Quench dynamics of the 2d XY model

We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and tem-perature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble-Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results to a host of physical systems with vortex excitations including planar ferro-magnets and liquid crystals as well as the Ginzburg-Landau approach to bidimensional freely decaying turbulence.