On the free energy within the mean-field approximation

TL;DR

This paper compares two common mean-field approximation methods using the antiferromagnetic Ising model, revealing that one formulation can fail to guarantee a valid variational principle, leading to incorrect results.

Contribution

It identifies a critical flaw in one formulation of the mean-field free energy approach, emphasizing the importance of the underlying variational principle.

Findings

One formulation lacks a guaranteed variational principle.

Straightforward minimization can produce incorrect results.

Highlights the need for careful formulation in mean-field approximations.

Abstract

We compare two widespread formulations of the mean-field approximation, based on minimizing an appropriately built mean-field free energy. We use the example of the antiferromagnetic Ising model to show that one of these formulations does not guarantee the existence of an underlying variational principle. This results in a severe failure where straightforward minimization of the corresponding mean-field free energy leads to incorrect results.