. We also note that density-density
correlation between odd sites (such as $O_{12}$) would show similar behavior
for the initial state $|0\rangle$; however, it would have vanishing
oscillation amplitude for the $|{\mathbb{Z}}_{2}\rangle$ initial state instead
of $|{\bar{\mathbb{Z}}_{2}}\rangle$.

##  V Crossover from weak to strong staggered detuning

In this section, we shall study the behavior of the quasienergy eigenstates as
$\delta E_{2}/w_{r}$ is gradually increased. We note at the outset that this
behavior can be reproduced by studying the properties of eigenstates of

$\displaystyle H_{m}$ $\displaystyle=$
$\displaystyle-(\Delta^{\prime}/2)\sum_{j}(-1)^{j}\sigma_{j}^{z}-\sum_{j}\tilde{\sigma}_{j}^{x}$
(34)

such that $\Delta^{\prime}\equiv 2\delta E_{2}/w_{r}$. This is due to the fact
that a variation of $\omega_{D}$ in $H_{F}^{(1)}$ can take us close to both
the commensuration point where $\delta E_{2}\simeq 0$ (for example, for
$\Delta\simeq 1.2$ and $\hbar\omega_{d}\simeq 3.6$) and the dynamical freezing
point where $w_{r}\simeq 0$ (for example, around $\lambda T/\hbar=4\pi$). We
therefore expect this analysis to reveal the nature of the crossover between
Hilbert space structure having quantum scars ($\Delta^{\prime}=0$) to that
having clustering ($\Delta^{\prime}\gg 1$). Our numerical results are
discussed in Sec. V.1 while a FSA based calculation for scars at small and
intermediate value of $\Delta$ is presented in Sec. V.2.

###  V.1 Numerical results for scars

Figure 13: Plot of the half chain entanglement $S_{L/2}$ of the eigenstates of
$H_{m}$ as a function of energy $E$ for (a) $\Delta^{\prime}=0.005$, (b)
$0.05$, (c) $0.5$, and (d) $2$. The green and the red circles indicate states
with high overlap with $|{\bar{\mathbb{Z}}_{2}}\rangle$ and
$|{\mathbb{Z}}_{2}\rangle$ respectively. All plots have same Y axes range and
correspond to $L=24$. See text for details.

In this section, we first show the entanglement entropy of the eigenstates of
$H_{m}$ as a function of dimensionless energy $E$ for four representative
values of $\Delta^{\prime}$ in Fig. 13. The top left panel of Fig. 13 shows
the existence of quantum scars for small $\Delta^{\prime}\ll 1$. These scars
which occurs close to the commensuration point where $x\ll 1$, have, in
accordance with standard expectation, almost equal overlap with
$|{\mathbb{Z}}_{2}\rangle$ (red circles) and $|{\bar{\mathbb{Z}}_{2}}\rangle$
(green circles) states. We find as we increases $\Delta^{\prime}$, the scars
with $E_{F}<0$ develops higher overlap with $|{\mathbb{Z}}_{2}\rangle$ while
those with $E_{F}>0$ exhibits higher overlap with
$|\bar{\mathbb{Z}}_{2}\rangle$. This asymmetry can be understood from the fact
that the presence of a large $\Delta^{\prime}$ pushes the states with high
$|{\mathbb{Z}}_{2}\rangle$ ($|{\bar{\mathbb{Z}}_{2}}\rangle$) overlap to lower
(higher) quasienergy. For $\Delta^{\prime}\gg 1$, there are almost no mid-
spectrum states which has high overlap with either $|{\mathbb{Z}}_{2}\rangle$
or $|{\bar{\mathbb{Z}}_{2}}\rangle$. Remarkably, even in this regime, we find
existence of scar like states as seen from bottom right panel of Fig. 13;
these do not show high overlap with either $|{\bar{\mathbb{Z}}_{2}}\rangle$ or
$|{\mathbb{Z}}_{2}\rangle$. The crossover between these two situation happens
between $0.75\leq\Delta^{\prime}\leq 1$ as can be seen from Fig. 14. Thus at
these values of $\Delta^{\prime}$, an almost continuous band of scar states
remain although their nature evolves with increasing $\Delta^{\prime}$.

