# Dynamical route to ergodicity and quantum scarring in kicked coupled top

Debabrata Mondal, Sudip Sinha, and Subhasis Sinha  Indian Institute of Science
Education and Research-Kolkata, Mohanpur, Nadia-741246, India

###### Abstract

Unlike classical system, understanding ergodicity from phase space mixing
remains unclear for interacting quantum systems due to the absence of phase
space trajectories. By considering an interacting spin model known as kicked
coupled top, we elucidate the manifestation of phase space dynamics on local
ergodic behavior of its quantum counterpart and quantum scarring phenomena. A
transition to chaos occurs by increasing the kicking strength, and in the
mixed phase space, the islands of regular motions within the chaotic sea
clearly exhibit deviation from ergodicity, which we quantify from entanglement
entropy and survival probability. Interestingly, the reminiscence of unstable
orbits and fixed points can be identified as scars in quantum states,
exhibiting athermal behavior and violation of Berry’s conjecture for ergodic
states. We also discuss the detection of quantum scars by a newly developed
method of ‘out-of-time-order correlators’, which has experimental relevance.

##  I Introduction

Ergodicity of quantum many body system is a complex phenomena which has
attracted significant interest in the recent years due to the advancement in
cold atom experiments to study the non-equilibrium dynamics of many
interacting particles Polkovnikov . Quantum ergodicity is one of the key
ingredients for understanding thermalization of isolated quantum system, which
is the foundation of statistical mechanics, although not been fully
understood. Unlike the case of classical ergodicity, which can be explained
from chaotic dynamics leading to phase space mixing Zaslavsky ; Eckmann ;
Casati , the route to ergodicity in an isolated quantum system has been a long
standing open problem. In this context, the eigenstate thermalization
hypothesis (ETH) Deutsch ; Srednicki  was proposed to explain ergodicity at
the level of individual eigenstates, and its connection with random matrix
theory (RMT) has also been explored in various quantum systems Izrailev_1 ;
Izrailev_2 ; Kafri . However, such mechanism does not provide a clear picture
of the underlying phase space mixing leading to the ergodic behavior of closed
quantum system in presence of interaction. Moreover, it is an important
question to ask, how the non uniform mixing in mixed phase space region
manifests in ergodicity of its quantum counterpart? The answer to this
question is important for understanding the route to the deviation from
ergodicity in an interacting quantum system. There are various examples of
interacting quantum systems, which fail to thermalize and exhibit such
deviation from ergodicity, the most popular being the systems showing many
body localization (MBL) MBL1 ; MBL2 ; MBL3 . Besides MBL, the non ergodic
phases have also been identified due to the presence of multifractal states
Altshuler ; Deng ; Fazio_bhm , which can also give rise to the anomalous
thermalization Luitz . A recent experiment on a chain of Rydberg atoms Bernien
revealed absence of thermalization and periodic revival for some special
initial state, which has been attributed to many body quantum scar (MBQS)
Turner ; Motrunich ; M_Lukin1 ; abanin . In addition, similar long lived
nonthermal excited states have also been observed in a recent experiment on
ultracold dipolar gas Lev . At the level of wavefunctions, the MBQS have also
been identified in different interacting quantum systems spin1 ; onsager_scars
; correlated_hopping ; fracton ; optical_lattice ; K_saito ; scars_mps ;
sengupta_scars ; AKLT ; AKLT2 ; mondal ; sinha1 . Originally, quantum scar has
been identified as reminiscence of classically unstable trajectories in a non
interacting system of chaotic billiards Heller . However, the deviation from
ergodicity due to the quantum scarring phenomenon and its connection with the
underlying dynamical behavior in an interacting quantum system is not very
clear and deserves more attention mondal ; sinha1 .

In this work, by considering a periodically driven collective spin model, we
explore the mixed phase dynamics and its manifestation in the ergodic behavior
of its quantum counterpart. The kicked coupled top (KCT) model consists of two
large spins interacting periodically among themselves, which has a suitable
classical limit, allowing us to study the phase space dynamics. By increasing
the kicking strength, KCT undergoes a smooth transition to chaos. Although,
there are several methods to detect the quantum signature of chaos from
eigenspectrum, they only provide an overall behavior without finer resolution
due to the absence of its direct correspondence with classical phase space. In
this context, it is a pertinent issue to investigate the ergodic behavior of a
quantum system corresponding to mixed phase space dynamics with coexistence of
regular region and chaotic sea, rather than featureless deep chaotic region.
In KCT model, using the spin coherent states, describing the phase space
semiclassically, we probe the local ergodic behavior of its quantum
counterpart from entanglement entropy and survival probability, revealing the
dynamical route to deviation from ergodicity. It is expected that with
increasing degree of chaos, the dynamics of two spins become more entangled
yielding enhanced entanglement entropy. Another property of ergodic evolution
is the loss of memory of initial state, which can be quantified from the
survival probability. In ergodic regime, the quantum states resemble the
properties of random states and both entanglement entropy and survival
probability converge to ergodic limit, independent of system parameters. From
these quantities, we detect the local ergodic behavior as well its deviation
due to the formation of quantum scar as a reminiscence of unstable dynamics.
To elucidate the dynamical route to quantum scarring in this model, we
simplified the dynamics into two classes corresponding to well known kicked
top (KT) model haake_original ; haake_book , and the instability generated
from mixing between them can lead to the formation of scars. We also
demonstrate detection of scars by using the method of ‘out-of-time-order
correlator’ (OTOC), which is a newly developed tool to diagnose chaos in
quantum systems stanford2 ; maldacena ; K_hashimoto ; Rozenbaum ; Garttner1 ;
butterfly_effect ; Swingle1 ; Fazio_otoc ; A_M_rey2 ; Santos_otoc ;
Garcia_mata ; sray1 ; lakshminarayan ; sudip_otoc  and implemented
experimentally otoc_exp_1 ; otoc_exp_2 . Such method has also been applied in
the context of black hole thermalization stanford2 ; maldacena , and
information scrambling Swingle1 , connecting the interdisciplinary areas of
research.

The rest of the paper is organized as follows. In Sec. II, we introduce the
Hamiltonian of KCT and discuss the Floquet formalism for stroboscopic
evolution of the corresponding operators. Next, in Sec. III, we derive the
classical map for large spin limit and analyze the model classically. The
fixed points and their stability is analyzed in subsection III.1. The
classification of the dynamics on reduced phase space and their correspondence
with effective KT model is discussed in subsection III.2. The onset of chaos
in KCT, as well the manifestation of phase space dynamics on quantum
ergodicity are presented in Sec. IV. We investigate the scarring phenomena in
Sec. V: by identifying the scarred eigenstates, discussed in details in
subsection V.1, and their detection using the newly developed technique of
OTOC in subsection V.2. Finally, in Sec. VI, we summarize the results and
discuss the possible experimental detection of scars. The detailed derivation
of stroboscopic evolution of spin operators is presented in appendix A. The
instability exponents of the unstable fixed points of both KCT and effective
KT model are compared in appendix B. In appendix C, the quantum scarring
phenomena in the effective KT corresponding to a dynamical class of KCT is
discussed in details.

##  II model

The periodically kicked coupled top (KCT) model ballentine ; KCT  is described
by the following Hamiltonian,

$\displaystyle\hat{\mathcal{H}}(t)$ $\displaystyle=$
$\displaystyle\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{c}(t)$ (1a)
$\displaystyle\hat{\mathcal{H}}_{0}$ $\displaystyle=$
$\displaystyle-\hbar\,\omega_{0}\,(\hat{S}_{1x}+\hat{S}_{2x})$ (1b)
$\displaystyle\hat{\mathcal{H}}_{c}(t)$ $\displaystyle=$
$\displaystyle-\hbar\,\frac{\mu}{S}\,\hat{S}_{1z}\,\hat{S}_{2z}\sum_{n=-\infty}^{\infty}\delta(t-nT)$
(1c)

where $\hat{S}_{ia}$ ($a=x,y,z$) represents the components of the spin
operators corresponding to two large spins ($i=1,2$) of equal magnitude $S$.
The Hamiltonian $\hat{\mathcal{H}}_{0}$ in Eq.(1b) describes the precession of
two non interacting spins around the $x$-axis with angular frequency
$\omega_{0}$, while the periodic kicking term with kicking strength $\mu$,
represented by $\hat{\mathcal{H}}_{c}(t)$ in Eq.(1c) periodically generates a
ferromagnetic interaction between them with time period $T$. In rest of the
paper and in all the figures, we scale energy (time) by $\omega_{0}$
($1/\omega_{0}$) and set $\hbar=1$, $T=1$.

