Stochastic Discrete Time Crystals: Entropy Production and Subharmonic Synchronization

Discrete time crystals are periodically driven systems that display spontaneous symmetry breaking of translation invariance in the form indefinite subharmonic oscillations. We introduce a thermodynamically consistent model for discrete crystal and analyze it using framework stochastic thermodynamics. In particular, we evaluate rate energy dissipation this many-body system interacting noisy oscillators contact with heat bath. The mean-field displays phenomenon synchronization, which corresponds to collective oscillations individual units. 2D does not synchronization but show time-crystalline phase, is characterized by power-law behavior number coherent size. This result demonstrates emergence possible even absence synchronization.