# Periodically driven model with quasiperiodic potential and staggered hopping
amplitudes: engineering of mobility gaps and multifractal states

Sreemayee Aditya1, K. Sengupta2 and Diptiman Sen1 1Centre for High Energy
Physics, Indian Institute of Science, Bengaluru 560012, India
2School of Physical Sciences, Indian Association for the Cultivation of
Science, Jadavpur, Kolkata 700032, India

###### Abstract

We study if periodic driving of a model with a quasiperiodic potential can
generate interesting Floquet phases which have no counterparts in the static
model. Specifically, we consider the Aubry-André model which is a one-
dimensional time-independent model with an on-site quasiperiodic potential
$V_{0}$ and a nearest-neighbor hopping amplitude which is taken to have a
staggered form. We add a uniform hopping amplitude which varies in time either
sinusoidally or as a square pulse with a frequency $\omega$. Unlike the static
Aubry-André model which has a simple phase diagram with only two phases (only
extended or only localized states), we find that the driven model has four
possible phases for the Floquet eigenstates: a phase with gapless quasienergy
bands and only extended states, a phase with multiple mobility gaps separating
different quasienergy bands, a mixed phase with coexisting extended,
multifractal, and localized states, and a phase with only localized states.
The multifractal states have generalized inverse participation ratios which
scale with the system size with exponents which are different from the values
for both extended and localized states. In addition, we observe intricate re-
entrant transitions between the different kinds of states when $\omega$ and
$V_{0}$ are varied. The appearance of such transitions is confirmed by the
behavior of the Shannon entropy. Many of our numerical results can be
understood from an analytic Floquet Hamiltonian derived using a Floquet
perturbation theory which uses the inverse of the driving amplitude as the
perturbation parameter. In the limit of high frequency and large driving
amplitude, we find that the Floquet quasienergies match the energies of the
undriven system, but the Floquet eigenstates are much more extended. We also
study the spreading of a one-particle wave packet and find that it is always
ballistic but the ballistic velocity varies significantly with the system
parameters, sometimes showing a non-monotonic dependence on $V_{0}$ which does
not occur in the static model. Finally, we compare the results for the driven
model with a static staggered hopping amplitude with a model with a static
uniform hopping amplitudes, and we find some significant differences between
the two cases. All our results are found to be independent of the driving
protocol, either sinusoidal or square pulse. We conclude that the interplay of
quasiperiodic potential and driving produces a rich phase diagram which does
not appear in the static model.

##  I Introduction

Periodically driven systems have been studied extensively for the last several
years because of the multitude of unusual phenomena that they can exhibit rev1
; rev2 ; rev3 ; rev4 ; rev5 ; rev6 ; rev7 ; rev8  which have no analogs in
time-independent systems. For instance, periodic driving can be used to
engineer topological states of matter topo1 ; topo2 ; topo3 ; topo4 ; topo5 ;
topo6 , Floquet time crystals tc1 ; tc2 ; tc3  and other novel steady states
russomanno ; lazarides , produce dynamical localization dynloc1 ; dynloc2 ;
dynloc3 ; dynloc4 , dynamical freezing dynfreez1 ; dynfreez2 ; dynfreez3  and
other dynamical transitions heyl1 ; nandy ; heyl2 ; karrasch ; kriel ; sirker
; canovi ; dutta ; sarkar ; arze ; aditya , tune a system into ergodic or non-
ergodic phases erg1 ; erg2 ; erg3 , and generate emergent conservation laws
haldar .

The effects of periodic driving on localization-delocalization transitions
have been relatively less studied sarkar1 ; sarkar2 . A well-studied time-
independent model in one dimension model with a localization-delocalization
transition is the Aubry-André model aubry ; soko . This is a tight-binding
model with uniform nearest-neighbor hopping amplitude $\gamma$ and a
quasiperiodic on-site potential with strength $V_{0}$. Depending on the value
of $V_{0}$, this model is known to have two phases, with all states being
extended (localized) for $V_{0}$ smaller (larger) than some critical value;
the critical value of $V_{0}$ is known to be equal to $2\gamma$. Studies of
other variants of this system have shown that multiple localization
transitions and states with multifractal properties can appear li ; mishra .
Given these different possibilities for an undriven system, it would be
interesting to study what can happen if a system with both a quasiperiodic
potential and a staggered hopping amplitude is driven periodically in time. In
particular, we would like to understand if the driving can generate mobility
edges or gaps, and whether states which are neither extended nor localized can
appear roy . (See Ref. grover,  for a recent experiment on a kicked system
with a quasiperiodic potential).

The plan of this paper is as follows. In Sec. II we present our model which is
essentially the Aubry-André model driven periodically in a particular way. The
Hamiltonian consists of a staggered time-independent hopping with values
$\gamma_{1}$ and $\gamma_{2}$ on alternating sites, and a uniform time-
periodic hopping with driving amplitude $a$ and frequency $\omega$ (both
hopping amplitudes are between nearest neighbors only), and a quasiperiodic
on-site potential with strength $V_{0}$. In the limit where both $a$ and
$\omega$ are much larger than the time-independent hopping amplitudes and
$V_{0}$ (i.e., $a,~{}\omega\gg\gamma_{1},~{}\gamma_{2},~{}V_{0}$), we use a
Floquet perturbation theory to analytically derive an effective time-
independent Hamiltonian $H_{F}$ called the Floquet Hamiltonian. Both
sinusoidal driving and driving by a periodic square pulse are considered. In
Sec. III we numerically study the properties of the eigenstates of the Floquet
operator $U$ (in particular, their scaling with the system size $L$ to see if
they are localized, extended or multifractal), and whether there are any
mobility edges or gaps in the Floquet quasienergies. We compare the results
obtained numerically from the Floquet operator with those obtained from the
Floquet Hamiltonian. We also study the Shannon entropy and the structure of
the Floquet Hamiltonian in real space to shed more light on the degree of
localization of the Floquet eigenstates. The form of the Floquet eigenstates
and eigenvalues in the high-frequency limit is studied. In Sec. IV we consider
the spreading of a wave packet and compare the cases where the time-
independent hopping amplitudes are uniform ($\gamma_{1}=\gamma_{2}$) and
staggered ($\gamma_{2}=-\gamma_{1}$) respectively. We use van Vleck
perturbation theory to gain an understanding of the results for the case of
uniform hopping. In Sec. V we summarize our results and point out some
directions for future studies.

Our main results are as follows. We find that in contrast to the Aubry-André
model which has only extended or only localized states depending on the value
of $V_{0}$, the periodically driven model has four possible phases: a phase
with gapless quasienergy bands and only extended states, a phase with multiple
mobility gaps separating different quasienergy bands, a mixed phase with
coexisting extended, multifractal, and localized states, and a phase with only
localized states. The multifractal states appear at intermediate driving
frequencies $\omega$ and large values of $V_{0}$; these are eigenstates of the
Floquet operator which are neither extended nor localized. The multifractal
nature is inferred from the scaling with the system size of the generalized
inverse participation ratio of these states; the scaling exponents are found
to be different from the values of both extended and localized states. We find
multiple re-entrant transitions between the different kinds of states for
certain ranges of values of $\omega$ and $V_{0}$; such transitions have no
counterparts in the Aubry-André model. All this is found to be true for both
sinusoidal driving and square pulse driving. In the high-frequency limit, we
find that the spectrum of Floquet quasienergies approaches the energy spectrum
of the model with no driving. However, when the driving amplitude is also
large, we find that the Floquet eigenstates are significantly different from
those of the undriven system; typically, driving makes the states much more
extended. We find that one-particle wave packets always spread ballistically
but the ballistic velocity varies considerably depending on the system
parameters. Interestingly, for the case of uniform time-independent hopping,
we sometimes find a non-monotonic dependence of the ballistic velocity on
$V_{0}$. In conclusion, we find that a combination of quasiperiodic potential,
periodic driving of the uniform hopping amplitude and a time-independent
staggered hopping hopping gives rise to a wide range of unusual properties of
the Floquet eigenstates.

##  II Model Hamiltonian and Floquet perturbation theory

In this section, we will study a one-dimensional model with both uniform and
staggered nearest-neighbor hopping amplitudes and a quasiperiodic on-site
potential. We will first consider what happens when the uniform hopping
amplitude is driven sinusoidally in time with a frequency $\omega$ and an
amplitude $a$. Later we will study what happens when the driving is given by a
periodic square pulse. The Hamiltonian of the system is given by

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle\sum_{j}~{}[(a\sin(\omega
t)+\gamma_{1})~{}(a_{j}^{\dagger}b_{j}+b_{j}^{\dagger}a_{j})$
$\displaystyle~{}~{}~{}~{}~{}~{}+(a\sin(\omega
t)+\gamma_{2})~{}(a_{j}^{\dagger}b_{j-1}+b_{j-1}^{\dagger}a_{j})$
$\displaystyle~{}~{}~{}~{}~{}~{}+V_{0}~{}\\{\cos(2\pi\beta(2j-1))~{}a_{j}^{\dagger}a_{j}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\cos(2\pi\beta(2j))~{}b_{j}^{\dagger}b_{j}\\}],$

where the quasiperiodic potential has strength $V_{0}$, and we take
$\beta=(\sqrt{5}-1)/2$ so that it is irrational. We have taken the unit cells,
labeled by an integer $j$, to consist of two sites labeled as $a_{j}$ and
$b_{j}$. We will set the nearest-neighbor spacing to be equal to 1; hence the
size of a unit cell is 2. We will impose periodic boundary conditions and take
the system to have a total of $L$ sites. As we will discuss later, for our
numerical calculations we will allow $\beta$ to deviate in a minimal way from
$(\sqrt{5}-1)/2$ in order to have periodic boundary conditions. We will set
$\hbar=1$ in this paper.

It is possible to do a unitary transformation which changes $b_{j}\to-b_{j}$
but keeps $a_{j}$ unchanged for all $j$ in Eq. (II). This allows us to change
the relative signs of both the time-independent and time-dependent hoppings on
alternating bonds. Hence there are two possibilities for these two hoppings:
both can be uniform, or one can be uniform and the other staggered. We find it
convenient to take the the driving term to be uniform rather than staggered.
[This may also be easier to realize experimentally. One way to drive the
hopping is to apply a time-dependent periodic pressure on the system. This
would make all the bond lengths increase and decrease alternately with time.
As a result all the nearest-neighbor hoppings will change periodically with
time. This corresponds to a uniform term in the time-dependent hopping].
Assuming the time-dependent hopping to be uniform, the time-independent
hoppings ($\gamma_{1}$ and $\gamma_{2}$) can be either uniform or staggered.
In this paper, we will mainly study the staggered hopping model in which
$\gamma_{2}=-\gamma_{1}$. We will also briefly study the case of uniform
hopping with $\gamma_{2}=\gamma_{1}$ and compare the results found there with
those obtained for $\gamma_{2}=-\gamma_{1}$. Finally, we note that if there is
no driving, i.e., $a=0$, then the cases of uniform and staggered time-
independent hoppings ($\gamma_{2}=\gamma_{1}$ and $\gamma_{2}=-\gamma_{1}$
respectively) are identical to each other since they are related by the
unitary transformation described above; both cases reduce to the Aubry-André
model.

