THE CLASSICAL SPECTRAL DENSITY METHOD AT WORK: THE HEISENBERG FERROMAGNET

In this article we review a less known unperturbative and powerful many-body method in the framework of classical statistical mechanics and then we show how it works by means of explicit calculations for a nontrivial cla ssical model. The formalism of two-time Green functions in classical statistical mecha nics is presented in a form parallel to the well known quantum counterpart, focusing on the spectral properties which involve the important concept of spectral density. Furthermore, the general ingredients of the classical spectral density method (CSDM) are presented with insights for systematic nonperturbative approximations to study conveniently the macroscopic properties of a wide variety of classical many-body systems also involving phase transitions. The method is implemented by means of key ideas for exploring the spectrum of elementary excitations and the damping effects within a unified formalism. Then, the effectiveness of the CSDM is tested with explicit calculations for the classical d-dimensional spin-S Heisenberg ferromagnetic model with long-range exchange interactions decaying as r −p (p > d) with distance r between spins and in the presence of an external magnetic field. The analysis of the thermodyna mic and critical properties, performed by means of the CSDM to the lowest order of approximation, shows clearly that nontrivial results can be obtained in a relativ ely simple manner already to this lower stage. The basic spectral density equations for t he next higher order level are also presented and the damping of elementary spin excitations in the low temperature regime is studied. The results appear in reasonable agr eement with available exact ones and Monte Carlo simulations and this supports the CSDM as a promising method of investigation in classical many-body theory.