Synchronization and phase transition of two-dimensional self-rotating clock models

We explore possible synchronization in two-dimensional (2D) locally coupled discrete-state oscillators under thermal fluctuations, using the self-rotating $q$-state clock model as a prototype. Large-scale Monte Carlo simulations reveal that for $q \ge q_c$ (with $q_c = 5$), the system undergoes two-step Berezinskii-Kosterlitz-Thouless (BKT) transitions: first from a disordered phase to a critical synchronized phase, and then to a spatiotemporal pattern phase. The latter includes oscillatory droplet states that survive in finite systems and a thermodynamically stable spiral wave state. Notably, the synchronized phase features algebraically decaying spatial correlations, alongside divergent coherence time, thus realizing a continuous time crystal; while it vanishes when $q < q_c$. Mean-field theory supports the existence of the synchronized phase, but predicts a lower critical value $q_c^{MF} = 4$.