We now compute the Floquet Hamiltonian using first-order Floquet perturbation
theory (FPT) for $a,~{}\omega\gg\gamma_{1},~{}\gamma_{2},~{}V_{0}$. rev8 ;
erg1 ; haldar ; soori  FPT is a method for deriving the Floquet Hamiltonian in
the limit where the driving amplitude is much larger than the coefficients of
the time-independent terms in the Hamiltonian. In that limit, FPT is more
powerful than the Magnus expansion which is often used in the high-frequency
limit rev4 ; rev6 ; unlike the Magnus expansion, FPT does not involve
expanding in powers of $1/\omega$. To develop FPT for our model, we write the
Hamiltonian as $H(t)=H_{0}(t)+V_{1}+V_{2}$, where, in terms of a momentum $k$
(which lies in the range $[-\pi/2,\pi/2]$),

$\displaystyle H_{0}(t)~{}=~{}\sum_{k}~{}[a\sin(\omega
t)(1+e^{-2ik})a_{k}^{\dagger}b_{k}~{}+~{}{\rm H.c.}],$ (2)

and the perturbation has two parts,

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\sum_{k}~{}[(\gamma_{1}+\gamma_{2}e^{-2ik})a_{k}^{\dagger}b_{k}+{\rm
H.c.}],$ $\displaystyle V_{2}$ $\displaystyle=$
$\displaystyle\sum_{k,k^{\prime}}~{}[f_{1}(k,k^{\prime})a_{k}^{\dagger}a_{k^{\prime}}+f_{2}(k,k^{\prime})b_{k}^{\dagger}b_{k^{\prime}}],$
(3)

with

$\displaystyle f_{1}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2V_{0}}{L}\sum_{j}\cos[4\pi\beta(j-\frac{1}{2})]~{}e^{-2i(k-k^{\prime})j},$
$\displaystyle f_{2}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2V_{0}}{L}\sum_{j}\cos[4\pi\beta
j]~{}e^{-2i(k-k^{\prime})j}.$ (4)

We note that the Fourier transform of the quasiperiodic potential couples
different momenta which is a consequence of the fact that such a potential
breaks translation symmetry. More specifically, $f_{1}(k,k^{\prime})$ and
$f_{2}(k,k^{\prime})$ are non-zero whenever $k-k^{\prime}=\pm 2\pi\beta$ mod
$\pi$.

The instantaneous eigenvalues of $H_{0}(t)$ are given by

$\displaystyle E_{k\pm}$ $\displaystyle=$ $\displaystyle\pm~{}2a~{}\sin(\omega
t)\cos(k).$ (5)

These satisfy the condition

$\displaystyle e^{i\int_{0}^{T}dt[E_{k+}(t)-E_{k-}(t)]}=1,$ (6)

where $T=2\pi/\omega$ is the time period of the drive. We therefore have to
carry out degenerate FPT. The eigenfunctions corresponding to $E_{k\pm}$ are
given by

$\displaystyle|k\pm\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1\\\ \pm
e^{ik}\end{array}\right),$ (9)

We begin with the Schrödinger equation

$i\frac{d|\psi\rangle}{dt}~{}=~{}\left(H_{0}+V\right)|\psi\rangle,$ (10)

where $V=V_{1}+V_{2}$. We assume that $|\psi(t)\rangle$ has the expansion

$\displaystyle|\psi(t)\rangle~{}=~{}\sum_{n}~{}c_{n}(t)~{}e^{-i\int_{0}^{t}dt^{\prime}E_{n}(t^{\prime})}|n\rangle.$
(11)

Eq. (10) then implies that

$\frac{dc_{m}}{dt}~{}=~{}-i~{}\sum_{n}~{}\langle
m|V|n\rangle~{}e^{i\int_{0}^{t}dt^{\prime}(E_{m}(t^{\prime})-E_{n}(t^{\prime}))}c_{n}.$
(12)

Integrating this equation, and keeping terms only to first order in $V$, we
find that

$\displaystyle c_{m}(T)$ $\displaystyle=$ $\displaystyle
c_{m}(0)~{}-~{}i~{}\sum_{n}\int_{0}^{T}dt\langle m|V|n\rangle$ (13)
$\displaystyle\times~{}e^{i\int_{0}^{t}\left(E_{m}(t^{\prime})-E_{n}(t^{\prime})\right)~{}dt^{\prime}}c_{n}(0).$

This can be written as

$\displaystyle
c_{m}(T)~{}=~{}\sum_{n}\left(I-iH_{F}^{(1)}T\right)_{mn}c_{n}(0),$ (14)

where $I$ denotes the identity matrix and $H_{F}^{(1)}$ is the Floquet
Hamiltonian to first order in $V$.

The calculation proceeds as follows. First, we consider
$H_{F1}^{(1)}=\sum_{k}H_{Fk1}^{(1)}$ which is proportional to $V_{1}$. This
term has non-zero matrix elements between states with the same momenta, and it
yields

$\displaystyle\langle k\pm|H_{Fk1}^{(1)}|k\pm\rangle$ $\displaystyle=$
$\displaystyle\pm\left(\gamma_{1}+\gamma_{2}\right)~{}\cos(k),$
$\displaystyle\langle k+|H_{Fk1}^{(1)}|k-\rangle$ $\displaystyle=$
$\displaystyle-i\left(\gamma_{1}-\gamma_{2}\right)\sin(k)~{}J_{0}\left(\mu_{k}\right)~{}e^{i\mu_{k}},$
$\displaystyle\mu_{k}$ $\displaystyle=$
$\displaystyle\frac{4a\cos(k)}{\omega}.$ (15)

Using Eq. (15), we write $H_{Fk1}^{(1)}$ as

$\displaystyle H_{Fk1}^{(1)}$ $\displaystyle=$
$\displaystyle(\gamma_{1}+\gamma_{2})\cos(k)~{}\sigma_{z}$ (16)
$\displaystyle+~{}(\gamma_{1}-\gamma_{2})J_{0}\left(\mu_{k}\right)\cos(\mu_{k})\sin(k)~{}\sigma_{y}$
$\displaystyle+~{}(\gamma_{1}-\gamma_{2})J_{0}\left(\mu_{k}\right)\sin(\mu_{k})\sin(k)~{}\sigma_{x}.$

where we have defined the Pauli matrices $\sigma_{x,y,z}$ in the
$|k+\rangle,~{}|k-\rangle$ basis.

Next, we change basis to

$\displaystyle|k\uparrow\rangle$ $\displaystyle=$ $\displaystyle
a_{k}^{\dagger}|0\rangle,\quad|k\downarrow\rangle=b_{k}^{\dagger}|0\rangle,$
(17)

so that

$\displaystyle|k\pm\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|k\uparrow\rangle\pm
e^{ik}|k\downarrow\rangle\right).$ (18)

Using a two-component operator $\psi_{k}=(a_{k},b_{k})$, we can write
$H_{F1}^{(1)}$ in the new basis as

$\displaystyle H_{F1}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}~{}\sum_{k}\psi_{k}^{\dagger}\left(\begin{array}[]{cc}\alpha_{2k}&\alpha_{1k}\\\
\alpha_{1k}^{\ast}&-\alpha_{2k}\end{array}\right)\psi_{k}$ (21)
$\displaystyle\alpha_{1k}$ $\displaystyle=$ $\displaystyle
g_{1k}~{}[1+J_{0}(\mu_{k})~{}\cos(\mu_{k})]$
$\displaystyle+~{}g_{2k}~{}[1-J_{0}(\mu_{k})~{}\cos(\mu_{k})],$ $\displaystyle
g_{1k}$ $\displaystyle=$ $\displaystyle\gamma_{1}+\gamma_{2}e^{-2ik},\quad
g_{2k}=\gamma_{2}+\gamma_{1}e^{-2ik},$ $\displaystyle\alpha_{2k}$
$\displaystyle=$
$\displaystyle(\gamma_{1}-\gamma_{2})~{}\sin(k)~{}J_{0}(\mu_{k})~{}\sin(\mu_{k}).$
(22)

We note that if $\gamma_{2}=\gamma_{1}$, we obtain
$\alpha_{1k}=2\gamma_{1}(1+e^{-i2k})$ and $\alpha_{2k}=0$. Hence
$H_{F1}^{(1)}$ does not depend on the driving parameters $\omega$ and $a$. It
is possible that higher order terms will depend on the driving parameters when
$\gamma_{2}=\gamma_{1}$ but these terms will be smaller than the first-order
term derived here. We therefore expect that driving will have a smaller effect
when $\gamma_{2}=\gamma_{1}$ compared to $\gamma_{2}=-\gamma_{1}$.

Next, we compute the first-order contribution to the Floquet Hamiltonian,
$H_{F2}^{(1)}$, arising from $V_{2}$. To this end, we first define

$\displaystyle p_{\pm}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(f_{1}(k,k^{\prime})\pm
f_{2}(k,k^{\prime})e^{-i(k-k^{\prime})}\right),$ $\displaystyle
q_{\pm}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(f_{2}(k,k^{\prime})\pm
f_{1}(k,k^{\prime})e^{i(k-k^{\prime})}\right),$ $\displaystyle
r_{\pm}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(f_{2}(k,k^{\prime})e^{-ik}\pm
f_{1}(k,k^{\prime})~{}e^{-ik^{\prime}}\right),$ $\displaystyle
t_{\pm}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(f_{1}(k,k^{\prime})e^{ik}\pm
f_{2}(k,k^{\prime})~{}e^{ik^{\prime}}\right),$
$\displaystyle\mu_{kk^{\prime}}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}~{}(\mu_{k}~{}\pm~{}\mu_{k^{\prime}}).$ (23)

We then find that in the $|k\pm\rangle$ basis, the matrix elements of
$H_{F2}^{(1)}$ are given by

$\langle
ks|H_{F2}^{(1)}|k^{\prime}s^{\prime}\rangle~{}=~{}p_{ss^{\prime}}(k,k^{\prime})~{}J_{0}(\mu_{kk^{\prime}}^{-ss^{\prime}})~{}e^{is\mu_{kk^{\prime}}^{-ss^{\prime}}},$
(24)

where $ss^{\prime}=+1$ if $s=s^{\prime}=+1$ or $s=s^{\prime}=-1$, and
$ss^{\prime}=-1$ if $s=+1,~{}s^{\prime}=-1$ or $s=-1,~{}s^{\prime}=+1$.

