Thermodynamic Behavior of the Heisenberg Ferromagnet

Abstract

A finite-temperature perturbation theory is presented for the Heisenberg model with the object of providing a formalism in which contact can be made with the low-temperature treatment by Dyson, with the random phase approximation of Englert, and, above the Curie point, with high-density treatments of the Ising model. A linked cluster expansion is set up and a simple high-density classification, valid above the Curie point, is applied. The first two terms in the high-density series, tree graphs and ring graphs, yield, respectively, molecular field theory and a form reducing to spin-wave results at low temperatures. A low-temperature classification is then developed which leads to an expansion of the free energy in powers of $T$ in which the terms have the form of those describing bosons with an effective interaction similar to Dyson's $\Gamma _{\rho \sigma }^{}{}_{}{}^{\lambda }$. The first two terms are the low-temperature approximations of trees and rings, respectively, which justifies the use of the high-density expansion below the Curie point. The next term, including all the effects of spin-wave interactions up to $T^4$ in the free energy, contains the Born approximation series presented by Dyson. In particular, the cancellation of $T^3$ terms in the leading Born approximation is demonstrated. A renormalized version of the high-density expansion necessary to treat the region of the Curie point is then considered, and its approximation by an "excluded volume" sum is shown to yield the Curie point of the spherical model, in common with the random phase approximation and with high-density approximations to the Ising model. The extent to which the high-density theory misrepresents the effect of spin-wave interactions is then discussed. In an Appendix an equations-of-motion approach to the random phase approximation and to the interactions between spin waves is presented.