# Scaling laws of the out-of-time-order correlators at the transition to the
spontaneous $\cal{PT}$-symmetry breaking in a Floquet system

Wen-Lei Zhao<EMAIL_ADDRESS>School of Science, Jiangxi University of
Science and Technology, Ganzhou 341000, China  Ru-Ru Wang  School of Science,
Jiangxi University of Science and Technology, Ganzhou 341000, China  Han Ke
School of Science, Jiangxi University of Science and Technology, Ganzhou
341000, China  Jie Liu<EMAIL_ADDRESS>Graduate School, China Academy of
Engineering Physics, Beijing 100193, China  HEDPS, Center for Applied Physics
and Technology, and College of Engineering, Peking University, Beijing 100871,
China

###### Abstract

We investigate both numerically and analytically the dynamics of out-of-time-
order correlators (OTOCs) in a non-Hermitian kicked rotor model, addressing
the scaling laws of the time dependence of OTOCs at the transition to the
spontaneous $\mathcal{PT}$ symmetry breaking. In the unbroken phase of
$\mathcal{PT}$ symmetry, the OTOCs increase monotonically and eventually
saturate with time, demonstrating the freezing of information scrambling. Just
beyond the phase transition points, the OTOCs increase in the power-laws of
time, with the exponent larger than two. Interestingly, the quadratic growth
of OTOCs with time emerges when the system is far beyond the phase transition
points. Above numerical findings have been validated by our theoretical
analysis, which provides a general framework with important implications for
Floquet engineering and the information scrambling in chaotic systems.

###### pacs:

03.65.-w, 03.65. YZ, 05.45.-a, 05.45.Mt

##  I Introduction

Non-Hermiticity has been regarded as a fundamental modification to the
conventional quantum mechanics Markum99 ; Stephanov99 ; Berry2004 ; Graefe2008
; Rotter2009 ; Ott2013 ; Harsh2014 , a subclass of which with $\cal{PT}$
symmetry even displays the transition from the real energy spectrum to complex
one. Such intrinsic spontaneous $\cal{PT}$-symmetry breaking occurs at the
exceptional points (EP), at which both the eigenstates and eigenvalues
coalesce Bender1998 ; Bender2002 ; Mosta2002 ; Bender2007 ; Mosta2010 ;
Harsh2010 ; Ganainy2018 ; Klaiman2008 ; Musslimani2008 . The existence of EP
leads to rich physics, such as the enhancement of precision in quantum sensors
Cai20 , the topological phase transition Bergholtz21 ; ZhongWang18 ; Mzhao1 ;
FYu ; Mzhao2 , the nonadiabatic transition FengYu21 ; WYWang22 , and the
unidirectional propagation of light YinHuang , just to name a few. Theoretical
advances have enabled exponential realizations of $\cal{PT}$ symmetric systems
in various fields, such as optical settings Sergi2014 ; Makris2008 ;
Ganainy2007 ; Guo2009 ; Regensburger2012 ; Hodaei2014 ; Feng2014 ; Jiahua2016
; Longhi2009 ; Longhi2010 ; Ruter2010 ; YongmeiXue , electronic circuits
Stegmaier2021 , and optomechanical systems Luxy15 . Moreover, the extension of
Floquet-driven systems to the $\mathcal{PT}$ symmetric regime has opened up
unique opportunities for understanding fundamental concepts such as quantum
chaos Bender09  and quantum-classical transition Bender10 ; wlzhao23a .
Interestingly, chaos is found to facilitate the scaling law of the spontaneous
$\mathcal{PT}$ symmetry breaking in a $\mathcal{PT}$ symmetric kicked rotor
(PTKR) model West2010 . This system even displays ballistic energy diffusion
Longhi2017  and the quantized acceleration of momentum current Zhao19 , which
enriches our understanding on the unique transport phenomena in the presence
of chaos.

The dynamics of OTOCs, originally introduced by Lakin et al., in the study of
quasiclassical theory of superconductivity Larkin1969 , has received extensive
studies in the fields of high energy physics Roberts2016 ; Maldacena2016 ;
Polchinski2016 , condensed matter physics Bohrdt2016 ; Garttner2017 ;
Banerjee2017 ; Shen2017 ; Fan2017 ; Huang2017  and quantum information Li2017
; Weinstein22 ; Hu23 . It has been found that OTOCs can effectively detect
quantum chaos Pappalardi22 ; GMata18 ; JiaoziWang21 ; Kidd2021 , quantum
thermalization Balachandran21 , and information scrambling Harris2022 ;
Roberts2022 ; Zhang2019 ; Yan2020 ; Patel2017 ; WLZhao23 . In the
semiclassical limit, the exponential growth of OTOCs is governed by the
Lyapunov exponent of classical chaos, which demonstrates a route of quantum-
classical correspondence Hashimoto2017 . In Floquet-driven systems, OTOCs has
been used to diagnose dynamical quantum phase transition Zamani2022  and
entanglement Martin2018 ; Lewis-Swan2019 . Intrinsically, we previously found
a quantized response of OTOCs when varying the kicking potential of the PTKR
model WlZhao22 . State-of-art experimental advances have observed different
kinds of OTOCs in the setting of nuclear magnetic resonance Wei2018 ; Nie2019
, trapping ions Landsman2019  and qubit under Floquet engineering Zhao2021 .

In this context, we both numerically and analytically investigate the dynamics
of OTOCs when the PTKR model is in different phases of $\cal{PT}$-symmetry. We
use a machine learning method, namely a long short-term memory network (LSTM),
to classify the phase diagram of $\cal{PT}$-symmetry breaking and extract the
phase boundary in a wide range of system parameters. We find that in the
unbroken phase of $\mathcal{PT}$ symmetry, OTOCs increase monotonically with
time evolution and eventually saturate, demonstrating the freezing of operator
growth. We analytically prove that the saturation of OTOCs is a power-law
function of the real part of the kicking potential. In the broken phase of the
$\mathcal{PT}$ symmetry, we find a power-law increase of OTOCs with time, for
which the characteristic exponent is larger than two when the system is just
beyond the phase transition point, and is equal to two for the system far
beyond the phase transition point. Through the detailed analysis of the
wavepacket’s dynamics in the time reversal process, we uncover the mechanisms
of both the dynamical localization and the power-law increase of OTOCs. Our
investigations reveal that the dynamics of OTOCs can be utilized to diagnose
spontaneous $\cal{PT}$-symmetry breaking.

The paper is organized as follows. In Sec. II, we describe the PTKR model and
show the scaling-law of spontaneous $\cal{PT}$-symmetry breaking. In Sec. III,
we show the scaling-laws of the dynamics of OTOCs at the transition to the
$\cal{PT}$-symmetry breaking. Sec. IV contains the theoretical analysis of the
scaling-laws of OTOCs. The conclusion and discussion are presented in Sec. V.

##  II Transition to spontaneous $\cal{PT}$-symmetry breaking in Floquet
systems

###  II.1 Model

The Hamiltonian of the PTKR model in dimensionless unites reads

$\textrm{H}=\frac{p^{2}}{2}+V(\theta)\sum_{n}\delta(t-t_{n})\;,$ (1)

where the kicking potential
$V(\theta)=K\left[\cos(\theta)+i\lambda\sin(\theta)\right]$ satisfies the
$\cal{PT}$-symmetric condition $V(\theta)=V^{*}(-\theta)$ Longhi2017 ;
West2010 ; Zhao19 . The parameters $K$ and $\lambda$ indicate the strength of
the real and imaginary parts of the kick potential, respectively. The
$p=-i\hbar_{\text{eff}}{\partial}/{\partial\theta}$ is angular momentum
operator, $\theta$ is the angle coordinate, and $\hbar_{\text{eff}}$ denotes
the effective planck constant. The time $t_{n}(=0,1,2\ldots)$ is integer,
indicating kicking numbers. The eigenequation of angular momentum operator is
$p|\varphi_{n}\rangle=n\hbar_{\text{eff}}|\varphi_{n}\rangle$ with eigenstate
$\langle\theta|\varphi_{n}\rangle=e^{in\theta}/\sqrt{2\pi}$ and eigenvalue
$p_{n}=n\hbar_{\text{eff}}$. On the basis of $|\varphi_{n}\rangle$, an
arbitrary quantum state can be expanded as
$|\psi\rangle=\sum_{n}\psi_{n}|\varphi_{n}\rangle$.

For time-periodic systems, i.e., $\textrm{H}(t+T)=\textrm{H}(t)$, the Floquet
theory predicts the eigenequation of the evolution operator
$U|\psi_{\varepsilon}\rangle=e^{-i\varepsilon}|\psi_{\varepsilon}\rangle$,
where the eigenphase $\varepsilon$ is referred to as quasienergy. One-period
time evolution of a quantum state of the PTKR system is given by
$|\psi(t_{j+1}\rangle=U|\psi(t_{j}\rangle$ with the Floquet operator

$U=U_{f}U_{K}=\exp\left(-\frac{i}{\hbar_{\text{eff}}}\frac{p^{2}}{2}\right)\exp\left[-\frac{i}{\hbar_{\text{eff}}}V(\theta)\right]\;.$
(2)

This demonstrates that in numerical simulations, one period evolution is split
into two steps, namely the kicking evolution $U_{K}$ and the free evolution
$U_{f}$. The kick evolution is realized in angle coordinate space, i.e.,
$\psi^{\prime}(\theta)=U_{K}(\theta)\psi(\theta,t_{j})$. Then, one can utilize
the fast Fourier transform to change the state $\psi^{\prime}(\theta)$ to
angular momentum space, thereby obtaining its component $\psi^{\prime}_{n}$ on
the eigenstate $|n\rangle$. Finally, the free evolution is conducted in
angular momentum space, i.e.,
$\psi_{n}(t_{j+1})=U_{f}(p_{n})\psi^{\prime}_{n}$. By repeating the same
procedure, one can get the quantum state at arbitrary time Casati79 .

###  II.2 Spontaneous $\cal{PT}$-symmetry breaking

It is straightforward to prove that the Floquet operator of the PTKR satisfies
the $\cal{PT}$ symmetry
$U=(\mathcal{P}\mathcal{T})^{\dagger}U\mathcal{P}\mathcal{T}$, where $\cal{P}$
and $\cal{T}$ are the parity and time reversal operators, respectively. Based
on conventional understanding of quantum mechanics, one asserts that the two
operators, i.e., $U$ and $\cal{PT}$ have simultaneous eigenstates, that is to
say, the quasieigenstate $|\psi_{\varepsilon}\rangle$ is also the eigenstate
of the $\cal{PT}$ operator, i.e.,
$\cal{PT}|\psi_{\varepsilon}\rangle=\pm|\psi_{\varepsilon}\rangle$. This
conclusion is indeed valid for positive quasienergies $\varepsilon>0$.
However, a notable feature of the PTKR system is that complex quasienergies
$\varepsilon=\varepsilon_{r}\pm\varepsilon_{i}$ emerge when the strength of
the imaginary part of the complex potential exceeds a threshold value, i.e.,
$\lambda>\lambda_{c}$ Longhi2017 ; West2010 ; Zhao19 . The threshold value
$\lambda_{c}$ is just the exceptional points of the system. It can be proven
that the quasieigenstate $|\psi_{\varepsilon}\rangle$ is no longer an
eigenstate of the $\cal{PT}$ operator due to the complex quasienergies, thus
demonstrating the spontaneous $\mathcal{PT}$-symmetry breaking. An intrinsic
quality of the PTKR system is that $\cal{PT}$ symmetry is helpful in
protecting the real spectrum of the Floquet operator.

