Exchange Renormalization and Phase Transition in the Classical Three‐Dimensional Anisotropic Ferromagnet

Abstract

The O(2) ( = XU) model has the interesting and important feature that it supports topological excitations which in two dimensions are vortices. By now the Kosterlitz-Thouless transition in 2d is quite well understood [l].The three-dimensional O(2) classical Heisenberg model (planar model) is known to have a second-order phase transition at TJJ = 2.17 from an ordered phase with nonzero magnetization to a disordered phase [2]. Banks and Myerson [3] have shown that above the critical temperature the effective potential for this model becomes linear, guaranteeing exponential fall-off of the correlation function. They postulated then that the mechanism which forces the potential to become linear was the appearance for large enough temperature of vortex rings whose size would grow without limit, the long vortex rings destroying the long-range spin-spin correlations. They estimated the upper bound for T,, using the Villain approximation, as TJJ z 6.3.Recently, Kohring et al.[4] using Monte Carlo techniques have shown that vortex strings are responsible for the phase transition in the three-dimensional planar model. However, the task of taking these topological excitations into account, as Kosterlitz-Thouless theory did for the two-dimensional O(2) model, remains an open problem.In this note we will extend a self-consistent approximation, developed to study the two-dimensional model [5], to study the classical 3D anisotropic ferromagnet with nearest-neighbor interaction, described by the following Hamiltonian:where J > 0, S,. is a classical spin, Y + u labels the six nearest-neighbor sites of Y in a simple cubic lattice (or eight in a body-centered lattice) and 0 5 , I < 1 describes an exchange easy-plane anisotropy.The classical XY model is given by (1) with i = 0, but considering S, = (S:, Sy, S:). The planar rotator model has a Hamiltonian of the same form, but the spins have only two components, S,. = (S:, Sy).In order to use the self-consistent approximation we start by writing Hamiltonian (1) in terms of the polar representation for the spin at site Y, *) CP 702, 30.161-970 Belo Horizonte, MG Brazil. ') Partially supported by CNPq and FINEP, Brazil.