Properties of some mean-field like approximations for the triangular Ising antiferromagnet

TL;DR

This paper critically examines various mean-field like approximations for the triangular Ising antiferromagnet, revealing their limitations and proposing improved methods that better capture the model's physical properties.

Contribution

It provides a detailed analysis of Bethe, cluster variation, and hard-spin mean-field approximations, highlighting their strengths and weaknesses for the triangular Ising antiferromagnet.

Findings

Bethe approximation predicts unphysical negative entropy at low temperatures.

Cluster variation results converge to exact values with larger clusters.

A new cluster variation approach yields a stable disordered phase down to zero temperature.

Abstract

Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B {\bf 56}, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field like approximations for this model, namely the Bethe approximation itself, the cluster variation method and the hard-spin mean-field theory. We show: (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered phase which is stable down to zero temperature; (iv) a few, so far unknown, zero temperature results (entropy and internal energy) of the hard-spin mean-field theory.