The quantum dynamics of a periodically kicked system is governed by its
Floquet operator, describing the unitary time evolution between two successive
kicks. The Floquet operator $\hat{\mathcal{F}}$ can be constructed from the
free evolution $\hat{U}_{0}=e^{-\imath\hat{\mathcal{H}}_{0}T}$ governed by the
time independent Hamiltonian $\hat{\mathcal{H}}_{0}$ within a time period $T$,
followed by an unitary operator
$\hat{U}_{c}=e^{\imath\frac{\mu}{S}\hat{S}_{1z}\hat{S}_{2z}}$ describing the
instantaneous kicking. Therefore, the Floquet operator can be written as
haake_book ,

$\displaystyle\hat{\mathcal{F}}=\hat{U}_{c}\hat{U}_{0}=e^{\imath\frac{\mu}{S}\hat{S}_{1z}\hat{S}_{2z}}e^{\imath(\hat{S}_{1x}+\hat{S}_{2x})T}$
(2)

For a periodically driven quantum system, the stroboscopic time evolution of
an initial state $\ket{\psi(0)}$ can be written in terms of the Floquet
operator $\hat{\mathcal{F}}$ as,

$\displaystyle\ket{\psi(n)}=\hat{\mathcal{F}}^{n}\ket{\psi(0)}$ (3)

where $\ket{\psi(n)}$ is the state after $n^{\rm th}$ kick at time $t=nT$.
Using the Heisenberg picture, the stroboscopic time evolution of an operator
$\hat{\mathcal{A}}$ can also be written in terms of $\hat{\mathcal{F}}$ as,

$\displaystyle\hat{\mathcal{A}}^{(n+1)}=\hat{\mathcal{F}}^{\dagger
n+1}\hat{\mathcal{A}}\hat{\mathcal{F}}^{n+1}=\hat{\mathcal{F}}^{\dagger}\hat{\mathcal{A}}^{(n)}\hat{\mathcal{F}}$
(4)

where $\hat{\mathcal{A}}^{(n)}$ denotes the operator at time $t=nT$.
Similarly, for the present system, we obtain the stroboscopic evolution the
spin components (see Appendix A for derivation) by setting
$\hat{\mathcal{A}}=\hat{S}_{ia}$, which can be written as,

$\displaystyle\left[\hat{S}^{(n+1)}_{1x,1y,1z},\hat{S}^{(n+1)}_{2x,2y,2z}\right]^{\text{T}}=\hat{\mathcal{R}}\left[\hat{S}^{(n)}_{1x,1y,1z},\hat{S}^{(n)}_{2x,2y,2z}\right]^{\text{T}}$
(5)

where $\hat{S}_{ix,iy,iz}$ represents the array of corresponding spin
operators $(\hat{S}_{ix},\hat{S}_{iy},\hat{S}_{iz})$ and T denotes the
transpose of the vector. The matrix $\hat{\mathcal{R}}$ generating time
evolution can be represented in the block diagonal form as,

$\hat{\mathcal{R}}=\left(\begin{array}[]{cc}\hat{\mathcal{R}}_{1}&0\\\
0&\hat{\mathcal{R}}_{2}\\\ \end{array}\right)$ (6)

The different blocks $\hat{\mathcal{R}}_{i}$ corresponding to the two spins
(for $i=1,2$) are given by,

$\hat{\mathcal{R}}_{i}=\left(\begin{array}[]{ccc}\cos\hat{Q}_{\bar{i}}^{(n)}&\sin\hat{Q}_{\bar{i}}^{(n)}\cos
T&\sin\hat{Q}_{\bar{i}}^{(n)}\sin T\\\
-\sin\hat{Q}_{\bar{i}}^{(n)}&\cos\hat{Q}_{\bar{i}}^{(n)}\cos
T&\cos\hat{Q}_{\bar{i}}^{(n)}\sin T\\\ 0&-\sin T&\cos T\\\ \end{array}\right)$
(7)

where $\hat{Q}_{\bar{i}}^{(n)}=\bar{\mu}(\hat{S}^{(n)}_{\bar{i}z}\cos
T-\hat{S}^{(n)}_{\bar{i}y}\sin T)$ with $\bar{i}\neq i$ and $\bar{\mu}=\mu/S$.
In the large spin limit, using the above stroboscopic evolution, we can obtain
the classical map for the spins, which we discuss in the next section.

##  III Classical Analysis

The classical limit of the above model can be achieved for large spin of
magnitude $S$, and the corresponding classical dynamics can be studied for the
appropriately scaled variables $\hat{s}_{ia}=\hat{S}_{ia}/S$, which behave
classically, since the commutator
$[\hat{s}_{ia},\hat{s}_{jb}]=i\epsilon_{abc}\delta_{ij}\hat{s}_{ic}/S$
vanishes in the limit $S\rightarrow\infty$. Classically, the spin vectors can
be written as
$\vec{s_{i}}\equiv(s_{ix},s_{iy},s_{iz})=(\sin\theta_{i}\cos\phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i})$,
where $\theta$ and $\phi$ denote its orientation, alternatively which can also
be represented by the canonically conjugate variables $\phi_{i}$ and
$z_{i}=\cos{\theta_{i}}$. By using Eq.(5), the stroboscopic time evolution of
the corresponding classical spin variables can be written as a classical map,

$\displaystyle\left[s^{(n+1)}_{1x,1y,1z},s^{(n+1)}_{2x,2y,2z}\right]^{\text{T}}=\mathcal{R}\left[s^{(n)}_{1x,1y,1z},s^{(n)}_{2x,2y,2z}\right]^{\text{T}}$
(8)

where $\mathcal{R}$ is the same matrix defined in Eq.(6) and Eq.(7), except
the fact that the operators $\hat{S}_{ia}$ are now replaced by the classical
variables $s_{ia}$ and the coupling $\bar{\mu}$ becomes $\mu$. As a result,
the stroboscopic map in Eq.(8) becomes independent of $S$ and the condition
$s^{2}_{ix}+s^{2}_{iy}+s^{2}_{iz}=1$ is preserved for both the spins $(i=1,2)$
in the stroboscopic evolution.

###  III.1 Fixed points and their stability

The overall dynamical behavior is captured by analyzing the fixed points (FPs)
and their stability, which is also important for understanding the ergodic
behavior of the present model. The FPs can be obtained from the condition,
$s^{(n)}_{ia}=s_{ia}^{*}$ (for all $n$). By analyzing the classical map given
in Eq.(8), two trivial FPs T± are obtained and are given by,

$\displaystyle\\{s_{1x}^{*},s_{1y}^{*},s_{1z}^{*},s_{2x}^{*},s_{2y}^{*},s_{2z}^{*}\\}=\\{\pm
1,0,0,\pm 1,0,0\\}$ (9)

which remain stable for small kicking strengths $\mu<\mu_{b}=2\tan(T/2)$, as
shown in Fig.1(a) and 1(d). At the critical kicking $\mu_{b}$, both the T±
undergo a pitchfork bifurcation and eventually become unstable, each giving
rise to two new stable non trivial FPs with $s^{*}_{ia}\neq 0$, which are
denoted by NT${}^{\rm L(R)}_{\pm}$ (see Fig.1(a) and 1(e)), where the
superscripts $\rm L(R)$ represent two new branches after bifurcation. The non
trivial FPs NT${}^{\text{L}(\text{R})}_{\pm}$, and their corresponding spin
components can be obtained from the following equations,