Figure 14: Plot of the half chain entanglement $S_{L/2}$ of the eigenstates of
$H_{m}$ as a function energy $E$ for (a) $\Delta^{\prime}=0.75$, (b) $1$, (c)
$1.25$, and (d) $1.75$. The red circles indicate states with high overlap with
$|{\mathbb{Z}}_{2}\rangle$. All plots have same Y axes range and correspond to
$L=24$. See text for details.

This structure of scars is preserved at higher values of $\Delta^{\prime}$
where the Hilbert space is strongly clustered. The mid-spectrum states here do
not have significant overlap with either $|{\mathbb{Z}}_{2}\rangle$ and
$|{\bar{\mathbb{Z}}_{2}}\rangle$. Instead, as we find numerically and as shown
in the left panel of Fig. 15, two of these states have high overlap with
$|{\mathbb{Z}}_{4}\rangle=|\uparrow,\downarrow,\downarrow,\downarrow,\uparrow,\downarrow,\downarrow,\downarrow...\rangle$
states. Other mid spectrum scar states have strong overlap either with the
single Rydberg excited state
$|1\rangle=|\uparrow,\downarrow,\downarrow....\rangle$ or the vacuum state
$|0\rangle$ as shown in the left panel of Fig. 15.

Figure 15: Left: Plot of $S_{L/2}$ for eigenstates of $H_{m}$ showing the scar
states having large overlap with $|0\rangle$ (black square), $|1\rangle$ (blue
square), and $|{\mathbb{Z}}_{4}\rangle$ (green square). The red dashed line
indicates the ETH predicted Page value of $S_{L/2}$. Right Panel: The dynamics
of $\langle N_{d}\rangle$ starting from state $|1\rangle$ (top) and the
fidelity $F(t)=|\langle\psi(t)|1\rangle|^{2}$ (bottom) as a function of time
showing scar induced long-time coherent oscillations. For all plots $L=24$ and
$\Delta^{\prime}=5$. See text for details. Figure 16: Left: The dynamics of
$\langle N_{d}\rangle$ (top) starting from the initial state $|0\rangle$ and
the fidelity $F(t)=|\langle\psi(t)|0\rangle|^{2}$ (bottom) as a function of
time. Right: Similar plots for the initial $|{\mathbb{Z}}_{4}\rangle$ state.
All left panel plots have same Y axes range as their right panel counterparts.
All plots correspond to $L=16$ and $\Delta^{\prime}=5$. See text for details.

These mid-spectrum scar states have no analogue in their counterparts for
small $\Delta^{\prime}$. Indeed this can be seen by studying the dynamics of
$\langle N_{d}\rangle=\langle\sum_{j}(\sigma_{j}^{z}+1)/2\rangle$ as a
function of time. The fidelity $F_{1}=|\langle 1|\psi(t)\rangle|^{2}$ of these
oscillations indicate perfect revivals as can be seen from the bottom right
panel of Fig. 15. The presence of such oscillations starting from single spin-
up state could not have been due to the $|{\mathbb{Z}}_{2}\rangle$ scars.
Similar oscillatory coherent dynamics with near perfect fidelity revivals is
also seen in this regime for initial states $|0\rangle$ and
$|{\mathbb{Z}}_{4}\rangle$ as shown in Fig. 16.

Figure 17: Plot of $\ln|\langle{\mathbb{Z}}_{4}|n\rangle|^{2}$ as a function
of $E_{F}$ for $|{\mathbb{Z}}_{4}\rangle$ states which hosts Rydberg
excitations on even (top panel) and odd (bottom panel) sites. For all plots
$\Delta^{\prime}=5$ and $L=16$. See text for details.

The frequencies of these oscillations and the nature of the dynamics starting
from the $|{\mathbb{Z}}_{4}\rangle$ state can be understood as follows. From
the plot of the overlap of the initial state with the Floquet eigenstates,
shown in Fig. 17, we find that the overlap of $|{\mathbb{Z}}_{4}\rangle$ is
maximal with the one of the mid-spectrum scar states. This state has $E_{F}>0$
($E_{F}<0$) if the $|{\mathbb{Z}}_{4}\rangle$ hosts Rydberg excitations on
odd(even) sites; this is due to the presence of a large $\Delta^{\prime}$ in
the Hamiltonian which breaks the particle-hole symmetry. The next largest
overlap corresponds to two states on both sides of the
$|{\mathbb{Z}}_{4}\rangle$ states; these have slightly different energies and
belong to primary clusters characterized by $Z_{\pi}=5$ and $7$ respectively.
The difference in energy between the $Z_{4}$ scar and these two states leads
to the main oscillation frequency $\omega\simeq\Delta^{\prime}/(2\hbar)$; in
addition, the small difference in frequencies of these two states lead to a
long-time beating phenomenon as can be seen from the right panel Fig. 16. The
analysis of the dynamics starting from $|1\rangle$ or $|0\rangle$ reveals a
qualitatively similar picture and we do not discuss them in details here.