We assume that the initial state is expanded as
$|\psi(t_{0})\rangle=\sum_{\varepsilon}\rho_{\varepsilon}|\psi_{\varepsilon}\rangle$.
Then, after the $n$th kick, the quantum state has the expression
$|\psi(t_{n})\rangle=U^{t_{n}}|\psi(t_{0})\rangle=\sum_{\varepsilon}\rho_{\varepsilon}e^{-i\varepsilon_{r}t_{n}}e^{\varepsilon_{i}t_{n}}|\psi_{\varepsilon}\rangle$,
whose norm $\mathcal{N}=\langle\psi(t_{n})|\psi(t_{n})\rangle$ exponentially
increases with time due to positive $\varepsilon_{i}$. We numerically
investigate the time evolution of $\cal{N}$ for different $\lambda$. Without
loss of generality, we choose a Gaussian wavepacket, i.e.,
$\psi{(\theta,t_{0})}=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$ with
$\sigma=10$ as the initial state in numerical simulations. Figure 1(a) shows
that for very small $\lambda$ (e.g., $\lambda=0.01$ and 0.05), the value of
$\mathcal{N}$ equals almost to unity with time evolution, which implies that
quasienergies are all real. Interestingly, for sufficiently large $\lambda$
(e.g., $\lambda=0.15$), $\mathcal{N}$ increases exponentially with time, i.e.,
$\mathcal{N}=e^{\mu t}$, and the growth rate $\mu$ increases with the increase
of $\lambda$. The non-unitary feature of the Floquet operator,
$U_{K}=\exp[K\lambda\sin(\theta)/\hbar]$, leads to the growth of the norm. A
rough estimation of the norm yields a time dependence of the form
$\mathcal{N}\propto\exp(K\lambda t/\hbar)$, indicating the relation of the
growth rate $\mu\propto\lambda$, which is confirmed by our numerical results
[see inset in Figure 1(a)]. We further investigate the long-time average value
of the norm, $\bar{\mathcal{N}}=\sum_{n=1}^{N}\mathcal{N}(t_{n})/N$, for a
wide range of $\lambda$. Figure 1(b) shows that, for a specific
$\hbar_{\text{eff}}$ (e.g., $\hbar_{\text{eff}}=0.1$), $\bar{\mathcal{N}}$
remains at unity for $\lambda$ smaller than a threshold value $\lambda_{c}$,
beyond which it monotonically increases with $\lambda$. It is reasonable to
believe that the threshold value $\lambda_{c}$ corresponds to the emergence of
spontaneous $\cal{PT}$-symmetry breaking.

Figure 1: (a) Norm $\cal{N}$ versus time for $\lambda=0.01$ (squares), 0.05
(circles), 0.15 (triangles), 0.2 (diamonds), and 0.3 (pentagrams). The
parameters are $K=5$ and $\hbar_{\text{eff}}=1$. Solid lines indicate the
exponential increase, i.e., $\mathcal{N}(t)=e^{\mu t}$, while dash-dotted line
denotes $\mathcal{N}=1$. Inset: the $\mu$ versus $\lambda$. Solid line
indicates the linear increase $\mu\propto\lambda$. (b) Average value
$\bar{\cal{N}}$ versus $\lambda$ for $K=5$ with $\hbar_{\text{eff}}=0.1$
(squares), 0.5 (circles), and 1 (triangles). Solid line denotes
$\mathcal{N}=1$. (c) Phase diagram of the spontaneous $\cal{PT}$-symmetry
breaking for $\hbar_{\text{eff}}=1$. One can see clearly a phase boundary
$\lambda_{c}$. (d) The value of $\lambda_{c}$ in the parameter space
$(K,\hbar_{\text{eff}})$.

Recently, the long short-term memory (LSTM) network has been exploited to
extract the character of time series and thus to predict the phase diagram of
quantum diffusion Mano21 . Based on the character of the time evolution of
$\mathcal{N}$, we conducted supervised training on the LSTM network and used
it to evaluate the feature of $\mathcal{N}(t)$, namely, whether
$\mathcal{N}(t)=e^{\mu t}$ or not, for different system parameters. This
highly effective machine learning method outputs the probability $\rho$ of the
time series $\mathcal{N}(t)$ to be exponentially increasing or not, which can
predict the phase diagram of spontaneous $\mathcal{PT}$-symmetry breaking.
Interestingly, our results show that the $\rho$ increases with the increase of
both $K$ and $\lambda$ [see Fig. 1(c)]. We identify two phases in the
parameters space $(K,\lambda)$, the boundary of which is clearly visible in
Fig. 1(c). We further investigate the $\lambda_{c}$ for different $K$ and
$\hbar_{\text{eff}}$. Our results demonstrate that the critical parameter
$\lambda_{c}$ increases with the increase of $\hbar_{\text{eff}}$ and
decreases with the increase of $K$ [see Fig. 1(d)]. This behavior is rooted in
the fact that the mean spacing level $\Delta$ of the quasienergies of the
quantum kicked rotor (QKR) model is proportional to $\hbar/K$ West2010 . The
smaller the $\Delta$ is, the easier it is for the non-Hermitian parameter
$\lambda$ to cause the coalescence of two quasienergies, implying the relation
$\lambda_{c}\propto\hbar/K$.

##  III Scaling laws of the OTOCs at the transition to the spontaneous
$\cal{PT}$-symmetry breaking

The OTOCs are defined by
$C(t_{n})=-\langle[\hat{A}(t_{n}),\hat{B}]^{2}\rangle$, with the operators
$\hat{A}(t_{n})=U^{\dagger}(t_{n})AU(t_{n})$ and $B$ being evaluated in the
Heisenberg picture Chen2017 ; Hashimoto2017 ; Dora2017 ; Mata2018 ;
Alavirad2018 ; Swingle2016 ; Hafezi2016 ; Li2017 ; Garttner2017 . The average,
i.e., $\langle\cdots\rangle=\langle\psi(t_{0})|\cdots|\psi(t_{0})\rangle$, is
taken over an initial state $|\psi(t_{0})\rangle$ Heyl2018 . In this work, we
consider the case where both $\hat{A}$ and $\hat{B}$ are angular momentum
operators, i.e., $C(t)=-\langle[{p}(t),{p}]^{2}\rangle$. We use a Gaussian
wavepacket, i.e.,
$\psi{(\theta,t_{0})}=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$ with
$\sigma=10$ as the initial state. It is worth noting that, as opposed to
static-lattice systems, periodically-driven systems have no thermal states, as
the temperature grows to infinity with time evolution D'Alessio14 . Thus,
there is no need to average over the initially thermal states in the
definition of $C(t_{n})$ in our system WlZhao22 ; WLZhao21 .

Straightforward derivation yields the equivalence

$\displaystyle C(t_{n})=$ $\displaystyle
C_{1}(t_{n})+C_{2}(t_{n})-2\text{Re}\left[C_{3}(t_{n})\right]\;,$ (3)

where the two-points correlators, namely, the first two terms in right side
are defined by

$C_{1}(t_{n})=\langle\psi_{R}(t_{0})|{p}^{2}|\psi_{R}(t_{0})\rangle\;,$ (4)
$C_{2}(t_{n})=\langle\varphi_{R}(t_{0})|\varphi_{R}(t_{0})\rangle\;,$ (5)

and the four-points correlator is

$C_{3}(t_{n})=\langle\psi_{R}(t_{0})|p|\varphi_{R}(t_{0})\rangle\;,$ (6)

with
$|\psi_{R}(t_{0})\rangle={U}^{\dagger}(t_{0},t_{n})pU(t_{0},t_{n})|\psi(t_{0})\rangle$
and
$|\varphi_{R}(t_{0})\rangle={U}^{\dagger}(t_{0},t_{n})pU(t_{0},t_{n})p|\psi(t_{0})\rangle$
Ueda2018 . The symbol $\text{Re}[\cdots]$ denotes the real part of a complex
variable.

To obtain the state $|\psi_{R}(t_{0})\rangle$, three steps must be carried
out: i) the forward evolution from $t_{0}$ to $t_{n}$, i.e.
$|\psi(t_{n})\rangle=U(t_{0},t_{n})|\psi(t_{0})$, ii) the action of the
operator $p$ on the state $|\psi(t_{n})\rangle$, i.e.
$|\tilde{\psi}(t_{n})\rangle=p|\psi(t_{n})\rangle$, and iii) the backward
evolution from $t_{n}$ to $t_{0}$, i.e.
$|\psi_{R}(t_{0})\rangle={U}^{\dagger}(t_{0},t_{n})|\tilde{\psi}(t_{n})\rangle$.
The expectation value of the square of the momentum can then be calculated
using $|\psi_{R}(t_{0})\rangle$ to obtain $C_{1}(t_{n})$ [see Eq. (4)]. To
numerically simulate $C_{2}(t)$, the operator $p$ should first be applied to
the initial state $|\psi(t_{0})\rangle$, yielding the new state
$|\varphi(t_{0})\rangle=p|\psi(t_{0})\rangle$. Then, the forward evolution is
conducted, i.e.,
$|\varphi(t_{n})\rangle=U(t_{0},t_{n})|\varphi(t_{0})\rangle$. Subsequently,
the action of $p$ is performed on $|\varphi(t_{n})\rangle$, obtaining
$|\tilde{\varphi}(t_{n})\rangle=p|\varphi(t_{n})\rangle$, afterwards, the
time-reversal is applied to $|\tilde{\varphi}(t_{n})\rangle$, resulting in
$|\varphi_{R}(t_{0})\rangle={U}^{\dagger}(t_{0},t_{n})|\tilde{\varphi}(t_{n})\rangle$.
Using Eq. (5), $C_{2}(t_{n})$ can then be calculated by evaluating the norm of
$|\varphi_{R}(t_{0})\rangle$. Lastly, the term $C_{3}(t_{n})$ [seen in Eq.
(6)] can be determined using the two states $|\psi_{R}(t_{0})\rangle$ and
$|\varphi_{R}(t_{0})\rangle$, which is usually complex since they are not
identical.