$\displaystyle
s_{iz}^{*2}\left[1+\tan^{2}\left(\frac{T}{2}\right)\text{cosec}^{2}\left(\frac{\mu}{2}s_{\bar{i}z}^{*}\right)\right]=1$
(10a) $\displaystyle
s_{ix}^{*}=\tan\left(\frac{T}{2}\right)\text{cot}\left(\frac{\mu}{2}s_{\bar{i}z}^{*}\right)s_{iz}^{*}$
(10b) $\displaystyle s_{iy}^{*}=-\tan\left(\frac{T}{2}\right)s_{iz}^{*}$ (10c)

The sign of $s^{*}_{iy}$, $s^{*}_{iz}$ differs for the two bifurcated branches
L and R, whereas the sign of $s^{*}_{ix}$ differs for the $\pm$ branches (see
Fig.1(a)). As the kicking strength increases, the FPs
NT${}^{\text{L}(\text{R})}_{\pm}$ become unstable at $\mu_{u}$, after that a
period-doubling bifurcation at $\mu_{\text{\tiny TC}}=\pi/\cos(T/2)>\mu_{u}$
occurs, which leads to the formation of 2-cycles denoted by
TC${}^{\text{L}(\text{R})}_{2\pm}$ (see Fig.1(a) and 1(f)). A 2-cycle
describes the periodic oscillation between two specific phase space points,
which can be obtained from the condition $s^{(n+2)}_{ia}=s^{(n)}_{ia}$ for
large $n$. The spin configuration corresponding to the pair of points of the
2-cycles TC${}^{\text{L}(\text{R})}_{2\pm}$ are given by,

$\displaystyle
s^{\prime}_{1x}=-s^{\prime\prime}_{1x}=\sqrt{1-\left(\frac{\pi}{\mu}\right)^{2}\sec^{2}\frac{T}{2}}$
(11a) $\displaystyle
s^{\prime}_{1y}=s^{\prime\prime}_{1y}=\pm\tan\left(\frac{T}{2}\right)\frac{\pi}{\mu}$
(11b) $\displaystyle s^{\prime}_{1z}=s^{\prime\prime}_{1z}=\mp\frac{\pi}{\mu}$
(11c)

where the upper(lower) signs (in Eq.(11)(b,c)) represents the 2-cycles
originated from L(R) branches of the non trivial FPs. The components of the
other spin of the same 2-cycles TC${}^{\text{L}(\text{R})}_{2\pm}$ can be
obtained from the conditions $s_{2x}=s_{1x},\,s_{2y}=\pm s_{1y},\,s_{2z}=\pm
s_{1z}$, where $\pm$ denotes the 2-cycles originated from ${\rm NT}_{\pm}$. It
is important to note that, apart from kicking strength $\mu$, the structure of
FPs, 2-cycles and their stability strongly depend on the driving period $T$.
For example, from Eq.(11) it is evident that, TC2± exist only for $T<\pi$. We
point out that in the present work, we restrict our discussion only for $T=1$.

Further increasing the coupling $\mu$, another pair of 2-cycles denoted by
${\rm TC}_{1\pm}$ emerge from the FPs ${\rm T}_{\pm}$. The spin components
corresponding to one of the two points of the 2-cycles ${\rm TC}_{1\pm}$ are
given by,

$\displaystyle
s_{iz}^{{}^{\prime}2}\left[1+\cot^{2}\left(\frac{T}{2}\right)\text{cosec}^{2}\left(\frac{\mu}{2}s_{\bar{i}z}^{\prime}\right)\right]=1$
(12a) $\displaystyle
s_{ix}^{\prime}=-\cot\left(\frac{T}{2}\right)\cot\left(\frac{\mu}{2}s_{\bar{i}z}^{\prime}\right)s_{iz}^{\prime}$
(12b) $\displaystyle
s_{iy}^{\prime}=\cot\left(\frac{T}{2}\right)s_{iz}^{\prime}$ (12c)

Another point of these 2-cycles can be obtained from the condition,
$s^{\prime\prime}_{ix}=s^{\prime}_{ix}$,
$s^{\prime\prime}_{iy}=-s^{\prime}_{iy}$,
$s^{\prime\prime}_{iz}=-s^{\prime}_{iz}$. It is important to note that, for
KCT, these pair of 2-cycles remain always unstable and are not very important,
however they have significance in the context of dynamical classes, which is
discussed in the next subsection and in appendix C.

Apart from these, there is another pair of FPs denoted by FP-$\pi$, since the
relative angle between the two spins is $\pi$, and are given by,

$\displaystyle\\{s_{1x}^{*},s_{1y}^{*},s_{1z}^{*},s_{2x}^{*},s_{2y}^{*},s_{2z}^{*}\\}=\\{\pm
1,0,0,\mp 1,0,0\\}$ (13)

which remain unstable for all kicking strengths. The FP structure and their
stability with increasing kicking strength are summarized in Fig.1(a). The
details of the stability analysis and the instability exponents of unstable
FPs and 2-cycles are given in appendix B. Here we only focus on these FPs
which capture the essential features of the phase space, however with
increasing kicking strength, more structure in the FPs and periodic cycles can
be formed within a narrow range of $\mu$, that are not relevant for the
present analysis.

###  III.2 Dynamical classes and effective kicked top model

Since the model consist of two identical spins, the Hamiltonian and the
classical dynamics remains invariant under the exchange of spins
$\vec{S}_{1}\leftrightarrow\vec{S}_{2}$. As a consequence, in terms of the
redefined variables, $s_{a\pm}=(s_{1a}\pm s_{2a})/2$ $(a=x,y,z)$ or
equivalently, $z_{\pm}=(z_{1}\pm z_{2})/2$ and
$\phi_{\pm}=(\phi_{1}\pm\phi_{2})/2$, the dynamics can be categorised into two
subclasses with reduced phase space, namely: class I. for which
{$s_{x-}=0,s_{y+}=0,s_{z+}=0$} and class II. for which
{$s_{x-}=0,s_{y-}=0,s_{z-}=0$} holds, or equivalently both of them can be
written as $\\{z_{\pm}=0,\phi_{\pm}=0\\}$. Note that, for both the classes,
$s_{1x}=s_{2x}$. It can be verified from the classical map in Eq.(8), either
of the above conditions (class I or II) remain valid for all coupling constant
and the dynamics of the remaining variables reduces to that of the well known
kicked top (KT) model haake_original ; haake_book , with (anti)ferromagnetic
interaction corresponding to class (I)II.

Here we derive the equations of motion (EOM) corresponding to dynamical class
I and show its correspondence with the effective antiferromagnetic KT model.
Using the full dynamical equations (given in Eq.(8)) and the constraints of
the dynamical class I, the EOM of remaining variables ($s_{x+},s_{y-},s_{z-}$)
are given by,

$\displaystyle\left[s^{(n+1)}_{x+,y-,z-}\right]^{\rm
T}=\mathcal{R}_{-}\left[s^{(n)}_{x+,y-,z-}\right]^{\rm T}$ (14)

where the time evolution matrix $\mathcal{R}_{-}$ can be written as follows,

$\mathcal{R}_{-}=\left(\begin{array}[]{ccc}\cos Q_{-}^{(n)}&-\sin
Q_{-}^{(n)}\cos T&-\sin Q_{-}^{(n)}\sin T\\\ \sin Q_{-}^{(n)}&\cos
Q_{-}^{(n)}\cos T&\cos Q_{-}^{(n)}\sin T\\\ 0&-\sin T&\cos T\\\
\end{array}\right)$ (15)

with $Q^{(n)}_{-}=\mu(s^{(n)}_{z-}\cos T-s^{(n)}_{y-}\sin T)$. The above EOM
can be shown to be the same as that of an effective antiferromagnetic KT
model, which is described by the Hamiltonian,

$\displaystyle\hat{\mathcal{H}}(t)$ $\displaystyle=$
$\displaystyle-\hat{S}_{x}+\frac{\mu}{2S}\,\hat{S}^{2}_{z}\sum_{n=-\infty}^{\infty}\delta(t-nT)$
(16)

where $(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$ are the components of the spin
operator, which reduces to the classical variables
$s_{a}=\hat{S}_{a}/S\,\,(a=x,y,z)$ in the limit of $S\rightarrow\infty$. The
EOM corresponding to dynamical class I given in Eq.(14) can be obtained from
the classical map of the KT model given in Eq.(16), under the change of
classical variables $(s_{x},s_{y},s_{z})\rightarrow(s_{x+},s_{y-},s_{z-})$. In
a similar manner, one can also show, for the dynamical class II, the
correspondence $(s_{x+},s_{y+},s_{z+})\rightarrow(s_{x},s_{y},s_{z})$ yields
the EOM of a ferromagnetic KT model, where $\mu$ flips its sign in Eq.(16).