The nature of the scar states with large overlap with $|1\rangle$ states may
be qualitatively understood by noting the form of $H_{4}^{(1)}$ given in Eq.
46. For strong $\Delta^{\prime}$, where the analysis of Ref. ethv05b,  holds,
$H_{4}^{(1)}$ is the leading term in $1/\Delta^{\prime}$ which induces spin
dynamics. Here we note that such a term acting on a state $|1_{j}\rangle$
(which represents a single Rydberg excitation on the $j^{\rm th}$ site)
connects it to $|1_{j+2}\rangle$. Thus it creates an almost closed subspace of
$L/2$ such states; the corresponding Floquet eigenstates is therefore expected
to have low entanglement and could be a candidate for the scar state having
large overlap with $|1\rangle$.

A similar, but more complicated holds for scars having large overlap with
$|{\mathbb{Z}}_{4}\rangle$. We note that $H^{(1)}_{4}$ acting on
$|{\mathbb{Z}}_{4}\rangle$ connects it to a series of $L/4$ states where any
up-spin state is shifted by two sites. These states are denoted by
$|{\mathbb{Z}}_{4j}\rangle$ and has repeated block of one up-spin on site
$4\ell$ for integer $\ell$ and three subsequent down-spins. Now let us
consider the subsequent action of $H_{4}^{(1)}$ on this state; it is easy to
see that a further shift of the $j^{\rm th}$ up-spin is either forbidden by
the constraint or gives back $|{\mathbb{Z}}_{4}\rangle$. This ensures that
action of $H^{(1)}_{4}$ on $|{\mathbb{Z}}_{4j}\rangle$ leads to
${}^{L/4}C_{2}$ new states. Following this argument we find that the dimension
of the almost closed subspace formed by the action of $H_{4}^{(1)}$ on
$|{\mathbb{Z}}_{4}\rangle$ is $\sim\sum_{n=1..L/4}\,^{L/4}C_{n}$. For finite
$L\leq 24$ studied in our numerics, this leads to a small, almost closed,
subspace of states. Thus qualitatively one may expect to have a low-entangled
eigenstate formed out of their superposition which have significant overlap
with $|{\mathbb{Z}}_{4}\rangle$. We conjecture that such a state may be the
$Z_{4}$ scar that we find. We also note that this suggests that scars having
large overlap with $|1\rangle$ (for which the closed subspace always has ${\rm
O}(L)$ states) would be more stable for large $L$ than their counterparts
having large overlap with $|{\mathbb{Z}}_{4}\rangle$.

###  V.2 FSA calculations

To understand the nature of these scars at low and intermediate
$\Delta^{\prime}$ and the change in their properties as $\Delta^{\prime}$ is
increased, we use a semi-analytic FSA treatment. To this end, we first note
that the presence of the staggered detuning term suggests that such a
construction should distinguish between odd and even sites in the lattice.
Therefore we adapt the formalism of Ref. turpap1,  which allows for such a
sublattice resolved formulation. In this method, the states are labeled by a
set of two integers $n_{1}$ and $n_{2}$ which denotes number of Rydberg
excitations (up-spins) on even and odd sites respectively. In what follows we
shall construct the matrix elements of $H_{m}$ between two such states and
then diagonalize this matrix numerically. Such a class of states constitutes a
symmetric subspace defined from a set of equivalence classes $(n_{1},n_{2})$;
all elements of these classes are invariant under shuffling of excitations in
each sublattice turpap1 . Thus in this scheme, the action of translation by an
even number of lattice states leads to an equivalent state. The assumption
here is that the key information about properties of the states reside in the
number of excitation in each sublattice. We note that this procedure naturally
yield states which have significant overlap with $|{\mathbb{Z}}_{2}\rangle$
and $|{\bar{\mathbb{Z}}_{2}}\rangle$ since these Neel states are included as
elements in the subspace turpap1 .

