Dynamical relaxation of correlators in periodically driven integrable quantum systems

TL;DR

This paper investigates how correlation functions in periodically driven integrable quantum systems relax over multiple drive cycles, revealing universal decay behaviors, phase transitions, and oscillatory phenomena linked to Floquet spectrum properties.

Contribution

It introduces a detailed analysis of the decay exponents of correlators in driven integrable models, connecting these to Floquet spectrum symmetries and identifying critical drive frequencies.

Findings

Correlation functions decay as n^{-(α+1)/β} with specific β values.

Decay exponents change at critical drive frequencies, indicating dynamical phase transitions.

Oscillatory long-time behavior occurs near critical frequencies, linked to stationary points in Floquet spectrum.

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.