Solving the Triangular Ising Antiferromagnet by Simple Mean Field

TL;DR

This paper introduces a simple loopwise mean field approach that accurately captures the absence of ordering in the triangular antiferromagnetic Ising model and finds a non-zero critical temperature for the ferromagnetic case.

Contribution

The paper presents a novel loopwise mean field scheme that preserves Hamiltonian symmetry and accurately solves the triangular Ising antiferromagnet without adjustable parameters.

Findings

Reproduces Wannier's exact result of no ordering at non-zero temperature for the antiferromagnet.

Obtains a non-zero critical temperature for the ferromagnetic case.

Provides a new approach for tackling random systems.

Abstract

Few years ago, application of the mean field Bethe scheme on a given system was shown to produce a systematic change of the system intrinsic symmetry. For instance, once applied on a ferromagnet, individual spins are no more equivalent. Accordingly a new loopwise mean field theory was designed to both go beyond the one site Weiss approach and yet preserve the initial Hamitonian symmetry. This loopwise scheme is applied here to solve the Triangular Antiferromagnetic Ising model. It is found to yield Wannier's exact result of no ordering at non-zero temperature. No adjustable parameter is used. Simultaneously a non-zero critical temperature is obtained for the Triangular Ising Ferromagnet. This simple mean field scheme opens a new way to tackle random systems.