In the $\cal{PT}$-symmetry breaking phase, the norm of the quantum state
$\mathcal{N}_{\psi}(t_{n})=\langle\psi(t_{n})|\psi(t_{n})\rangle$ increases
exponentially with time regardless of the forward or backward evolution. To
address this issue and eliminate its contribution to the OTOCs, we normalize
the time-evolved state. For the forward evolution $t_{0}\rightarrow t_{n}$ of
$|\psi(t_{0})\rangle$, we set the norm of the quantum state to be the same as
that of the initial state, i.e.,
$\mathcal{N}_{\psi}(t_{j})=\langle\psi(t_{0})|\psi(t_{0})\rangle$ with $0\leq
j\leq n$. The backward evolution starts from the state
$|\tilde{\psi}(t_{n})\rangle$, whose norm
$\mathcal{N}_{\tilde{\psi}}(t_{n})=\langle\psi(t_{n})|p^{2}|\psi(t_{n})\rangle$
is the mean energy of the state $|\psi(t_{n})\rangle$. Thus, it is reasonable
to take the norm of the quantum state during the backward evolution
$t_{n}\rightarrow t_{0}$ to be $\mathcal{N}_{\tilde{\psi}}(t_{n})$, i.e.,
$\mathcal{N}_{\psi_{R}}(t_{j})=\mathcal{N}_{\tilde{\psi}}(t_{n})$. In short,
the norm of the time-evolved state for both the forward and backward evolution
is equal to that of the state it starts from. If the same normalization
procedure is applied to the evolution of $|\varphi(t_{n})\rangle$, then we
will have
$\mathcal{N}_{\varphi}(t_{j})=\langle\varphi(t_{0})|\varphi(t_{0})\rangle$ and
$\mathcal{N}_{\varphi_{R}}(t_{j})=\langle\tilde{\varphi}(t_{n})|\tilde{\varphi}(t_{n})\rangle$
($0\leq j\leq n$) for the forward and time reversal evolutions, respectively.

Figure 2: (a) Time dependence of $C$ for $K=6$ with $\lambda=10^{-5}$
(squares), $0.022$ (circles), $0.2$ (diamonds), $0.5$ (triangles), and $0.9$
(pentagrams). Solid lines in red denote theoretical prediction in Eqs. (12)
and (13), i.e., $C(t)\propto t^{\eta}$. (b) The $\eta$ versus $\lambda$ for
$\hbar_{\text{eff}}=0.3$ (squares) and 0.5 (circles). Arrow marks the phase
transition point $\lambda_{c}\approx 0.001$ for $\hbar_{\text{eff}}=0.3$.

In order to understand the effects of $\cal{PT}$-symmetry breaking on the
dynamics of $C(t_{n})$, we numerically investigated the time evolution of
$C(t_{n})$ for different $\lambda$. Figure 2(a) shows that, for values of
$\lambda$ smaller than the phase transition point (e.g.,
$\lambda=10^{-5}\ll\lambda_{c}$), the $C(t_{n})$ increases gradually up to
saturation. Interestingly, for $\lambda$ slightly larger than $\lambda_{c}$,
the $C(t_{n})$ increases in a power-law of time, i.e., $C(t_{n})\propto
t^{\eta}$ with $\eta>2$ [see $\lambda=0.022$ with $\eta=3.4$ in Fig. 2(a)]. We
dub this phenomenon as a super-quadratic growth (SQG) of $C(t_{n})$. When the
value of $\lambda$ is much larger than the phase transition point, i.e.,
$\lambda\gg\lambda_{c}$ [e.g., $\lambda=0.5$ and 0.9 in Fig.2(a)], the
quadratic growth (QG) of OTOCs $C(t_{n})\propto t^{2}$ emerges. We further
investigate the exponent $\eta$ for different $\lambda$. Our results show that
$\eta$ is zero for $\lambda<\lambda_{c}$, increases abruptly to a maximum
value greater than two for $\lambda$ slightly larger than $\lambda_{c}$, and
finally saturates to two for sufficiently large $\lambda$ [e.g., see
$\hbar_{\text{eff}}=0.3$ in Fig. 2(b)]. It is evident that the scaling-law of
OTOCs reveals the emergence of the spontaneous $\mathcal{PT}$-symmetry
breaking and unveils the correlation between information scrambling and the
$\mathcal{PT}$-symmetry phase transition.

##  IV Theoretical analysis of the dynamics of OTOCs

###  IV.1 Mechanism of the saturation of $C(t)$ for $\lambda<\lambda_{c}$

We numerically investigate the time evolution of the three parts of the OTOCs,
i.e., $C_{1}$, $C_{2}$, and $C_{3}$ for $\lambda<\lambda_{c}$. Figure 3(a)
shows that the time dependence of $C_{1}$ and $C$ almost overlap, displaying
rapid growth up to saturation. Since the real part of $C_{3}$, i.e.,
$\textrm{Re}(C_{3})$ fluctuates between positive and negative values, we plot
the absolute value $|\textrm{Re}(C_{3})|$ in Fig. 3(b). Both $C_{2}$ and
$|\textrm{Re}[C_{3}]|$ saturate after a very short time evolution.
Importantly, $C_{1}$ is at least 4 orders of magnitude larger than both
$C_{2}$ and $|\textrm{Re}[C_{3}]|$, leading to a perfect consistency between
$C_{1}$ and $C$. Consequently, based on Eq. (3), we can safely use the
approximation

$\displaystyle
C(t_{n})\approx\langle\psi_{R}(t_{0})|{p}^{2}|\psi_{R}(t_{0})\rangle=\langle
p^{2}(t_{0})\rangle_{R}\mathcal{N}_{\psi_{R}}(t_{0})\;,$ (7)

where $\langle
p^{2}(t_{0})\rangle_{R}=\langle\psi_{R}(t_{0})|{p}^{2}|\psi_{R}(t_{0})\rangle/\mathcal{N}_{\psi_{R}}(t_{0})$
denotes the exceptional value of energy of the state $|\psi_{R}(t_{0})\rangle$
divided by its norm
$\mathcal{N}_{\psi_{R}}(t_{0})=\langle\psi_{R}(t_{0})|\psi_{R}(t_{0})\rangle$.

Figure 3: (a) Time evolution of $C$ (circles) and $C_{1}$ (squares). Note that
$C$ almost fully overlaps with $C_{1}$. (b) Dependence of $C_{2}$ (squares)
and $|\textrm{Re}(C_{3})|$ (circles) on time. The parameters are $K=6$,
$\lambda=10^{-5}$, and $\hbar_{\text{eff}}=0.3$. (c) The $\bar{C}$ versus $K$
with $\lambda=10^{-5}$ for $\hbar_{\text{eff}}=1$ (squares), 5 (circles), 10
(triangles), and 15 (diamonds). Solid lines indicates our theoretical
prediction in Eq. (8).

The normalization procedure for time reversal yields the equivalence
$\mathcal{N}_{\psi_{R}}(t_{0})=\mathcal{N}_{\tilde{\psi}}(t_{n})=\langle\psi(t_{n})|p^{2}|\psi(t_{n})\rangle$,
which shows that the value of $\mathcal{N}_{\psi_{R}}(t_{0})$ is just the mean
energy of the state $|\psi(t_{n})\rangle$ at the time $t=t_{n}$. For
$\lambda<\lambda_{c}$, the quasienergies are all real, thus the dynamics of
the PTKR is the same as that of the Hermitian QKR. A noteworthy characteristic
of the QKR’s energy diffusion is the phenomenon of DL, i.e. the mean energy
$\langle p^{2}\rangle$ gradually approaches to saturation level with
increasing time due to quantum coherence. It is reasonable to believe that the
mechanism of DL suppresses the growth of both $\mathcal{N}_{\psi_{R}}(t_{0})$
and $\langle p^{2}(t_{0})\rangle_{R}$, and therefore leads to the saturation
of $C(t_{n})$.

To confirm this conjecture, we consider a specific time, i.e., $t=t_{n}$, and
numerically trace the evolution of $\langle p^{2}\rangle$ for both the forward
($t<t_{n}$) and backward ($t>t_{n}$) evolution. Figure 4(a) shows that for
$t_{n}=2500$, $\langle p^{2}\rangle$ increases rapidly to saturation during
forward time evolution from $t_{0}$ to $t_{2500}$, then jumps to a specific
value at the start of the time reversal (i.e., at $t=t_{2500}$) before finally
saturating for the backward evolution from $t_{2500}$ to $t_{0}$. This clearly
demonstrates the emergence of the DL, which is also reflected by the
probability density distribution in momentum space. We compare the momentum
distributions at the end of the forward evolution (i.e., $t=t_{2500}$) and the
end of time reversal (i.e., $t=t_{0}$) in Fig. 4(b). One can see that the two
quantum states almost overlap with each other, both of which are exponentially
localized in momentum space, i.e., $|\psi(t_{n})|^{2}\sim\exp(-|p|/L)$ [see
Fig. 4(b)]. A rough estimation yields
$\mathcal{N}_{\psi_{R}}(t_{0})=\langle\psi(t_{n})|p^{2}|\psi(t_{n})\rangle\sim
L^{2}$ and $\langle
p^{2}(t_{0})\rangle_{R}\sim\int_{-\infty}^{\infty}p^{2}\exp(-|p|/L)dp\sim
L^{2}$. Plugging the two relations into Eq. (7), we can immediately get the
estimation of the OTOCs, i.e., $C(t_{n})\sim L^{4}$. It is known that the
localization length is in a quadratic function of $K$, i.e., $L\propto K^{2}$
Izrailev90 , which results in the relation

$C\propto K^{8}\;.$ (8)

This clearly demonstrates that the $C$ is time-independent after the long term
evolution, verifying our numerical results in Fig. 2(a) and Fig. 3(a).

To provide evidence of our analytical prediction, we investigate the time-
averaged value of OTOCs, i.e., $\bar{C}=\sum_{j=1}^{N}C(t_{j})/N$, numerically
for different $K$. In the numerical simulations, we ensure that $N$ is large
enough for the long-term saturation of $C(t)$ to be well quantified by
$\bar{C}$. Our numerical results show that for a specific $\hbar$, $\bar{C}$
increases in a power-law of $K$ [see Fig. 3(c)], which is well described by
our theoretical prediction in Eq. (8). This is a strong indication of the
validity of our analytical analysis. Our findings of the dependence of the
OTOCs on the kick strength provide an opportunity to control the operator
growth with an external driven potential.

Figure 4: (a) Time trace of $\langle p^{2}\rangle$ (squares) and $\mathcal{N}$
(circles) during the forward evolution $t_{0}\rightarrow t_{2500}$, the action
of $p$ at the time $t=t_{2500}$, and the time reversal $t_{2500}\rightarrow
t_{5000}$. Green dashed line marks $t_{n}=2500$. (b) Momentum distributions
for the state $|\psi(t)\rangle$ (squares) at the time $t_{n}=2500$ and the
state $|\psi_{R}(t_{0})\rangle$ (circles) at the end of time reversal. Solid
line indicates the exponentially-localized shape $|\psi(p)|^{2}\propto
e^{-|p|/L}$ with $L\approx 46$. The parameters are $K=6$, $\lambda=10^{-5}$,
and $\hbar_{\text{eff}}=0.3$.

The discontinuous jump in the mean square momentum, $\langle p^{2}\rangle$, at
$t=t_{2500}$, the beginning of time reversal, is due to the action of the
operator $p$ on the quantum state $|\psi(t_{2500})\rangle$. This action
generates the quantum state
$|\tilde{\psi}(t_{2500})\rangle=p|\psi(t_{2500})\rangle$, for which the mean
value is given by
$\langle\tilde{\psi}(t_{2500})|p^{2}|\tilde{\psi}(t_{2500})\rangle=\langle\psi(t_{2500})|p^{4}|\psi(t_{2500})\rangle$.
The exponentially-localized shape of the quantum state,
$|\psi(t_{2500})|^{2}\sim\exp(-|p|/L)$ [see Fig. 4(b)], allows us to obtain
the expectation values $\langle\psi(t_{2500})|p^{2}|\psi(t_{2500})\rangle\sim
L^{2}\hbar_{\text{eff}}^{2}$ and
$\langle\tilde{\psi}(t_{2500})|p^{2}|\tilde{\psi}(t_{2500})\rangle=\langle\psi(t_{2500})|p^{4}|\psi(t_{2500})\rangle\sim
L^{4}\hbar_{\text{eff}}^{4}$, which quantitatively explains the discontinuous
increase in the mean energy from $L^{2}\hbar_{\text{eff}}^{2}$ to
$L^{4}\hbar_{\text{eff}}^{4}$.

