Floquet dynamical quantum phase transitions in periodically quenched systems

In momentum presentation, the time-dependent Hamiltonian $H(k,t)$ of our periodically quenched lattice takes the form

$$H(k,t)=\begin{cases}h_{x}(k)\sigma_{x}&t\in[2\ell,2\ell+1),\\ h_{y}(k)\sigma_{y}&t\in[2\ell+1,2\ell+2).\end{cases}$$

(1)

Here $k\in[-\pi,\pi)$ is the quasimomentum defined in the first Brillouin zone. $\ell\in\mathbb{Z}$ counts the number of driving periods. $\sigma_{x,y,z}$ are Pauli matrices in their usual representations, and we will denote the $2\times 2$ identity matrix as $\sigma_{0}$. It is clear that $H(k,t)$ is periodic in time with the period $T=2$, i.e., $H(k,t)=(k,t+2)$. We will also use the symbol $s$ to denote the time within a driving period, i.e., the micromotion time, so that any physical time $t$ can be decomposed as

$$t=s+2\ell,\qquad\ell\in\mathbb{Z},\qquad s\in[0,2).$$

(2)

Meanwhile, according to Eq. (1), $H(k,t)$ has the chiral symmetry $\Gamma=\sigma_{z}$ at any instant of time, i.e., $\Gamma H(k,t)\Gamma=-H(k,t)$ and $\Gamma^{2}=\sigma_{0}$. This allows one to characterize the topological phases of the system by integer winding numbers AsbothSTF1 ; ZhouSQKR ; ZhouDSCL . Note in passing that Eq. (1) represents one of the typical quench protocols for the realization of one-dimensional (1D) Floquet systems with chiral (sublattice) symmetry and nontrivial topological properties. The Pauli matrices in Eq. (1) can be replaced by any two of the three Pauli spins $\sigma_{x,y,z}$, and $h_{x,y}$ can be arbitrary functions of the quasimomentum $k$. Various other periodic kicking or quenching protocols for 1D chiral symmetric lattice models can be reduced to the form of Eq. (1), such as those reported in Refs. AsbothSTF1 ; ZhouSQKR . Therefore, the theoretical framework developed here is expected to be appliable to Floquet DQPTs in a broad range of periodically quenched and continuously driven DQPTExp11 chiral symmetric two-band systems.

We are interested in such a dynamical process, in which the system is initialized in a uniformly filled Floquet band and then undergoes nonequilibrium evolution governed by the piecewise quenched $H(k,t)$. The initial state of the system is obtained by solving the Floquet eigenvalue equation $U(k)|\psi(k)\rangle=e^{-iE(k)}|\psi(k)\rangle$, where $|\psi(k)\rangle$ is the Floquet eigenstate with eigenphase $E(k)$. The Floquet operator $U(k)$, referring to the evolution operator of the system over a complete driving period $T$ (e.g., from $t=0^{-}$ to $t=0^{-}+2$), takes the form

$$U(k)=e^{-ih_{y}(k)\sigma_{y}}e^{-ih_{x}(k)\sigma_{x}},$$

(3)

according to Eq. (1). Taking the Taylor expansion of the two terms on the right hand side of Eq. (3), we can express $U(k)$ equivalently as

	$\displaystyle U(k)$	$\displaystyle=\cos[h_{x}(k)]\cos[h_{y}(k)]\sigma_{0}$
		$\displaystyle-i\sin[h_{x}(k)]\cos[h_{y}(k)]\sigma_{x}$
		$\displaystyle-i\sin[h_{y}(k)]\cos[h_{x}(k)]\sigma_{y}$
		$\displaystyle+i\sin[h_{x}(k)]\sin[h_{y}(k)]\sigma_{z},$		(4)

which can be further recast into a single exponential form as

$$U(k)=e^{-iE(k)\vec{n}(k)\cdot\vec{\sigma}}=e^{-i\vec{d}(k)\cdot\vec{\sigma}}.$$

(5)

Here the eigenphase dispersion reads

$$E(k)=\arccos\{\cos[h_{x}(k)]\cos[h_{y}(k)]\},$$

(6)

$\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$, and the components of unit vector $\vec{n}=[n_{x}(k),n_{y}(k),n_{z}(k)]$ are explicitly given by

	$\displaystyle n_{x}(k)$	$\displaystyle=\sin[h_{x}(k)]\cos[h_{y}(k)]/\sin[E(k)],$
	$\displaystyle n_{y}(k)$	$\displaystyle=\sin[h_{y}(k)]\cos[h_{x}(k)]/\sin[E(k)],$		(7)
	$\displaystyle n_{z}(k)$	$\displaystyle=-\sin[h_{x}(k)]\sin[h_{y}(k)]/\sin[E(k)].$

