Building continuous time crystals from rare events

Symmetry-breaking dynamical phase transitions (DPTs) abound in the fluctuations of nonequilibrium systems. Here we show that the spectral features of a particular class of DPTs exhibit the fingerprints of the recently discovered time-crystal phase of matter. Using Doob's transform as a tool, we provide a mechanism to build time-crystal generators from the rare event statistics of some driven diffusive systems. An analysis of the Doob's smart field in terms of the order parameter of the transition then leads to the time-crystal exclusion process (tcEP), a stochastic lattice gas subject to an external packing field which presents a clear-cut steady-state phase transition to a time-crystalline phase which breaks continuous time-translation symmetry and displays rigidity and long-range spatio-temporal order. A hydrodynamic analysis of the tcEP transition uncovers striking similarities, but also key differences, with the Kuramoto synchronization transition. Possible experimental realizations of the tcEP are also discussed.