###  IV.2 Mechanism of the SQG of $C(t)$ for $\lambda\gtrsim\lambda_{c}$

Figure 5(a) shows the time evolution of $C_{1}$, $C_{2}$,
$|\textrm{Re}(C_{3})|$, and $C$ for $\lambda$ just larger than the
$\cal{PT}$-symmetry phase transition point $\lambda\gtrsim\lambda_{c}$. It is
clear that the time dependence of $C_{1}$ corresponds perfectly to that of
$C$, both of which increase following the SQG $C\propto t^{3.4}$. The time
evolution of $C_{2}$ displays the QG, i.e., $C_{2}(t)\propto t^{2}$, while
$|\textrm{Re}(C_{3})|$ follows the SQG $|\textrm{Re}(C_{3})|\sim t^{2.9}$. In
addition, one can see that $C_{1}$ is larger than both $C_{2}$ and
$|\textrm{Re}(C_{3})|$ by approximately four orders of magnitude. Therefore,
it is sufficient to analyze the time evolution of the term $C_{1}$ to uncover
the mechanism of the SQG of $C$.

Figure 5: Time evolution of $C_{1}$ (squares), $C_{2}$ (diamonds),
$|\textrm{Re}(C_{3})|$ (triangles), and $C$ (circles) for $\lambda=0.022$ (a)
and 0.9 (b). In (a): Red lines indicates the power-law fitting. In (b): Red
lines indicate the quadratic function $C_{1}\approx 44t^{2}$, $C_{2}\approx
17t^{2}$, and $|\textrm{Re}(C_{3})|\approx 2t^{2}$. Inset: The $\Delta_{12}$
(squares) and $\Delta_{13}$ (circles) versus time. Solid (dashed-dotted) line
denotes $\Delta_{12}\approx 2.5$ ($\Delta_{13}\approx 22$). Other parameters
are same as Fig. 4.

Since the value of $C(t)$ at a specific time $t=t_{n}$ is dependent on both
the mean energy $\langle p^{2}(t_{0})\rangle_{R}$ and the norm
$\mathcal{N}_{\psi_{R}}(t_{0})$ of the state $|\psi_{R}(t_{0})\rangle$ at the
end of time reversal [see Eq. (7)], we numerically calculate the forward and
backward time evolution of $\langle p^{2}\rangle$, $\langle p\rangle$, and
$\mathcal{N}$ with a fixed $t_{n}$ (e.g., $t_{n}=2500$ in Fig. 6). Figure 6(a)
demonstrates that the mean energy diffuses ballistically with time $\langle
p^{2}\rangle\approx\gamma^{2}t^{2}$ during the forward evolution
$t_{0}\rightarrow t_{2500}$, and it displays the intrinsic time reversal
during $t_{2500}\rightarrow t_{0}$. Meanwhile, the mean momentum $\langle
p\rangle$ linearly increases for $t_{0}<t<t_{2500}$, and linearly decays for
$t_{2500}\rightarrow t_{0}$ [see Fig. 6(b)]. Moreover, Fig. 6(a) reveals that
the norm remains unity, i.e., $\mathcal{N}(t)=1$, during the forward evolution
and equals the mean energy at the time $t=t_{2500}$, i.e.,
$\mathcal{N}_{\psi_{R}}(t_{j})=\langle p^{2}(t_{2500})\rangle$ during the time
reversal. Taking the ballistic diffusion of energy into account, the following
equivalence can be derived

$\mathcal{N}_{\psi_{R}}(t_{0})=\gamma^{2}t_{n}^{2}\;.$ (9)

In order to measure the degree of time reversal for a fixed $t_{n}$, we define
the ratio of mean energy between forward ($t<t_{n}$) and backward ($t>t_{n}$)
time evolution as

$\mathcal{R}(t_{j})=\frac{\langle p^{2}(2t_{n}-t_{j})\rangle_{R}}{\langle
p^{2}(t_{j})\rangle}\;,$ (10)

where $\langle p^{2}(2t_{n}-t_{j})\rangle_{R}$ and $\langle
p^{2}(t_{j})\rangle$ ($0\leq j\leq n$) denote the mean square of momentum for
the forward evolution and time reversal, respectively. The inset in Fig. 6(b)
shows that $\mathcal{R}$ is very large (i.e., $\mathcal{R}\gtrsim 10^{4}$ for
$t=t_{0}$) and approaches almost one with time evolution. This reveals that
the mean energy at the end of time reversal is much greater than that at the
initial time, i.e., $\langle p^{2}(t_{0})\rangle_{R}\gg\langle
p^{2}(t_{0})\rangle$. We further investigate the time evolution of $\langle
p^{2}(t_{0})\rangle_{R}$, and find [see Fig. 6(c)] that the $\langle
p^{2}(t_{0})\rangle_{R}$ increases in the power-law of time

$\langle p^{2}(t_{0})\rangle_{R}\propto t^{1.4}\;.$ (11)

Substituting Eqs. (9) and (11) into Eq. (7) yields the SQG of OTOCs

$C(t)\propto t^{\eta}\quad\text{with}\quad\eta\approx 3.4\;.$ (12)

Figure 6: Top two panels: Time trace of $\langle p^{2}\rangle$ (squares) in
(a), $\mathcal{N}$ (circles) in (a), and $\langle p\rangle$ in (b) during the
forward evolution $t_{0}\rightarrow t_{2500}$, the action of $p$ at the time
$t=t_{2500}$, and the time reversal $t_{2500}\rightarrow t_{0}$. Green dashed
lines makes $t=t_{2500}$. In (a): Red line indicates the quadratic function
$\langle p^{2}\rangle=\gamma^{2}t^{2}$ with $\gamma\approx 3.2$. In (b): Red
line indicates the linear growth $\langle p\rangle=\gamma t$. Inset: $\cal{R}$
versus time. Red line marks $\mathcal{R}=1$. In (c): Time dependence of
$\langle p^{2}(t_{0})\rangle_{R}$. Red line indicates the power-law fitting
$\langle p^{2}(t_{0})\rangle_{R}\propto t^{1.4}$. The parameter is
$\lambda=0.022$. Other parameters are same as Fig. 4.

Figure 7 shows the probability density distribution of the state at forward
$|\psi(t_{j})\rangle$ and backward $|\psi_{R}(t_{j})\rangle$ evolution in both
the real and momentum space. The initial state is a Gaussian wavepacket
$\psi{(\theta,t_{0})}=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$ centered at
$\theta=0$ and $p=0$ [see Figs. 7(a) and (b)]. Interestingly, one can observe
that the quantum state is mainly distributed in the region $0<\theta<\pi$ for
both the forward and backward evolution. This is due to the fact that the
action of the Floquet operator of the kicking term
$U_{K}(\theta)=\exp[K\lambda\sin(\theta)/\hbar_{\text{eff}}]\exp[-iK\cos(\theta)/\hbar_{\text{eff}}]$
on a quantum state, i.e., $U_{K}(\theta)\psi(\theta)$ helps to amplify the
state within the region $0<\theta<\pi$ as $K\lambda\sin(\theta)>0$. Assuming
that the real part of the kicking potential provides the driven force
$F=K\sin(\theta)$, the PTKR experiences a positive magnitude force $F>0$
during the forward evolution, thus the momentum grows with time, as shown in
Fig. 6(b). For the time reversal, the sign of kick strength $K$ flips, i.e.,
$K\rightarrow-K$, so the mean momentum decreases with time evolution. Our
conjecture is supported by the numerical results of momentum distributions.
Figures 7(b), (d) and (f) show that the wavepacket, like a soliton, moves to
the positive direction in momentum space for $t_{0}\rightarrow t_{2500}$,
resulting in $\langle p\rangle=\gamma t$ [in Fig. 6(b)], and moves back to the
opposite direction for $t_{2500}\rightarrow t_{0}$. In addition, the width of
the wavepacket in momentum space is so narrow that one can safely use the
approximation $\langle p^{2}\rangle\sim(\langle p\rangle)^{2}$, which is
verified by our numerical results in Figs. 6(a).

Figure 7: Probability density distributions in real (left panels) and momentum
(right) space for $\lambda=0.022$. (a)-(d) Black and blue lines separately
correspond to the states at forward $|\psi(t)\rangle$ and backward
$|\psi_{R}(t)\rangle$ evolution with $t=t_{0}$ (top panels) and $t=t_{1250}$
(middle panels). Inset in (b) shows a magnified view of the momentum
distribution for the initial Gaussian wavepacket around $p=0$. Bottom panels:
Probability density distributions in real (e) and momentum (f) space at the
time $t=t_{2500}$. Black and blue lines indicate the state
$|\psi(t_{2500})\rangle$ and
$|\tilde{\psi}(t_{2500})\rangle=p|\psi(t_{2500})\rangle$, respectively. Other
parameters are same as Fig. 4.

###  IV.3 Mechanism of the QG of $C(t)$ for $\lambda\gg\lambda_{c}$

We numerically investigate the time evolution of $C_{1}$, $C_{2}$, $C_{3}$,
and $C$ for $\lambda\gg\lambda_{c}$. As shown in Fig. 5(b), all of them
increase in the way of QG (i.e., $\propto t^{2}$). We use the ratios
$\Delta_{12}=C_{1}/C_{2}$ and $\Delta_{13}=C_{1}/|\textrm{Re}[C_{3}]|$ to
quantify the differences among $C_{1}$, $C_{2}$, and $|\textrm{Re}[C_{3}]|$.
Our investigation show that both of them are larger than one, specifically
$\Delta_{12}\approx 2.5$ and $\Delta_{13}\approx 22$ [see the inset in Fig.
5]. This suggests that $C_{1}$ contributes mainly to the $C$, which is
verified by the good agreement between $C_{1}$ and $C$ [see Fig. 5(b)]. To
reveal the mechanism of the QG of $C$, we proceed to analyze the time
evolution of $C_{1}$ by thoroughly investigating both the forward and backward
evolution of the mean values $\langle p^{2}\rangle$, $\langle p\rangle$, and
the norm $\cal{N}$ for a given $t_{n}$.

