We study many-body localised quantum systems subject to periodic driving. We find that the presence of a mobility edge anywhere in the spectrum is enough to lead to delocalisation for any driving strength and frequency. By contrast, for a fully local
ised many-body system, a delocalisation transition occurs at a finite driving frequency. We present numerical studies on a system of interacting one-dimensional bosons and the quantum random energy model, as well as simple physical pictures accounting for those results.
When a closed quantum system is driven periodically with period $T$, it approaches a periodic state synchronized with the drive in which any local observable measured stroboscopically approaches a steady value. For integrable systems, the resulting b
ehaviour is captured by a periodic version of a generalized Gibbs ensemble. By contrast, here we show that for generic non-integrable interacting systems, local observables become independent of the initial state entirely. Essentially, this happens because Floquet eigenstates of the driven system at quasienergy $omega_alpha$ consist of a mixture of the exponentially many eigenstates of the undriven Hamiltonian which are thus drawn from the entire extensive undriven spectrum. This is a form of equilibration which depends only on the Hilbert space of the undriven system and not on any details of its Hamiltonian.
The nature of the behaviour of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics. After an initial transient, such a system is known to synchronize with the driving; in
contrast to the non-driven case, no fundamental principle has been proposed for constructing the resulting non-equilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which we explicitly identify. This result constitutes a generalisation of the concepts of equilibrium statistical mechanics to a class of far-from-equilibrium-systems, up to now mainly accessible using ad-hoc methods.
