Dynamics of Topological Defects in the noisy Kuramoto Model in two dimensions

TL;DR

This study investigates the out-of-equilibrium dynamics of the 2D noisy Kuramoto model, revealing domain growth, free topological defects, and super-diffusive vortex motion, independent of inertia effects.

Contribution

It demonstrates that the 2D noisy Kuramoto model lacks a phase transition and exhibits unique defect dynamics distinct from the 2D XY model.

Findings

Correlation length remains finite and scales inversely with frequency distribution width.

Domain growth initially follows the 2D XY model law but diverges in defect behavior.

Vortices perform super-diffusive random walks with an exponent of 3/2.

Abstract

We consider the two-dimensional (2D) noisy Kuramoto model of synchronization with short-range coupling and a Gaussian distribution of intrinsic frequencies, and investigate its ordering dynamics following a quench. We consider both underdamped (inertial) and over-damped dynamics, and show that the long-term properties of this intrinsically out-of-equilibrium system do not depend on the inertia of individual oscillators. The model does not exhibit any phase transition as its correlation length remains finite, scaling as the inverse of the standard deviation of the distribution of intrinsic frequencies. The quench dynamics proceeds via domain growth, with a characteristic length that initially follows the growth law of the 2D XY model, although is not given by the mean separation between defects. Topological defects are generically free, breaking the Berezinskii-Kosterlitz-Thouless scenario of the 2D XY model. Vortices perform a random walk reminiscent of the self-avoiding random walk, advected by the dynamic network of boundaries between synchronised domains; featuring long-time super-diffusion, with the anomalous exponent $\alpha=3/2$.