Figure 8(a) shows that the mean energy exhibits ballistic diffusion $\langle
p^{2}\rangle\approx\gamma^{2}t^{2}$, during the forward evolution from $t_{0}$
to $t_{n}$, and decays as the inverse of a quadratic function, with $\langle
p^{2}\rangle\propto t^{-2}$, during the backward evolution from $t_{n}$ to
$t_{0}$. This decay is symmetric with respect to the $\langle p^{2}\rangle$ of
$t<t_{n}$. The dynamics of the mean momentum also exhibits perfect time
reversal, namely it linearly increases as $\langle p\rangle=\gamma t$ during
$t_{0}\rightarrow t_{2500}$ and decreases linearly during $t_{2500}\rightarrow
t_{0}$. The ratio $\mathcal{R}$ remains close to one throughout the time
evolution, except at the end, i.e., $\mathcal{R}(t=2500)\approx 2.5$ [see the
inset in Fig. 8(b)], providing a clear evidence of time reversal. For the
forward evolution from $t_{0}$ to $t_{2500}$, the norm $\mathcal{N}(t_{j})$ is
equal to unity, while for the interval $t_{2500}\rightarrow t_{0}$, it is
equal to the value of $\langle p^{2}(t_{2500})\rangle$, i.e.,
$\mathcal{N}_{\psi_{R}}(t_{j})=\langle p^{2}(t_{2500})\rangle$ [see Fig.
8(a)]. By utilizing the ballistic diffusion of mean energy, we establish the
relationship
$\mathcal{N}_{\psi_{R}}(t_{0})\approx\mathcal{N}_{\psi_{R}}(t_{n})\approx\gamma^{2}t_{n}^{2}$,
where $t_{n}$ is an arbitrary time. We further evaluate the behavior of
$\langle p^{2}(t_{0})\rangle_{R}$ for different $t_{n}$. As shown in Fig.
8(c), the $\langle p^{2}(t_{0})\rangle_{R}$ remains almost constant at a value
of one, indicating that it is independent of time. Plugging in the values of
$\mathcal{N}_{\psi_{R}}(t_{0})$ and $\langle p^{2}(t_{0})\rangle_{R}$ into Eq.
(7), we obtain the QG of OTOCs

$C(t)\approx\gamma^{2}t^{\eta}\quad\text{with}\quad\eta=2.$ (13)

Figure 8: Same as in Fig. 6 but for $\lambda=0.9$. In (a): Red line indicates
the quadratic function $\langle p^{2}\rangle=\gamma^{2}t^{2}$ with
$\gamma\approx 6.3$. In (b): Red line denotes the linear growth $\langle
p\rangle=\gamma t$. In (c): Red line indicates $\langle
p(t_{0})\rangle_{R}\approx 1$.

The time reversal of a wavepacket’s dynamics is clearly seen in the evolution
of its momentum distributions. For the forward time evolution, the quantum
state is localized at the point $\theta_{c}=\pi/2$ [seen in Figs. 9(a), (c)
and (e)], which is the result of the localization-effect of the imaginary part
of the Floquet operator $U_{K}(\theta)$. With the wavepacket mimicking a
classical particle, it experiences a kicking force of magnitude
$F=K\sin(\theta_{c})=K$, resulting in a constant acceleration of momentum
$\Delta p=K$, which is reflected in the linear growth of momentum. This
phenomenon of the directed current is also seen in the propagation of momentum
distributions in Figs. 9(b), (d) and (f), where a soliton can be observed
moving unidirectionally towards the positive direction in momentum space.

During the backward evolution, the wavepacket of the real space remains
centered at $\theta_{c}=\pi/2$, with a width much smaller than the
corresponding state at the time of forward evolution [see Figs. 9(a), (c) and
(e)]. As the particle is exposed to the kicking force with $F=-K$ during time
reversal, its momentum decreases linearly in time, which is also reflected in
the propagation of the wavepackets in momentum space [Figs. 9(b), (d) and
(f)]. It is evident that the $|\psi(t_{n})|^{2}$ is in perfect overlap with
the $|\psi_{R}(t_{n})|^{2}$, apart from the initial state [see Figs. 9(b)].
The width of $|\psi_{R}(t_{0})|^{2}$ is considerably larger than that of
$|\psi(t_{0})|^{2}$, leading to the ratio of energy being larger than one,
i.e., $\mathcal{R}=\langle p^{2}(t_{5000})\rangle_{R}/\langle
p^{2}(t_{0})\rangle\approx 2.5$ [see the inset in Fig. 8(b)].

Figure 9: Same as in Fig. 7 but for $\lambda=0.9$.

##  V Conclusion and discussion

In this work, we investigate the dynamics of OTOCs in a PTKR model and achieve
its scaling laws in different phases of $\cal{PT}$ symmetry. We use the time
series of the norm to train a LSTM, which enables us to extract a clear phase
diagram of $\cal{PT}$-symmetry breaking, with a phase boundary at
$\lambda_{c}$. For $\lambda<\lambda_{c}$, we find that the DL of energy
diffusion suppresses the growth of OTOCs, and prove analytically the
dependence of OTOCs on the kicking strength, i.e., $C\propto K^{8}$. At the
vicinity of the phase transition points, i.e., $\lambda\gtrsim\lambda_{c}$, we
observe a SQG of OTOCs, i.e., $C(t)\propto t^{\eta}$ with an exponent
$\eta>2$. Interestingly, a QG of OTOCs, i.e., $C(t)\propto t^{2}$ emerges for
$\lambda\gg\lambda_{c}$. We elucidate the mechanisms of both the SQG and QG by
analyzing the time-reversed wavepacket’s dynamics. Our results demonstrate
that the spontaneous $\cal{PT}$-symmetry breaking profoundly affects the
dynamics of OTOCs, providing an unprecedented opportunity for diagnosing the
spontaneous $\cal{PT}$-symmetry breaking with OTOCs.

In recent years, the OTOCs have been widely used to investigate the operator
growth in quantum mapping systems Moudgalya19 , the information scrambling in
spin chains TianciZhou20 , and the quantum thermalization in many-body chaotic
systems KenXuanWei19 . Theoretical studies have demonstrated that the QKR
model is mathematically equivalent to the kicked Heisenberg spin XXZ chain
Boness10 , indicating a connection between the magnon dynamics and quantum
diffusion of chaotic systems. Our findings therefore bridge the gap between
the information scrambling in condensed matter physics and the operator growth
in quantum chaotic systems. This also paves the way for the experimental
observation of OTOCs dynamics in chaotic systems using spin chain platforms.

## ACKNOWLEDGMENTS

Wen-Lei Zhao is supported by the National Natural Science Foundation of China
(Grant Nos. 12065009), the Natural Science Foundation of Jiangxi province
(Grant Nos. 20224ACB201006 and 20224BAB201023) and the Science and Technology
Planning Project of Ganzhou City (Grant No. 202101095077). Jie Liu is
supported by the NSAF (Contract No. U1930403).

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is a Nash diffeomorphism, we conclude that $\lim_{t\to0^+}\alpha_i(t)\cdot (g_i\circ\alpha_i)(t)=0$ (because $p_i\not\in\Ss_i^{*,(d_i)}$), which is a contradiction because $\lim_{t\to0^+}(g_i\circ\alpha_i)(t)$ exists and it is non-zero. Consequently, $G_i(\Ss_i^{*,(d_i)})$ is non-compact. Again, as the restriction $G_i|_{\Ss_i^{*,(d_i)}\setminus\{g_i=0\}}$ is a Nash diffeomorphism,

is pure dimensional of dimension $d_i$. As $\Ss_i^{*,(d_i)}$ is pure dimensional of dimension $d_i$ and $\dim(\{g_i=0\})\leq d_i-1$, we deduce $q_i\in\cl(\Ss_i^{*,(d_i)}\setminus\{g_i=0\})$. As $q_i\in\Ss_i^{*,(d_i)}\cap\{g_i=0\}$, we conclude $0\in G_i(\cl(\Ss_i^{*,(d_i)}\setminus\{g_i=0\}))\subset\cl(G_i(\Ss_i^{*,(d_i)}\setminus\{g_i=0\}))$, so $G_i(\Ss_i^{*,(d_i)})$ is pure dimensional of dimension $d_i$. In addition,

0\in G_i(\Ss_i^{*,(d_i)})=G_i(\Ss_i^{*,(d_i)})\cup\{0\}=G_i(\Ss_i^{*,(d_i)}\cup(\Ss_i^*\cap\{g_i=0\}))=G_i(\Ss_i^*).

Case 2. If $Q_i(p_i)=0$, we have $Q_i\circ\alpha_i\in\R[[\t]]_{\rm alg}$ is a non-zero series. Let $(Y_i,\phi_i)$ be the blow-up of $\R^m$ at $p_i$. The restriction $\phi_i:Y_i\setminus\{\phi_i^{-1}(p_i)\}\to\R^m\setminus\{p_i\}$ is a Nash diffeomorphism and $p_i\not\in\Ss\cup X_i$, so $\phi_i^{-1}(\Ss)$ is Nash diffeomorphic to $\Ss$ and $\phi_i^{-1}(X_i)$ is Nash diffeomorphic to $X_i$. The series $Q_i\circ\alpha_i=(Q_i\circ\phi_i)\circ(\phi_i^{-1}\circ\alpha_i)$ and let $(Q_i\circ\phi_i)^*$ be the strict transform of $(Q_i\circ\phi_i)$. The order of the series $(Q_i\circ\phi_i)^*\circ(\phi_i^{-1}\circ\alpha_i)$ is strictly smaller than the order of $Q_i\circ\alpha_i$, because we have eliminated from $(Q_i\circ\phi_i)$ a power of an equation of the exceptional divisor. Let $p_i':=\lim_{t\to0^+}(\phi_i^{-1}\circ\alpha_i)(t)$. If $(Q_i\circ\phi_i)^*(p_i')\neq0$, we have finished with this index $i$. Otherwise, we repeat the previous process with the point $p_i'$. In each step the order of the strict transform of the corresponding polynomial substituted in the corresponding curve has strictly smaller order, so in finitely many steps we achieve order $0$ and the corresponding polynomial does not vanish at the limit point. This allows, after finitely many steps, to reduce Case 2 to Case 1.

After composing all the involved blow-ups (corresponding to all the indices $i=1,\ldots,s$ that are under the assumptions of Case 2) and taking the corresponding strict transforms, we find a Nash manifold $M\subset\R^p$, a Nash diffeomorphism $\varphi:M\to U$ and Nash functions $g_i:M\to\R$ such that $\varphi^{-1}(X_i)\subset\{g_i=0\}$ and the corresponding Nash map

G_i:M\to\R^{p+1},\ x\mapsto(x\cdot g_i(x),g_i(x))

satisfies $0\in G_i(\varphi^{-1}(\Ss_i^*))=G_i(\varphi^{-1}(\Ss_i^{*,(d_i)}))$ and $G_i(\varphi^{-1}(\Ss_i^{*,(d_i)}))$ is pure dimensional of dimension $d_i$ and non-compact for $i=1,\ldots,s$, as required.

We are ready to prove Theorem <ref>.