We could then obtain the Floquet eigenstates by solving the eigenvalue equation of the effective Floquet Hamiltonian, i.e.,

$$H_{{\rm eff}}(k)|\psi_{\pm}(k)\rangle=E_{\pm}(k)|\psi_{\pm}(k)\rangle,$$

(8)

where the effective Hamiltonian $H_{{\rm eff}}(k)$ and eigenphase bands $E_{\pm}(k)$ are given by

$$H_{{\rm eff}}(k)=\vec{d}(k)\cdot\vec{\sigma},\qquad E_{\pm}(k)=\pm E(k),$$

(9)

and the explicit expressions of Floquet eigenstates $|\psi_{\pm}(k)\rangle$ are found to be

$$|\psi_{v}(k)\rangle=\frac{1}{\sqrt{2E_{v}(k)[E_{v}(k)-d_{z}(k)]}}\begin{bmatrix}d_{x}(k)-id_{y}(k)\\ E_{v}(k)-d_{z}(k)\end{bmatrix},$$

(10)

where $v=\pm$ will be used to denote the signs of the eigenphase dispersions $E_{\pm}(k)$. With these considerations, the time evolution operator of the system from time $t=0$ to a later time $t=s+2\ell$ [see Eq. (2)] can be written as $U(k,t)=U(k,s)U^{\ell}(k)$, where

$$U(k,s)=\begin{cases}e^{-ish_{x}(k)\sigma_{x}},&s\in[0,1)\\ e^{i(2-s)h_{y}(k)\sigma_{y}}U(k).&s\in[1,2)\end{cases}$$

(11)

Here $U(k)$ is the Floquet operator of the system over a complete driving period, as given by Eq. (3) Note1 . The evolution of a Floquet eigenstate $|\psi_{v}(k)\rangle$ ($v=\pm$) from $t=0$ to $t=s+2\ell$ is then described by $|\psi_{v}(k,t=s+2\ell)\rangle=U(k,s)U^{\ell}(k)|\psi_{v}(k)\rangle$, which can be expressed more explicitly as

$$|\psi_{v}(k,t)\rangle=e^{-i\ell E_{v}(k)}e^{-ish_{x}(k)\sigma_{x}}|\psi_{v}(k)\rangle$$

(12)

for $s\in[0,1)$ and

$$|\psi_{v}(k,t)\rangle=e^{-i(\ell+1)E_{v}(k)}e^{i(2-s)h_{y}(k)\sigma_{y}}|\psi_{v}(k)\rangle$$

(13)

for $s\in[1,2)$, respectively. Since $\{|\psi_{v}(k)\rangle|v=\pm\}$ in Eq. (10) form a complete and orthonormal basis, the dynamics of our periodically quenched system is completely solved by Eqs. (12)-(13) once the explicit expressions of $h_{x}(k)$ and $h_{y}(k)$ are specified.

With these preparations, we can now investigate the Floquet DQPTs in our class of piecewise quenched system. The central object in the study of DQPTs is the return amplitude (or return probability), which may be viewed as a dynamical version of the partition function DQPTRev1 . For an evolution from time $t=0$ to $t=s+2\ell$ [see Eq. (2)], the return amplitude of a single Floquet-Bloch state $|\psi_{v}(k)\rangle$ ($v=\pm$) is given by

$$G_{v}(k,t)=\langle\psi_{v}(k)|U(k,t)|\psi_{v}(k)\rangle=e^{-i\ell E_{v}(k)}G_{v}(k,s),$$

(14)

where

$$G_{v}(k,s)=\langle\psi_{v}(k)|U(k,s)|\psi_{v}(k)\rangle$$

(15)

describes the return amplitude of initial state $|\psi_{v}(k)\rangle$ at a given $k\in[-\pi,\pi)$ in micromotion time $s\in[0,2)$. Following the theory of Floquet DQPTs DQPTExp11 , if a state $|\psi_{v}(k)\rangle$ with $k=k_{c}$ happens to evolve to its orthogonal state at a critical time $s=s_{c}$, we will have $G_{v}(k_{c},s_{c})=0$, and the rate function of return probability (to be defined later) will become nonanalytic in the thermodynamic limit, signifying Floquet DQPTs at time $t_{c}=s_{c}+2\ell$ for all $\ell\in\mathbb{Z}$.

