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 $S_n(l) sim l^{alpha(n,omega)}$ are generated, where $l$ is the linear dimension of the subsystem, and $d-1 le alpha(n,omega) le d$. We identify and analyze the crossover phenomenon from an area ($S sim l^{ d-1}$ for $dgeq1$) to a volume ($S sim l^{d}$) law and provide a criterion for their occurrence which constitutes a generalization of Hastings 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 eigensprectrum 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.
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