Figure 1: Classical analysis of kicked coupled top (KCT): (a-c) Bifurcation
diagram for different fixed points (FPs) and 2-cycles with increasing kicking
strength $\mu$ for (a) KCT, which is compared with that of effective kicked
top model with (b) antiferromagnetic (dynamical class I) and (c) ferromagnetic
(class II) interaction. The stable (unstable) branches of FPs are denoted by
solid black (red) lines, and the stable (unstable) branches of 2-cycles are
denoted by dashed black (red) lines. The different bifurcations and
instabilities are marked by the arrowheads. (d-f) The phase portrait of KCT in
$z_{1}$-$\phi_{1}$ plane for increasing $\mu$, pointing the various FPs and
2-cycles.

It is important to note that the actual phase space is not restricted by the
constraints of the dynamical classes and the presence of initial perturbations
violating the corresponding constraints leads to the mixing between the
classes. Even when the dynamics is restricted to a particular dynamical class,
the instabilities generated by the initially present small fluctuation can
lead to the deviation from the corresponding class. As a result, the actual
dynamics of the KCT model can deviate from that of the effective KT model. For
clarification, we have shown the FPs and their stability for both the
antiferromagnetic and ferromagnetic kicked top model (class I and II) in
Fig.1(b,c), and are compared with that of the KCT model in Fig.1(a). It is
evident from Fig.1(b,c), for dynamical classes I and II, the FPs and their
stability exhibit complementary behavior. As a result, the FPs which are not
present in the dynamical class I, such as NT+, are present in class II.
However, the FPs of both the effective KT models are present in the KCT model.
Due to the presence of perturbations violating the constraints, the stable FPs
and 2-cycles of a particular dynamical class become unstable in the KCT model
for certain range of kicking strength, such as the unstable 2-cycles TC1± in
KCT appear as stable 2-cycles in the corresponding KT model (see Fig.1(b,c)).
Such constraint violating fluctuations leading to the instability of the FPs
plays a crucial role in the ergodic properties and formation of quantum scars,
which is discussed in the later part of this work.

##  IV Onset of chaos and ergodic behavior

After the bifurcation of trivial FPs at $\mu_{b}$, more FP structures in the
phase space appear, however the regular region around them shrinks and the
trajectories in the remaining part become more irregular, as a result, a mixed
phase space behavior is observed for intermediate kicking strengths (see
Fig.2(b)). Further increasing the kicking strength, the stable islands become
unstable gradually, and the whole phase space is filled up with chaotic
trajectories eventually, as shown in Fig.2(c). Classically, the local chaotic
behavior in the phase space can be identified by non vanishing Lyapunov
exponent, which signals the exponential growth of initial perturbation with
time lichtenberg ; strogatz . In the present analysis, the Lyapunov exponent
$\lambda_{l}$ is numerically obtained by the method discussed in qr . Since,
the Lyapunov exponent in general can depend on the initial phase space point,
to quantify the overall chaotic behavior, we compute the averaged Lyapunov
exponent $\bar{\lambda}_{l}$ by averaging $\lambda_{l}$ over $\sim$ 4000
different initial phase space points. The onset of chaos in KCT is signalled
from the sharp growth of $\bar{\lambda}_{l}$ with increasing kicking strength
$\mu$ above $\mu_{b}$, as depicted in Fig.2(d). The onset of chaos triggers
the mixing in phase space, which is a key ingredient for classical ergodicity.

Figure 2: (a-c) Phase portraits on Bloch sphere for increasing $\mu$,
exhibiting onset of chaos. (d) Variation of average Lyapunov exponent
$\bar{\lambda}_{l}$ with increasing $\mu$. (e) Level spacing distribution of
eigenphases of Floquet operator $\hat{\mathcal{F}}$ for two different values
of $\mu$, exhibiting quantum signature of chaos. The solid red (dashed blue)
lines denote the Poisson (Wigner-Dyson) statistics. (f) Average ratio of
consecutive level spacings $\langle r_{\nu}\rangle$ with increasing $\mu$,
showing a crossover from Poisson to Wigner-Dyson statistics. In this and all
other figures, the quantum calculations are done for $S=20$ in KCT.

Usually, the quantum signature of chaos can be detected from spectral
statistics of the corresponding Hamiltonian. According to Berry Tabor’s
conjecture BT , Poisson distribution of energy level spacing implies regular
classical dynamics, whereas Bohigas-Giannoni-Schmit (BGS) conjecture BGS
suggests, Wigner-Dyson distribution of level spacing for a classically chaotic
system. For periodically driven quantum systems, one can analyze the spectral
statistics of eigenphases of the Floquet operator $\hat{\mathcal{F}}$. The
eigenspectrum of $\hat{\mathcal{F}}$ is obtained from diagonalization,
$\hat{\mathcal{F}}\ket{\phi_{\nu}}=e^{\imath\phi_{\nu}}\ket{\phi_{\nu}}$,
where $\phi_{\nu}$ and $\ket{\phi_{\nu}}$ are eigenphases and corresponding
eigenvectors of $\hat{\mathcal{F}}$, which contain relevant information
related to the dynamics and ergodic properties. Numerically, the
diagonalization of Floquet operator $\hat{\mathcal{F}}$ is done in the basis
of $\hat{S}_{iz}$.

In order to perform the spectral analysis corresponding to a particular
symmetry sector, we identify two types of symmetries in the KCT model. The
Hamiltonian in Eq.(1) remains invariant under the action of parity
$\hat{\Pi}=e^{\imath\pi(\hat{S}_{1x}+\hat{S}_{2x})}$ and spin exchange
($S_{1}\leftrightarrow S_{2}$) operator $\hat{\mathcal{O}}$ mondal , which
flips the indices of basis states $\ket{m_{1z},m_{2z}}$, where $m_{iz}$ are
the quantum numbers of $\hat{S}_{iz}$. Both the operators posses two
eigenvalues $\pm 1$, which we call as even (+1) and odd (-1). For spectral
statistics, we only consider the eigenphases of the Floquet operator, for
which the eigenvalue of $\hat{\Pi}(\hat{\mathcal{O}})$ are +1(+1). Next, the
eigenphases are arranged within the range $[-\pi$,$\pi]$ in ascending order to
compute the corresponding level spacings,
$\delta_{\nu}=\phi_{\nu+1}-\phi_{\nu}$. We calculate the normalized level
spacing distribution keeping the mean to be unity, following the procedure as
outlined in haake_book .

Figure 3: Reflection of phase space dynamics on local ergodic behavior of KCT
quantified in terms of entanglement entropy and survival probability. Vertical
columns: (a) Phase portraits for initial conditions belonging to dynamical
class II (ferromagnetic KT). (b) Phase portraits in presence of small initial
perturbations violating dynamical class II. (c) Color scaled plots of time
averaged deviation of entanglement entropy $\Delta S_{\rm en}$ and (d)
survival probability $\Delta F$ from their ergodic limit. Initial coherent
states correspond to dynamical class II (as in (a)). Due to the presence of
intrinsic quantum fluctuations in coherent states, the constraint of dynamical
class II is not maintained in quantum dynamics, thus the phase portraits in
(b) are reflected on (c) and (d). The circles correspond to the unstable FPs
in color scaled plots ((c) and (d)). The different rows correspond to (i)
$\mu=0.5$, (ii) $\mu=1.5$, (iii) $\mu=3.22$, (iv) $\mu=3.8$, (v) $\mu=4.34$.
In the quantum dynamics, the time averaging is done from $n=50$ to $n=70$.