The only if conditions are obtained straightforwardly. The proof of the converse is conducted in several steps:

Step 1. Suppose $\Ss_i^*$ is non-compact for $i=1,\ldots,s$ and $\Ss_i^*$ is compact for $i=s+1,\ldots,r$. For each $i=1,\ldots,r$ let $f_i:\Ss\to\R$ be a Nash function on $\Ss$ such that $\Ss_i^*=\{f_i=0\}$ (see <cit.>). Let $U$ be an open semialgebraic neighborhood of $\Ss$ in $\R^m$ to which all the Nash functions $f_i$ extend as Nash functions $F_i:U\to\R$. Define $X_i:=\bigcup_{j\neq i}\{F_j=0\}$ and observe that $\Ss_i^*\cap X_i=\Ss_i^*\cap\bigcup_{j\neq i}\Ss_j^*$ is a semialgebraic subset of $\Ss_i^*$ of dimension strictly smaller than $d_i$. We distinguish two cases:

Case 1. Non-compact irreducible components. By Lemma <ref> we may assume (up to a suitable Nash diffeomorphism) that for each $i=1,\ldots,s$ there exist a Nash function $g_i:U\to\R$ whose respective zero set $\{g_i=0\}$ contains $X_i$ and the corresponding Nash map

G_i:U\to\R^{m+1},\ x\mapsto(x\cdot g_i(x),g_i(x))

satisfies $0\in G_i(\Ss_i^*)=G_i(\Ss_i^{*,(d_i)})$ and $G_i(\Ss_i^{*,(d_i)})$ is non-compact and pure dimensional of dimension $d_i$ for $i=1,\ldots,s$. In addition,

G_i(\Ss)=G_i(\Ss_i^*\cup\bigcup_{j\neq i}\Ss_j^*)=G_i(\Ss_i^*)\cup\bigcup_{j\neq i}G_i(\Ss_j^*)=G_i(\Ss_i^*)\cup\{0\}=G_i(\Ss_i^{*,(d_i)}).

Case 2. Compact irreducible components. For each $i=s+1,\ldots,r$ let $q_i\in\Ss_i^{*,(d_i)}$ and $h_i$ be a polynomial whose zero set is the union of $\{q_i\}$ and the Zariski closure $Y_i$ of $\cl(\Ss_i^*\setminus\Ss_i^{*,(d_i)})$. Define $g_i:=h_i\prod_{j\neq i}F_j:U\to\R$ and observe that $\{g_i=0\}=\{q_i\}\cup(Y_i\cap\Ss_i^*)\cup\bigcup_{j\neq i}\Ss_j^*$. As $\Ss_i^*$ is irreducible and $g_i$ does not vanish identically on $\Ss_i^*$, the intersection $\{g_i=0\}\cap\Ss_i^*$ has dimension $<d_i:=\dim(\Ss_i^*)$. Consider the Nash map

G_i:U\to\R^{m+1},\ x\mapsto (x\cdot g_i(x),g_i(x)),

whose restriction to $U\setminus\{g_i=0\}$ is a Nash diffeomorphism between $U\setminus\{g_i=0\}$ and $G_i(U)\setminus\{0\}$. Observe that $G_i(\Ss_j^*)=\{0\}$ if $i\neq j$ and let us check: $\Ss_i':=G_i(\Ss)=G_i(\Ss_i^{*,(d_i)})$ is pure dimensional of dimension $d_i$.

As $G_i|_{\Ss_i^{*,(d_i)}\setminus\{g_i=0\}}:\Ss_i^{*,(d_i)}\setminus\{g_i=0\}\to G_i(\Ss_i^{*,(d_i)})\setminus\{0\}$ is a Nash diffeomorphism, $G_i(\Ss_i^{*,(d_i)})\setminus\{0\}=G_i(\Ss_i^{*,(d_i)}\setminus\{g_i=0\})$ is pure dimensional of dimension $d_i$. As $\Ss_i^{*,(d_i)}$ is pure dimensional of dimension $d_i$ and $\dim(\{g_i=0\})\leq d_i-1$, we deduce $q_i\in\cl(\Ss_i^{*,(d_i)}\setminus\{g_i=0\})$. As $q_i\in\Ss_i^{*,(d_i)}\cap\{g_i=0\}$, we conclude $0\in G_i(\cl(\Ss_i^{*,(d_i)}\setminus\{g_i=0\}))\subset\cl(G_i(\Ss_i^{*,(d_i)}\setminus\{g_i=0\}))$, so $G_i(\Ss_i^{*,(d_i)})$ is pure dimensional of dimension $d_i$. In addition,

0\in G(\Ss_i^{*,(d_i)})=G(\Ss_i^{*,(d_i)})\cup\{0\}=G(\Ss_i^{*,(d_i)}\cup(\Ss_i^*\cap\{g_i=0\}))=G(\Ss_i^*).

G_i(\Ss)=G_i(\Ss_i^*\cup\bigcup_{j\neq i}\Ss_j^*)=G_i(\Ss_i^*)\cup\bigcup_{j\neq i}G_i(\Ss_j^*)=G_i(\Ss_i^*)\cup\{0\}=G_i(\Ss_i^{*,(d_i)})

for $i=s+1,\ldots,r$.

Step 2. Define $\Ss_i':=G_i(\Ss)$ for $i=1,\ldots,r$ and

G:\Ss\to\R^{(m+1)r},\ x\mapsto(G_1(x),\ldots,G_r(x)).

Observe that

and $G(\Ss)=\bigcup_{i=1}^sG(\Ss_i^*)$. In addition, $G(\Ss_i^*)\cap G(\Ss_j^*)=\{(0,\ldots,0)\}$ if $i\neq j$.

We distinguish two cases:

Case 1. If $\Ss_i^*$ is non-compact, $\Ss_i'$ is non-compact. By Theorem <ref> there exists a Nash map $H_i:\R^{m+1}\to\R^{d_i}$ such that $H_i(\Ss_i')=\R^{d_i}$. We may assume in addition $H_i(0)=0$.

Case 2. If $\Ss_i$ is compact, also $\Ss_i'$ is compact and there exists by Theorem <ref> a Nash map $H_i:\R^{m+1}\to\R^{d_i}$ such that $H_i(\Ss_i')=\ol{\Bb}_{d_i}$. Following the proof of Theorem <ref>, the reader can check that we may assume $H_i(0)=0$.

Step 3. Let $q\in\bigcap_{i=1}^r\Tt_i$ and assume $q$ is the origin of $\R^n$. Let $F_i:\R^{d_i}\to\R^n$ be a Nash map such that $F_i(\R^{d_i})=\Tt_i$ for $i=1,\ldots,s$ and $F_i(\ol{\Bb}_{d_i})=\Tt_i$ for $i=s+1,\ldots,r$. Define $E_i=\R^{d_i}$ if $\Ss_i'$ is non-compact ($i=1,\ldots,s$) and $E_i=\ol{\Bb}_{d_i}$ if $\Ss_i'$ is compact ($i=s+1,\ldots,r$).

We may assume in addition $F_i(0)=0$ for $i=1,\ldots,r$. We have

(F_i\circ H_i\circ G_i)(\Ss_j^*)=\begin{cases}
F_i(E_i)=\Tt_i&\text{if $j=i$},\\
F_i(\{0\})=\{0\}&\text{if $j\neq i$,}
\end{cases}

so $(F_i\circ H_i\circ G_i)(\Ss)=\Tt_i$. Observe that

\begin{equation*}
\begin{split}
((F_1\circ H_1,\ldots,F_r\circ H_r)&\circ G)(\Ss_i^*)=(F_1\circ H_1\circ G_1,\ldots,F_r\circ H_r\circ G_r)(\Ss_i^*)\\
&=(F_1\circ H_1)(\{0\})\times\cdots\times(F_{i-1}\circ H_{i-1})(\{0\})\times(F_i\circ H_i)(\Ss_i')\\
&\hspace{3.842cm}\times(F_{i+1}\circ H_{i+1})(\{0\})\times\cdots\times(F_r\circ H_r)(\{0\})\\
&=F_1(\{0\})\times\cdots\times F_{i-1}(\{0\})\times F_i(E_i)\times F_{i+1}(\{0\})\times\cdots\times F_r(\{0\})\\
\end{split}
\end{equation*}

Thus, if

F:=\sum_{i=1}^r(F_i\circ H_i\circ G_i):\Ss\to\Tt,

we have $F(\Ss_i^*)=\Tt_i$ for $i=1,\ldots,r$, so

as required.

Recall that the analytic path-connected components of $\Ss$ are irreducible semialgebraic sets. Thus, each of them is contained in an irreducible component of $\Ss$. If $\Ss_i^*$ is the irreducible component of $\Ss$ that contains $\Ss_i$ for $i=1,\ldots,r$, it may happen that $\Ss_i^*=\Ss_j^*$ for some $i\neq j$ or $\Ss_i^*\neq\Ss_j^*$, whereas $\Ss_i\subsetneq\Ss_i^*$ and $\Ss_j\subsetneq\Ss_j^*$.

(i) Define $\Ss:=\Ss_1\cup\Ss_2\cup\Ss_3\subset\R^2$ where $\Ss_1:=\{\x\geq1\}$, $\Ss_2:=\{\y=0\}$ and $\Ss_3:=\{\x\leq-1\}$. Observe that $\Ss_1$, $\Ss_2$ and $\Ss_3$ are the analytic path-connected components of $\Ss$, whereas $\Ss$ is irreducible. Thus, $\Ss_1^*=\Ss_2^*=\Ss_3^*$.

(ii) Define $\Ss:=\Ss_1\cup\Ss_2\subset\R^3$ where $\Ss_1:=\{\x=0,\y\geq0\}$ and $\Ss_2:=\{\y\leq0,\z=0\}$. Observe that $\Ss_1$ and $\Ss_2$ are the analytic path-connected components of $\Ss$, whereas $\Ss_1^*=\Ss_1\cup\{\x=0,\z=0\}$ and $\Ss_2^*=\Ss_2\cup\{\x=0,\z=0\}$ are the irreducible components of $\Ss$.

(i) Let $\Ss\subset\R^m$ be a semialgebraic set and $\{\Ss_i\}_{i=1}^r$ the analytic path-connected components of $\Ss$. Let $\Ss_i^*$ be the irreducible component of $\Ss$ that contains $\Ss_i$ for $i=1,\ldots,r$ and assume $\Ss_i^*\neq\Ss_j^*$ for $1\leq i<j\leq r$. Denote $d_i:=\dim(\Ss_i^*)$. We claim:

(1) $\{\Ss_i^*\}_{i=1}^r$ is the collection of the irreducible components of $\Ss$.

(2) $\Ss_i^{*,(d_i)}=\Ss_i$ for $i=1,\ldots,r$.

As $\Ss=\bigcup_{i=1}^r\Ss_i\subset\bigcup_{i=1}^r\Ss_i^*\subset\Ss$, we deduce that $\{\Ss_i^*\}_{i=1}^r$ is the collection of the irreducible components of $\Ss$, because $\Ss_i^*\neq\Ss_j^*$ if $i\neq j$. Thus, (1) holds.

Let us check (2). To that end, we prove first: $\dim(\Ss_j\cap\Ss_i^*)<\dim(\Ss_i^*)$ if $j\neq i$.

Otherwise, there exists $\Ss_j$ with $j\neq i$ such that $\dim(\Ss_j\cap\Ss_i^*)=\dim(\Ss_i^*)$, so $\dim(\Ss_j^*\cap\Ss_i^*)=\dim(\Ss_i^*)$ and each Nash function that vanishes identically on $\Ss_j^*$ vanishes also identically on $\Ss_i^*$. Thus, $\Ss_i^*\subset\Ss_j^*$ and $i=j$, which is a contradiction.

