Floquet time crystals in clock models

We construct a class of period-n-tupling discrete time crystals based on Z₍ clock variables, for all the integers n. We consider two classes of systems where this phenomenology occurs: disordered models with short-range interactions and fully connected models. In the case of short-range models, we provide a complete classification of time-crystal phases for generic n. For the specific cases of n=3 and n=4, we study in detail the dynamics by means of exact diagonalization. In both cases, through an extensive analysis of the Floquet spectrum, we are able to fully map the phase diagram. In the case of infinite-range models, the mapping onto an effective bosonic Hamiltonian allows us to investigate the scaling to the thermodynamic limit. After a general discussion of the problem, we focus on n=3 and n=4, representative examples of the generic behavior. Remarkably, for n=4 we find clear evidence of a crystal-to-crystal transition between period n-tupling and period n/2-tupling.