  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 temperature 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
ferromagnets and liquid crystals as well as the Ginzburg-Landau approach to
bidimensional freely decaying turbulence.
