Entanglement generation in periodically driven integrable systems: Dynamical phase transitions and steady state

Abstract

We study a class of periodically driven $d$-dimensional integrable models and show that after $n$ drive cycles with frequency $\ensuremath{\omega}$, pure states with non-area-law entanglement entropy ${S}_{n}(l)\ensuremath{\sim}{l}^{\ensuremath{\alpha}(n,\ensuremath{\omega})}$ are generated, where $l$ is the linear dimension of the subsystem, and $d\ensuremath{-}1\ensuremath{\le}\ensuremath{\alpha}(n,\ensuremath{\omega})\ensuremath{\le}d$. The exponent $\ensuremath{\alpha}(n,\ensuremath{\omega})$ eventually approaches $d$ (volume law) for large enough $l$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We identify and analyze the crossover phenomenon from an area $(S\ensuremath{\sim}{l}^{d\ensuremath{-}1}$ for $d\ensuremath{\ge}1$) to a volume $(S\ensuremath{\sim}{l}^{d})$ law and provide a criterion for their occurrence which constitutes a generalization of Hastings's theorem to driven integrable systems in one dimension. We also find that ${S}_{n}$ generically decays to ${S}_{\ensuremath{\infty}}$ as ${(\ensuremath{\omega}/n)}^{(d+2)/2}$ for fast and ${(\ensuremath{\omega}/n)}^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigenspectrum of the Floquet Hamiltonian. This dynamical transition manifests itself in the temporal behavior of all local correlation functions and does not require a critical point crossing during the drive. We find that these dynamical phases show a rich re-entrant behavior as a function of $\ensuremath{\omega}$ for $d=1$ models and also discuss the dynamical transition for $d>1$ models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in $d=1$) appear in ${S}_{\ensuremath{\infty}}$ as a function of $\ensuremath{\omega}$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.