As seen from Fig.2(e), the resulting level spacing ($\delta$) distribution of
eigenphases follows Poisson statistics, ${\rm P}_{\rm P}(\delta)=e^{-\delta}$
for smaller values of kicking strength, on the contrary, the spacing
distribution shows level repulsion and approaches to Wigner-surmise, ${\rm
P}_{\rm WS}(\delta)=(\pi\delta/2)e^{-\pi\delta^{2}/4}$ corresponding to
orthogonal class of RMT for larger values of kicking strength above $\mu_{b}$,
where the underlying phase space becomes fully chaotic. In addition, the
average ratio of consecutive level spacings, $\langle
r_{\nu}\rangle=\langle\rm min(\delta_{\nu},\delta_{\nu+1})/\rm
max(\delta_{\nu},\delta_{\nu+1})\rangle$ bogomonly_2  also exhibits crossover
from Poisson statistics with $\langle r_{\nu}\rangle\sim 0.386$ to that of
circular orthogonal ensemble (COE) of RMT with $\langle r_{\nu}\rangle\sim
0.527$ rigol_alessio  (see Fig.2(f)).

Although, the spectral statistics reveals the underlying signature of chaos at
the quantum level, the information about local chaotic behavior is still
missing due to the absence of phase space description in quantum mechanics. To
probe the local chaotic behavior, we use the prescription of spin coherent
states radcliffe ,

$\displaystyle\ket{\theta,\phi}=\left(\cos{\frac{\theta}{2}}\right)^{2S}\exp\left(\tan{\frac{\theta}{2}}e^{\imath\phi}\hat{S}_{-}\right)\ket{S_{z}=S}$
(17)

with $\theta$ and $\phi$ representing the orientation of the spin vector
$\vec{S}$, which provides a semiclassical description of phase space. To
investigate the local degree of ergodicity, we evolve a coherent state,
$\ket{\psi_{c}}\equiv\ket{\theta_{1},\phi_{1}}\otimes\ket{\theta_{2},\phi_{2}}$
corresponding to a particular phase space point for a sufficiently long time,
and analyze the different properties of the final state $\ket{\psi(n)}$.
First, we compute the reduced density matrix of the final state,
$\hat{\rho}_{S}={\rm
Tr}_{\bar{\mathcal{S}}}\left(\ket{\psi(n)}\bra{\psi(n)}\right)$ by integrating
out one of the spin sectors, which yields the entanglement entropy $S_{\rm
en}$ as,

$\displaystyle S_{\rm en}=-{\rm Tr}\hat{\rho}_{\mathcal{S}}{\rm
log}\hat{\rho}_{\mathcal{S}}$ (18)

It is expected that in the chaotic regime, the entanglement entropy increases
with enhanced degree of chaos vidmar_rigol ; Lewenstein ; S_ghose1 , and in
the extreme limit, it attains a maximum value $S_{\rm max}$ corresponding to a
completely random state page , which is given by,

$\displaystyle S_{\rm max}={\rm log}(2S+1)-1/2$ (19)

Another characteristic feature of ergodic evolution is the loss of memory of
the initial state, which can be quantified from survival probability. It is
defined as the overlap of the time evolved state $\ket{\psi(n)}$ with the
initial state $\ket{\psi(0)}$,

$\displaystyle F(n)=|\langle\psi(n)|\psi(0)\rangle|^{2}$ (20)

In the ergodic evolution, $F(n)$ decreases and at long time, saturates to
limit $F_{\rm COE}=3/\mathcal{N}$ obtained from RMT Izrailev_2 , with Hilbert
space dimension $\mathcal{N}=(2S+1)^{2}$. To probe the non ergodic behavior,
we focus on the deviation of entanglement entropy $\Delta S_{\rm
en}=|\bar{S}_{\rm en}-S_{\rm max}|$ and survival probability $\Delta
F=|\bar{F}-F_{\rm COE}|$ from their ergodic limit. To eliminate the effect of
temporal fluctuations, we obtain the time averaged value of the corresponding
quantities denoted by $\bar{S}_{\rm en}$ and $\bar{F}$, where the time
averaging is done over a certain interval towards the end of the stroboscopic
evolution. For understanding the local ergodic behavior in phase space and to
unveil its connection with the underlying classical dynamics, we compare
$\Delta S_{\rm en}$ and $\Delta F$ with the corresponding classical phase
portraits to investigate the dynamical route to local deviation from
ergodicity. For clarity, here we only consider the dynamical class II (or
equivalently the ferromagnetic KT model) defined by the constraint
{$z_{-}=0,\phi_{-}=0$}, as well for quantum evolution, we choose the initial
coherent states representing this class. We also investigate the changes in
the phase portrait due to the presence of the small perturbations violating
this constraint (see column 2 of Fig.3) and compare them with the phase
portrait of dynamical class II (see column 1 of Fig.3). This comparison of
classical phase portraits is important in the present context since such
violation of constraints is inevitable in quantum evolution of coherent states
of corresponding dynamical class, due to the presence of inherent quantum
fluctuations. The manifestation of classical phase space dynamics on ergodic
behavior of its quantum counterpart is evident from the comparison of $\Delta
S_{\rm en}$ and $\Delta F$ with the phase portraits for different values of
$\mu$, as depicted in Fig.3. The regular regions of phase space around the
stable FPs leads to the strong deviation from ergodic behavior, which is
quantified by enhancement of $\Delta S_{\rm en}$ and $\Delta F$, as seen in
Fig.3(c,d). In the mixed phase space, the regular regions correspond to
smaller $S_{\rm en}$ and larger $F$ values compared to the chaotic regions. It
is also evident from Fig.3, the dynamics in presence of the constraint
violating perturbations captures the ergodic behavior of its quantum
counterpart more accurately. Such local behavior of ergodicity quantified from
$\Delta S_{\rm en}$ and $\Delta F$ elucidates its underlying connection with
the corresponding dynamics.

##  V Quantum scars

In this section, we discuss the dynamical route to formation of quantum scars
and their identification. From comparison of first two columns in Fig.3, it is
clearly visible that, due to the presence of constraint violating
perturbations of dynamical class II, certain FPs such as ${\rm T}_{-}$ become
unstable and vanish from the phase portrait, however their reminiscence are
still visible in $\Delta S_{\rm en}$ and $\Delta F$ as a scar of corresponding
unstable FPs. As seen from Fig.3(d), $\Delta F$ is more capable of detecting
scars compared to $\Delta S_{\rm en}$, as the chaotic region increases. It is
important to note that, even when the FPs become unstable, the phase space
trajectories still have a tendency to localize around them leading to the
formation of quantum scars (see Fig.5(e,f)). The mixed phase space region
gives rise to fascinating ergodic behavior, since stable FPs surrounded by the
chaotic sea can coexist with the scars of unstable FPs, however the deviation
from ergodicity is more prominent for stable FPs than that of quantum scars.

