Finite temperature dynamics of vortices in the two dimensional anisotropic Heisenberg model

TL;DR

This paper investigates how finite temperature influences the motion of non-planar vortices in a two-dimensional anisotropic Heisenberg model, combining analytical theory with simulations to understand vortex dynamics near the Kosterlitz-Thouless transition.

Contribution

The authors develop a parameter-free collective variable theory incorporating stochastic forces to analytically describe vortex motion at finite temperatures.

Findings

Analytical equations match Langevin simulations up to 25% of the transition temperature.

Vortex variance exhibits non-standard time dependence perpendicular to the driving force.

Discreteness effects become significant at higher temperatures, limiting the approach's applicability.

Abstract

We study the effects of finite temperature on the dynamics of non-planar vortices in the classical, two-dimensional anisotropic Heisenberg model with XY- or easy-plane symmetry. To this end, we analyze a generalized Landau-Lifshitz equation including additive white noise and Gilbert damping. Using a collective variable theory with no adjustable parameters we derive an equation of motion for the vortices with stochastic forces which are shown to represent white noise with an effective diffusion constant linearly dependent on temperature. We solve these stochastic equations of motion by means of a Green's function formalism and obtain the mean vortex trajectory and its variance. We find a non-standard time dependence for the variance of the components perpendicular to the driving force. We compare the analytical results with Langevin dynamics simulations and find a good agreement up to temperatures of the order of 25% of the Kosterlitz-Thouless transition temperature. Finally, we discuss the reasons why our approach is not appropriate for higher temperatures as well as the discreteness effects observed in the numerical simulations.