Dynamical relaxation of correlators in periodically driven integrable quantum systems

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