The conditions of Floquet DQPTs are determined by the expressions of critical momentum $k_{c}$ and critical time $t_{c}$. For our periodically quenched system, the forms of $k_{c}$ and $t_{c}$ can be found analytically. Using Eqs. (12)-(15) and the fact that $\langle\psi_{v}(k)|\sigma_{j}|\psi_{v}(k)\rangle=vn_{j}(k)$ for $j=x,y,z$ [see Eq. (7) for $n_{x,y,z}(k)$], we find that

$$G_{v}(k,s)=\begin{cases}\cos[sh_{x}(k)]-i\sin[sh_{x}(k)]vn_{x}(k),\\ e^{-iE_{v}(k)}\{\cos[s^{\prime}h_{y}(k)]+i\sin[s^{\prime}h_{y}(k)]vn_{y}(k)\},\end{cases}$$

(16)

where $s\in[0,1)$, $s^{\prime}\equiv 2-s\in[0,1)$ and $[n_{x}(k),n_{y}(k)]$ are the components of unit vector $\vec{n}(k)$ in Eq. (7). The expressions of $k_{c}$ and $s_{c}$ are then determined by the solutions of $G_{v}(k,s)=0$, yielding the following two sets of critical conditions

$$\begin{cases}sh_{x}(k)=\frac{2p-1}{2}\pi\quad\&\quad\sin[h_{x}(k)]\cos[h_{y}(k)]=0,\\ s^{\prime}h_{y}(k)=\frac{2q-1}{2}\pi\quad\&\quad\sin[h_{y}(k)]\cos[h_{x}(k)]=0,\end{cases}$$

(17)

for $s\in[0,1)$ and $s^{\prime}\equiv 2-s\in[0,1)$. Here $p,q\in\mathbb{Z}$ and Eq. (7) has been used to reach the second equality in each set of condition. Note that these conditions do not depend on the values of $v$ ($=\pm$), implying that we would have the same Floquet DQPT conditions for the initial state being prepared in either Floquet band. By straightforward calculations, we see that in each half of the driving period, the condition in Eq. (17) generates two possible sets of solutions for the critical momenta and time. For $s\in[0,1)$, we have $G_{v}(k_{c},s_{c})=0$ if

$$\begin{cases}h_{x}(k_{c})=m\pi,&m\in\mathbb{Z}_{\neq 0}\\ s_{c}=\frac{2p-1}{2m}\in(0,1),&p\in\mathbb{Z}\end{cases}$$

(18)

or

$$\begin{cases}h_{y}(k_{c})=\frac{2m-1}{2}\pi,&m\in\mathbb{Z}\\ s_{c}=\frac{2p-1}{2h_{x}(k_{c})}\in(0,1).&p\in\mathbb{Z}\end{cases}$$

(19)

For $s\in[1,2)$, we have $G_{v}(k_{c},s_{c})=0$ if

$$\begin{cases}h_{y}(k_{c})=n\pi,&n\in\mathbb{Z}_{\neq 0}\\ s^{\prime}_{c}=\frac{2q-1}{2n}\in(0,1),&q\in\mathbb{Z}\end{cases}$$

(20)

or

$$\begin{cases}h_{x}(k_{c})=\frac{2n-1}{2}\pi,&n\in\mathbb{Z}\\ s^{\prime}_{c}=\frac{2q-1}{2h_{y}(k_{c})}\in(0,1),&q\in\mathbb{Z}\end{cases}$$

(21)

where $s^{\prime}_{c}=2-s_{c}$. Given the explicit forms of $h_{x}(k)$ and $h_{y}(k)$, these conditions then determine all possible critical momenta $k_{c}$ and critical time $t_{c}=s_{c}+2\ell$ ($\ell\in\mathbb{Z}$) of Floquet DQPTs in the class of periodically quenched systems considered in this work. Note in passing that the critical time $s_{c}$ in Eqs. (18) and (20) take universal forms and independent of the explicit expressions of $h_{x}(k)$ and $h_{y}(k)$, which indicates their connections with the global nature of the underlying Floquet states. In the meantime, we notice that when the Floquet eigenphase spectrum of the system becomes gapless, i.e., when $E_{\pm}(k)=0$ or $E_{\pm}(k)=\pm\pi$ in Eq. (6), we would obtain

$$\begin{cases}h_{x}(k_{0})=m\pi,&m\in\mathbb{Z}\\ h_{y}(k_{0})=n\pi,&n\in\mathbb{Z}\end{cases}$$

