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 $t\lesssim {\tau }_{*}$, eventually heating up to an infinite-temperature state after times $t\sim {\tau }_{*}$. 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.