Using Eq. (18), we find that $H_{F2}$ is diagonal in the $|k\uparrow\rangle$,
$|k\downarrow\rangle$ basis and takes the form

$\displaystyle H_{F2}^{(1)}$ $\displaystyle=$
$\displaystyle~{}~{}\left[\cos(\mu_{kk^{\prime}}^{-})~{}J_{0}(\mu_{kk^{\prime}}^{-})~{}p_{+}(k,k^{\prime})~{}+~{}\cos(\mu_{kk^{\prime}}^{+})~{}J_{0}(\mu_{kk^{\prime}}^{+})~{}p_{-}(k,k^{\prime})\right]~{}a_{k}^{\dagger}a_{k^{\prime}}$
(25)
$\displaystyle+~{}\left[\cos(\mu_{kk^{\prime}}^{-})~{}J_{0}(\mu_{kk^{\prime}}^{-})~{}q_{+}(k,k^{\prime})~{}+~{}\cos(\mu_{kk^{\prime}}^{+})~{}J_{0}(\mu_{kk^{\prime}}^{+})~{}q_{-}(k,k^{\prime})\right]~{}b_{k}^{\dagger}b_{k^{\prime}}$
$\displaystyle+~{}\left[\sin(\mu_{kk^{\prime}}^{-})~{}J_{0}(\mu_{kk^{\prime}}^{-})~{}r_{+}(k,k^{\prime})~{}+~{}\sin(\mu_{kk^{\prime}}^{+})~{}J_{0}(\mu_{kk^{\prime}}^{+})~{}r_{-}(k,k^{\prime})\right]~{}a_{k}^{\dagger}b_{k^{\prime}}$
$\displaystyle+~{}\left[\sin(\mu_{kk^{\prime}}^{-})~{}J_{0}(\mu_{kk^{\prime}}^{-})~{}t_{+}(k,k^{\prime})~{}+~{}\sin(\mu_{kk^{\prime}}^{+})~{}J_{0}(\mu_{kk^{\prime}}^{+})~{}t_{-}(k,k^{\prime})\right]~{}b_{k}^{\dagger}a_{k^{\prime}}.$

This completes our derivation of the first-order Floquet Hamiltonian within
FPT.

We can follow the same procedure to find the Floquet Hamiltonian for square
pulse driving. We consider a driving protocol having the form

$\displaystyle f(t)~{}$ $\displaystyle=$ $\displaystyle~{}a~{}~{}~{}{\rm
for}~{}~{}~{}0\leq t\leq T/2,$ (26) $\displaystyle=$
$\displaystyle-a~{}~{}~{}{\rm for}~{}~{}~{}T/2\leq t\leq T.$

The form of the first-order Floquet Hamiltonian for square pulse driving is
similar to the form we obtained for sinusoidal driving except that $J_{0}(x)$
has to be replaced by $\sin(x)/x$ and $\mu_{k}=4a\cos(k)/\omega$ has to be
replaced by $\mu_{k}=aT\cos(k)$. A square pulse driving is easier to study
numerically since the Floquet operator $U$ which time-evolves the system for
one time period $T$ can be obtained by simply multiplying two operators, one
which time-evolves from $t=0$ to $T/2$ and the other which evolves from
$t=T/2$ to $T$.

We now look at the form of $H_{F}^{(1)}$ in the limits $\omega\gg
a\gg\gamma_{1},~{}\gamma_{2},~{}V_{0}$ and
$a\gg\omega\gg\gamma_{1},~{}\gamma_{2},V_{0}$, which we will refer to as the
high- and intermediate-frequency regimes respectively. In both cases, we are
assuming that $\omega\gg\gamma_{1},~{}\gamma_{2},V_{0}$. If this condition is
not satisfied (i.e., if we are in the low-frequency regime), the Floquet
Hamiltonian $H_{F}$ cannot be uniquely defined since the eigenvalues
$e^{-i\theta_{m}}$ of the Floquet operator $U$ will not satisfy
$|\theta_{m}|\ll\pi$ for all the Floquet eigenstates $m$. Hence
$H_{F}=(i/T)\ln U$ will suffer from branch cut ambiguities. This necessitates
folding back of Floquet eigenstates into the first Floquet Brillouin zone erg3
, a technical complication which we will avoid in this work.

We first consider the case of sinusoidal driving and look at the two limits
separately.

(i) The high-frequency limit can be studied using either the FPT described
above or the Magnus expansion. In the FPT, we have
$\mu_{k}=(4a/\omega)\cos(k)\to 0$ for all $k$ lying in the range
$[-\pi/2,\pi/2]$. Since the Bessel function satisfies $J_{0}(z)\to 1$ when
$z\to 0$, we find that $H_{F1}^{(1)}+H_{F2}^{(2)}$ approaches the time-
independent part, $V_{1}+V_{2}$, of the Hamiltonian $H(t)$ given in Eq. (3).
Hence, the high-frequency limit should give the same results as the undriven
system with $a=0$. However, we will see later that although this is true for
the Floquet eigenvalues, it does not seem to hold for the Floquet eigenstates.

(ii) The intermediate-frequency limit cannot be studied using the Magnus
expansion, and we have to use the FPT. We now use the fact that $J_{0}(z)$
goes to zero as $\sqrt{2/(\pi z)}\cos(z-\pi/4)$ when $|z|\to\infty$.
Furthermore, $\mu_{k}\to\pm\infty$ except near the special point $k=\pi/2$.
Similarly,
$\mu_{kk^{\prime}}^{\pm}=(1/2)(\mu_{k}\pm\mu_{k^{\prime}})\to\pm\infty$ except
near the point $(k,k^{\prime})=(\pi/2,\pi/2)$ and the line $k=\pm k^{\prime}$.
However, we saw after Eq. (4) that the functions $f_{1}(k,k^{\prime})$ and
$f_{2}(k,k^{\prime})$ are non-zero only if $k-k^{\prime}=\pm 2\pi\beta$ mod
$\pi$. Since $\beta=(\sqrt{5}-1)/2\simeq 0.618$, and $k,~{}k^{\prime}$ lie in
the range $[-\pi/2,\pi/2]$, we take $2\pi\beta$ mod $\pi$ to be equal to
either $(\sqrt{5}-2)\pi\simeq 0.742$ or $(\sqrt{5}-3)\pi\simeq-2.400$, so that
$k-k^{\prime}\simeq 0.742$ or $-2.400$. Putting together the conditions
arising from $\mu_{kk^{\prime}}^{\pm}$, $f_{1}(k,k^{\prime})$ and
$f_{2}(k,k^{\prime})$, we see that in Eq. (25), the terms involving products
of $J_{0}(\mu_{kk^{\prime}}^{\pm})$ and $f_{1}(k,k^{\prime})$ or
$f_{2}(k,k^{\prime})$ will go to zero at all values of $k,~{}k^{\prime}$
except near the four points $(k,k^{\prime})\simeq\pm(0.742/2)(1,-1)$ or
$\pm(2.400/2)(1,-1)$. Ignoring these special points, we see that in the
intermediate-frequency limit, $H_{F2}^{(1)}\to 0$ in Eq. (25), which implies
that the effect of the quasiperiodic potential goes to zero. Similarly, we see
that except near the special point $k=\pi/2$, $J_{0}(\mu_{k})\to 0$, and
$H_{F1}^{(1)}$ in Eq. (22) approaches the form

$H_{F1}^{(1)}~{}=~{}\frac{1}{2}~{}(\gamma_{1}+\gamma_{2})~{}\sum_{k}\psi_{k}^{\dagger}\left(\begin{array}[]{cc}0&1+e^{-i2k}\\\
1+e^{i2k}&0\end{array}\right)\psi_{k}.$ (27)

This describes a system with uniform hopping amplitude
$(1/2)(\gamma_{1}+\gamma_{2})$ on all bonds. We thus see that in the
intermediate-frequency limit, the system approaches a simple limit in which
the hopping amplitude is uniform and there is no quasiperiodic potential. The
quasienergies will then be given by $(\gamma_{1}+\gamma_{2})\cos(k)$ which is
a gapless spectrum.

We reach similar conclusions in the case of square pulse driving. Namely, the
high-frequency limit is the same as an undriven system with staggered hopping
amplitudes $\gamma_{1},~{}\gamma_{2}$ and a quasiperiodic potential $V_{0}$,
while the intermediate-frequency limit is a system with a uniform hopping
amplitude $(1/2)(\gamma_{1}+\gamma_{2})$ and no quasiperiodic potential.

Finally, we discuss some exact symmetries of the Floquet Hamiltonian. We can
show that the Floquet Hamiltonian for the present problem can only have terms
proportional to odd powers of $V_{1}$ and $V_{2}$ for a drive satisfying the
condition $f(-t)=-f(t)$ (which is satisfied if $f(t)$ is either given by
$a\sin(\omega t)$ or by the square pulse form in Eq. 26). To see this, we
first show that the Floquet operator $U$ satisfies

$\displaystyle
U^{-1}(a,\gamma_{1},\gamma_{2},V_{0})~{}=~{}U(a,-\gamma_{1},-\gamma_{2},-V_{0}).$
(28)

Hence the Floquet Hamiltonian $H_{F}$, which satisfies $U=e^{-iH_{F}T}$, must
satisfy

$\displaystyle
H_{F}(a,\gamma_{1},\gamma_{2},V_{0})=-H_{F}(a,-\gamma_{1},-\gamma_{2},-V_{0}).$
(29)

Hence $H_{F}$ can only have odd powers of $\gamma_{1},~{}\gamma_{2}$ and
$V_{0}$. As a result, the next correction after the terms of first-order in
$\gamma_{1},~{}\gamma_{2}$ and $V_{0}$ will be of third order and will
therefore be small. Hence, in the following sections we will only show the
perturbative results based on first-order FPT.

##  III Results for the model with $~{}\gamma_{2}=-\gamma_{1}$

The Floquet Hamiltonian obtained in Sec. II allows us to study the properties
of the Floquet eigenstates within first-order FPT. To this end, we numerically
diagonalize $H_{F}^{(1)}=H_{F1}^{(1)}+H_{F2}^{1)}$ to obtain its eigenvectors
and eigenvalues

$\displaystyle H^{(1)}_{F}|\psi_{m}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle E_{Fm}^{(1)}|\psi_{m}^{(1)}\rangle.$ (30)

On the other hand, the Floquet operator $U$ will have eigenvectors
$|\psi_{m}\rangle$ and corresponding eigenvalues $\exp(-i\theta_{m})$, where
$\theta_{m}=E_{Fm}T$ can be chosen to lie in the range $[-\pi,\pi]$. Then the
quasienergies $E_{Fm}$ obtained from $\theta_{m}$ will lie in the range
$[-\omega/2,\omega/2]$.