Figure 18: A plot of $\log_{10}|\langle{\mathbb{Z}}_{2}|\psi\rangle|^{2}$ for
states obtained via FSA (large red circles) and those obtained from exact
numerics (small black circle) for $\Delta^{\prime}=0.5$ and $L=24$. See text
for details.

The first step towards construction of the matrix elements of $H_{m}$ is to
construct an orthonormal basis for the symmetric subspace. To this end, we use
the prescription of Ref. turpap1,  and write, for a chain with $2L$ sites

$\displaystyle|n_{1},n_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}}\sum_{a}|\alpha_{a}\rangle$ (35)
$\displaystyle{\mathcal{N}}$ $\displaystyle=$
$\displaystyle\frac{(L-n_{1}-n_{2})L}{(L-n_{1})(L-n_{2})}\left(\begin{array}[]{c}L-n_{1}\\\
n_{2}\end{array}\right)\left(\begin{array}[]{c}L-n_{2}\\\
n_{1}\end{array}\right)$ (40)

where any state $|\alpha_{a}\rangle$ contain $n_{1}$ and $n_{2}$ total Rydberg
excitations on even and odd sites respectively. The number of such states,
${\mathcal{N}}$, depends on specific values of $n_{1}$ and $n_{2}$ and also
the Hilbert space constraint of having no two Rydberg excitation on
neighboring sites.

The matrix element of $H_{m}$ between these states can be easily obtained
following the prescription of Ref. turpap1, . To this end, we write, using
$|0\rangle$ as the reference state,

$\displaystyle\langle n_{1},n_{2}|H_{m}|m_{1},m_{2}\rangle=F_{1}+F_{2}$
$\displaystyle
F_{1}=\Delta^{\prime}(n_{2}-n_{1})\delta_{n_{1},m_{1}}\delta_{n_{2},m_{2}}$
(41) $\displaystyle
F_{2}=-\Big{[}M(n_{1},n_{2})\delta_{m_{1},n_{1}+1}\delta_{m_{2},n_{2}}$
$\displaystyle\quad\quad+M(n_{2},n_{1})\delta_{m_{1}n_{1}}\delta_{m_{2},n_{2}+1}\Big{]}+{\rm
h.c.}$ $\displaystyle
M(n_{1},n_{2})=\sqrt{\frac{(L-n_{1}-n_{2})(L-n_{1}-n_{2}+1)n_{1}}{L-n_{1}}}$

We note that the structure of the FSA Hamiltonian suggests that a large
$\Delta^{\prime}$ essentially separates out states with large $n_{1}$ from
those with large $n_{2}$. Thus the eigenvalues of these Hamiltonian, for large
$\Delta^{\prime}$, is expected to have large overlap with either
$|{\mathbb{Z}}_{2}\rangle$ or $|{\bar{\mathbb{Z}}_{2}}\rangle$ but not both.

Figure 19: A plot of $\log_{10}|\langle{\mathbb{Z}}_{2}|\psi\rangle|^{2}$
(blue circles) and $\log_{10}|\langle{\bar{\mathbb{Z}}_{2}}|\psi\rangle|^{2}$
(red circles) for states obtained via FSA for (a) $\Delta^{\prime}=0.005$, (b)
$0.05$, (c) $0.5$, and (d) $2$. All plots have same Y axes range and
correspond to $L=24$. See text for details.

The diagonalization of the FSA matrix constructed from Eq. 41 leads to the
eigenvalues of $H_{m}$ for a fixed $\Delta^{\prime}$. A comparison of the
$|{\mathbb{Z}}_{2}\rangle$ overlap of these states with those of their
counterparts obtained via exact numerics for $L=24$ and $\Delta^{\prime}=0.5$,
plotted in Fig. 18, shows a near-exact match between the scar eigenstates
obtained by these two methods; the eigenstates with $E_{F}<0$ has a much
higher overlap with $|{\mathbb{Z}}_{2}\rangle$ compared to those with
$E_{F}>0$ in both cases. This allows us to surmise that the present FSA
captures the essence of the scar eigenstates in the presence of finite
$\Delta$.