###  V.1 quantum scars of unstable fixed points and 2-cycles

Figure 4: Identification of different types of scars in Floquet eigenstates.
(a-d) Overlap $|\langle\psi_{c}|\phi_{\nu}\rangle|^{2}$ of the coherent states
$\ket{\psi_{c}}$ corresponding to different unstable FPs and 2-cycles
(mentioned in the figure) with the Floquet eigenstates. The scarred
eigenstates with maximum overlap are marked by the arrowheads. (e-h) Husimi
distribution of the eigenstates having maximum overlap (as marked by the
arrowheads in (a-d)) respectively, revealing the scar of the corresponding
unstable FPs and 2-cycles. Figure 5: (a-b) The Shannon entropy $S_{\rm Sh}$ of
different Floquet eigenstates for two different values of $\mu$. The
horizontal dashed line represents the COE limit and the eigenstates containing
the scar of NT+ are marked by arrowheads. (c) The probability distribution
$P(\eta)$ for scarred eigenstates of NT+ (marked by arrows in (a-b)) for two
different $\mu$ values. The black dashed line corresponds to the Porter-Thomas
(PT) distribution. (d) The overlap $|\langle\psi_{c}|\phi_{\nu}\rangle|^{2}$
of coherent state corresponding to NT+ with different Floquet eigenstates for
different values of $\mu$ . The maximum overlap are marked by the arrowheads.
(e) Unstable trajectories around NT+ in presence of perturbations violating
class II. (f) Husimi distribution of the deviated state (marked by triangle in
(a)) depicting the scar of such unstable trajectory.

The scars of the unstable FPs, as identified from the deviation in
entanglement entropy $\Delta S_{\rm en}$ and survival probability $\Delta F$
shown in Fig.3(c,d)), can also be detected in the Floquet eigenstates. The
scarred eigenstates $\ket{\phi_{\nu}}$ can be identified from the large
overlap $|\langle\psi_{c}|\phi_{\nu}\rangle|^{2}\gg 1/\mathcal{N}$ abanin ;
sinha1 ; mondal  with the coherent state $\ket{\psi_{c}}$ representing
semiclassically the unstable FP of corresponding scar (see Fig.4(a-d)). On the
contrary, it is expected that such overlap becomes $\sim 1/\mathcal{N}$ in the
ergodic regime indicating complete delocalization. To visualise the scars, we
compute the Husimi distribution of the reduced density matrix
$\hat{\rho}^{\nu}_{\mathcal{S}}$ obtained from the scarred eigenstates
$\ket{\phi_{\nu}}$,

$\displaystyle
Q(\theta,\phi)=\frac{1}{\pi}\bra{\theta,\phi}\hat{\rho}^{\nu}_{\mathcal{S}}\ket{\theta,\phi}$
(21)

which describes the semiclassical phase space distribution. As shown in
Fig.4(e-h), the Husimi distribution of such eigenstates exhibit maximum
density around the unstable FPs, indicating a localization in phase space.
Note that, we have plotted the Husimi distributions in the
$z_{i}=\cos{\theta_{i}}$ and $\phi_{i}$ plane, to compare it with the
classical phase portraits. Following this prescription, we identify the scars
of trivial FPs T±, non trivial FPs NT±, FP-$\pi$ and 2-cycles TC2± shown in
Fig.1(a). Because of the complementary behavior of the dynamical classes, we
only show the scars corresponding to class II in Fig.4. Here we emphasize
that, these scars can also be observed in KT model with corresponding
dynamical class except the scar of FP-$\pi$. In appendix C, we discuss in
details such scarring phenomena in KT model, since the experimental
realization of this model opens up the possibility to detect the scars.

Rather than the entanglement entropy, the statistical analysis of the Floquet
eigenstates provides an effective way to distinguish the scarred states. For
this purpose, we decompose the Floquet eigenstates
$\ket{\phi_{\nu}}=\sum_{i}\phi^{i}_{\nu}\ket{i}$ in the computational basis
$\ket{i}$. In the chaotic regime, according to Berry’s conjecture berry_conj ,
the eigenstates behave as random states and the probability distribution of
their components $\eta=|\phi^{i}_{\nu}|^{2}\mathcal{N}$, follows the well
known Porter-Thomas (PT) distribution ${\rm
P}(\eta)=(1/\sqrt{2\pi\eta})\exp(-\eta/2)$ haake_book . Consequently, the
Shannon entropy $S_{\rm Sh}=-\sum_{i}|\phi^{i}_{\nu}|^{2}{\rm
log}|\phi^{i}_{\nu}|^{2}$ of such ergodic states attains the value ${\rm
log}(0.48\mathcal{N})$ corresponding to its COE limit Izrailev_2 ;
rigol_alessio . As the system approaches to chaos, the Shannon entropy $S_{\rm
Sh}$ of the Floquet eigenstates forms a band like structure around the COE
limit, however for some eigenstates, $S_{\rm Sh}$ is found to be significantly
lower than this limit, which we identify as eigenstates bearing scar, as shown
in Fig.5(a,b). Apart from the scars of the FPs and 2-cycles, we find other
type of scars, which resemble the shape of unstable orbits around such FPs, as
shown in Fig.5(e,f). As seen from Fig.5(c), unlike the ergodic states, the
eigenstates bearing scar deviate from the PT distribution, leading to the
violation of Berry’s conjecture mondal ; sinha1 . However, magnitude of such
deviation, as well as the overlap with the corresponding coherent state,
depends on the degree of scarring, which decreases with enhanced instability
of the underlying dynamics, as shown in Fig.5(d). Consequently, the scars
gradually disappear as the system enters into deep chaotic regime and
eventually becomes uniformly ergodic.

Figure 6: Dynamical signature of quantum scar corresponding to a 2-cycle
(TC2+) of KCT: (a) Stroboscopic time evolution ($t=nT$) of the Husimi
distribution exhibiting periodic oscillation of phase space density between
the fixed points of unstable TC2+ at $\mu=4.34$. The pink dashed lines denote
the two fixed points of this 2-cycle represented by $s^{\prime}$ ,
$s^{\prime\prime}$ and we choose the initial coherent state representing one
of them. (b) Overlap (survival probability) of the stroboscopically evolved
state $\ket{\psi(n)}$ with the coherent states
$\ket{\psi_{c}}=\ket{s^{\prime}}$ (red line) and with
$\ket{\psi_{c}}=\ket{s^{\prime\prime}}$ (blue line), corresponding to two
points of 2-cycle. Complementary behavior of both the overlaps reflects
periodic oscillation as observed in (b). The pink dashed line in (b)
represents the COE limit of survival probability.

Unlike the scar of FPs, the quantum scarring of 2-cycles such as TC2± has an
interesting dynamical feature, since it is the shortest orbit representing the
oscillation between two phase space points $s^{\prime}$ and
$s^{\prime\prime}$. Here, we discuss the dynamical manifestation of the
unstable 2-cycles TC2±. Starting from the initial coherent state representing
one of these points of the 2-cycle, we obtain the Husimi distribution of the
stroboscopically evolved state $\ket{\psi(n)}$ successively, exhibiting the
oscillation of phase space density between these two points. In Fig.6 (a),
such oscillations of Husimi distribution is shown for the 2-cycle TC2+. As a
result of the instability of this 2-cycle, the Husimi distribution spreads
out, however, the quantum scar can still be identified from the accumulation
of the phase space density around these points of TC2+. We also calculate the
overlap of the time evolved state $\ket{\psi(n)}$ with the coherent states
$\ket{s^{\prime}}$ and $\ket{s^{\prime\prime}}$ corresponding to two fixed
points of this 2-cycle. As depicted in Fig.6(b), the complementary behavior of
the oscillations of the overlaps clearly captures the dynamics between the two
points of unstable TC2+.

