  The out-of-equilibrium dynamics of many body systems has recently received a
burst of interest, also due to experimental implementations. The dynamics of
both observables, such as magnetization and susceptibilities, and quantum
information related quantities, such as concurrence and entanglement entropy,
have been investigated under different protocols bringing the system out of
equilibrium. In this paper we focus on the entanglement entropy dynamics under
a sinusoidal drive of the transverse magnetic field in the 1D quantum Ising
model. We find that the area and the volume law of the entanglement entropy
coexist under periodic drive for an initial non-critical ground state.
Furthermore, starting from a critical ground state, the entanglement entropy
exhibits finite size scaling even under such a periodic drive. This
critical-like behaviour of the out-of-equilibrium driven state can persist for
arbitrarily long time, provided that the entanglement entropy is evaluated on
increasingly subsystem sizes, whereas for smaller sizes a volume law holds.
Finally, we give an interpretation of the simultaneous occurrence of critical
and non-critical behaviour in terms of the propagation of Floquet
quasi-particles.