Figure 1: (a-d) Plots of $I_{m}^{(2)}L^{\eta}$ (sorted in increasing order)
versus $m/L$ for $\gamma_{1}=1$, $\gamma_{2}=-1$, $V_{0}=2.5$, system sizes
$L=3000,~{}4000,~{}5000$ and 6000, and sinusoidal driving. (a)
$I_{m}^{(2)}L^{\eta}$ for Floquet eigenstates with $\omega=5$ and $a=5$. The
exponent $\eta=1$, and we see that all states are extended. (b)
$I_{m}^{(2)}L^{\eta}$ with $\omega=40$ and $a=5$. Here $\eta=0$ showing that
all states are localized. (c) and (d) $I_{m}^{(2)}L^{\eta}$ with $\omega=15.1$
and $a=5$. Plots (c) and (d) correspond to $\eta=0$ and 1 respectively, and
they clearly show a jump in the IPR value around $m/L\simeq 0.76$. (e) Plot of
$I_{m}^{(2)}$ versus quasienergy $E_{F}$ for $\omega=15$, $a=5$, and $L=3000$.
We see several gaps in $E_{F}$.

Figure 2: (a-d) Plots of $I_{m}^{(2)}L^{\eta}$ versus $m/L$ and (e) plot of
$I_{m}^{(2)}$ versus $E_{F}$ for the same parameter values as in Fig. 1 but
obtained using first-order FPT. All the plots agree qualitatively with the
plots from exact numerics shown in Fig. 1.

The results obtained using FPT will be compared with those obtained from an
exact numerical calculation of the Floquet operator. To this end, we divide
the time period $T$ into $N$ steps so that $H(t)$ does not vary appreciably
within the time step $\delta t=T/N$. For the present case, we find that $N\sim
500$ achieves this task; a further increase in $N$ does not change the
numerical results appreciably. This allows us to use the standard Suzuki-
Trotter decomposition to write $U$ as

$\displaystyle U(T,0)$ $\displaystyle=$
$\displaystyle\prod_{j=1,N}U(t_{j},t_{j-1}),$ $\displaystyle U_{j}\equiv
U(t_{j},t_{j-1})$ $\displaystyle=$ $\displaystyle
e^{-iH[(t_{j}+t_{j-1})/2]\delta t},$ (31)

where $t_{0}=0$ and $t_{N}=T$. We can then numerically diagonalize $U$ to
obtain its eigenvectors $|\psi_{m}\rangle$ and eigenvalues $e^{-i\theta_{m}}$.
We can then write

$U(T,0)~{}=~{}\sum_{m}e^{-i\theta_{m}}|\psi_{m}\rangle\langle\psi_{m}|.$ (32)

In this section, we will study and compare the properties of the Floquet
eigenstates obtained using first-order FPT and exact numerics. To this end, we
compute the inverse participation ratio (IPR) of these eigenstates given, for
a normalized Floquet eigenstate $m$, as

$I_{m}^{(2)}~{}=~{}\sum_{j=1}^{L}|\psi_{m}(j)|^{4}.$ (33)

It is well-known that $I_{m}^{(2)}$ can distinguish between localized and
extended states; $I_{m}^{(2)}\sim L^{-\eta}$ where $\eta=0~{}(1)$ for
localized (extended) eigenstates.

###  III.1 Localized, extended and multifractal states

We now present our numerical results. Figures 1 (a-d) show plots of
$I_{m}^{(2)}L^{\eta}$ (sorted in increasing order) versus $m/L$ for systems
with $\gamma_{1}=1$, $\gamma_{2}=-1$, $V_{0}=2.5$, different values of
$\omega$ and $a$, and sinusoidal driving. For generating this and all
subsequent figures, the quasiperiodic potential at site $j$ is taken to be
$V_{j}=V_{0}\cos(2\pi\alpha j)$, where $\alpha$ is taken to be a rational
approximant for $(\sqrt{5}-1)/2$ by choosing $\alpha=N/L$, where $L$ is the
system size, and $N$ is the integer closest to $L(\sqrt{5}-1)/2$. We have
chosen $L=3000,~{}4000,~{}5000$ and 6000. The plots clearly indicate a jump in
the IPR value for certain ranges of values of $\omega$ and $a$. From Figs. 1
(a) and (b), we find that at relatively low (high) driving frequencies, all
the Floquet eigenstates are extended (localized). This can be seen from the
collapse of the $I_{m}^{(2)}L^{\eta}$ curves for $\eta=1$ at $\omega=5$ (Fig.
1 (a)) and $\eta=0$ at $\omega=40$ (Fig. 1 (b)) for different $L$. In
contrast, at an intermediate driving frequency, $\omega=15.1$, as shown in
Figs. 1 (c) and (d), a jump in the IPR between localized and extended
eigenstates around $m/L\simeq 0.76$. The data for different $L$ collapses upon
scaling with $L^{\eta}$ for two different values of $\eta$; the collapse
happens for $\eta=0$ for $m/L\gtrsim 0.764$ (Fig. 1 (c)) and for $\eta=1$ for
$m/L\lesssim 0.764$ (Fig. 1 (d)). This demonstrates the appearance of an IPR
jump around $m/L\simeq 0.76$ at this driving frequency. Figure 1 (e) shows a
plot of $I_{m}^{(2)}$ versus the quasienergy $E_{F}$ for $\omega=5$, $a=5$,
and $L=3000$. This clearly shows that there are gaps in the quasienergy
spectrum; we call these mobility gaps. (In this paper, we use the term
mobility gap regardless of whether the states in the quasienergy bands on the
two sides of a gap are both extended or both localized or one extended and the
other localized). The states with $E_{F}$ around zero have large IPRs and
correspond to localized states, whereas states with $E_{F}$ away from zero (on
either the positive or the negative side) have very small IPRs and are
extended states. [It is important to note that sorting in increasing order of
$I_{m}^{(2)}$ (as done in Figs. 1 (a-d)) and in increasing order of $E_{F}$
(as done in Fig. 1 (e)) are quite different from each other. The presence of
mobility gaps becomes apparent only in the latter way of sorting].

In Fig. 2, similar plots are shown for Floquet eigenstates obtained from
first-order FPT. These correspond to the same parameter values as their
counterparts in Fig. 1. A comparison between Figs. 1 and 2 shows that first-
order FPT provides a reasonable match to the exact numerics, although the IPR
values found from exact numerics are significantly smaller than the ones found
from first-order FPT. Importantly, both exact numerics (Fig. 1 (e)) and first-
order FPT (Fig. 2 (e)) show mobility gaps at intermediate driving frequencies
like $\omega=5$. We conclude that first-order FPT can provide a good
understanding of the driven system when $a\gg\gamma_{1},\gamma_{2}$.

To determine if any of the eigenstates exhibit multifractal behavior, we
calculate a generalized IPR defined as

$I_{m}^{(q)}~{}=~{}\sum_{j}~{}|\psi_{m}(j)|^{2q},$ (34)

and study its scaling with the system size $L$. sarkar1 ; caste ; evers ;
rodri  If $I_{m}^{(q)}$ scales as $L^{-\eta_{q}}$ and $\eta_{q}=(q-1)D_{q}$,
then $D_{q}=1$ for all $q$ for extended states (since $|\psi_{m}(j)|^{2}$ is
of the order of $1/L$ for all $j$ for such states), and $D_{q}=0$ for all $q$
for localized states (since $|\psi_{m}(j)|^{2}$ for such states is of order 1
over a finite region whose size remains constant as $L\to\infty$).
Multifractal states typically have $0<D_{q}<1$ for all $q$. Figure 3 shows
plots of $I_{m}^{(q)}L^{\eta}$ for $q=3$ and 4 for systems with
$\gamma_{1}=1$, $\gamma_{2}=-1$, $\omega=15.1$, $a=5$, $V_{0}=2.5$, different
system sizes, and sinusoidal driving. Figures 3 (a) and (c) show that states
with $m/L\gtrsim 0.764$ scale with $L$ with the powers $\eta_{3}=\eta_{4}=0$,
showing that $D_{q}=0$; hence these states are localized. Figures 3 (b) and
(d) show that states with $m/L\lesssim 0.764$ scale with powers $\eta_{3}=2$
and $\eta_{4}=3$, showing that $D_{q}=1$; hence these states are extended.
Thus there is no evidence of multifractal states for this set of parameter
values. However, we will show below that multifractal behavior appears for a
different range of values of $\omega$.

Figure 3: Plots showing $I_{m}^{(3)}L^{\eta}$ and $I_{m}^{(4)}L^{\eta}$
(sorted in increasing order) versus $m/L$ obtained by exact numerical
calculations for $L=3000,~{}4000,~{}5000$ and 6000, with
$\gamma_{1}=1,~{}\gamma_{2}=-1,~{}a=5,~{}\omega=15.1$, $V_{0}=2.5$, and
sinusoidal driving. Plot (a) shows a data collapse of $I_{m}^{(3)}$ with
$\eta=0$ for $m/L\gtrsim 0.764$. Plot (b) shows a data collapse of
$I_{m}^{(3)}$ with $\eta=2$ for $m/L\lesssim 0.764$. Plot (c) shows a data
collapse of $I_{m}^{(4)}$ with $\eta=0$ for $m/L\gtrsim 0.764$. Plot (d) shows
a data collapse of $I_{m}^{(4)}$ with $\eta=3$ for $m/L\lesssim 0.764$. These
plots show that $I_{m}^{(q)}$ scales with $\eta=0$ for states with $m/L\gtrsim
0.764$ (localized states) and with $\eta=q-1$ for states with $m/L\lesssim
0.764$ (extended states).

In Fig. 4, we show plots of $I_{m}^{(2)}$ and $I_{m}^{(3)}$ versus $m/L$, and
their scaling exponents with $L$, $\eta_{2}$ and $\eta_{3}$, versus $\omega$
and $m/L$ for systems with $\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$,
$V_{0}=2.5$, and square pulse driving. Figures 4 (a-b) show $I_{m}^{(2)}$ and
$I_{m}^{(3)}$ for a system with $L=3000$, while Figs. 4 (c-d) show the scaling
exponents $\eta_{2}$ and $\eta_{3}$ for these two quantities extracted from
the results for $L=3000,~{}4000,~{}5000$ and 6000. We see that most states
have $\eta_{2}=\eta_{3}=0$ (localized) at lower frequencies and have
$\eta_{2}=1$ and $\eta_{3}=2$ (extended) at higher frequencies. Interestingly,
however, we see some states for which $\eta_{2}$ and $\eta_{3}$ are different
from the values of both localized and extended states. Hence these states have
a multifractal nature. In Fig. 4 (e) we show $I_{m}^{(2)}$ versus the
quasienergy $E_{Fm}$ for $\omega=30$. For this value of $\omega$, all three
kinds of states coexist, extended, multifractal and localized, as is evident
from the scaling exponents $\eta_{2}$ and $\eta_{3}$ shown in Figs. 4 (c-d).
We have also studied what happens if the driving is sinusoidal instead of
square pulse and we find similar results which we do not show here. This
demonstrates that the generation of multifractal states does not depend
significantly on the driving protocol.