###  V.2 Signature of scars from FOTOC dynamics

In recent years, a technique known as ‘out-of-time-order correlator’ (OTOC)
has been extensively studied to probe quantum many body chaos and scrambling
phenomena stanford2 ; maldacena ; K_hashimoto ; Rozenbaum ; Garttner1 ;
butterfly_effect ; Swingle1 ; Fazio_otoc ; A_M_rey2 ; Santos_otoc ;
Garcia_mata ; sray1 ; lakshminarayan ; sudip_otoc . The OTOC for two operators
$\hat{W}$ and $\hat{V}$ is defined as,

$\displaystyle O(t)={\rm
Tr}\hat{\rho}_{0}\hat{W}^{\dagger}(t)\hat{V}^{\dagger}(0)\hat{W}(t)\hat{V}(0)$
(22)

where $\hat{W}(t)$ denotes the operator at time $t$ and $\hat{\rho}_{0}$ is
the initial density matrix. For unitary operators $\hat{W}$ and $\hat{V}$, the
growth rate of $1-{\rm Re}(O(t))$ can yield the Lyapunov exponent in quantum
systems Rozenbaum ; A_M_rey2 ; Santos_otoc , moreover its saturation value can
provide an alternate measure to quantify the degree of ergodicity Garcia_mata
; sray1 ; lakshminarayan ; sudip_otoc . For the pure states, the OTOC can be
generalized to ‘Fidelity-OTOC’ (FOTOC) $\mathcal{F}_{G}$ for a hermitian
operator $\hat{G}$, which is defined for $\hat{W}=e^{\imath\delta\phi\hat{G}}$
and $\hat{V}=\hat{\rho}_{0}=\ket{\psi(0)}\bra{\psi(0)}$ corresponding to the
initial state $\ket{\psi(0)}$ A_M_rey2 ; Santos_otoc . In the limit
$\delta\phi\ll 1$, the FOTOC can be written in terms of the fluctuation
$f_{G}$ of the corresponding operator $\hat{G}$,

$\displaystyle
1-\mathcal{F}_{G}\approx\delta\phi^{2}\left(\langle\hat{G}^{2}\rangle-\langle\hat{G}\rangle^{2}\right)\equiv\delta\phi^{2}f_{G}.$
(23)

which simplifies the computation of $\mathcal{F}_{G}$ and makes it suitable
for collective systems. In the perturbative regime ($\delta\phi\ll 1$), the
dynamics of FOTOC, as well the growth rate of $1-\mathcal{F}_{G}$ can be
captured from the time evolution of the corresponding fluctuation $f_{G}$,
which can successfully capture the instability exponent Santos_otoc  and
scrambling A_M_rey2  in quantum system.

Figure 7: (a-b) Comparison of FOTOC dynamics starting from initial coherent
states corresponding to different stable (solid) and unstable (dashed) FPs for
two different values of $\mu$. As NT+ becomes unstable, the effect of scarring
is reflected from the growth and oscillation of FOTOC (shown in (b)), which is
contrasted with its behavior when NT+ is stable (shown in (a)). (c-d) Dynamics
of FOTOC starting from the initial state representing one of the fixed points
of TC2+ (red line) for two different values of $\mu$ corresponding to stable
(solid) and unstable (dashed) TC2+. In all the cases, the green line denotes
FOTOC dynamics for a random initial coherent state belonging to the chaotic
region.

The dynamics of $1-\mathcal{F}_{G}$ for collective spin systems like KCT can
alternatively be studied from the fluctuation $f_{G}$ for suitable spin
operators with $\hat{G}=\hat{S}_{ia}/S$. To detect the dynamical signature of
quantum scars, we investigate the dynamics of total fluctuation $f_{is}$ of
all the components of a particular spin sector, which is given by,

$\displaystyle
f_{is}=\sum_{a=x,y,z}f_{isa}=\sum_{a=x,y,z}(\langle\hat{S}^{2}_{ia}\rangle-\langle\hat{S}_{ia}\rangle^{2})/S^{2}.$
(24)

In the mixed phase space region, we study the dynamics of total spin
fluctuations $f_{is}$ starting from the initial coherent states representing
the stable (unstable) FPs and 2-cycles. For stable FPs surrounded by the
regular regions of phase space, the $f_{is}$ exhibits oscillatory behavior
with very small amplitude, whereas for initial coherent state belonging to the
chaotic region, $f_{is}$ grows rapidly and saturates to unity. On the other
hand, the $f_{is}$ for unstable FPs exhibits an intermediate behavior with
slower growth rate and large oscillations (see Fig.7(a,b)) indicating the
scarring phenomena. Such behavior can be contrasted with that of stable FPs,
which can have relevance in experiments to distinguish quantum scars from
unstable FPs in the mixed phase space region. Similarly, we also study the
dynamics of $f_{is}$ for the stable and unstable 2-cycle TC2+ shown in Fig.7
(c,d), which exhibits larger growth rate as the 2-cycle becomes unstable. The
reduction of degree of scarring due to enhanced dynamical instability can also
be captured from FOTOC dynamics, since both the growth rate and saturation
corresponding to a scarred state increases as the system approaches to a more
chaotic regime with increasing kicking strength $\mu$.

##  VI Conclusion

In the present work, we investigated the local ergodic behavior of a coupled
top model subjected to periodic kicking and unveil its connection with the
underlying phase space dynamics, which plays a crucial role in the formation
of quantum scars. With increasing the kicking strength, the system undergoes a
crossover from regular to chaotic dynamics. In the mixed phase space, the
regular regions around the stable fixed points (FPs) and 2-cycles give rise to
strong deviation of entanglement entropy and survival probability from their
ergodic limit, revealing the local ergodic behavior. As the unstable FPs and
2-cycles disappear from the phase portrait, their reminiscence can still be
visible through deviation from the ergodic limit, exhibiting quantum scarring
phenomena. Also, we discuss the methods for identification of scars in Floquet
eigenstates from their statistical properties and Shannon entropy. Such
eigenstates carrying the scars exhibit violation of Berry’s conjecture in
contrast to the ergodic states. However, even after instability, the
trajectories remain localized near such unstable FPs, which essentially gives
rise to phase space localization in scarred states, as visible in Husimi
distribution. Apart from the FPs, we have also identified the scars of
2-cycles, giving rise to oscillation between two phase space points.

We have shown how quantum scars in mixed phase space can be distinguished from
both the stable FPs and ergodic states, by the FOTOC dynamics, which can serve
as an efficient method for its experimental detection. The implementation of
FOTOC has already been done in trapped ion simulators A_M_rey2 , which can
also serve as a platform to engineer collective spin models otoc_exp_2 . The
experimental realization of kicked top model in cold atom setup KT_exp , and
in superconducting qubits Neill  has opened up the immediate possibility to
investigate the quantum scarring phenomena.

## ACKNOWLEDGMENT

We thank Hans Kroha and Sayak Ray for comments and discussion.

##  Appendix A Derivation of stroboscopic evolution of spin operators

In the Heisenberg picture, the stroboscopic time evolution of the spin
operators can be written in terms of $\hat{\mathcal{F}}$ as,
$\hat{S}_{ia}^{(n+1)}=\hat{\mathcal{F}}^{\dagger
n+1}\hat{S}_{ia}\hat{\mathcal{F}}^{n+1}=\hat{\mathcal{F}}^{\dagger}\hat{S}^{(n)}_{ia}\hat{\mathcal{F}}$,
where $i=1,2$; $a=x,y,z$ and $\hat{S}_{ia}^{(n)}$ denotes the operator at time
$t=nT$. Following this prescription, here we only derive the equation of
motion for the $z$ component of spin $\hat{S}_{1z}$,

$\displaystyle\hat{S}_{1z}^{(n+1)}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{F}}^{\dagger
n}(\hat{\mathcal{F}}^{\dagger}\hat{S}_{1z}\hat{\mathcal{F}})\hat{\mathcal{F}}^{n}$
(25) $\displaystyle=$ $\displaystyle\hat{\mathcal{F}}^{\dagger
n}(e^{-\imath(\hat{S}_{1x}+\hat{S}_{2x})T}e^{-\imath\frac{\mu}{S}\hat{S}_{1z}\hat{S}_{2z}}\hat{S}_{1z}$
$\displaystyle
e^{\imath\frac{\mu}{S}\hat{S}_{1z}\hat{S}_{2z}}e^{\imath(\hat{S}_{1x}+\hat{S}_{2x})T})\hat{\mathcal{F}}^{n}$
$\displaystyle=$ $\displaystyle\hat{\mathcal{F}}^{\dagger
n}(e^{-\imath(\hat{S}_{1x}+\hat{S}_{2x})T}\hat{S}_{1z}e^{\imath(\hat{S}_{1x}+\hat{S}_{2x})T})\hat{\mathcal{F}}^{n}$
$\displaystyle=$ $\displaystyle\hat{\mathcal{F}}^{\dagger
n}(\hat{S}_{1z}\cos{T}-\hat{S}_{1y}\sin{T})\hat{\mathcal{F}}^{n}$
$\displaystyle=$
$\displaystyle\hat{S}^{(n)}_{1z}\cos{T}-\hat{S}^{(n)}_{1y}\sin{T}$

where we have used the commutation relation
$[\hat{S}_{ia},\hat{S}_{jb}]=\imath\epsilon_{abc}\delta_{ij}\hat{S}_{ic}$ and
the following operator identity,

