Magnetisation and mean field theory in the Ising model

TL;DR

This paper provides a comprehensive, pedagogical tutorial on applying mean field theory to the two-dimensional Ising model, including derivations, interpretations, and discussions of phase transition thermodynamics.

Contribution

It offers a complete, self-contained overview with novel interpretations of the self-consistency condition and detailed analysis of the Ising model's features.

Findings

Derivation of magnetisation function using mean field theory

Graphical and physical interpretation of results

New perspective on the self-consistency condition

Abstract

In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.