Classical stochastic discrete time crystals

We describe a general and simple paradigm for discrete time crystals (DTCs), systems with a stable subharmonic response to an external driving field, in a classical thermal setting. We consider, specifically, an Ising model in two dimensions, as a prototypical system with a phase transition into stable phases distinguished by a local order parameter, driven by thermal dynamics and periodically kicked with a noisy protocol. By means of extensive numerical simulations for large sizes-allowed by the classical nature of our model-we show that the system features a true disorder-DTC order phase transition as a function of the noise strength, with a robust DTC phase extending over a wide parameter range. We demonstrate that, when the dynamics is observed stroboscopically, the phase transition to the DTC state appears to be in the equilibrium two-dimensional Ising universality class. However, we explicitly show that the DTC is a genuine nonequilibrium state. More generally, we speculate that systems with thermal phase transitions to multiple competing phases can give rise to DTCs when appropriately driven.