$\displaystyle
e^{t\hat{X}}\hat{Y}e^{-t\hat{X}}=\hat{Y}+t[\hat{X},\hat{Y}]+\frac{t^{2}}{2}[\hat{X},[\hat{X},\hat{Y}]]+...$
(26)

In similar manner, the equations of motion for other components can be
derived. To obtain the classical map (see Eq.(8) of Sec. III), we have
redefined the operators $\hat{S}_{ia}$ as $\hat{s}_{ia}=\hat{S}_{ia}/S$, which
can be treated as classical variables in the limit $S\rightarrow\infty$, since
the commutation relation [$\hat{s}_{ia},\hat{s}_{jb}$] vanishes as $1/S$.

Figure B1: Dynamical instability of different FPs and 2-cycles (as mentioned
in the figure). The instability exponent $\lambda_{\rm I}$ with increasing
kicking strength $\mu$ for (a) KCT and (b) ferromagnetic KT model
corresponding to dynamical class II. Figure C2: Identification of different
scars in ferromagnetic KT model and their dynamical signature. (a1-a5) Overlap
of Floquet eigenstates $\ket{\phi_{\nu}}$ with coherent state representing
different FPs and 2-cycles of KT model. (b1-b5) Husimi distribution of the
eigenstate with maximum overlap (marked by arrowheads in (a1-a5)) revealing
the quantum scars of corresponding unstable FPs and 2-cycles. (c1-c5) Survival
probability $|\langle\psi(n)|\psi(0)\rangle|^{2}$ where $\ket{\psi(0)}$
represents the initial coherent state of the above mentioned unstable FPs
exhibiting revival phenomena due to scarring. The last column (d1-d3) shows
the scarring of TC1- in KCT, where the same quantities are compared with that
of the ferromagnetic KT model shown in column (a5-c5).

##  Appendix B Stability analysis

The stability of the fixed points (FPs) and 2-cycles can be analyzed by
linearizing the classical map given in Eq.(8) for small fluctuation around
them. Following the standard procedure in lichtenberg ; strogatz , we
construct the Jacobian matrix ${\rm J}$, whose matrix elements are given by
${\rm J}_{\alpha\beta}=\partial s^{(n+1)}_{\alpha}/\partial s^{(n)}_{\beta}$,
where $n$ is the stroboscopic time and $\alpha,\beta=1,2...,6$ are the indices
of the array $s=\\{s_{1x},s_{1y},s_{1z},s_{2x},s_{2y},s_{2z}\\}$ representing
the phase space point of the two spin system. We calculate the instability of
an unstable FP represented by $s^{*}$ from the instability exponent
$\lambda_{I}=\ln(|j_{m}|)>0$, where $j_{m}$ is the eigenvalue of the Jacobian
matrix J($s^{*}$) with maximum magnitude, evaluated at $s^{*}$. Similarly, the
stability of a 2-cycle can be obtained from the matrix $\tilde{\rm J}={\rm
J}(s^{\prime}){\rm J}(s^{\prime\prime})$, where the Jacobian matrices ${\rm
J}$ are evaluated at the corresponding fixed points $s^{\prime}$ and
$s^{\prime\prime}$ of the 2-cycle. The corresponding instability exponent of
the 2-cycle is given by $\lambda_{\text{I}}=(1/2)\ln(|\tilde{j}_{m}|)>0$,
where $\tilde{j}_{m}$ is the eigenvalue of matrix ${\tilde{\rm J}}$ with
maximum magnitude. The stability of FP (2-cycles) is ensured if the magnitude
of all eigenvalues of ${\rm J}$ ($\tilde{\rm J}$) are unity lichtenberg . In
the present case, as a result of the constraint,
$s^{2}_{ix}+s^{2}_{iy}+s^{2}_{iz}=1$ (for $i=1,2$), the magnitude of two
eigenvalues of ${\rm J}$ and $\tilde{\rm J}$ always remain unity. We compute
the instability exponents $\lambda_{\text{I}}$ of the FPs and 2-cycles of KCT,
that we discussed in Sec.III, with increasing kicking strength $\mu$ (see
Fig.B1(a)). We also compare them with the instability of the FPs and 2-cycles
of ferromagnetic KT model (corresponding to dynamical class II), shown in
Fig.B1(b). Here it is important to note that the 2-cycle TC1± is present in
KCT but remains always unstable.

Figure C3: Signature of scars of 2-cycles in ferromagnetic KT model from FOTOC
dynamics: (a-b) Comparison of FOTOC dynamics starting from initial coherent
states corresponding to 2-cycles TC2+ (red line) and TC1- (blue line) for two
different values of $\mu$. The stable (unstable) 2-cycles are shown by solid
(dashed) line. In both the figures, the green line represents the same for the
initial coherent state belonging to the chaotic regime. Scarring of TC1- is
captured from the larger growth rate of FOTOC as it becomes unstable (shown in
(b)), which can be contrasted to its behavior when it is stable (shown in
(a)). With increasing $\mu$, the FOTOC for unstable TC2+ (shown in (b))
becomes almost similar to that of an ergodic state showing the reduction of
degree of scarring.

##  Appendix C quantum scars in ferromagnetic kicked top model

As shown in subsection III.2, the dynamics of kicked coupled top (KCT) can be
divided into two classes (I)II corresponding to (anti)ferromagnetic kicked top
(KT) model. Similar scarring phenomena can also be observed in the KT model,
which we discuss in this appendix. Here we analyze the scars of unstable FPs
and 2-cycles corresponding to the ferromagnetic KT model, which is shown in
Fig.C2. The scarred eigenstates $\ket{\phi_{\nu}}$ are identified from the
large overlap with the coherent states $\ket{\psi_{c}}$ representing the
unstable FPs and 2-cycles (see Fig.C2(a1-a5)). The scars in such eigenstates
can also be visualized from the Husimi distribution localized around those FPs
and 2-cycles, as depicted in Fig.C2(b1-b5). Here we have identified scars of
trivial FPs T±, non trivial FPs NT+ and the 2-cycles TC2+, TC1- (see Fig.1(c)
for the fixed points of ferromagnetic KT model), which also manifest revivals
in the corresponding survival probabilities $F(n)$ (see Fig.C2(c1-c5)). We
also emphasize, although the 2-cycle TC1- is present in both the ferromagnetic
KT as well as KCT model, the degree of scarring in KCT is weaker due to larger
instability generated because of mixing between the two dynamical classes
(class I and II), as reflected from the comparison of Husimi distributions
shown in Fig.C2(b5) and C2(d2). Such scars can also be identified from FOTOC
dynamics, which is shown for 2-cycles TC1- and TC2+. As shown from the
comparison between Fig.C3(a,b), it is evident that the onset of dynamical
instability of TC1- leads to a rapid enhancement in growth and magnitude of
FOTOC, which is however slower than an ergodic state. As a result of enhanced
instability, the degree of scarring of TC2+ reduces and corresponding FOTOC
becomes almost indistinguishable from that of an ergodic state, which is shown
in Fig.C3(b). Here we point out that the KT model has already been realized
experimentally in cold atom systems KT_exp , as well in superconducting qubits
Neill , which opens up an immediate possibility to investigate such quantum
scarring phenomena, particularly the scar of 2-cycles can also be diagnosed
from the FOTOC dynamics.

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