\Ss_i\setminus\bigcup_{j\neq i}\Ss_j\subset\Ss_i^*\setminus\bigcup_{j\neq i}\Ss_j\subset\Ss\setminus\bigcup_{j\neq i}\Ss_j=\Ss_i\setminus\bigcup_{j\neq i}\Ss_j,

so $\Ss_i\setminus\bigcup_{j\neq i}\Ss_j=\Ss_i^*\setminus\bigcup_{j\neq i}\Ss_j$ is non-empty and has dimension $d_i$. As $\Ss_i$ is pure dimensional of dimension $d_i$ and $\bigcup_{j\neq i}\Ss_i^*\cap\Ss_j$ has dimension $<d_i$, we deduce that $\Ss_i\setminus\bigcup_{j\neq i}\Ss_j$ is dense in $\Ss_i$. In addition, $\Ss_i\subset\Ss_i^{*,(d_i)}\subset\Ss_i^*$ (because $\Ss_i$ is pure dimensional of dimension $d_i$), so $\Ss_i\setminus\bigcup_{j\neq i}\Ss_j=\Ss_i^{*,(d_i)}\setminus\bigcup_{j\neq i}\Ss_j$, which is dense in $\Ss_i^{*,(d_i)}$. As

\Ss_i^{*,(d_i)}\setminus\bigcup_{j\neq i}\Ss_j\subset\Ss_i^*\setminus\bigcup_{j\neq i}\Ss_j=\Ss_i\setminus\bigcup_{j\neq i}\Ss_j,

we conclude taking closures in $\Ss$ that $\Ss_i^{*,(d_i)}=\Ss_i$ (because both $\Ss_i^{*,(d_i)}$ and $\Ss_i$ are closed in $\Ss$).

(ii) Observe that Theorems <ref> and <ref> are particular cases of Theorem <ref> when $\Tt$ is connected by analytic paths.

As a straightforward consequence of Theorem <ref> and Remark <ref>(i), we have the following:

Let $\Ss\subset\R^m$ and $\Tt\subset\R^n$ be semialgebraic sets, let $\{\Ss_i\}_{i=1}^r$ be the family of analytic path-connected components of $\Ss$ and let $\Ss_i^*$ be the irreducible component of $\Ss$ that contains $\Ss_i$ for $i=1,\ldots,r$. Assume $\Ss_i^*\neq\Ss_j^*$ for $1\leq i<j\leq r$. Let $\{\Tt_i\}_{i=1}^r$ be a family of (non-necessarily distinct) semialgebraic subsets of $\Tt$ connected by analytic paths and assume $\bigcap_{i=1}^r\Tt_i\neq\varnothing$ and $\bigcup_{i=1}^r\Tt_i=\Tt$. Then there exists a surjective Nash map $f:\Ss\to\Tt$ such that $f(\Ss_i)=\Tt_i$ for $i=1,\ldots,r$ if and only if $e_i:=\dim(\Tt_i)\leq\dim(\Ss_i)=:d_i$ and $\Tt_i$ is compact in case $\Ss_i$ is compact for $i=1,\ldots,r$.

§ TWO APPLICATIONS OF THE MAIN RESULTS

In this section we present two remarkable consequences of Theorem <ref>. The first one about representation of pure dimensional compact irreducible arc-symmetric semialgebraic sets as Nash images of closed balls. As a second consequence we show that a compact semialgebraic set is the projection of a non-singular compact algebraic set with the simplest possible topology (a disjoint union of spheres).

§.§ Representation of arc-symmetric compact semialgebraic sets.

It follows from Theorem <ref> and <cit.> that a pure dimensional compact irreducible arc-symmetric semialgebraic set is a Nash image of $\ol{\Bb}_d$ where $d:=\dim(\Ss)$.

Let $X$ be the Zariski closure of $\Ss$ and $\pi:\widetilde{X}\to X$ a resolution of the singularities of $X$ (see [31]). Assume $\widetilde{X}\subset\R^p$ and $\pi$ is the restriction to $\widetilde{X}$ of a polynomial map $\Pi:\R^p\to\R^n$. By <cit.> applied to the irreducible arc-symmetric set $\Ss$ there exists a connected component $E$ of $\widetilde{X}$ such that $\pi(E)=\cl(\Reg(\Ss))=\Ss$ (recall that $\Ss$ is pure dimensional and compact). As $\pi$ is proper and $\Ss$ is compact, also $E$ is compact (because it is a closed subset of the compact set $\pi^{-1}(\Ss)$). Thus, $E$ is a connected compact Nash manifold. By Theorem <ref> there exists a Nash map $f_0:\R^{d}\to\R^p$ such that $f_0(\ol{\Bb}_d)=E$. Consequently, the Nash map $f:=\pi\circ f_0:\R^{d}\to\R^n$ satisfies $f(\ol{\Bb}_d)=\pi(f_0(\ol{\Bb}_d))=\pi(E)=\Ss$, as required.

§.§ Elimination of inequalities.

To prove Corollary <ref> we recall first the following well-known separation result, that we include here for the sake of completeness.

Let $\Ss_1,\Ss_2\subset\R^n$ be semialgebraic sets such that $\Ss_1$ is compact, $\Ss_2$ is closed and $\Ss_1\cap\Ss_2=\varnothing$. Then there exists $f\in\R[\x]$ such that $\Ss_1\subset\{f<0\}$ and $\Ss_2\subset\{f>0\}$.

We may assume $\Ss_1\subset\Bb_n(0,\frac{1}{2})$. Let $g:\R^n\to\R$ be a continuous function such that $\Ss_1\subset\{g<0\}$ and $\Ss_2\subset\{g>0\}$. Let

\veps:=\dist(\Ss_1,\Ss_2):=\min\{\dist(x_1,x_2):\ x_1\in\Ss_1,x_2\in\Ss_2\}>0.

By Weierstrass' approximation theorem there exists a polynomial $f_0\in\R[\x]$ such that

\max\{|g(x)-f_0(x)|:\ x\in\ol{\Bb}_n(0,1)\}<\frac{\veps}{3}.

By <cit.> there exists a constant $c>0$ and $m\geq1$ such that $|f_0(x)|<c(1+\|x\|^2)^m$ on $\R^n$. Thus, $|f_0(x)|<2^mc\|x\|^{2m}$ on $\R^n\setminus\ol{\Bb}_n(0,1)$. Denote $c':=2^mc$ and let $k\geq m$ be such that
$\frac{c'}{2^{2k}}<\frac{\veps}{3}$. Define $f:=f_0+c'\|\x\|^{2k}\in\R[\x]$. The reader can check that $\Ss_1\subset\{f<0\}$ and $\Ss_2\subset\{f>0\}$, as required.

(i) By Theorem <ref> and using the fact that the closed unit ball $\ol{\Bb}_d$ is the projection of the sphere $\sph^d$, there exists a Nash map $f:\R^{d+1}\to\R^n$ such that $f(\sph^d)=\Ss$. By Artin-Mazur's description of Nash maps <cit.> there exist $s\geq1$ and a non-singular irreducible algebraic set $Z\subset\R^{d+1+n+s}$ of dimension $d$, a connected component $M$ of $Z$ and a Nash diffeomorphism $g:\sph^d\to M$ such that the following diagram is commutative.

\xymatrix{
M_\epsilon\ar<EMAIL_ADDRESS>M_-\ar@{^{(}->}[r]\ar[dd]^{\pi|_Y}&Z\times\R\ar@{^{(}->}[r]\ar[d]^{\pi|_{Z\times\R}}&\R^{m+1}\ar[d]^{\pi}\\

We denote the projection of $\R^{d+1}\times\R^n\times\R^s$ onto the first space $\R^{d+1}$ with $\pi_1$ and the projection of $\R^{d+1}\times\R^n\times\R^s$ onto the second space $\R^n$ with $\pi_2$. Write $m:=d+1+n+s$. As $M$ is compact, there exists by Lemma <ref> a polynomial $f:\R^m\to\R$ such that $M=Z\cap\{f>0\}$. Observe that $M$ is the projection of the algebraic set

Y:=\{(z,t)\in Z\times\R:\ f(z)t^2-1=0\}

under the projection $\pi:\R^m\times\R\to\R^m,\ (z,t)\mapsto z$. Fix $\epsilon=\pm1$ and let $M_\epsilon:=Y\cap\{\epsilon t>0\}$. Consider the Nash diffeomorphism

\varphi_\epsilon:M\to M_\epsilon,\ x\mapsto\Big(x,\epsilon\frac{1}{\sqrt{f(x)}}\Big)

whose inverse map is the restriction of the projection $\pi$ to $M_\epsilon$.

Observe that $\{M_\epsilon\}_{\epsilon\in\{-1,1\}}$ is the collection of the connected components of $Y$. As $\pi(M_\epsilon)=M$ and using the diagram above, we deduce

In addition, each $M_\epsilon$ is Nash diffeomorphic to $\sph^d$ and for $\veps\neq\veps'$ the polynomial map

\phi:\R^m\times\R\to\R^m\times\R,\ (x,t)\mapsto(x,-t)

induces an involution of $Y$ such that $\phi(M_\epsilon)=M_{\epsilon'}$. As $Z$ is non-singular, also $Y$ is non-singular. Let $X$ be the irreducible component of $Y$ that contains $M_{+1}$. Observe that either $X=M_{+1}$ or $X=Y$.

Then $k:=d+s+2$ and the non-singular algebraic set $X$ satisfies the requirements in the statement.

In addition, $X$ has at most two connected components and each of them is Nash diffeomorphic to $\sph^d$. Thus, $X$ is Nash diffeomorphic to $\sph^d\times\{1,s\}$, where $s=1,2$ is the number of connected components of $X$.

(ii) Let $\Ss_1,\ldots,\Ss_r$ be the (compact) analytic path-connected components of $\Ss$, which satisfy $\Ss=\bigcup_{i=1}^r\Ss_i$. By (i) there exist $m\geq1$ and for each $i=1,\ldots,r$ a non-singular algebraic set $X_i\subset\R^m$ that is Nash diffeomorphic to a disjoint union of at most two spheres of $\R^{d+1}$ (each of them isometric to $\sph^{d_i}$ where $d_i:=\dim(\Ss_i)\leq d=\dim(\Ss)$) and satisfies $\pi(X_i)=\Ss_i$, where

\pi:\R^n\times\R^{m-n}\to\R^n,\ (x,y)\mapsto x

is the projection onto the first $n$ coordinates. Consider the pairwise disjoint union $X:=\bigsqcup_{i=1}^rX_i\times\{i\}\subset\R^{m+1}$ and the projection

\pi':\R^n\times\R^{m+1-n}\times\R\to\R^n,\ (x,y,t)\mapsto x.

Then $X$ is a non-singular algebraic set, which is Nash diffeomorphic to a finite pairwise disjoint union of spheres of dimension $\leq d$ and satisfies $\pi(X)=\Ss$, as required.

The following lemma together with Example <ref> shows that Corollary <ref> is sharp.

Let $Z\subset\R^m$ be a non-singular irreducible algebraic set and $M$ one of its connected components of maximal dimension $d$. Suppose there exists an irreducible algebraic set $Y\subset\R^p$ of dimension $d$ and a rational map $\varphi:\R^p\dashrightarrow\R^m$ such that $\varphi|_Y:Y\to M$ is bijective. Then $M$ is the unique connected component of $Z$ of dimension $d$.

