Prethermalization in aperiodically kicked many-body dynamics

Driven many-body systems typically experience heating due to the lack of energy conservation. Heating may be suppressed for time-periodic drives, but little is known for less regular drive protocols. In this paper, we investigate the heating dynamics in aperiodically kicked systems, specifically those driven by quasiperiodic Thue-Morse or a family of random sequences with $n$-multipolar temporal correlations. We demonstrate that multiple heating channels can be eliminated even away from the high-frequency regime. The number of eliminated channels increases with multipolar order $n$. We illustrate this in a classical kicked rotor chain where we find a long-lived prethermal regime. When the static Hamiltonian only involves the kinetic energy, the prethermal lifetime $t^{*}$ can strongly depend on the temporal correlations of the drive, with a power-law dependence on the kick strength $t^{*}\sim K^{-2n}$, for which we can account using a simple linearization argument.

Figure 1

(a) Time evolution of the averaged kinetic energy for $n$-random multipolar drive (RMD) and Thue-Morse (TM) drive with $K=0.03$ in a log-log scale. (b) Dependence of the prethermal lifetime $t^{*}$ on $1/K$ in a log-log scale. Dashed lines ($K^{-2n}$ for $n>0$ and $K^{-2}$ for $n=0$) are plotted to guide the eyes. (c) Prethermal lifetime $t^{*}$ scaling for TM drives.

Figure 2

(a) Trajectories in phase space. The black orbit is obtained by the area-preserving map ${\overline{M}}_n^{\prime }$. The magenta curve with a constant expansion rate is generated via ${\overline{M}}_1$. The blue curve is a single realization obtained by stochastically applying the matrix $M_1^{\pm }$. (b) The averaged radius $\langle r_h\rangle$ matches well with theoretical predictions (dashed lines). $F=0.08$ is used in both panels.

Figure 3

(a) Time evolution of the averaged kinetic energy density for different multipolar order $n$ with $K=0.005$ in a log-log scale. (b) Prethermal lifetime $t^{*}$ as a function of $1/K$ in a log-log scale. The system starts from the initial angular momentum $p_j\left(0\right)=0$ with $B=0.01$. Other numerical parameters are the same as in Fig. 1. Dashed lines correspond to the scaling $K^{-2}$.

Figure 4

Time evolution of the averaged kinetic energy for different kick periods $\tau$. Reducing $\tau$ clearly increases the heating rate. We use $n=1$ random multipolar drive (RMD) and a fixed kick strength $K=0.5$. $q_j\left(0\right)$ is generated from a random uniform sampling within $[0,2\pi ]$, and $p_j\left(0\right)$ is randomly chosen from a Gaussian distribution with zero mean and standard deviation $\sigma =1$. Different values of the kick period $\tau$ are shown in the legend.

Figure 5

The averaged prethermal kinetic energy density $\langle E_{\text{kin}}^{*}\rangle$ for $n=1$ random multipolar drive (RMD) for (a) $B=0$ with $\tilde{p}=0.1$, (b) $B=0$ with $\tilde{p}=0$, and (c) $B=0.01$ with $\tilde{p}=0$. The prethermal temperature scales as $T\sim K^2$ for (a) and (b) when the kick is weak. In the last case, it is fixed by $B$, so all the plateau parts of energy curves are at the same value $\langle E_{\text{kin}}^{*}\rangle =0.0048$. The rest numerical settings are the same as in Fig. 1. All plots use a log-log scale.

Figure 6

Prethermalization for the periodically driven case: (a) averaged kinetic energy vs time in a log-log scale, and (b) prethermal lifetime $t^{*}$ vs $1/K$ in a semilog scale.

Figure 7

(a) and (b) depict the power-law and exponential fitting of the prethermal lifetime $t^{*}(1/K)$ for the Thue-Morse (TM) drive.

Figure 8

(a) Momentum distribution at the end of the prethermal stage for a Gaussian initial momentum distribution with a zero mean and a standard deviation $\sigma$ indicated in different colours, and (b) corresponding prethermal lifetime $t^{*}$ as a function of $1/K$ for $n=20$ in log-log. The rest numerical settings are the same as in Fig. 1.

Figure 9

Time evolution of the averaged kinetic energy for the TM drive. The simulation results converge for large systems, and $N=500$ is already sufficient to produce thermodynamically large systems. Here, we use the kick strength $K=0.07$, and the results are averaged over 200 random realizations. Initial states are the same as in Fig. 1.