Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems

We establish some general dynamical properties of quantum many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasiconserved extensive quantity H-*, which plays the role of an effective static Hamiltonian. The dynamics of the system (e.g., evolution of any local observable) is well approximated by the evolution with the Hamiltonian H-* up to time tau(*), which is exponentially large in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where H-* is ergodic, the driven system prethermalizes to a thermal state described by H-* at intermediate times t less than or similar to tau(*), eventually heating up to an infinite-temperature state after times t similar to tau(*). Our results indicate that rapidly driven many-body systems generically exhibit prethermalization and very slow heating. We briefly discuss implications for experiments which realize topological states by periodic driving.