  Despite being forbidden in equilibrium, spontaneous breaking of time
translation symmetry can occur in periodically driven, Floquet systems with
discrete time-translation symmetry. The period of the resulting discrete time
crystal is quantized to an integer multiple of the drive period, arising from a
combination of collective synchronization and many body localization. Here, we
consider a simple model for a one dimensional discrete time crystal which
explicitly reveals the rigidity of the emergent oscillations as the drive is
varied. We numerically map out its phase diagram and compute the properties of
the dynamical phase transition where the time crystal melts into a trivial
Floquet insulator. Moreover, we demonstrate that the model can be realized with
current experimental technologies and propose a blueprint based upon a one
dimensional chain of trapped ions. Using experimental parameters (featuring
long-range interactions), we identify the phase boundaries of the
ion-time-crystal and propose a measurable signature of the symmetry breaking
phase transition.