Figure 4: (a-b) Plots of $I_{m}^{(2)}$ and $I_{m}^{(3)}$ (sorted in increasing
order of quasienergy $E_{Fm}$) versus $\omega$ and $m/L$ for $\gamma_{1}=1$,
$\gamma_{2}=-1$, $a=5$, $V_{0}=2.5$, $L=3000$, and square pulse driving. (c-d)
Plots of $\eta_{2}$ and $\eta_{3}$ (extracted from the IPRs sorted in
increasing order for $L=3000,~{}4000,~{}5000$ and 6000) versus $\omega$ and
$m/L$, with the parameter values same as in plots (a-b). Multiple jumps in
$I_{m}^{(2)}$ and $I_{m}^{(3)}$ observed in plots (a-b) indicate multiple
localization-delocalization transitions. This is more evident in plots (c-d)
where $\eta_{2}$ and $\eta_{3}$ are extracted from the scaling analysis of IPR
values. We note that for $30\lesssim\omega\lesssim 45$ in plots (c-d), the
values of $\eta_{2}$ and $\eta_{3}$ for a small fraction of states are
different from the usual scaling exponents of both extended states
($\eta_{2}=1$ and $\eta_{3}=2$) and localized states ($\eta_{2}=\eta_{3}=0$).
Plot (e) shows $I_{m}^{(2)}$ as a function of $E_{Fm}$ (sorted in increasing
order) for $\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$, $V_{0}=2.5$, $\omega=30$
and $L=3000$.

Figure 5: (a) Plot of $I_{m}^{(2)}$ (sorted in increasing order of
quasienergy) versus $\omega$ and $m/L$ with $\gamma_{1}=\gamma_{2}=1$ (uniform
hopping), $a=5$, $V_{0}=2.5$, $L=3000$, and square pulse driving. (b) Plot of
$\eta_{2}$ (extracted from the IPRs sorted in increasing order for
$L=3000,~{}4000,~{}5000$ and 6000) versus $\omega$ and $m/L$.

Figure 5 shows plots of $I_{m}^{(2)}$ and the scaling exponent $\eta_{2}$
versus $\omega$ and $m/L$ for $L=3000,~{}4000,~{}5000$ and 6000, for
$\gamma_{1}=\gamma_{2}=1$ (uniform hopping), $a=5$, $V_{0}=2.5$, and square
pulse driving. The results looks somewhat similar to those shown in Fig. 4 for
$\gamma_{1}=-\gamma_{2}=1$ (staggered hopping) in that both extended and
localized states appear in the two cases. However, extended states persist up
to a larger values of $\omega$ for $\gamma_{1}=-\gamma_{2}=1$ compared to
$\gamma_{1}=\gamma_{2}=1$.

It is useful to look at the average values of both $I_{m}^{(2)}$ and the
normalized participation ratio (NPR) which, up to a factor of $L$, is the
inverse of $I_{m}^{(2)}$. li  For the $m$-th Floquet eigenstate, the NPR is
defined as ${\rm NPR}_{m}=1/(LI_{m}^{(2)})$. The average values of these two
quantities are then defined as $\langle{\rm
IPR}\rangle=(1/L)\sum_{m}I_{m}^{(2)}$ and $\langle{\rm
NPR}\rangle=(1/L)\sum_{m}{\rm NPR}_{m}$. For a large system size $L$, we have
$I_{m}^{(2)}\sim 1/L~{}(1)$ and ${\rm NPR}_{m}\sim 1~{}(1/L)$ for an extended
(localized) state $m$, respectively. Hence, the quantity
$\phi=\log_{10}(\langle{\rm IPR}\rangle\langle{\rm NPR}\rangle)$ will be large
and negative (of the order of $-\log_{10}L$) either if all states are extended
or if all states are localized. But if $\phi$ is not large and negative, this
would indicate that there are some states which are neither extended nor
localized. We will also look at the average value of the Shannon entropy as
another measure of the degree of localization dynloc4 . For the $m$-th Floquet
eigenstate, the Shannon entropy is defined as
$S_{m}=-\sum_{n}|\psi_{m}(n)|^{2}\ln(|\psi_{m}(n)|^{2})$, and its average is
then given by $\langle S\rangle=(1/L)\sum_{m}S_{m}$). For an extended
(localized) state $m$, we have $S_{m}\sim\ln L~{}(0)$, respectively. Hence the
average value $\langle S\rangle$ will be of order $\ln L$ if all states are
extended and will be close to zero if all states are localized.

Figure 6 shows plots of $\langle{\rm IPR}\rangle$, $\phi=\log_{10}(\langle{\rm
IPR}\rangle\langle{\rm IPR}\rangle$), and $\langle S\rangle$ versus $\omega$
and $V_{0}$, for a system with $\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$ and
square pulse driving. Figure 6 (a) indicates that all states are extended when
both $\omega$ and $V_{0}$ are small and are localized when both $\omega$ and
$V_{0}$ are large. For $2\lesssim V_{0}\lesssim 3.5$, we see that only
extended states low values of $\omega$, coexisting states of different types
for intermediate values of $\omega$, and only localized states for large
values of $\omega$. Interestingly, when $V_{0}\gtrsim 3.5$, we find multiple
re-entrant transitions between regions with only extended and only localized
states as $\omega$ is increased. These observations are confirmed
qualitatively by Figs. 6 (b) and (c). We note that the range of values of
$V_{0}$ which supports re-entrant transitions is the same range where the
time-independent part of the model exhibits only localized states.

Figure 6: (a) Plot on a log scale of the average of $I_{m}^{(2)}$, denoted as
$\langle IPR\rangle$, versus $V_{0}$ and $\omega$ for
$\gamma_{1}=1,\gamma_{2}=-1$, $a=5$, $L=3000$, and square pulse driving. (b)
Plot of $\phi=\log_{10}({\langle\text{IPR}\rangle\langle\text{NPR}\rangle})$
versus $V_{0}$ and $\omega$. (c) Plot of the average Shannon entropy $\langle
S\rangle$ versus $V_{0}$ and $\omega$. Plots (a-c) suggest that all the
Floquet eigenstates below $V_{0}\simeq 2$ are extended for all values of
$\omega$. For $2\lesssim V_{0}\lesssim 3.5$, we find three distinct regions,
with only extended states for small $\omega$, coexisting states of different
types for intermediate $\omega$, and only localized states for large $\omega$.
A different behavior appears for $V_{0}\gtrsim 3.5$, where we see multiple re-
entrant transitions between regions with only extended and only localized
states as $\omega$ is increased.

Figure 7 shows plots of $I_{m}^{(2)}$, the scaling exponent $\eta_{2}$ and the
real part of the Floquet eigenvalues versus $\omega$ and $m/L$ for systems
with $\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$, $V_{0}=5.5$ and different sizes,
for square pulse driving. This figure confirms the multiple re-entrant
transitions between regions which are completely extended, completely
localized, or have both extended and localized states, that we see in Fig. 6
when $V_{0}\gtrsim 3.5$. (We note that similar re-entrant transitions have
been observed in a kicked quasicrystal grover ). We would like to mention here
that the plots in Fig. 7 are difficult to explain by any perturbation theory
since the values of $a$, $\omega$ and $V_{0}$ are all comparable to each
other.

Figure 7: (a) Plot of $I_{m}^{(2)}$ (sorted in increasing order of
quasienergy) versus $\omega$ and $m/L$, with $\gamma_{1}=1$, $\gamma_{2}=-1$,
$a=5$, $V_{0}=5.5$, $L=3000$, and square pulse driving. (b) Plot of $\eta_{2}$
versus $\omega$ and $m/L$. We extract $\eta_{2}$ using system sizes
$L=3000,~{}4000,~{}5000$ and 6000. The plots confirm the multiple transitions
seen in Fig. 6 for $V_{0}\gtrsim 3.5$ as $\omega$ is varied. Plot (b) shows
several transitions. A transition to a phase with all extended states occurs
at $\omega\simeq 5.1$, and this phase then continues up to $\omega\simeq 7.1$.
For $\omega\gtrsim 7.1$, we get both extended and localized states till
$\omega\simeq 9.1$. Another phase of extended states appears for
$9.2\lesssim\omega\lesssim 12.1$. Beyond $\omega\simeq 12.1$, plot (b)
suggests that the system moves towards complete localization as $\omega$
increases.

Fig. 8 shows the average $\langle{\rm IPR}\rangle$ and average Shannon entropy
$\langle S\rangle$ as functions of $V_{0}$ and of $\omega$ for a system with
uniform hopping, $\gamma_{1}=\gamma_{2}=1$, $a=5$ and $L=3000$, for square
pulse driving. The results deviate significantly from the case of staggered
hopping, $\gamma_{2}=-\gamma_{1}$, shown in Fig. 6, particularly for
$V_{0}\gtrsim 3.5$.

It is known that there are no re-entrant transitions in the Aubry-André model
(which is the time-independent part of our model) if there is no driving.
Furthermore, there are no such transitions in a driven model with no
quasiperiodic potential. Comparing Fig. 6 (where $\gamma_{2}=-\gamma_{1}$) and
Fig. 8 (where $\gamma_{2}=\gamma_{1}$), we see that staggered hopping, driving
and quasiperiodic potential are all necessary to clearly see re-entrant
transitions between delocalized states and other kinds of states.

Figure 8: (a) Plot on a log scale of the average of $I_{m}^{(2)}$,
$\langle{\rm IPR}\rangle$, versus $\omega$ and $V_{0}$, with
$\gamma_{1}=\gamma_{2}=1$ (uniform hopping), $a=5$, $L=3000$, and square pulse
driving. (b) Plot of average Shannon entropy, $\langle S\rangle$, versus
$\omega$ and $V_{0}$. For $V_{0}\gtrsim 3.5$, we see that the case of uniform
hopping shown here deviates significantly from the case of non-uniform hopping
shown in Fig. 6.

###  III.2 High-frequency limit

In the high-frequency limit $\omega\gg a\gg\gamma_{1},~{}\gamma_{2},~{}V_{0}$,
we expect the effects of driving to disappear as we have argued in Sec. II
Surprisingly, however, we find that although the quasienergies for large
$\omega$ match the energies for the undriven system, the IPRs do not agree in
the two cases.

