Very slow heating for weakly driven quantum many-body systems

It is well understood that many-body systems driven at high frequency heat up only exponentially slowly and exhibit a long prethermalization regime. We prove rigorously that a certain relevant class of systems heat very slowly under weak periodic driving at intermediate frequency as well. This class of systems are those whose time-dependent, possibly translation-invariant, Hamiltonian is a weak perturbation of a sum of mutually commuting terms. This condition covers several periodically kicked systems that have been considered in the literature recently, in particular kicked Ising models. In contrast to the high-frequency regime, the prethermalization dynamics of our systems is in general not related to any time-independent effective Floquet Hamiltonian. Our results also have non-trivial implications for closed (time-independent) systems. We use the example of an Ising model with transversal and longitudinal field to show how they imply confinement of excitations. More generally, they show how "glassy" kinetically constrained models emerge naturally from simple many-body Hamiltonians, thus connecting to the topic of 'translation-invariant localization'.