Phase Synchronization in 2D Kuramoto Model

We study a system of identical Kuramoto oscillators in the presence of Gaussian white noise. The oscillators are arranged on a two-dimensional periodic lattice and interact with their nearest neighbors only [1-2]. In the thermodynamic limit, the stationary states at very low temperatures (limit T→0) are well described by the presence of topological defects (vortices and anti-vortices) in the phase-field of the oscillators. We apply duality transformation on a Hamiltonian [3] to explore the underlying vortex structure.
The system exhibits a phase transition from stochastic synchronization to de-synchronization as temperature varies. We investigate the nature of the phase transition and calculate the critical temperature numerically. The relaxation dynamics is also studied. In the low-temperature regime, the system relaxes to equilibrium algebraically whereas, at high temperature, the decay is exponential. Moreover, the system, at very low temperature belongs to the EW universality class yielding the dynamic exponent z=2 [4].

[1] Hong H, Park H and Choi M Y 2005 Phys. Rev. E 72 036217
[2] Lee T E, Tam H, Refael G, Rogers L and Cross M C 2010 Phys. Rev. E 82 036202
[3] Witthaut D and Timme M 2014 Phys. Rev. E 90 032917
[4] Forrest B M and Tang L-H 1990 J. Stat. Phys. 60 1-2