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

Abstract

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)-like transitions: first from a disordered phase to a critical synchronized phase, and then to a spatiotemporal pattern phase. Notably, the synchronized phase features algebraically decaying spatial correlations and divergent coherence time, realizing an effective continuous time crystal across macroscopic yet finite scales; while it vanishes when $q < q_c$. A dynamic renormalization group analysis shows this behavior arises from an emergent U(1) symmetry for $q \ge q^{RG}_c=5$, and indicates a crossover scale to Kardar-Parisi-Zhang (KPZ) universality diverges double-exponentially with $q$, ensuring the pre-asymptotic stability of the synchronized phase. Mean-field theory predicts a lower critical value $q_c^{MF} = 4$.