Coarsening of two dimensional XY model with Hamiltonian dynamics: Logarithmically divergent vortex mobility

TL;DR

This paper studies the coarsening dynamics of a Hamiltonian XY model, revealing a logarithmically diverging vortex mobility that leads to super-diffusive growth of the characteristic length scale, with implications for nonequilibrium relaxation and correlation functions.

Contribution

It introduces a phenomenological model explaining super-diffusive growth via vortex mobility divergence and analyzes nonequilibrium relaxation to estimate equilibrium correlation exponents.

Findings

Vortex mobility diverges logarithmically with vortex-antivortex pair size.

Characteristic length scale grows as $L(t) o ((t+t_0) \, \ln(t+t_0))^{1/2}$.

Correlation functions follow scaling laws with dynamic exponent $z_{nr} = 1$.

Abstract

We investigate the coarsening kinetics of an XY model defined on a square lattice when the underlying dynamics is governed by energy-conserving Hamiltonian equation of motion. We find that the apparent super-diffusive growth of the length scale can be interpreted as the vortex mobility diverging logarithmically in the size of the vortex-antivortex pair, where the time dependence of the characteristic length scale can be fitted as $L(t) \sim ((t+t_{0}) \ln(t+t_{0}))^{1/2}$ with a finite offset time $t_0$. This interpretation is based on a simple phenomenological model of vortex-antivortex annihilation to explain the growth of the coarsening length scale $L(t)$. The nonequilibrium spin autocorrelation function $A(t)$ and the growing length scale $L(t)$ are related by $A(t) \simeq L^{-\lambda}(t)$ with a distinctive exponent of $\lambda \simeq 2.21$ (for $E=0.4$) possibly reflecting the strong effect of propagating spin wave modes. We also investigate the nonequilibrium relaxation (NER) of the system under sudden heating of the system from a perfectly ordered state to the regime of quasi-long-range order, which provides a very accurate estimation of the equilibrium correlation exponent $\eta (E) $ for a given energy $E$. We find that both the equal-time spatial correlation $C_{nr}(r,t)$ and the NER autocorrelation $A_{nr}(t)$ exhibit scaling features consistent with the dynamic exponent of $z_{nr} = 1$.