Stable time rondeau crystals in dissipative many-body systems

TL;DR

Driven many-body systems can host non-equilibrium temporal orders like time rondeau crystals (TRCs), but heating typically destroys them. This work introduces dissipation and multistability via a dipolar random kicking protocol in a dissipative kicked-rotor model, enabling TRCs to persist indefinitely in the thermodynamic limit. Synchronization in the non-interacting limit stabilizes the long-time order, while finite interactions $J$ induce a de-synchronization phase transition at a finite critical coupling $J_c$, captured by linear stability analysis and supported by large-scale simulations and Lyapunov-exponent statistics. The results point to a versatile route for stabilizing partial temporal order in dissipative many-body systems and open avenues for quantum realizations and experimental tests with Rydberg atoms or superconducting circuits.

Abstract

Driven systems offer the potential to realize a wide range of non-equilibrium phenomena that are inaccessible in static systems, such as the discrete time crystals. Time rondeau crystals with a partial temporal order have been proposed as a distinctive prethermal phase of matter in systems driven by structured random protocols. Yet, heating is inevitable in closed systems and time rondeau crystals eventually melt. We introduce dissipation to counteract heating and demonstrate stable time rondeau crystals, which persist indefinitely, in a many-body interacting system. A key ingredient is synchronization in the non-interacting limit, which allows for stable time rondeau order without generating excessive heating. The presence of many-body interaction competes with synchronization and a de-synchronization phase transition occurs at a finite interaction strength. This transition is well captured via a linear stability analysis of the underlying stochastic processes.

Figures (14)

• Figure 1: (a) Basins of fixed points in the dissipative kicked rotor model. Transition rules at stroboscopic times can be determined by tuning the random kick strength $K_r$. For instance, after a positive kick $+K_r$, the rotor can suddenly jump from point $A$ to $A'$, which is located in the basin of the fixed point $P_+$ (blue), realizing $P_0{\to} P_+$. (b) Exact transition rules. (c) A dipolar kick induces a time rondeau order without generating unwanted heating.
• Figure 2: Red and blue trajectories of a single rotor quickly synchronize, exhibiting the time rondeau order where the long-time order at $t{=}2mT$ coexists with the short-time disorder. The gray line depicts the average momentum in a many-rotor system, which follows the single-rotor trajectory. We use $J{=}0.25,$$K_r{=}5.5$ for the numerical simulation. The initial $\theta_i$ is randomly sampled within $[0, 2\pi]$ and $p_i$ is sampled around $P_+$ according to a Gaussian distribution of a standard deviation $0.1$.
• Figure 3: (a) Spatial-temporal distribution of momentum in the synchronized phase for weak $J{=}0.25$. (b) Defects persist for stronger interaction $J{=}0.35$. Both (a) and (b) use the same random kick sequence. (c) and (d) Momentum variance and time rondeau order parameter for different $J$. Ensemble averages over different initial states and random drive realizations are performed. The system demonstrates synchronized TRC for a weak interaction (black and green) while a larger interaction induces de-synchronization (red). The initial momentum is sampled from a Gaussian distribution with a standard deviation $6$ and zero average. We use parameters $K_r=5.5, L=256$ and a periodic boundary condition for numerical simulations.
• Figure 4: (a) Dependence of order parameters on $J$ for $K_r{=}5.5,\gamma{=}0.8$. The phase transition occurs at $J_c{\approx}0.32$. (b) Dependence of order parameters on the dissipation rate $1{-}\gamma$ for $K_r{=}5.5,J{=}0.2$. Regions I and IV correspond to the de-synchronization phase. In Regions II and III, the system is synchronized, $\langle\bar{\sigma}_p^2\rangle{=}0$. However, only for $1{-}\gamma{>}0.16$ (gray dashed line in panels (b) and (d)), the single kicked rotor has three fixed points and the many-body system exhibits the desired synchronized TRC. (c) Phase diagram for different $K_r$ and $J$. (d) Phase diagram for different $1{-}\gamma$ and $J$. The color in both (c) and (d) denotes the value of $\langle\bar{\sigma}_p^2\rangle$ where we truncate its maximum value to $8$. White and blue regions correspond to the synchronization and de-synchronization phases, respectively. The purple curve corresponds to $\langle\bar{\sigma}_p^2\rangle{=}10^{-5}$. A linear stability analysis (LSA) captures the synchronization phase transition (red). The $T{\to}\infty$ assumption predicts the critical interaction strength, orange dashed lines in (c) and (d), which notably overestimate the phase boundary. The black dots denote the phase boundary obtained by $\bar{\mathcal{O}}_{\bar{p}}$. Initial state distribution is the same as in Fig. \ref{['fig:avar-t']}.
• Figure 5: Distribution of largest Lyapunov exponents (LLEs). We consider a de-synchronization phase transition occurs when 5% of LLEs become positive. Numerically we use $K_r{=}5.5, m{=}300$ and 1000 different mean-field trajectories. Convergence of the distribution is discussed in Sec. SM 2.4 supplement.