Prethermal Phases of Matter Protected by Time-Translation Symmetry

In a periodically driven (Floquet) system, there is the possibility for new phases of matter, not present in stationary systems, protected by discrete time-translation symmetry. This includes topological phases protected in part by time-translation symmetry, as well as phases distinguished by the spontaneous breaking of this symmetry, dubbed Floquet time crystals. We show that such phases of matter can exist in the prethermal regime of periodically driven systems, which exists generically for sufficiently large drive frequency, thereby eliminating the need for integrability or strong quenched disorder, which limited previous constructions. We prove a theorem that states that such a prethermal regime persists until times that are nearly exponentially long in the ratio of certain couplings to the drive frequency. By similar techniques, we can also construct stationary systems that spontaneously break continuous time-translation symmetry. Furthermore, we argue that for driven systems coupled to a cold bath, the prethermal regime could potentially persist to infinite time.