Time crystallinity in dissipative Floquet systems

We investigate the conditions under which periodically driven quantum systems subject to dissipation exhibit a stable subharmonic response. Noting that coupling to a bath introduces not only cooling but also noise, we point out that a system subject to the latter for the entire cycle tends to lose coherence of the subharmonic oscillations, and thereby the long-time temporal symmetry breaking. We provide an example of a shortranged two-dimensional system which does not suffer from this and therefore displays persistent subharmonic oscillations stabilized by the dissipation. We also show that this is fundamentally different from the disordered discrete time crystal previously found in closed systems, both conceptually and in its phenomenology. The framework we develop here clarifies how fully connected models constitute a special case where subharmonic oscillations are stable in the thermodynamic limit.