Floquet thermalization: Symmetries and random matrix ensembles

We investigate the role of symmetries in determining the random matrix class describing quantum thermalization in a periodically driven many-body quantum system. Using a combination of analytical arguments and numerical exact diagonalization, we establish that a periodically driven "Floquet" system can be in a different random matrix class from the instantaneous Hamiltonian. A periodically driven system can thermalize even when the instantaneous Hamiltonian is integrable. A Floquet system that thermalizes in general can display integrable behavior at commensurate driving frequencies. When the instantaneous Hamiltonian and the Floquet operator both thermalize, the Floquet problem can be in the unitary class while the instantaneous Hamiltonian is always in the orthogonal class, and vice versa. We extract general principles regarding when a Floquet problem can thermalize to a different symmetry class from the instantaneous Hamiltonian. A (finite-sized) Floquet system can even display crossovers between different random matrix classes as a function of driving frequency.