We study a class of periodically driven $d$-dimensional integrable models and show that after $n$ drive cycles with frequency $\omega$, pure states with non-area-law entanglement entropy $S_n\left(l\right)\sim l^{\alpha (n,\omega )}$ are generated, where $l$ is the linear dimension of the subsystem, and $d-1\le \alpha (n,\omega )\le d$. The exponent $\alpha (n,\omega )$ eventually approaches $d$ (volume law) for large enough $l$ when $n\to \infty$. We identify and analyze the crossover phenomenon from an area $(S\sim l^{d-1}$ for $d\ge 1$) to a volume $\left(S\sim l^d\right)$ 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 $S_{\infty }$ as ${(\omega /n)}^{(d+2)/2}$ for fast and ${(\omega /n)}^{d/2}$ 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 $\omega$ 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 $S_{\infty }$ as a function of $\omega$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.

• Received 1 December 2015
• Revised 14 November 2016

DOI:https://doi.org/10.1103/PhysRevB.94.214301