The overlap between these eigenstates with $|{\mathbb{Z}}_{2}\rangle$ (red
dots) and $|{\bar{\mathbb{Z}}_{2}}\rangle$ (blue dots) is shown in Fig. 19.
From these, we find that the FSA eigenstates qualitatively reproduce the
characteristics of scars seen in exact numerics. For low $\Delta^{\prime}$,
the scar states have near exact overlap with both the Neel states (top left
panel of Fig. 19); in contrast, as $\Delta^{\prime}$ is increased, states with
$E_{F}>0$ develop a large overlap with $|{\bar{\mathbb{Z}}_{2}}\rangle$ while
those with $E_{F}<0$ overlaps strongly with $|{\mathbb{Z}}_{2}\rangle$. Thus
the crossover of the nature of the scars as a function of $\Delta^{\prime}$ is
well captured by the FSA. This behavior can be understood to be a consequence
of the presence of the diagonal elements of the FSA matrix for finite
$\Delta^{\prime}$ (Eq. 41); their presence pushes states having large overlap
with $|{\mathbb{Z}}_{2}\rangle$ ($|{\bar{\mathbb{Z}}_{2}}\rangle$) to opposite
ends of the Floquet spectrum. The mid-spectrum scar states at large
$\Delta^{\prime}$, however, can not be captured by the present FSA formalism.
A more detailed quantitative understanding of such states and their
description in terms of a modified FSA approach is left for future studies.

##  VI Discussion

In this work, we have studied a periodically driven Rydberg chain with both
uniform and staggered detuning. We have primarily focussed on the limit of
large drive amplitude and a square pulse protocol; the uniform detuning term
has been made time dependent. We find that the presence of a staggered
detuning term leads to several features of the driven system that have no
analogue in Floquet dynamics of Rydberg atoms without such a term.

The first of such features involves ergodicity violation of such system via
clustering of Floquet eigenstates. We find that when $\Delta\gg w_{r}$, the
eigenstates of the Floquet Hamiltonian arrange themselves into several
discrete well-separated clusters in the Hilbert space. Each of these clusters
belong to definite values of $Z_{\pi}$ and the number of such clusters depend
on the ratio of $\hbar\omega_{D}/\Delta$. We dub this phenomenon as primary
clustering and recognize this to be consequence of near-integrability of
$H_{F}$ for large $\Delta/w_{r}$. In addition, we also find a secondary
clustering which leads to discrete set of states within each of these primary
clusters. We tie the presence of such secondary clustering to the existence of
emergent conserved quantity, $Y$, which was noted earlier in the context of
non-driven Rydberg chains with such staggered detuning in Refs. ethv05a, ;
ethv05b, . We note that these clusterings leading to violation of ETH do not
stem from standard Hilbert space fragmentation hsf1 ; hsf2 ; hsf3 ; hsf4
since the Floquet Hamiltonian of the staggered Rydberg chain do not have
dynamically disconnected sector at any finite $w_{r}$. Furthermore such
clustering is not expected to survive in the thermodynamic limit; however,
they will be a feature of finite-sized chains which are typically realized in
standard experiments exp3 ; exp4 .

We also show that the existence of $Y$ and hence secondary clustering is
contingent on incommensuration of the drive frequency and $\Delta$ which leads
to a non-zero $\delta E_{2}$. Thus at commensurate drive frequencies where
$\delta E_{2}=0$, the secondary clustering is destroyed and states within each
primary cluster become ergodic. This is reflected in the values of local
observables such as $O_{22}$ near these commensurate drive frequencies which
reach very close to their ETH predicted values. This leads to a possibility of
tuning ergodicity properties of the driven system using the drive frequency.

We also study the behavior of the correlation function $O_{22}$ for the driven
chain starting from the vacuum and the Neel states. We find, for initial
$|0\rangle$ and $|{\mathbb{Z}}_{2}\rangle$ dynamical freezing of $O_{22}$ at
specific drive frequencies whose values are well approximated by the analytic
Floquet Hamiltonian obtained using FPT. When driven close to these freezing
frequencies, $O_{22}$ starting from $|0\rangle$ or $|{\mathbb{Z}}_{2}\rangle$
displays oscillations with perfect revivals. The amplitude of these
oscillations decreases with increasing $\Delta$ in the large or intermediate
$\Delta/w_{r}$ regime; in contrast, their frequencies are pinned to
$\Delta/\hbar$ for both the initial states. We provide a simple qualitative
explanation of these features using the structure of the Floquet eigenstates
in this regime. In contrast such oscillations are completely suppressed when
the dynamics starts from $|{\bar{\mathbb{Z}}_{2}}\rangle$; this leads to a
range of drive frequencies where one encounters dynamic freezing. This range
increases with $\Delta$. We explain this phenomenon qualitatively and point
out that the dynamics of $O_{j2}$, where $j$ is odd, would show exactly
opposite behavior; it would show range of freezing frequencies for dynamics
starting from $|{\mathbb{Z}}_{2}\rangle$ state. This dichotomy is a direct
consequence of breaking of the sublattice symmetry by the staggered detuning
term.

