Dynamical relaxation of correlators in periodically driven integrable quantum systems

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.