Figure 9: (a) Plot of $I_{m}^{(2)}$ versus quasienergy $E_{F}$ (sorted in
increasing order) for a system with $\gamma_{1}=1$, $\gamma_{2}=-1$,
$\omega=35$, $a=5$, $V_{0}=2.5$, $L=3000$, and square pulse driving. (b) Plot
of $I_{m}^{(2)}$ versus quasienergy for a system with $\gamma_{1}=1$,
$\gamma_{2}=1$ (uniform hopping), $\omega=35$, $a=5$, and $V_{0}=2.5$. (c)
Plot of $I_{m}^{(2)}$ versus energy $E$ (sorted in increasing order) for an
undriven system ($a=0$) with $\gamma_{1}=1$, $\gamma_{2}=1$, and $V_{0}=2.5$.
(d) Plot of $I_{m}^{(2)}$ versus energy $E$ (sorted in increasing order) for
an undriven system ($a=0$) with $\gamma_{1}=1$, $\gamma_{2}=-1$, and
$V_{0}=1.5$. Note that the range of IPR values in plot (d) is much smaller
than in plots (a-c). In all cases we have considered system size $L=3000$ and
square pulse driving.

In Figs. 9 (a) and (b), we show plots of $I_{m}^{(2)}$ versus quasienergy
$E_{F}$ (sorted in increasing order) for systems with $\gamma_{1}=1$,
$\gamma_{2}=-1$ and $+1$ respectively, $\omega=35$, $a=5$, $V_{0}=2.5$,
$L=3000$, and square pulse driving. Fig. 9 (c) shows a plot of $I_{m}^{(2)}$
versus energy $E$ (sorted in increasing order) for an undriven system with
$\gamma_{1}=1$, $\gamma_{2}=1$ (the relative sign of $\gamma_{1}$ and
$\gamma_{2}$ is unimportant in the absence of driving), $V_{0}=2.5$, and
$L=3000$. We have chosen $V_{0}=2.5$ so that $V_{0}>2\gamma_{1}$ and all the
states of the undriven system are localized. We note that the quasienergies in
plots (a) and (b) and the energies in plot (c) have almost the same values.
This is expected since driving at high frequencies should give the same
Floquet Hamiltonian as the undriven part of the Hamiltonian. Surprisingly,
however, the IPR values are not the same in the three plots. The IPR values
for the driven system with uniform hopping amplitudes differ only a little
from the IPRs of the undriven system, but the IPRs of the driven staggered
model ($\gamma_{2}=-\gamma_{1}$) differ substantially from those of the
undriven system. (This may be because driving has a smaller effect when
$\gamma_{2}=\gamma_{1}$ compared to $\gamma_{2}=-\gamma_{1}$, for reasons
explained after Eq. (22)). We observe that the undriven system (plot (c))
shows only localized states, but the driven staggered model (plot (a)) shows
that some of the states have very low IPR values and are therefore extended
states. Thus driving even at a high frequency seems to convert some of the
localized states to extended states, provided that
$a\gg\gamma_{1},~{}\gamma_{2}$ and $V_{0}$. (We have checked numerically that
if $a\ll\gamma_{1},~{}\gamma_{2}$ and $V_{0}$, then all the states remain
localized). This suggests that although terms of order $1/\omega$ and higher
in the Floquet Hamiltonian are small for large $\omega$, they have a
significant effect on the IPR values of the Floquet eigenstates. It is
possible that these small terms have a negligible effect on the quasienergies,
but they couple localized states which lie close to each other and this
hybridization produces extended states.

Figure 10: Plots of $E_{Fm}$ versus for $\omega$ for (a) $V_{0}=1.5$, (b)
$V_{0}=2.5$, and (c) $V_{0}=5.5$, for systems with $\gamma_{1}=1$,
$\gamma_{2}=-1$, $a=5$, size $L=2000$, and square pulse driving. The spectrum
of $E_{Fm}$ is gapless for intermediate $\omega$ and develops several gaps as
$\omega$ increases.

On the other hand, if we choose $V_{0}<2\gamma_{1}$ and all the other
parameters are the same as in Figs. 9 (a-c), we find that there is no
discernible differences between the spectra of quasienergies (or energies) and
IPR values in the three cases. All the states are extended in all three cases
(driven with $\gamma_{2}=\pm\gamma_{1}$ and undriven). Fig. 9 (d) shows a plot
of the energies and corresponding IPR values for an undriven system with
$\gamma_{1}=1$, $\gamma_{2}=-1$, and $V_{0}=1.5$. The IPR versus quasienergy
plots for the driven systems with $\gamma_{2}=\pm\gamma_{1}$ look the same and
are not shown here.

In Fig. 10, we show plots of all the quasienergies $E_{Fm}$ for systems with
$\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$, size $L=2000$, square pulse driving,
and (a) $V_{0}=1.5$, (b) $V_{0}=2.5$, and (c) $V_{0}=5.5$. We have chosen
these three values of $V_{0}$ for the following reason. For $V_{0}=1.5$, all
states are found to be extended. For $V_{0}=2.5$, various kinds of states can
coexist depending on the values of $\omega$, as we see in Fig. 4. For
$V-0=5.5$, we find re-entrant transitions as shown in Fig. 7. The plots in
Fig. 10 show that the spectrum of $E_{Fm}$ is generally gapless for
intermediate frequencies but several gaps appear as $\omega$ is increased.
This can be qualitatively understood from the FPT as follows. In the limit
that $\gamma_{1},~{}\gamma_{2},~{}V_{0}\ll\omega\ll a$, we saw in the
discussion given towards the end of Sec. II that the model reduces to one in
which the hopping amplitude is uniform and the quasiperiodic potential is
absent. This system clearly has no gaps in the spectrum. On the other hand,
when $\gamma_{1},~{}\gamma_{2},~{}V_{0}\ll a\ll\omega$, we saw that the model
reduces to an undriven system ($a=0$). Hence the quasienergies will approach
the energies of the undriven system. For the undriven Aubry-André model, it is
known that there is a large number (in fact, an infinite number) of energy
gaps for any non-zero value of $V_{0}$. aubry ; soko  It would be useful to
understand precisely how the different gaps appear successively as $\omega$ is
increased from intermediate to large values. [We have checked numerically that
for $V_{0}=0$, there are no gaps for any value of $\omega$ or $a$. Hence the
gaps seen in Fig. 10 are entirely due to the presence of a quasiperiodic
potential].

###  III.3 Floquet Hamiltonian in real space

To visualize the effects of localization, it is instructive to look at the
matrix elements of the Floquet Hamiltonian. Figure 11 shows the absolute
values of the matrix elements of $H_{F}(X,Y)$ in real space for systems with
$\gamma_{1}=1$, $\gamma_{2}=-1$, $V_{0}=2.5$, size $L=2000$, and sinusoidal
driving. $H_{F}(X,Y)$ is obtained by doing a Fourier transform of $H_{F}$ in
momentum space as found numerically from the Floquet operator $U$ (Figs. 11
(a-c)). The plots are shown as a function of $Y-X$ and $X$. If the system was
translation invariant, the plot would depend only on the relative coordinate
$Y-X$ and not on $X$. Indeed we see that the plots do not vary with $X$ at
length scales much larger than $1/\beta\simeq 1.62$ (which is the length scale
of variation of the quasiperiodic potential $\cos(2\pi\beta j)$). In terms of
the relative coordinate $Y-X$, the plots show a large spread in Fig. 11 (a)
corresponding to all states being extended, a smaller spread in Fig. 3 (b)
corresponding to a case where there are both localized and extended states,
and a very small spread in Fig. 11 (c) corresponding to all states being
extended. We find that $H_{F}(X,Y)$ obtained by doing a Fourier transform of
$H_{F}$ in momentum space as found from the first-order FPT gives very similar
results, and we do not show those plots here.

A more detailed explanation of the results shown in Fig. 11 is as follows. We
know that although the Hamiltonian of the undriven Aubry-André model is local
in real space, it exhibits a delocalized phase below a critical disorder
strength. We now look at the structure of matrix elements of the Floquet
Hamiltonian $H_{F}(X,Y)$ in real space. In all the plots in Fig. 11, we have
taken we consider $\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$, and $V_{0}=2.5$.
$V_{0}=2.5$ necessarily implies that the static model exhibits a localized
phase in the absence of driving. However, in the presence of driving and with
diminishing values of $\omega$, we observe a significant decrease in the value
of the diagonal elements $H_{F}(X,X)$. The values of these elements is found
to be about 2 for $\omega$=40, about $1.2$ for $\omega$=15.1, and about $0.35$
for $\omega$=5. The decreasing value of the diagonal elements implies that the
effective strength of the quasiperiodic potential is decreasing. Furthermore,
we find that there is a rapid spreading of the off-diagonal matrix elements
signifying a growing non-locality of $H_{F}$ as $\omega$ decreases. The
combination of these two effects leads the system towards increasing
delocalization.

Figure 11: Plots of the absolute values of the matrix elements of the Floquet
Hamiltonian $H_{F}(X,Y)$ in real space ($X$ and $Y$ represent space
coordinates), where the axes are chosen to be $X$ and $Y-X$. In all cases, we
considered $\gamma_{1}=1,~{}\gamma_{2}=-1$, $V_{0}=2.5$, system size $L=2000$,
and sinusoidal driving. Figures (a-c) are obtained from exact numerical
calculations. The parameters used are (a) $a=10,~{}\omega=5$, (b)
$a=5,~{}\omega=15.1$, and (c) $a=5,~{}\omega=40$. We note that plot (a) and
(d) is for a system with only extended states, plot (b) is for a system with
both extended and localized states, and plot (c) is for a system with only
localized states.

###  III.4 Spreading of a one-particle wave packet

We now consider a measure of delocalization given by the spreading of a one-
particles state which is initially localized at the middle site of system of
size $L$, namely, the site $j_{0}=L/2$. This initial state, denoted
$|\psi_{in}\rangle$, is evolved over $n$ time periods by acting on it with the
Floquet operator $n$ times. This gives

$|\psi_{nT}\rangle~{}=~{}U^{n}~{}|\psi_{in}\rangle.$ (35)

The root mean squared displacement is then given by

$\sigma(n)~{}=~{}[\sum_{j}~{}(j-j_{0})^{2}~{}|\psi_{nT}(j)|^{2}]^{1/2}.$ (36)

The growth of $\sigma(n)$ with $n$ gives an idea of how delocalized the system
is. Fig. 12 shows plots of $\sigma(n)$ for systems with $\gamma_{1}=1$,
$\gamma_{2}=\pm 1$ (uniform and staggered hopping amplitudes respectively),
$a=5$, $\omega=5$ and 30, various values of $V_{0}$, and square pulse driving.
(The system size $L$ is taken to be 6000). We see that the spreading of the
wave packet is always ballistic. The ballistic velocity, given by the slope of
$\sigma(n)$ versus $n$, is smaller for $\omega=30$ compared to $\omega=5$.
This is because driving with a very large frequency is equivalent to not
driving at all. In the absence of driving, it is known that a system with a
quasiperiodic potential with strength $V_{0}<2\gamma_{1}$ has only extended
states, while a system with $V_{0}>2\gamma_{1}$ has only localized states
aubry ; soko ; these results hold regardless of whether
$\gamma_{2}=\gamma_{1}$ or $\gamma_{2}=-\gamma_{1}$ since we can change the
sign of $\gamma_{2}$ by doing a unitary transformation which does not affect
$\gamma_{1}$ and $V_{0}$. Indeed we see in Figs. 12 (b) and (d), that the wave
packet spreads ballistically if $V_{0}<2$ but remains completely localized if
$V_{0}>2$. The situation at the lower frequency, $\omega=5$, is more
interesting. The system with staggered hopping, $\gamma_{2}=-\gamma_{1}$,
shows ballistic spreading for $V_{0}=1.5,~{}3$ and 6, and no spreading for
$V_{0}=9$. However, the system with uniform hopping, $\gamma_{2}=\gamma_{1}$,
always shows some ballistic spreading for all values of $V_{0}$. Furthermore,
the ballistic velocity is a non-monotonic function of $V_{0}$, being smaller
for $V_{0}=1.5$ and 9 compared to $V_{0}=3$ and 6. A similar non-monotonic
behavior of the ballistic spreading is seen in Fig. 13 for driving at an
intermediate frequency with $\omega=12$.

