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

We study a class of periodically driven d-dimensional integrable models and show that after n drive cycles with frequency., pure states with non-area-law entanglement entropy S-n(l) similar to l(alpha(n,omega)) are generated, where l is the linear dimension of the subsystem, and d - 1 <= alpha(n,omega) <= d. The exponent a(n,omega) eventually approaches d (volume law) for large enough l when n -> 8. We identify and analyze the crossover phenomenon from an area (S similar to l(d) 1 for d >= 1) to a volume (S similar to 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 Sn generically decays to S infinity 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. for d = 1 models and also discuss the dynamical transition ford > 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 infinity as a function of. whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.