We have also studied the quantum scars in the Floquet eigenspectrum of such a
driven Rydberg chain. Such scars are known to exist for $\Delta^{\prime}=0$
ethv1  and are dubbed as PXP scars ethv01 ; they have identical overlap with
both the Neel states in this limit. We find that with increasing
$\Delta^{\prime}$, the scars with energy $E_{F}<0$ develop a stronger overlap
with $|{\mathbb{Z}}_{2}\rangle$ while those with $E_{F}>0$ overlap more with
$|{\bar{\mathbb{Z}}_{2}}\rangle$. This phenomenon clearly originates from the
breaking of the $Z_{2}$ sublattice symmetry due to presence of $\Delta$. In
fact, at large $\Delta^{\prime}$, we find that no near mid-spectrum scar
states have significant overlap with either of the Neel states. Instead, as
confirmed from quench dynamics studies in Sec. V, they have a high overlap
with either $|{\mathbb{Z}}_{4}\rangle$, vacuum ($|0\rangle$) or single dipole
($|1\rangle$) states. This indicates that the presence of large
$\Delta^{\prime}$ leads to scars which have no analog with the standard PXP
scars studied earlier in the literature. Our FSA analysis which explains the
property of the scars at weak or intermediate $\Delta^{\prime}$ fails to
capture these mid-spectrum scars at large $\Delta^{\prime}$.

In conclusion, we have shown that a driven Rydberg chain with staggered
detuning term leads to several interesting phenomena. These include ergodicity
violation via Floquet eigenstate clustering in the strong staggered detuning
limit, dynamical freezing, sustained coherent oscillations with perfect
revivals near the freezing frequency, existence of separate class of quantum
scars with large overlap with $|0\rangle$, $|1\rangle$ and
$|{\mathbb{Z}}_{4}\rangle$ states, and the possibility of tuning ergodicity
property of these chains with the drive frequency. The experimental
implementation of a Rydberg chain has already been achieved exp3 ; exp4 ; a
possible extension of some of these experiments with implementation of
staggered detuning may provide a suitable experimental platform testing our
theoretical predictions.

*

##  Appendix A Projection operator formalism

In this appendix, we detail the computation of $Y$ and $H_{2}$ using a
perturbative formalism developed in Ref. ethv05b, . To this end we shall start
from a Hamiltonian $H_{m}$ (Eq. 34) in the main text and treat
$1/\Delta^{\prime}$ as the perturbative parameter. In what follows, we write
$H_{m}=H_{0}+H_{1}$ in terms of $Z_{\pi}$ (Eq. 20). This yields

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle-\Delta^{\prime}Z_{\pi},\quad
H_{1}=-\sum_{j}\tilde{\sigma}_{j}^{x}.$ (42)

In the rest of this appendix, we shall treat $H_{1}$ perturbatively in the
regime where $\Delta^{\prime}>1$.

The formalism used in Ref. ethv05b,  constitutes construction of a
perturbative effective Hamiltonian for such a system. The method relies on the
presence of a canonical transformation, implemented through an operator $S$,
which yields

$\displaystyle H_{\rm eff}$ $\displaystyle=$ $\displaystyle
e^{iS}H_{m}e^{-iS}=H_{0}+H^{(1)}+H^{(2)}+...$ (43) $\displaystyle H^{(1)}$
$\displaystyle=$ $\displaystyle[iS,H_{0}]+H_{1}$ $\displaystyle H^{(2)}$
$\displaystyle=$ $\displaystyle
P^{\prime}[iS,H_{1}]+\frac{1}{2}[iS,[iS,H_{0}]]P^{\prime}$

where the ellipsis represents higher order terms and $P^{\prime}$ is a
projection operator which projects the Hamiltonian to the low-energy manifold
of $H_{0}$ ethv05b . The next step is to determine $S$ to first order in
perturbation theory by setting $H^{(1)}=0$. A straightforward calculation
using this condition yields

