Dynamical scaling: The two-dimensional $\mathrm{XY}$ model following a quench

To sensitively test scaling in the two-dimensional $\mathrm{XY}$ model quenched from high temperatures into the ordered phase, we study the difference between measured correlations and the (scaling) results of a Gaussian-closure approximation. We also directly compare various length scales. All of our results are consistent with dynamical scaling and an asymptotic growth law $L\sim (t/\ln [{t/t}_0]{)}^{1/2}$, though with a time scale $t_0$ that depends on the length scale in question. We then reconstruct correlations from the minimal-energy configuration consistent with the vortex positions, and find them significantly different from the “natural” correlations — though both scale with L. This indicates that both topological (vortex) and nontopological “spin-wave” contributions to correlations are relevant arbitrarily late after the quench. We also present a consistent definition of dynamical scaling applicable more generally, and emphasize how to generalize our approach to other quenched systems where dynamical scaling is in question. Our approach directly applies to planar liquid-crystal systems.