Vortices and the entrainment transition in the two-dimensional Kuramoto model

We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling K is larger than a threshold K-E, there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at K-E. We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase (K>K-E), vortices move in fixed paths around clusters, while in the unentrained phase (K