$\displaystyle iS$ $\displaystyle=$
$\displaystyle\frac{1}{2i\Delta^{\prime}}\sum_{j}\tilde{\sigma}_{j}^{y}(-1)^{j}$
(44)

Using Eq. 44, one can now compute $H^{(2)}$. While doing this, it is important
to remember that the final results needs to be projected in the low-energy
manifold of $H_{0}$. This consideration leads to a single term in the second
order which can be easily computed by substituting Eq. 44 in Eq. 43 given by

$\displaystyle H^{(2)}$ $\displaystyle=$
$\displaystyle\frac{-1}{\Delta^{\prime}}Y$ (45)

where the expression of $Y$ is given in Eq. 24 of the main text. We note that
as long as the perturbation theory is valid $Y$ emerges as a constant of
motion; however, this breaks down, along with the perturbative approach, when
$\Delta^{\prime}\to 0$.

The higher order terms in the perturbation theory can be computed
systematically using this approach and is detailed out in Ref. ethv05b, . This
yields $H^{(3)}=0$; the leading order non-zero term is given by $H^{(4)}$. As
shown in Ref. ethv05b, , this can be written as

$\displaystyle H^{(4)}$ $\displaystyle=$ $\displaystyle
H_{1}^{(4)}+H_{2}^{(4)}$ $\displaystyle H_{1}^{(4)}$ $\displaystyle=$
$\displaystyle\frac{-1}{2\Delta^{\prime
3}}\sum_{j}(-1)^{j}(\tilde{\sigma}_{j}^{+}\tilde{\sigma}_{j+2}^{-}+{\rm
h.c.})$ (46) $\displaystyle H_{2}^{(4)}$ $\displaystyle=$
$\displaystyle\frac{-1}{2\Delta^{\prime
3}}\sum_{j}P_{j-1}\sigma_{j}^{z}P_{j+1}P_{j+2}+P_{j-2}P_{j-}\sigma_{j}^{z}P_{j+1}$
$\displaystyle-2P_{j-1}\sigma_{j}^{z}P_{j+1}$

We note that $H_{1}^{(4)}$ is the leading order term introducing non-trivial
spin dynamics. The expression of $H_{1}^{(4)}$ has been used in Sec. V of the
main text for developing a qualitative understanding of the
$|{\mathbb{Z}}_{4}\rangle$ and single-dipole scar states.

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$x\in A$. We conclude that $A$ is closed.

By assumption, $A$ contains $\sigma\cap\mathbb{Z}^{r}$. By the first step, to
prove (ii) it will thus be enough to show that $A$ contains
$\sigma\cap\mathbb{Q}^{r}$, which is dense in the closed convex cone
$\overline{\sigma}$. Let $x\in\sigma\cap\mathbb{Q}^{r}$, and pick
$a\in\mathbb{Z}_{>0}$ such that $ax\in\mathbb{Z}^{r}$. For
$m\in\mathbb{Z}_{>0}$, write $m=aq_{m}+r_{m}$ with $q_{m}\in\mathbb{N}$ and
$r_{m}\in\\{0,\dots a-1\\}$. Then $\lfloor mx_{i}\rfloor-q_{m}ax_{i}=\lfloor
r_{m}x_{i}\rfloor$ remains bounded, and Lemma A.4 thus yields a constant
$C^{\prime}>0$ depending on $|x|$ such that

(A.6) $\left|h(\sumop\displaylimits_{i}\lfloor mx_{i}\rfloor
L_{i})-h(q_{m}\sumop\displaylimits_{i}ax_{i}L_{i})\right|\leq
C^{\prime}m^{s-1}$

for all $m\in\mathbb{Z}_{>0}$. Since $ax\in\sigma\cap\mathbb{Z}^{r}$, we have
by assumption
$q_{m}^{-s}h(q_{m}\sumop\displaylimits_{i}ax_{i}L_{i})\to\hat{h}(ax)=a^{s}\hat{h}(x)$.
Using (A.6), we get $m^{-s}f_{m}(x)\to\hat{h}(x)$, i.e. $x\in A$, which
concludes the proof of (ii). ∎

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