Callen-like method for the classical Heisenberg ferromagnet

TL;DR

This paper extends the quantum Callen method to classical Heisenberg ferromagnets, deriving a new magnetization formula that aligns well with numerical and analytical results across different dimensions and temperature ranges.

Contribution

The paper introduces a classical Callen-like formula for magnetization in Heisenberg ferromagnets, improving accuracy over previous methods.

Findings

Good agreement with transfer-matrix data in 1D

Consistent with high-temperature series in 3D

More accurate across various temperatures

Abstract

A study of the d-dimensional classical Heisenberg ferromagnetic model in the presence of a magnetic field is performed within the two-time Green function's framework in classical statistical physics. We extend the well known quantum Callen method to derive analytically a new formula for magnetization. Although this formula is valid for any dimensionality, we focus on one- and three- dimensional models and compare the predictions with those arising from a different expression suggested many years ago in the context of the classical spectral density method. Both frameworks give results in good agreement with the exact numerical transfer-matrix data for the one-dimensional case and with the exact high-temperature-series results for the three-dimensional one. In particular, for the ferromagnetic chain, the zero-field susceptibility results are found to be consistent with the exact analytical ones obtained by M.E. Fisher. However, the formula derived in the present paper provides more accurate predictions in a wide range of temperatures of experimental and numerical interest.