Eur. Phys. J. B 7, 607-618 (1999)
https://doi.org/10.1007/s100510050653

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

¹ Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
² Grupo Interdisciplinar de Sistemas Complicados, Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés (Madrid), Spain
³ Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, New Mexico 87545, USA
⁴ Grupo Interdisciplinar de Sistemas Complicados, Departamento de Física de Materiales, Facultad de Físicas, Universidad Complutense, 28040 Madrid, Spain

Received: 27 April 1998
Revised: 2 September 1998
Accepted: 10 September 1998
Published online: 15 February 1999

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.