Dynamical relaxation of correlators in periodically driven integrable quantum systems

Abstract

We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady state value as $n^{-(\alpha+1)/\beta}$, where $n$ is the number of drive cycles and $\alpha$ and $\beta$ denote positive integers. We find that generically $\beta=2$ within a dynamical phase characterized by a fixed $\alpha$; however, its value can change to $\beta=3$ or $\beta=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\beta$, and chart out the values of $\alpha$ and $\beta$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale $n_c$ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, $\omega_c$, is approached from below ($\omega < \omega_c$). We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.