  We consider a non-equilibrium extension of the two-dimensional (2D) XY model,
equivalent to the noisy Kuramoto model of synchronization with short-range
coupling, where rotors sitting on a square lattice are self-driven by random
intrinsic frequencies. We study the static and dynamic properties of
topological defects (vortices) and establish how self-spinning affects the
Berezenskii-Kosterlitz-Thouless phase transition scenario. The non-equilibrium
drive breaks the quasi-long-range ordered phase of the 2D XY model into a
mosaic of ordered domains of controllable size and results in self-propelled
vortices that generically unbind at any temperature, featuring superdiffusion
$\langle r^2(t)\rangle\sim t^{3/2}$ with a Gaussian distribution of
displacements. Our work provides a simple framework to investigate topological
defects in active matter and sheds new light on the problem of synchronization
of locally coupled oscillators.