(22)

where the set $\{k_{0}\}$ gives the locations at which the eigenphase band-gap closes in $k$-space, which is usually accompanied by a transition between different Floquet topological phases ZhouSQKR . From Eqs. (18) and (20), it is clear that once $m,n\neq 0$ in Eq. (22), the gapless quasimomenta $\{k_{0}\}$ of the Floquet spectrum are also critical momenta of the Floquet DQPTs, yielding critical times at $s_{c}$ and $2-s_{c}$ within a driving period. Since the set $\{k_{0}\}$ is generally different for different values of $(m,n)$ in Eq. (22), the possible critical momenta and critical times in Eqs. (18) and (20) will also be different in general when the system is initialized in different Floquet topological phases. This allows one to discriminate Floquet topological phases from the behaviors of Floquet DQPTs, thus extending the connection between Floquet DQPTs and Floquet topological matter found in Ref. DQPTExp11 to more general situations.

We next introduce the rate function of return probability, which is the observable to show the time-domain nonanalyticity of Floquet DQPTs explicitly. Following Eq. (14), the return probability of a Floquet state $|\psi_{v}(k)\rangle$ ($v=\pm$) evolved from time zero to $t$ is given by

$$g_{v}(k,t)=|\langle\psi_{v}(k)|U(k,t)|\psi_{v}(k)\rangle|^{2}=g_{v}(k,s),$$

(23)

where

$$g_{v}(k,s)=|\langle\psi_{v}(k)|U(k,s)|\psi_{v}(k)\rangle|^{2},$$

(24)

and $t=s+2\ell$ for $s\in[0,2)$, $\ell\in\mathbb{Z}$. It is clear that $g_{v}(k,t)$ possesses the same temporal periodicity as the system Hamiltonian, i.e., $g_{v}(k,t)=g_{v}(k,t+2)$. The complete behavior of $g_{v}(k,t)$ at any time $t$ is then determined once $g_{v}(k,s)$ is obtained. For our periodically quenched system, $g_{v}(k,s)$ takes the following explicit form according to Eq. (16)

$$g_{v}(k,s)=\begin{cases}\cos^{2}[sh_{x}(k)]+\sin^{2}[sh_{x}(k)]n_{x}^{2}(k),&s\in[0,1)\\ \cos^{2}[s^{\prime}h_{y}(k)]+\sin^{2}[s^{\prime}h_{y}(k)]n_{y}^{2}(k),&s\in[1,2)\end{cases}$$

(25)

where $s^{\prime}=2-s\in[0,1)$. Again, we notice that $g_{v}(k,s)$ is independent of $v$ ($=\pm$), i.e., in which band the system is initialized, and $g_{v}(k,s)=0$ at each given set of critical momentum and time $(k,s)=(k_{c},s_{c})$ in Eqs. (18)-(21). The latter suggests us to consider the rate function of return probability $f_{v}(t)$, defined as

$$f(t)=-\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k}\ln[g(k,t)]=-\int_{-\pi}^{\pi}\frac{dk}{2\pi}\ln[g(k,t)],$$

(26)

where $N$ is the number of degrees of freedom in the system (e.g., the number of unit cells), which is taken to be infinity in the thermodynamic limit. We have also denoted $g_{v}(k,s)$ by $g(k,s)$ in Eq. (26), as the former is independent of $v$. The rate function thus defined may then be viewed as a “free energy density” of the Floquet dynamics. Since $f(t)$ is a periodic function of time with the period $T=2$, i.e., $f(t)=f(t+2)$, we have for any time $t=s+2\ell$ with $s\in[0,2)$ and $\ell\in\mathbb{Z}$ that

$$f(t)=f(s)=-\int_{-\pi}^{\pi}\frac{dk}{2\pi}\ln[g(k,s)].$$

(27)

If $g(k,s)$ vanishes at certain critical time and momentum $(k_{c},t_{c})$, $\ln[g(k_{c},s_{c})]$ will diverge, and after the integration over $k$, $f(s)$ could become nonanalytic at time $t_{c}=s_{c}+2\ell$ for all $\ell\in\mathbb{Z}$, which are referred to as Floquet DQPTs. For 1D systems, a Floquet DQPT is usually manifested as a cusp in the profile of $f(t)$ around $t=t_{c}$ DQPTExp11 . Using Eqs. (25) and (27), we can further obtain the explicit expression of the return rate function $f(s)$ for our periodically quenched system, i.e.,

$$f(s)=-\int_{-\pi}^{\pi}\frac{dk}{2\pi}\cdot\begin{cases}\ln\{\cos^{2}[sh_{x}(k)]+\sin^{2}[sh_{x}(k)]n_{x}^{2}(k)\},\\ \ln\{\cos^{2}[s^{\prime}h_{y}(k)]+\sin^{2}[s^{\prime}h_{y}(k)]n_{y}^{2}(k)\},\end{cases}$$

(28)

for $s\in[0,1)$ and $s^{\prime}=2-s\in[0,1)$, respectively. The profile of $f(s)$ over a driving period $s\in[0,2)$ then contains all the information about the time-domain nonanalytic properties of Floquet DQPTs.

