Dynamics of the classical two dimensional XY ferromagnet at low temperatures: a memory function approach

Abstract

The spin dynamics of a two dimensional XY ferromagnet are reexamined at low temperatures in the framework of the Mori continued fraction formalism using the Gaussian approximation. In this formalism, the terms on denominators for the Laplace transform of the relaxation function Rt are related to the frequency moments ! n q of the relaxation shape function Rq;!. In the Gaussian approximation scheme, we truncate the continued fraction for Rt on the second stage. Adopting this approximation, we calculated up to the sixth moment. The moments are written in terms of the static spin correlation functions. In the estimation of the fourth and sixth moment at nite temperature, the four and sixspin correlation functions may be approximated by a sum of products of appropriate pair correlation functions mode-mode decoupling. In this work we calculated the static spin correlation functions for the expressions to the fourth and sixth moments, needed in the study of the dynamics. I IntroductionIt is well known that, in two dimensional 2D planar magnetic spin systems, a topological phase transition referred to generally as Kosterlitz-ThoulessKT transition is expected to occur at a temperature 1, 2 T kT 6 = 0. This transition is associated with the unbind of pair of vortices, which bind together to form vortex pairs of zero net helicity, as the temperature is raised above T KT . The classical two-dimensional ferromagnetic models, the XY and the planar model, described by the same Hamiltonian where J is the exchange constant, have been studied in literature. In the XY modelS r the spin at site r i s a three-dimensional classical rotor, while for the plane rotor modelS r is again a classical rotor but now i t i s constrained to two dimensions, having no z component, and therefore no dynamics. In this paper we will take jS r j = S = 1. This produces no loss of generality when a classical system is under investigation, as the case of generic spin S is easily brought back b y the rescaling of the exchange constant J ! JS 2 .For the XY model, as vortices and antivortices unbind they di use as a vortex gas 3, 4 . The autocorrelations of the di using vortices give rise to a characteristic central quasi elastic peak in the dynamic structure factor Sq;!. Above T KT the dynamics of Hamilto-