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The breaking of the continuous time-translation symmetry manifests, inMarkovian open quantum systems, through the emergence of non-stationarydynamical phases. Systems that display nonequilibrium transitions into thesephases are referred to as time-crystals, and they can be realized, for example,in many-body systems governed by collective dissipation and long-rangedinteractions. Here, we provide a complete analytical characterization of aboundary time-crystal phase transition. This involves exact expressions for theorder parameter and for the dynamics of quantum fluctuations, which, in thetime-crystalline phase, remains asymptotically non-Markovian as a consequenceof the time-translation symmetry breaking. We demonstrate that boundarytime-crystals are intrinsically critical phases, where fluctuations exhibit apower-law divergence with time. Our results show that a dissipativetime-crystal phase is far more than merely a classical non-linear andnon-stationary (limit cycle) dynamics of a macroscopic order parameter. It israther a genuine many-body phase where the properties of correlationsdistinctly differs from that of stationary ones.

Exact solution of a boundary time-crystal phase transition: time-translation symmetry breaking and non-Markovian dynamics of correlations