In previous studies, dynamical topological order parameters (DTOPs) have been introduced to characterize DQPTs following a sudden quench DTOP1 ; DTOP2 , or Floquet DQPTs induced by harmonic driving fields DQPTExp11 . The values of a DTOP, obtained from the winding number of the noncyclic (Pancharatnam) geometric phase of the evolving state over the first Brillouin zone at each time, usually show a quantized jump at every critical time of a DQPT or Floquet DQPT, and remain quantized otherwise. It thus plays the role of a topological order parameter of the dynamics. In this work, we extend the DTOP-characterization of Floquet DQPTs to our periodically quenched setting.

The noncyclic geometric phase is given by the difference between the total phase and dynamical phase of the return amplitude $G_{v}(k,t)$ ($v=\pm$). Following Eq. (15), the total phase is

$$\phi_{v}(k,t)=-i\ln\left[\frac{G_{v}(k,t)}{|G_{v}(k,t)|}\right]=-\ell E_{v}(k)+\phi_{v}(k,s),$$

(29)

where we have set $t=s+2\ell$ for $s\in[0,2)$ and $\ell\in\mathbb{Z}$ to reach the second equality. Using Eq. (16), we obtain the micromotion part of the total phase $\phi_{v}(k,s)$ for our periodically quenched setting as

$$\phi_{v}(k,s)=\begin{cases}-\arctan\{\tan[sh_{x}(k)]vn_{x}(k)\},\\ \arctan\{\tan[s^{\prime}h_{y}(k)]vn_{y}(k)\}-E_{v}(k),\end{cases}$$

(30)

for $s\in[0,1)$ and $s^{\prime}=2-s\in[0,1)$, respectively. The dynamical phase of the return amplitude is defined as

$$\phi_{v}^{{\rm D}}(k,t)=-\int_{0}^{t}dt^{\prime}\langle\psi_{v}(k,t^{\prime})|H(k,t^{\prime})|\psi_{v}(k,t^{\prime})\rangle.$$

(31)

Using the time-periodicity of Hamiltonian $H(k,t)$ and the direct result of Floquet theorem $|\psi_{v}(k,t+2)\rangle=e^{-iE_{v}(k)}|\psi_{v}(k,t)\rangle$, $\phi_{v}^{{\rm D}}(k,t)$ can be further expressed as

$$\phi_{v}^{{\rm D}}(k,t)=\ell\phi_{v}^{{\rm D}}(k,2)+\phi_{v}^{{\rm D}}(k,s),$$

(32)

where we have set $t=s+2\ell$ with $s\in[0,2)$ and $\ell\in\mathbb{Z}$ on the right hand side of the equality. For our periodically quenched system, we find with the help of Eqs. (1), (10), (12) and (13) that $\phi_{v}^{{\rm D}}(k,2)=-v\vec{h}(k)\cdot\vec{n}(k)$ and

$$\phi_{v}^{{\rm D}}(k,s)=\begin{cases}-vsh_{x}(k)n_{x}(k),&s\in[0,1)\\ -v[\vec{h}(k)\cdot\vec{n}(k)-s^{\prime}h_{y}(k)n_{y}(k)],&s\in[1,2)\end{cases}$$

(33)

where $s^{\prime}=2-s$. Note that the dynamical phase is closely related to the relative angle between the components $[h_{x}(k),h_{y}(k),0]$ of the time-dependent Hamiltonian $H(k,t)$ and the components $[n_{x}(k),n_{y}(k),n_{z}(k)]$ of the Floquet effective Hamiltonian in each half of the driving period. Taking the difference between the total phase and dynamical phase, we find the noncyclic geometric phase DQPTExp11 to be

$$\phi_{v}^{{\rm G}}(k,t)=\phi_{v}(k,t)-\phi_{v}^{{\rm D}}(k,t),$$

(34)

whose explicit expression for our periodically quenched system can be obtained directly from Eqs. (29)-(33). At a fixed time $t$, the winding number of $\phi_{v}^{{\rm G}}(k,t)$ in $k$-space then determines the DTOP $w_{v}(t)$, given by

$$w_{v}(t)=\int\frac{dk}{2\pi}\partial_{k}\phi_{v}^{{\rm G}}(k,t).$$

(35)