Let $\widetilde{Y}\subset\C^p$ be the complexification of $Y$ and $\widetilde{Z}\subset\C$ the complexification of $Z$. Observe that $\widetilde{Y},\widetilde{Z}$ are irreducible algebraic sets of (complex) dimension $d$. Consider the rational map $\widetilde{\varphi}:\C^p\dashrightarrow\C^m$ that extends $\varphi$. As $\varphi(Y)=M\subset Z\subset\widetilde{Z}$, the Zariski closure of $\varphi(Y)$ in $\C^m$ is contained in $\widetilde{Z}$. As $M$ has (real) dimension $d$ and $\widetilde{Z}$ is an irreducible algebraic set of $\C^n$ of (complex) dimension $d$, we deduce that $\widetilde{Z}$ is the Zariski closure of $\varphi(Y)$. As $\widetilde{\varphi}:\C^p\dashrightarrow\C^m$ is continuous for the Zariski topology, $\widetilde{Y}$ is the Zariski closure of $Y$ and $\widetilde{Z}$ is the Zariski closure of $\varphi(Y)=\widetilde{\varphi}(Y)$, we conclude $\widetilde{\varphi}(\widetilde{Y})\subset\widetilde{Z}$ and the Zariski closure of $\widetilde{\varphi}(\widetilde{Y})$ is $\widetilde{Z}$. Thus, $\widetilde{\varphi}|_{\widetilde{Y}}:\widetilde{Y}\dashrightarrow\widetilde{Z}$ is a dominant rational map. Denote with ${\mathcal R}(\widetilde{Y})$ the field of rational functions on $\widetilde{Y}$ and with ${\mathcal R}(\widetilde{Z})$ the field of rational functions on $\widetilde{Z}$. The map $\widetilde{\varphi}^*:{\mathcal R}(\widetilde{Z})\to{\mathcal R}(\widetilde{Y}),\ f\mapsto f\circ\widetilde{\varphi}$ is a homomorphism of fields of the same transcendence degree $d$ over $\C$. Consequently, ${\mathcal R}(\widetilde{Y})$ is an algebraic extension of ${\mathcal R}(\widetilde{Z})$ of finite degree $m$. By <cit.> the number of points in a general fiber of $\widetilde{\varphi}$ is equal to $m$. As $M$ has (real) dimension $d$, there exists a point $p\in M$ such that the fiber $\widetilde{\varphi}^{-1}(p)$ has exactly $m$ points. As $Y$ is a (real) algebraic set, $\widetilde{Y}\cap\R^p=Y$. As $\varphi$ is a real rational map and $\varphi|_Y:Y\to M$ is bijective, we conclude that $m$ is odd, because if $z\in\widetilde{Y}\setminus Y$ and $\widetilde{\varphi}(z)=p$, then $\ol{z}\in\widetilde{Y}\setminus Y$ and $\widetilde{\varphi}(\ol{z})=p$.

Suppose $Z$ has another connected component $M'$ of dimension $d$. Then there exists $q\in M'$ such that $\widetilde{\varphi}^{-1}(q)$ has exactly $m$ points. As $\widetilde{Y}\cap\R^p=Y$ and $\widetilde{\varphi}(Y)=M$, we conclude that $\widetilde{\varphi}^{-1}(q)\subset\widetilde{Y}\setminus Y$. As $q\in Z=\widetilde{Z}\cap\R^n$, we deduce that if $z\in\widetilde{\varphi}^{-1}(q)$, also $\ol{z}\in\widetilde{\varphi}^{-1}(q)$. Thus, $\widetilde{\varphi}^{-1}(q)$ consists of an even number of elements, which is a contradiction because $m$ is odd. Consequently, $M$ is the unique connected component of $Z$ of dimension $d$, as required.

Let $X:=\{\y^2=-(\x-1)(\x-2)(\x+1)\}\subset\R^2$, which is an irreducible non-singular cubic with two connected components of dimension $1$, one is bounded (that we denote with $C_1$) and the other one is unbounded (that we denote with $C_2$). Consider the polynomial $\x$, which satisfies $X\cap\{\x>0\}=C_1$ and $X\cap\{\x<0\}=C_2$. Let $Y:=\{(x,y,z)\in X\times\R:\ xz^2-1=0\}\subset\R^3$, which has exactly two connected components $M_1:=Y\cap\{\z>0\}$ and $M_2:=Y\cap\{\z<0\}$ and both have dimension $1$. A priori we do not know if $Y$ is irreducible, so we cannot apply Lemma <ref> directly to $Y$.

[->,thick] (-2, 0) – (2.5, 0) ;
[->,thick] (0, -2.5) – (0,2.5) ;
[thick, scale=1,domain=1:1.6, draw=black,samples=1000] plot (-, sqrt((+1)*(+2)*(-1)));
[thick, scale=1,domain=1:1.6, draw=black,samples=1000] plot (-, -sqrt((+1)*(+2)*(-1)));
(1.5,-1.25) node$C_1$;
[thick, scale=1,domain=-2:-1, draw=black,samples=1000] plot (-, sqrt((-1)*(+2)*(+1)));
[thick, scale=1,domain=-2:-1, draw=black,samples=1000] plot (-, -sqrt((-1)*(+2)*(+1)));
(-0.9,1.8) node$C_2$;

The cubic curve $\y^2=-(\x-1)(\x-2)(\x+1)$.

Suppose there exists an algebraic set $Z\subset\R^p$ of dimension $1$ and a polynomial map $\varphi:\R^p\to\R^3$ such that $\varphi|_Z:Z\to M_1$ is bijective. Let $\pi:\R^3\to\R^2,\ (x,y,z)\mapsto(x,y)$, which satisfies $\pi|_{M_1}:M_1\to C_1$ is bijective. Thus, the composition $\pi\circ f:\R^p\to\R^2$ is a polynomial map that satisfies $\pi|_{Z}:Z\to C_1$ is bijective, but this contradicts Lemma <ref>. Consequently, there does not exist the couple $(\varphi,Z)$.

The previous example suggests that in the statement of Corollary <ref>(i) two connected components Nash diffeomorphic to $\sph^d$ are needed.

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§.§ Nash interpolation.

As a consequence of Theorem <ref> and the existence of a decomposition of semialgebraic sets into analytic path-connected components we obtain the following interpolation result for Nash maps:

Let $\Ss\subset\R^n$ and $\Tt\subset\R^m$ be semialgebraic sets and let $\Tt^*$ be an analytic path-connected component of $\Tt$. Given $p_1,\ldots,p_k\in \Ss$ and $q_1,\ldots,q_k\in\Tt^*$ (non necessarily distinct), there exists a Nash map $F:\Ss\to \Tt$ such that $F(p_i)=q_i$ for each $i=1,\ldots,k$.

Let $d:=\dim(\Tt^*)$. As $\Tt^*$ is connected by analytic paths, there exists by Theorem <ref> a surjective Nash map $f:\R^d\to \Tt^*$. For each $i=1,\ldots,k$ fix $s_i\in f^{-1}(q_i)$. By Lagrange's interpolation there exists a polynomial map $g:\R^n\to \R^d$ such that $g(p_i)=s_i$. Thus, the Nash map $F:=f\circ g|_{\Ss}:\Ss\to \Tt$ satisfies $F(p_i)=q_i$ for each $i=1,\ldots,k$.

(i) Let $\Ss:=\{\z(\x^2+\y^2)-\x^3=0\}\subset\R^3$ be Cartan's umbrella. The analytic path-connected components of $\Ss$ are

\Ss_1:=(\Ss\setminus\{\x=0,\y=0\})\cup\{0\} \text{\, and \, } \Ss_2:=\{\x=0,\y=0\}.

In fact, $\Ss_1$ is image of $\R^2$ through the analytic map

\varphi:\R^2\to\R^3,\ (u,v)\mapsto(u\cos v, u\sin v, u\cos^3v).

(ii) Let $\Ss:=\Ss_1\cup\Ss_2\cup\Ss_3\subset\R^3$, where

\Ss_1:=[-1,1]\times\{0\}\times [-1,1],\, \Ss_2:=\{\z=0, \x\geq 1\},\, \Ss_3:=\{\z=0,\x\leq -1\}.

The analytic path-connected components of $\Ss$ are $\Ss_1, \Ss_2$ and $\Ss_3$. In contrast, $\Ss$ has two irreducible components, which are $\Ss\cap\{\y=0\}$ and $\Ss\cap\{\z=0\}$.

Recall that if $\Ss_1^*$ and $\Ss_2^*$ are irreducible components of a semialgebraic set $\Ss\subset\R^n$, then $\dim(\Ss_1^*\cap\Ss_2^*)<\min\{\dim(\Ss_1^*),\dim(\Ss_2^*)\}$ (see Remark <ref>). In the case of the analytic path-connected components of $\Ss$ this inequality is no longer true.

Let $W:=\{\x^2-\z\y^2=0\}\subset\R^3$ be Whitney's umbrella. Its analytic path-connected components are

W_1:=W\cap \{\z\geq 0\} \text{\, and \, } W_2:=\{\x=0,\y=0\}.

It holds $W_1\cap W_2=\{\x=0,\y=0,\z\geq0\}$, which has dimension $1$, so

\dim(W_1\cap W_2)=1=\dim(W_2).

However, we have the following result:

Let $\Ss\subset\R^n$ be a semialgebraic set and let $\{\Ss_i\}_{i=1}^r$ be its family of analytic path-connected components. Then

\dim(\Ss_i\cap\Ss_j)<\max\{\dim(\Ss_i),\dim(\Ss_j)\}

for $1\leq i<j\leq r$.

If $\dim(\Ss_i)<\dim(\Ss_j)$, then

\dim(\Ss_i\cap\Ss_j)\leq\dim(\Ss_i)<\dim(\Ss_j)\leq\max\{\dim(\Ss_i),\dim(\Ss_j)\}.

Suppose next $e:=\dim(\Ss_i)=\dim(\Ss_j)$ and $\dim(\Ss_i\cap\Ss_j)=e$. Let $\Ss_0:=\Ss_i\cup\Ss_j$ and observe that $\Sing(\Ss_0)$ has dimension $\leq e-1$. Thus, there exists $y\in(\Ss_1\cap\Ss_2)\setminus\Sing(\Ss_0)$. Let $x_k\in\Ss_k$ for $k=1,2$. As each $\Ss_k$ is connected by analytic paths, by Theorem <ref> each $\Ss_k$ is connected by Nash paths. Thus, there exist Nash paths $\alpha_k:[0,1]\to\Ss_k$ such that $\alpha_k(0)=x_k,\alpha_k(1)=y$. The continuous semialgebraic path $\alpha:=\alpha_1*\alpha_2^{-1}$ connects the points $x_1$ and $x_2$ and satisfies $\eta(\alpha)\subset\{y\}\subset\Reg(\Ss_0)$. Consequently, $\Ss_0$ is well-welded, so $\Ss_0$ is connected by analytic paths by Theorem <ref>. As $\Ss_i,\Ss_j$ are analytic path-connected components of $\Ss$, we conclude $\Ss_i=\Ss_0=\Ss_j$, which is a contradiction. Thus, $\dim(\Ss_i\cap\Ss_j)<e=\max\{\dim(\Ss_i),\dim(\Ss_j)\}$, as required.