Figure 12: (a-b) Root mean squared displacement, $\sigma(n)$, versus the
stroboscopic evolution number $n$ for $\gamma_{1}=1$, $\gamma_{2}=-1$, $a=5$,
and $\omega=5$ and $\omega=30$ respectively, for a one-particle state which is
initially located at the middle site of a large system. (c-d) Plot of
$\sigma(n)$ versus $n$ for $\gamma_{1}=\gamma_{2}=1$ (uniform hopping), $a=5$,
and $\omega=5$ and $\omega=30$, respectively. All the figures are for system
size $L=6000$ and square pulse driving. Plots (a-b) show that compared to the
high-frequency regime (where the behavior is similar to an undriven system),
the intermediate-frequency regime shows a ballistic behavior up to a larger
value of $V_{0}$ for the case of staggered hopping compared to uniform
hopping. For the uniform hopping system at the lower value of $\omega=5$, plot
(c) exhibits a non-monotonic ballistic velocity of $\sigma(n)$ as a function
of $V_{0}$ before showing a completely localized behavior for large $V_{0}$.
Such a non-monotonic behavior is not seen at the higher value of $\omega=30$
as shown in plot (d).

##  IV Perturbation theory for a model with uniform hopping

In this section we will study a system with a uniform hopping amplitude,
$\gamma_{1}=\gamma_{2}=1$, which is driven by a square pulse. For uniform
hopping, the unit cell consists of a single site and we will denote the
fermionic operator as $c_{j}$. The Hamiltonian is given by

$\displaystyle H(t)$ $\displaystyle=$
$\displaystyle(1+f(t))\sum_{j}~{}(c_{j}^{\dagger}c_{j+1}+{\rm H.c.})$ (37)
$\displaystyle+~{}V_{0}~{}\sum_{j}\cos(2\pi\beta j)~{}c_{j}^{\dagger}c_{j},$

where $f(t)$ has the square pulse form described in Eq. (26). Note that for
$V_{0}=0$, the periodic driving would have no effect since the time-
independent and driven parts of the Hamiltonian commute with each other in
that case, and the Floquet Hamiltonian $H_{F}$ would just be given by
$\sum_{j}(c_{j}^{\dagger}c_{j+1}+{\rm H.c.})$ independently of the driving
frequency.

Figure 13: (a) Root mean squared displacement, $\sigma(n)$, versus the
stroboscopic evolution number $n$ from the exact numerical calculation with
$\gamma_{1}=\gamma_{2}=1$ (uniform hopping), $a=5$, and $\omega=12$. (b)
$\sigma(n)$ versus $n$ as obtained from the second-order Floquet Hamiltonian
in Sec. IV. Both figures are for $L=3000$ and square pulse driving. The two
figures agree qualitatively and indicate a non-monotonic ballistic velocity of
$\sigma(n)$ as $V_{0}$ increases.

We will now find the Floquet Hamiltonian $H_{F}$ in the high-frequency regime
using van Vleck perturbation theory rev4 ; rev6 . This is a perturbative
expansion in powers of $1/\omega$, and it has the advantage over the Floquet-
Magnus expansion that it does not depend on the phase of the driving protocol,
i.e., it is invariant under $f(t)\to f(t+t_{0})$. Furthermore, unlike the FPT
which gives us $H_{F}$ in momentum space, the van Vleck perturbation theory
gives us $H_{F}$ in real space. This makes it more suitable for studying the
dynamics of a wave packet.

The zeroth-order Floquet Hamiltonian given by van Vleck perturbation theory
turns out to be

$\displaystyle H_{F}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{T}~{}\int_{0}^{T}dt~{}H(t)$ $\displaystyle=$
$\displaystyle\sum_{j}(c_{j}^{\dagger}c_{j+1}+{\rm
H.c.})+V_{0}\sum_{j}\cos(2\pi\beta j)~{}c_{j}^{\dagger}c_{j}.$

The $m$-th Fourier component of $H(t)$ is given by

$\displaystyle H_{m}$ $\displaystyle=$
$\displaystyle\frac{2ia}{m\pi}\sum_{j}(c_{j}^{\dagger}c_{j+1}+{\rm
H.c.})~{}~{}{\rm for}~{}~{}m~{}~{}\text{odd},$ (39) $\displaystyle=$
$\displaystyle 0~{}~{}{\rm for}~{}~{}m~{}~{}\text{even ~{}and}~{}\neq 0,$

and $H_{0}=H_{F}^{(0)}$. We then find that the first-order Floquet
Hamiltonian, $H_{F}^{(1)}=\sum_{m\neq 0}\left[H_{-m},H_{m}\right]/(2m\omega)$,
is zero since $H_{-m}=(-1)^{m}H_{m}$ for all $m$. The second-order Floquet
Hamiltonian consists of two terms

$\displaystyle H_{F}^{(2)}$ $\displaystyle=$ $\displaystyle\sum_{m\neq
0}\frac{[[H_{-m},H_{0}],H_{m}]}{2m^{2}\omega^{2}}$ (40)
$\displaystyle+\sum_{m\neq 0}\sum_{n\neq
0}\frac{[[H_{-m},H_{m-n}],H_{n}]}{3mn\omega^{2}}.$

The first term in Eq. (40) takes the form

$\displaystyle H_{F}^{(2)}$ $\displaystyle=$ $\displaystyle
2t_{2}\sum_{j}~{}[\cos(2\pi\beta(j+1))+\cos(2\pi\beta(j-1))$ (41)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2\cos(2\pi\beta
j)]~{}c_{j}^{\dagger}c_{j}$
$\displaystyle+t_{2}\sum_{j}~{}[2\cos(2\pi\beta(j+1))-\cos(2\pi\beta(j+2))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\cos(2\pi\beta
j)]~{}(c_{j}^{\dagger}c_{j+2}+{\rm H.c.}),$

where $t_{2}=a^{2}T^{2}V_{0}/96$. The second term in Eq. (40) turns out to be
zero. The third-order Floquet Hamiltonian vanishes due to the symmetry given
in Eq. (29). In principal we can calculate higher order terms but such
corrections are expected to be small compared to $H_{F}^{(0)}$ and
$H_{F}^{(2)}$.

Fig. 13 shows a comparison between the results obtained for the root mean
squared displacement $\sigma(n)$ versus the stroboscopic evolution number $n$
as obtained from exact numerics and the second-order Floquet Hamiltonian
derived above, for systems with $a=5$, $\omega=12$, various values of $V_{0}$,
and square pulse driving. We find a qualitative agreement between the two sets
of results.

We can now understand better the results for staggered hopping and uniform
hopping shown in Figs. 12 and 13. Naively, one should expect that the
spreading of the wave function should diminish as the value $V_{0}$ of the
quasiperiodic potential increases. However, in the uniform hopping case and in
the presence of driving, we observe in Figs. 12 and 13 that the ballistic
spreading velocity first increases and then decreases (Fig. 12) and vice versa
(Fig. 13) with increasing $V_{0}$ depending on the values of $\omega$. It is
difficult to obtain any analytical insight for the first case (Fig. 12) since
the parameter values do not allow us to perform a perturbative expansion. The
second case (Fig. 13) can, however, be explained using the Floquet van Vleck
Hamiltonian described above. The zeroth-order term in this Hamiltonian is just
the Aubry-André model as expected. But the second-order term contains a second
nearest-neighbor quasiperiodic hopping as well as a quasiperiodic on-site
potential. Furthermore, both these terms depend on $V^{2}_{0}/\omega$. We
believe that these competing terms in $H_{F}$ can give rise to a non-monotonic
dependence of the velocity of spreading as a function of $V_{0}$. To support
this statement, we compare the wave packet dynamics obtained from the exact
numerical calculation with the one obtained from the van Vleck Hamiltonian in
Fig. 13. We observe that the two results qualitatively agree with each other.

##  V Discussion

We will now summarize our results. We considered a one-dimensional model with
time-independent staggered hopping amplitudes $\gamma_{1}$ and $\gamma_{2}$
and a periodically driven uniform hopping amplitude between nearest-neighbor
sites, and a quasiperiodic potential with strength $V_{0}$. We have studied
the effects of both sinusoidal driving and driving by a periodic square pulse.
While we have mainly studied the case of staggered hopping amplitudes,
$\gamma_{2}=-\gamma_{1}$, we have also looked at the case with uniform hopping
amplitudes, $\gamma_{2}=\gamma_{1}$, for comparison. (In the absence of
driving, both these cases reduce to the Aubry-André model, but we find some
significant differences between the two cases when they are driven). In the
limit that the driving amplitude $a$ and frequency $\omega$ are much larger
than $\gamma_{1},~{}\gamma_{2}$ and $V_{0}$, we analytically derived a Floquet
Hamiltonian $H_{F}$ to first-order in Floquet perturbation theory. We then
numerically computed the Floquet operator $U$. The eigenstates and eigenvalues
of $U$ were found, and various properties of the Floquet eigenstates were then
examined as follows.

We studied a generalized IPR, called $I_{m}^{(q)}$, of the Floquet eigenstates
(labeled as $m$), and found the exponent of their scaling with the system size
$L$, namely, $I_{m}^{(q)}\sim 1/L^{\eta_{q}}$. For standard extended states we
expect $\eta_{q}=q-1$ for all $q$, while standard localized states have
$\eta_{q}=0$ for all $q$. We found that while most states are either extended
or localized in a standard way, there are some multifractal states which show
an intermediate behavior with $0<\eta_{q}<q-1$ (we have looked at the cases
$q=2,~{}3$ and 4). These states typically appear along with re-entrant
transitions at intermediate frequencies $\omega$ and large values of $V_{0}$.
A study of the average Shannon entropy also suggests that there can be
different kinds of states which have very different degrees of localization.
In addition, we find re-entrant transitions between regions with extended,
localized and multifractal states as $\omega$ is varied. We note that re-
entrant transitions have been studied earlier in periodically driven non-
Hermitian systems with quasiperiodic potentials longwen .

