Analysis of the exactness of mean-field theory in long-range interacting systems

TL;DR

This paper investigates the accuracy of mean-field theory in long-range interacting lattice systems, demonstrating its exactness for free energy calculations and identifying conditions where it fails, especially with conserved magnetization.

Contribution

It proves the exactness of mean-field theory for free energy in long-range systems and explores its limitations under magnetization conservation.

Findings

Mean-field free energy matches long-range system free energy.

Mean-field metastable states are preserved in long-range systems.

Mean-field theory fails to be accurate when magnetization is conserved in certain regions.

Abstract

Relationships between general long-range interacting classical systems on a lattice and the corresponding mean-field models (infinitely long-range interacting models) are investigated. We study systems in arbitrary dimension d for periodic boundary conditions and focus on the free energy for fixed value of the total magnetization. As a result, it is shown that the equilibrium free energy of the long-range interacting systems are exactly the same as that of the corresponding mean-field models (exactness of the mean-field theory). Moreover, the mean-field metastable states can be also preserved in general long-range interacting systems. It is found that in the case that the magnetization is conserved, the mean-field theory does not give correct property in some parameter region.