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

We establish some general dynamical properties of lattice many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasi-conserved extensive quantity $H_{\ast }$, 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 $H_{\ast }$ up to time ${\tau }_{\ast }$, which is exponentially long in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where $H_{\ast }$ is ergodic, the driven system prethermalizes to a thermal state described by $H_{\ast }$ at intermediate times $t\lesssim {\tau }_{\ast }$, eventually heating up to an infinite-temperature state at times $t\sim {\tau }_{\ast }$. 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.