Discrete time crystals in one-dimensional classical Floquet systems with nearest-neighbor interactions

TL;DR

The paper addresses the existence of prethermal discrete time crystals (PDTCs) in a one-dimensional, disorder-free classical Floquet system with nearest-neighbor interactions. It develops a 1D spin model driven by alternating half-period Hamiltonians $H_z$ and $H_x$ and a global flip $R_x(\pi)$, starting from finite-temperature-like initial states, and analyzes the stroboscopic dynamics through the zeroth-order Floquet Hamiltonian $\overline{H}_{\mathrm{eff}}$, magnetization $M^z$, and decorrelator $d$. The key findings show disorder-free DTC order persisting up to a prethermal-like regime, with the thermalization time $\tau^*$ growing exponentially with the driving frequency $\omega$ and depending on the initial energy density, and a robust subharmonic response to driving that tolerates imperfect spin flips. The results extend the landscape of PDTCs to classical 1D NN systems, highlight a slow-dynamics mechanism tied to $D_x$ that governs DTC lifetimes, and suggest experimental platforms for observing classical DTCs.

Abstract

Prethermal discrete time crystals (PDTCs), an emergent non-equilibrium phase of matter, have been studied in two- and higher-dimensional lattices with nearest-neighbor (NN) interactions and one-dimensional (1D) lattices with long-range interactions. However, different from prethermalization that can be observed in 1D Floquet classical spin systems with NN interactions, classical PDTCs in Floquet 1D systems with only NN interactions have not been proposed before. Here, we demonstrate the emergence of disorder-free discrete time crystals (DTCs) in 1D Floquet classic spin systems with NN interactions. We show that the thermalization time first grows exponentially as the driving frequency increases and is then saturated, which depends on the energy density of the initial state. Since thermalization of the effective Hamiltonian is slow, there is no typical prethermalization and PDTCs in the Floquet system before final thermalization. The robustness of DTC order is verified by introducing imperfect spin flip operations. Our work provides an exploration of quantum characteristics, when considering the classical counterparts of quantum phenomena, and will be helpful for further investigations of both classical and quantum prethermal systems and discrete time-crystalline order

Figures (7)

• Figure 1: Schematic diagram for the Floquet driving system. During the first half period and second half period, the dynamics of the system is governed by $H_z$ and $H_x$, respectively. At the beginning of each period, a global spin flip operation $R_x(\pi)$ along the $x$-axis is applied.
• Figure 2: Characterization of prethermalization-like DTCs in 1D classical systems with NN interactions. (a--c) Dynamics of the average energy density $\overline{H}_\mathrm{eff}$ (a), the magnetization along $z$-direction $M^z$ (b), and the normalized decorrelator $d$ (c) versus the evolution time $t/T$, for different driving frequencies $\omega$. Here, $J_z=0.399$, $J_x=0.011$, $b_z=-0.016$, $b_x=-0.3$, $N=100$, and the periodic boundary conditions (PBCs) are considered. For each driving frequency, two curves in the same color in (a, b) denote values at even and odd periods, respectively. The driving frequency $\omega$ ranges from $0.8\pi$ to $1.4\pi$. (d) Average thermalization time $\tau^{*}$ versus the driving frequency $\omega$, which is obtained as the time when $d$ reaches $0.9$, as illustrated in (c). The mean values shown with errorbars for the one standard derivation (1SD) are obtained with 100 random initial states. The black dashed line denotes the exponential fitting $e^{c\omega}$. Hereafter, the same set of parameters are used unless stated otherwise.
• Figure 3: Robustness of DTCs in 1D classical systems with NN interactions by introducing an imperfect global $x$-flip operation with a small error $\delta_r$. (a--c) Fourier transform (FT) signals of the average magnetization $M^z$ (c) are compared with those of the trivial driving without any interaction (a) and (b) for $\delta_r=0$ rad and 0.03 rad, respectively, with the driving frequency $\omega$ being set as $\pi$. The first 500 periods of $M_z$ are chosen to calculate the FT signals, with an accuracy of $10^{-3}$ rad in the spectrum. For $\delta_r=0.03$ rad, robust DTC order is observed in (c) with one peak of the FT signals, while the FT signals of the trivial imperfect global $x$-flip in (b) splits into 2 peaks. (d) Crystalline fraction $f$ as a function of $\delta_r$. The crystalline fraction $f$ approaching 1 indicates the robust subharmonic response of PDTC order. The subharmonic response is stable for $\delta_r / \pi \in[-0.1,0.27]$, with a maximum tolerance of the error of around $\pi/10$.
• Figure 4: Initial energy-density dependence of DTCs in 1D classical systems with NN interactions. (a--c) Time evolutions of the mean values of the energy density $\overline{H}_{\textrm{eff}}$ (a), the magnetization ${M}^z$ (b), and the normalized decorrelator $d$ (c) for different initial states, with $S_0^z$ denoting the $z$-component value of the spin vector.
• Figure 5: Time evolutions under the impact of $D^{(0)}$ and $D_x$ starting with different initial states with different initial average polar angles $\theta$. (a,b) Dynamics of the mean values of the magnetization $M^z$ (a) and the normalized decorrelator $d$ (b) under the impact of $D^{(0)}$. There exists flip of the magnetization for $\theta=\pi/6$. (c,d) Long-time behaviors of the magnetization $M^z$ (c) and the normalized decorrelator $d$ (d) in the toggling frame, during the dynamics under the impact of spin-flip-symmetric $D_x$. No magnetization flip is observed for dynamics with $D_x$. Here, the $\theta / \pi$ is chosen as $\pm \frac{1}{6}$, $\pm \frac{2}{6}$, $\pm \frac{3}{6}$, and $0$. Results are obtained from the dynamics starting with 100 random initial states.