  The breaking of the continuous time-translation symmetry manifests, in
Markovian open quantum systems, through the emergence of non-stationary
dynamical phases. Systems that display nonequilibrium transitions into these
phases are referred to as time-crystals, and they can be realized, for example,
in many-body systems governed by collective dissipation and long-ranged
interactions. Here, we provide a complete analytical characterization of a
boundary time-crystal phase transition. This involves exact expressions for the
order parameter and for the dynamics of quantum fluctuations, which, in the
time-crystalline phase, remains asymptotically non-Markovian as a consequence
of the time-translation symmetry breaking. We demonstrate that boundary
time-crystals are intrinsically critical phases, where fluctuations exhibit a
power-law divergence with time. Our results show that a dissipative
time-crystal phase is far more than merely a classical non-linear and
non-stationary (limit cycle) dynamics of a macroscopic order parameter. It is
rather a genuine many-body phase where the properties of correlations
distinctly differs from that of stationary ones.
