Dynamical Scaling: the Two-Dimensional XY Model Following a Quench

TL;DR

This study investigates the dynamical scaling behavior of the 2D XY model after a quench, confirming the asymptotic growth law and highlighting the roles of vortex and spin-wave contributions to correlations.

Contribution

It introduces a method to test scaling in the 2D XY model, compares correlations with Gaussian-closure approximations, and discusses the relevance of topological and non-topological effects.

Findings

Results are consistent with the growth law $L \\sim (t/\\ln[t/t_0])^{1/2}$.

Correlations reconstructed from vortex configurations differ from natural correlations but both scale with $L$.

Both vortex and spin-wave effects influence correlations even at late times.

Abstract

To sensitively test scaling in the 2D 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 non-topological (``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.