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

TL;DR

This study investigates synchronization phenomena in 2D self-rotating clock models, revealing two-step BKT transitions and the emergence of a continuous time crystal phase for q ≥ 5 through large-scale simulations.

Contribution

It demonstrates the existence of a synchronized phase with algebraic correlations and time crystal behavior in 2D clock models, supported by Monte Carlo simulations and mean-field theory.

Findings

Two-step BKT transitions for q ≥ 5

Existence of a continuous time crystal phase

Mean-field predicts lower critical q_c^{MF} = 4

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) 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 $q_c^{MF} = 4$.