Dynamics of Entanglement generation in periodically driven integrable systems

We study entanglement entropy $S_n$ of periodically driven $d-$dimensional integrable models after $n$ drive cycles with frequency $\omega$. We demonstrate that such a drive may be used for controlled generation of states with non-area-law scaling of $S_n$ and provide a criteria for their occurrence which constitutes a generalization of Hastings' theorem to driven integrable systems. We find that $S_n$ decays to $S_{\infty}$ as $(\omega/n)^{(d+2)/2}$ for fast and $(\omega/n)^{d/2}$ for slow drives; these two dynamical phases are separated by a transition associated with the change in topology of the spectrum of the system's Floquet Hamiltonian. We show that these dynamical phases show re-entrant behavior as a function of $\omega$ for $d=1$ (and a class of $d=2$) models and discuss experiments which can test our theory.