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) 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 $q_c^{MF} = 4$.

Figures (6)

• Figure 1: (a) Schematic of the self-rotating clock. Spin vector (blue arrow) points only along $q$ equidistant orientations (red dots). (b)-(d) Typical system trajectories in the $(m_x,m_y)$ plane before and after bifurcations, with $q=q_c^{MF}=4$ and $f = 0.12$ for $J=0.6$ (disordered phase), 0.7 (synchronized phase) and 0.8 (static ordered phase). Results are obtained from numerical solutions of Eq. \ref{['nonlinearmaster']}. Black points: stable fixed points; hollow points: saddle fixed points; red points: starting points of dynamical trajectories. Trajectory color variation indicates time progression. Insets show the corresponding temporal evolutions of spin probabilities $\{P_s\}$.
• Figure 2: (a) Phase diagram in $(J, qf)$ plane for $q=q_c=5$, obtained from simulations with the system size $L=400$. Different symbols represent distinct phases. The solid lines are phase boundaries still existing in the thermodynamic limit, while dotted lines exist only for finite systems. Along $J$-axis, the equilibrium quasi-long-range ordered phase (QO) and long-range ordered phase (LO) are also marked. (b1)-(b5) show typical configuration snapshots (different colors represent different spin states) and corresponding temporal evolutions of spin percentage $P_s$, respectively, for disordered phase (DO, $f=0.1,J=0.9$), critical synchronized phase (CS, $f=0.1,J=1.1$), spiral wave state (SW, $f=0.4,J=1.6$), multi-droplet state (MD, $f=0.1,J=1.5$) and single-droplet state (SD, $f=0.1,J=1.7$).
• Figure 3: Critical synchronized phase and BKT phase transitions. (a) Variation of $R$ and $\mathcal{L}$ with $J$ for different system sizes $L$. (b) Variation of $U$ and $\chi$ with $J$ around critical point $J_{1}$. (c) Scaling relation of $R$ versus $L$ on a double-logarithmic plot for different $J$. (d) The autocorrelation function $C_{M_{x}}(\tau)$ of synchronized phase for different $L$. (e) The fitted scaling relationship between coherence time $\tau_c$ and $L$. In (a)-(e) $q=5$ and $f=0.1$ remain fixed. (f) $U$ around critical point $J_{2}$ at fixed $q=5$ and $f =0.4$, obtained from disordered initial configurations. For intermediate $L$ and larger $J$, $U < 1/3$ stems from the heavy-tailed probability distribution $P(r)$, which is induced by the intermittent annihilation of spiral waves (see SM for details).
• Figure 4: (a) Rapid annihilation of spiral cores after pairwise excitation in critical synchronized phase ($q=5, f=0.2, J=1.25,L=400$). The red arrows indicate the rotation direction of spiral cores and the propagation direction of spherical waves. (b) Formation of spiral wave state ($q = 5,f = 0.5,J = 1.7,L = 400$): A pair of stochastically excited spiral cores ultimately develops into spiral turbulence, disrupting global phase synchronization.
• Figure A1: Projections of typical system trajectories in the $(m_x,m_y)$ plane before and after bifurcation, for different $q$ values at fixed $qf = 0.48$. (a)-(c) $q = 3$ and $J = 0.6, 0.65, 0.8$ (disordered, coexisting and static ordered phases). (d)-(f) $q=5$ and $J=0.5, 0.8, 1$ (disordered, synchronized and static ordered phases).