  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 $d\geq1$) 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.