When the evolution of the system passes a critical time $t_{c}$ of the Floquet DQPTs, the value of $w_{v}(t)$ is expected to show a quantized jump DQPTExp11 , i.e.,

$$w_{v}(t_{c}^{+})-w_{v}(t_{c}^{-})\in\mathbb{Z},$$

(36)

where the exact amount of jump depends on the configuration of the critical momenta set $\{k_{c}\}$ in the first Brillouin zone. For certain specific models, the integration range of $k$ in Eq. (36) could be chosen as a subset of the first Brillouin zone $k\in[-\pi,\pi)$ due to the possible translation and reflection symmetries of $\phi_{v}^{{\rm G}}(k,t)$ in $k$-space. For example, if we have $h_{j}(k\pm\pi)=-h_{j}(k)$ for $j=x,y$, both $\phi_{v}(k,t)$ and $\phi_{v}^{{\rm D}}(k,t)$ will be invariant under the translation over $\pi$ in $k$-space, and the same is true for $\phi_{v}^{{\rm G}}(k,t)$. So the range of integration in Eq. (36) can be confined to $k\in[0,\pi]$ in this case. More examples about the reduction of integration range in the calculation of $w_{v}(t)$ will be made clear in the next section.

To sum up, we have established a theoretical framework to analyze Floquet DQPTs in a typical class of periodically quenched lattice model. The critical conditions as summarized in Eqs. (18)-(21) clearly suggest that by enlarging the amplitudes of quench functions $h_{x}(k)$ and $h_{y}(k)$, the exact number of Floquet DQPTs that could occur in each driving periodic can be put under flexible control. This highlights one of the key advantages of Floquet engineering in the realization of DQPTs. Besides, we have observed the connection between Floquet DQPTs and the underlying Floquet topological phases of the periodically quenched system, and also introduced a DTOP to characterize the topological signatures of Floquet DQPTs. In the following section, we will demonstrate our results with a prototypical periodically quenched lattice model, which is rich in Floquet DQPTs and experimentally realizable at the same time.

Our explicit model is obtained by setting

$$h_{x}(k)=J_{x}\cos k,\quad h_{y}(k)=J_{y}\sin k,$$

(37)

in the general Hamiltonian Eq. (1), leading to the Floquet operator

$$U(k)=e^{-iJ_{y}\sin k\sigma_{y}}e^{-iJ_{x}\cos k\sigma_{x}}.$$

(38)

Here the parameters $J_{x}$ and $J_{y}$ control the amplitudes of the two quenches within each half of the driving period. Physically, they can be interpreted as the kicking strengths of a spin-$1/2$ quantum kicked rotor ZhouSQKR , or the hopping amplitudes of a particle on a tight-binding lattice ZhouDWN . In the former case, the system may be realized by the setup proposed in Refs. SQKRExp1 ; SQKRExp2 , which achieves the quantum walk in momentum space by ⁸⁷Rb BEC atoms. In the latter case, the system may be realized by replacing the harmonic driving fields with square-wave pulses in the NV center experimental setup of Refs. DQPTExp11 ; ZhouGTPExp . Theoretically, rich Floquet topological phases have been identified in this model, which are characterized by larger topological invariants and multiple topological edge modes ZhouSQKR . This further motivates us to choose this model as our working example to demonstrate the key features of Floquet DQPTs in periodically quenched systems.

From now on, we take the quench amplitudes $J_{x,y}>0$ without loss of generality. Using Eqs. (37) and (38), the Floquet operator can be written in a single exponential form as $U(k)=e^{-iE(k)\vec{n}(k)\cdot\vec{\sigma}}$, where $\cos[E(k)]=\cos(J_{x}\cos k)\cos(J_{y}\sin k)$ and the unit vector $\vec{n}(k)=[n_{x}(k),n_{y}(k),n_{z}(k)]$, with

	$\displaystyle n_{x}(k)$	$\displaystyle=\frac{\sin(J_{x}\cos k)\cos(J_{y}\sin k)}{\sin[E(k)]},$
	$\displaystyle n_{y}(k)$	$\displaystyle=\frac{\sin(J_{y}\sin k)\cos(J_{x}\cos k)}{\sin[E(k)]},$		(39)
	$\displaystyle n_{z}(k)$	$\displaystyle=-\frac{\sin(J_{x}\cos k)\sin(J_{y}\sin k)}{\sin[E(k)]}.$