Interestingly, we find that in the high-frequency limit where $\omega$ is much
larger than all the other parameters, $\gamma_{1},~{}\gamma_{2},V_{0}$ and
$a$, the quasienergies are almost the same as the energies for the undriven
system ($a=0$), but the corresponding IPRs can have quite different values
when $V_{0}$ is large and $\gamma_{2}=-\gamma_{1}$. Namely, when $V_{0}$ is
larger than the critical value $2\gamma_{1}$, we know that all the states of
the undriven system are localized. We then find that periodic driving of this
system at a high frequency does not change the spectrum of quasienergies
noticeably but significantly reduces some of the IPR values. Thus driving
seems to convert some of the localized states to extended states. The exact
mechanism by which this happens may be an interesting problem for future
studies.

In general, our numerical results show that there are many gaps in the
quasienergy which separate states with significantly different values of the
IPR (see Figs. 1 (e) and 2 (e) in particular). These mobility gaps appear due
to the presence of the quasiperiodic potential. The number of gaps increases
as the driving frequency is increased from intermediate to large values, as we
see in Fig. 10. It would be useful to study exactly why this happens.
Interestingly, our numerical results show only mobility gaps. We have not
found any examples of mobility edges which appear when there is a continuous
range of quasienergies, and somewhere within that range there is a particular
quasienergy which separates localized and extended states.

As a dynamical signature of localization, we have studied how a one-particle
state initialized at one particular site of the system evolves in time. For
all the parameter values that we have examined, we find that the mean squared
displacement always increases linearly in time. The ballistic velocity, given
by the slope of the displacement versus time, depends on the system
parameters. The velocity generally decreases as either $\omega$ or $V_{0}$ is
increased. However, for a system with uniform hopping amplitudes
($\gamma_{2}=\gamma_{1}$) and intermediate values of $\omega$, the velocity
shows a non-monotonic dependence on $V_{0}$.

There is some evidence of re-entrant transitions in a recent experiment in
which ultracold bosonic atoms are trapped in an optical lattice and the atoms
experience a quasiperiodic potential whose strength is given periodic
$\delta$-function kicks grover . The experiment finds regimes of parameter
space where there are multiple transitions between localized, extended and
multifractal states as the strength of the kicked quasiperiodic potential is
varied.

Acknowledgments

S.A. thanks MHRD, India for financial support through a PMRF. K.S. thanks DST,
India for support through SERB project JCB/2021/000030. D.S. thanks SERB,
India for funding through Project No. JBR/2020/000043.

## References

  * (1)
  * (2) S. Blanes, F. Casas, J. A. Oteo, and J. Ros, Physics Reports 470, 151 (2009).
  * (3) S. N. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1 (2010).
  * (4) L. D’Alessio and A. Polkovnikov, Ann. Phys. 333, 19 (2013).
  * (5) M. Bukov, L. D’Alessio and A. Polkovnikov, Adv. Phys. 64, 139 (2015).
  * (6) L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).
  * (7) T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. B 93, 144307 (2016).
  * (8) T. Oka and S. Kitamura, Annu. Rev. Condens. Matter Phys. 10, 387 (2019).
  * (9) S. Bandyopadhyay, S. Bhattacharjee and D. Sen, J. Phys. Condens. Matter 33, 393001 (2021); A. Sen, D. Sen, and K. Sengupta, J. Phys. Condens. Matter 33, 443003 (2021).
  * (10) T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010); T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Phys. Rev. B 84, 235108 (2011).
  * (11) N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490 (2011).
  * (12) M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, Phys. Rev. B 88, 155133 (2013); M. Thakurathi, K. Sengupta, and D. Sen, Phys. Rev. B 89, 235434 (2014).
  * (13) A. Kundu, H. A. Fertig, and B. Seradjeh, Phys. Rev. Lett. 113, 236803 (2014).
  * (14) F. Nathan and M. S. Rudner, New J. Phys. 17, 125014 (2015).
  * (15) B. Mukherjee, A. Sen, D. Sen, and K. Sengupta, Phys. Rev. B 94, 155122 (2016); B. Mukherjee, P. Mohan, D. Sen, and K. Sengupta, Phys. Rev. B 97, 205415 (2018).
  * (16) V. Khemani, A. Lazarides, R.Moessner, and S. L. Sondhi, Phys. Rev. Lett. 116, 250401 (2016).
  * (17) D. V. Else, B. Bauer, and C. Nayak, Phys. Rev. Lett. 117, 090402 (2016).
  * (18) J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe, Nature (London) 543, 217 (2017).
  * (19) A. Russomanno, A. Silva, and G. E. Santoro, Phys. Rev. Lett. 109, 257201 (2012).
  * (20) A. Lazarides, A. Das, and R. Moessner, Phys. Rev. E 90, 012110 (2014).
  * (21) T. Nag, S. Roy, A. Dutta, and D. Sen, Phys. Rev. B 89, 165425 (2014); T. Nag, D. Sen, and A. Dutta, Phys. Rev. A 91, 063607 (2015).
  * (22) A. Agarwala, U. Bhattacharya, A. Dutta, and D. Sen, Phys. Rev. B 93, 174301 (2016); A. Agarwala and D. Sen, Phys. Rev. B 95, 014305 (2017).
  * (23) D. J. Luitz, Y. Bar Lev, and A. Lazarides, SciPost Phys. 3, 029 (2017); D. J. Luitz, A. Lazarides, and Y. Bar Lev, Phys. Rev. B 97, 020303(R) (2018).
  * (24) R. Ghosh, B. Mukherjee, and K. Sengupta, Phys. Rev. B 102, 235114 (2020).
  * (25) A. Das, Phys. Rev. B 82, 172402 (2010); S. Bhattacharyya, A. Das, and S. Dasgupta, Phys. Rev. B 86, 054410 (2012); S. S. Hegde, H. Katiyar, T. S. Mahesh, and A. Das, Phys. Rev. B 90, 174407 (2014).
  * (26) S. Mondal, D. Pekker, and K. Sengupta, Europhys. Lett. 100, 60007 (2012); U. Divakaran and K. Sengupta, Phys. Rev. B 90, 184303 (2014).
  * (27) S. Lubini, L. Chirondojan, G.-L. Oppo, A. Politi, and P. Politi, Phys. Rev. Lett. 122, 084102 (2019).
  * (28) M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett. 110, 135704 (2013).
  * (29) A. Sen, S. Nandy, and K. Sengupta, Phys. Rev. B 94, 214301 (2016); S. Nandy, K. Sengupta, and A. Sen, J. Phys. A: Math. Theor. 51, 334002 (2018).
  * (30) For a review, see M. Heyl, Rep. Prog. Phys 81, 054001 (2018).
  * (31) C. Karrasch and D. Schuricht, Phys. Rev. B 87, 195104 (2013).
  * (32) J. N. Kriel and C. Karrasch, and S. Kehrein, Phys. Rev. B 90, 125106 (2014).
  * (33) F. Andraschko and J. Sirker, Phys. Rev. B 89, 125120 (2014).
  * (34) E. Canovi, P. Werner, and M. Eckstein, Phys. Rev. Lett. 113, 265702 (2014).
  * (35) S. Sharma, U. Divakaran, A. Polkovnikov, and A. Dutta, Phys. Rev. B 93, 144306 (2016); S. Bandyopadhyay, A. Polkovnikov, and A. Dutta, Phys. Rev. Lett. 126, 200602 (2021).
  * (36) M. Sarkar and K. Sengupta, Phys. Rev. B 102, 235154 (2020).
  * (37) S. E. Tapias Arze, P. W. Clayes, I. P. Castillo, and J.-S. Caux, SciPost Phys. Core 3, 001 (2020).
  * (38) S. Aditya, S. Samanta, A. Sen, K. Sengupta, and D. Sen, Phys. Rev. B 105, 104303 (2022).
  * (39) B. Mukherjee, S. Nandy, A. Sen, D. Sen, and K. Sengupta, Phys. Rev. B 101, 245107 (2020).
  * (40) B. Mukherjee, A. Sen, D. Sen, and K. Sengupta, Phys. Rev. B 102, 075123 (2020).
  * (41) B. Mukherjee, A. Sen, and K. Sengupta, Phys. Rev. B 106, 064305 (2022).
  * (42) A. Haldar, D. Sen, R. Moessner, and A. Das, Phys. Rev. X 11, 021008 (2021).
  * (43) M. Sarkar, R. Ghosh, A. Sen, and K. Sengupta, Phys. Rev. B 103, 184309 (2021).
  * (44) M. Sarkar, R. Ghosh, A. Sen, and K. Sengupta, Phys. Rev. B 105, 024301 (2022).
  * (45) S. Aubry and G. André, Ann. Israel Phys. Soc 3, 18 (1980).
  * (46) J. B. Sokoloff, Phys. Rep. 126, 189 (1985); J. M. Luck, J. Stat. Phys. 72, 417 (1993), and Phys. Rev. B 39, 5834 (1989); J. Bellissard, in From Number Theory to Physics, edited by M. Waldschmidt, P. Moussa, J. M. Luck, and C. Itzykson (Springer, Berlin, 1992).
  * (47) X. Li, X. Li, and S. Das Sarma, Phys. Rev. B 96, 085119 (2017); X. Li and S. Das Sarma, Phys. Rev. B 101, 064203 (2020).
  * (48) S. Roy, T. Mishra, B. Tanatar, and S. Basu, Phys. Rev. Lett. 126, 106803 (2021); A. Padhan, M. K. Giri, S. Mondal, and T. Mishra, Phys. Rev. B 105, L220201 (2022); S. Roy, S. Chattopadhyay, T. Mishra, and S. Basu, Phys. Rev. B 105, 214203 (2022).
  * (49) S. Roy, I. M. Khaymovich, A. Das, and R. Moessner, SciPost Phys. 4, 025 (2018).
  * (50) T. Shimasaki, M. Prichard, H. E. Kondakci, J. Pagett, Y. Bai, P. Dotti, A. Cao, T.-C. Lu, T. Grover, and D. M. Weld, arXiv:2203.09442v2.
  * (51) A. Soori and D. Sen, Phys. Rev. B 82, 115432 (2010).
  * (52) C. Castellani, and L. Peliti, J. Phys. A 19, L429 (1986).
  * (53) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
  * (54) J. Lindinger, A. Buchleitner, and A. Rodríguez, Phys. Rev. Lett. 122, 106603 (2019).
  * (55) L. Zhou, Phys. Rev. Research 3, 033184 (2021); L. Zhou and W. Han, Phys. Rev. B 106, 054307 (2022).