Pre-thermal Time Crystals and Floquet topological phases without disorder

We show that both discrete and continuous time-translation symmetry can be broken in the prethermal regime of quantum systems that eventually thermalize. We prove a theorem that states that such “time crystals” persist until times that are nearly exponentially-long in the couplings and, in driven systems, the drive frequency. After this thermalization time, the time-translational symmetry breaking oscillations fade away. However, during the time interval prior to that, a time crystal can exist even without disorder, and its properties are encapsulated by a field theory analogous to that of equilibrium spontaneous symmetry-breaking phases. When coupled to a cold bath, the pre-thermal regime could potentially persist to infinite time. Similar conclusions hold for topological phases of driven systems.