The Floquet effective Hamiltonian then takes the form $H_{{\rm eff}}=E(k)\vec{n}(k)\cdot\vec{\sigma}=\vec{d}(k)\cdot\vec{\sigma}$, with the corresponding eigenvalue equation $H_{{\rm eff}}(k)|\psi_{\pm}(k)\rangle=E_{\pm}(k)|\psi_{\pm}(k)\rangle$, whose explicit solutions are given by Eq. (10). Following Eq. (22), the gapless quasimomenta of the model satisfy $J_{x}\cos k_{0}=m\pi$ and $J_{y}\sin k_{0}=n\pi$ for $m,n\in\mathbb{Z}$, yielding the curves that describing the boundaries between different Floquet topological phases, i.e., $m^{2}\pi^{2}/J_{x}^{2}+n^{2}\pi^{2}/J_{y}^{2}=1$ for $|m|\leq J_{x}/\pi$ and $|n|\leq J_{y}/\pi$ ZhouSQKR . According to the discussion in the last section, the solutions of $k_{0}$ also generate a set of critical momenta $k_{c}$ for the Floquet DQPTs when $m,n\neq 0$.

Using Eqs. (18)-(21), we could obtain the solutions of critical momenta and critical times for the PQL model if the system is initialized in a Floquet eigenstate. In the first half of a driving period, i.e., for $s\in[0,1)$, the set of $(k_{c},s_{c})$ satisfy

$$\begin{cases}k_{c}=\arccos\left(\frac{m\pi}{J_{x}}\right),&m\in\mathbb{Z}_{\neq 0}\cap|m|\leq\frac{J_{x}}{\pi}\\ s_{c}=\frac{2p-1}{2m}\in[0,1),&p\in\mathbb{Z}\end{cases}$$

(40)

or

$$\begin{cases}k_{c}=\arcsin\left(\frac{2m-1}{2J_{y}}\pi\right),&m\in\mathbb{Z}\cap\left|m-\frac{1}{2}\right|\leq\frac{J_{y}}{\pi}\\ s_{c}=\frac{2p-1}{2J_{x}\cos k_{c}}\pi\in[0,1).&p\in\mathbb{Z}\end{cases}$$

(41)

Similarly, in the second half of a driving period, i.e., for $s\in[1,2)$, the set $(k_{c},s_{c}=2-s^{\prime}_{c})$ satisfy

$$\begin{cases}k_{c}=\arcsin\left(\frac{n\pi}{J_{y}}\right),&n\in\mathbb{Z}_{\neq 0}\cap|n|\leq\frac{J_{y}}{\pi}\\ s^{\prime}_{c}=\frac{2q-1}{2n}\in[0,1),&q\in\mathbb{Z}\end{cases}$$

(42)

or

$$\begin{cases}k_{c}=\arccos\left(\frac{2n-1}{2J_{x}}\pi\right),&n\in\mathbb{Z}\cap\left|n-\frac{1}{2}\right|\leq\frac{J_{x}}{\pi}\\ s^{\prime}_{c}=\frac{2q-1}{2J_{y}\sin k_{c}}\pi\in[0,1).&q\in\mathbb{Z}\end{cases}$$

(43)

Eqs. (40)-(43) provide us with analytical solutions for all possible critical momenta $k_{c}$ and critical times $t_{c}=s_{c}+2\ell$ ($\ell\in\mathbb{Z}$) of the Floquet DQPTs in the PQL model. The notable connection between the expressions of $k_{c}$ in Eqs. (40), (42) and the gapless quasimomentum $k_{0}$ satisfying $k_{0}=\arccos(m\pi/J_{x})=\arcsin(n\pi/J_{y})$ for $m,n\in\mathbb{Z}$ also allows one to obtain qualitatively different patterns of Floquet DQPTs in different Floquet topological phases of the PQL model, as will be demonstrated explicitly by numerical calculations.

In the following, we present typical examples of Floquet DQPTs in the PQL model by evaluating the rate function of return probability in Eq. (26), noncyclic geometric phase in Eq. (34) and DTOP in Eq. (35). We first evolve the system according to Eq. (23) for each Floquet eigenstate, and then obtain the return rate from Eq. (26) by numerically evaluating the integral over quasimomentum $k$. The geometric phase is obtained by taking the difference between the total phase and the dynamical phase of the return amplitude following Eqs. (29) and (31). After numerically obtaining the geometric phase, we compute the DTOP in terms of Eq. (35). In the computation of the integral, $N=300$ grids in $k$-space is taken, which is enough for the convergence of the numerical data in order to reproduce the nonanalytic signals of Floquet DQPTs in the thermodynamic limit (corresponding to $N\rightarrow\infty$). Note that for the PQL model, we have $h_{j}(k\pm\pi)=-h_{j}(k)$ for $j=x,y$ according to Eq. (37), implying that the geometric phase has a translational symmetry over $\pi$ in $k$-space. Furthermore, since $h_{x}(k)$ [$h_{y}(k)$] and $n_{x}(k)$ [$n_{y}(k)$] are even (odd) functions of $k$, the geometric phase also has a reflection symmetry with respect to $k=0$ according to Eqs. (30), (33) and (34). Therefore, in the calculation of DTOP in Eq. (35) for the PQL model, we could restrict the range of integration over $k$ to one quarter of the first Brillouin zone, e.g., $k\in[0,\pi/2]$. As a further simplification, we notice that the order of the two quenches are swapped under the combined effects of parameter exchange $J_{x}\leftrightarrow J_{y}$ and momentum shift $k\rightarrow\pi/2-k$. The result of such a swap is simply a shifting of the Floquet DQPTs from one half of the driving period to the other half, as directly deducible from Eq. (28). Therefore, for $J\neq J^{\prime}$, the Floquet DQPTs in the system with $(J_{x},J_{y})=(J,J^{\prime})$ and $(J_{x},J_{y})=(J^{\prime},J)$ can be directly related with each other. For $J=J^{\prime}$, each Floquet DQPT happens at the micromotion time $s_{c}$ has a mirror that happens at $2-s_{c}$ for all $t_{c}=s_{c}+2\ell$ with $\ell\in{\mathbb{Z}}$.

In the first set of results, we consider the initial state of the PQL model to be prepared in the Floquet band with quasienergy $E_{-}(k)$ at each quasimomentum $k\in[-\pi,\pi)$, and present the rate function of return probability $f(t)$ [Eq. (28)], noncyclic geometric phase $\phi_{-}^{{\rm G}}(k,t)$ [Eq. (34)] and DTOP $w_{-}(t)$ [Eq. (35)] versus time over a driving period $t=0\rightarrow 2$ in Figs. 1, 2 and 3, respectively. The system parameters are set as $J_{x}=0.5\pi$ for all figure panels and $J_{y}=1.1\pi,2.1\pi,3.1\pi,4.1\pi$ in the panels (a), (b), (c) and (d) of each figure, respectively. According to Ref. ZhouSQKR , the PQL model with these four sets of system parameters belong to four different Floquet topological phases, with larger topological invariants at larger values of $J_{y}$. In Fig. 1, we observe that with the increase of $J_{y}$, the system undergoes more and more Floquet DQPTs within a single driving period, with a cusp observed at every critical time $t_{c}$ of the DQPT. The possible values of critical times are predicted precisely by the critical condition in Eq. (42), whose solutions of $k_{c}$ are closely related to the gapless quasimomenta $k_{0}$ of the system. Specially, more Floquet DQPTs are observed when the initial phase of the PQL model carries larger topological invariants (see the phase diagram in Fig. 1 of Ref. ZhouSQKR ). This implies the possibility of discriminating different Floquet topological phases of the PQL model from Floquet DQPTs, and also introduces a route to engineer more Floquet DQPTs in a finite time window with the help of Floquet topological phases. Besides, the global profiles of $f(t)$ in Fig. 1 do not decay with time thanks to the applied driving field, which allow us to observe and detect the nonanalytic signatures of Floquet DQPTs in a larger time window compared with the case in conventional DQPTs following a single quench DQPTRev1 . The noncyclic geometric phase $\phi_{-}^{{\rm G}}(k,t)$, as presented in Fig. 2 also possesses a $2\pi$-jump around the critical momenta $k_{c}$ (red dashed lines) at each critical time $t_{c}$ (black dash-dotted lines), reflecting the discontinuity in the time-derivatives of the rate function $f(t)$ at the Floquet DQPTs. After the integration over $k\in[0,\pi/2]$, the $2\pi$-change of $\phi_{-}^{{\rm G}}(k,t)$ (corresponding to the sudden change of colors from dark blue to dark yellow or the opposite in each figure panel) finally leads to the integer quantized jump of DTOP at each critical time (black dash-dotted lines) of the Floquet DQPT, as presented in Fig. 3. Except at the critical times, the DTOP remain quantized over other time durations, and thus playing the role of a topological order parameter for Floquet DQPTs. Notably, in Figs. 3(c) and 3(d), $w_{-}(t)$ possesses a change $\Delta{w_{-}}=2$ at $t_{c}=1.5$. Such a jump of DTOP in a step bigger than $1$ is not observed in DQPTs following a single quantum quench, and may thus be a unique feature of DQPTs following time-periodic drivings or other more complicated nonequilibrium protocols.