Mean-field models with short-range correlations

TL;DR

This paper extends mean-field models by incorporating short-range correlations through a general Hamiltonian, revealing new phase transition phenomena and finite size effects, with implications for understanding complex many-body systems.

Contribution

It introduces a generalized mean-field equation for models with short-range correlations and analyzes phase transitions in such systems, including effects of external fields and finite sizes.

Findings

Generalized Curie-Weiss mean-field equation derived

First-order and inverse phase transitions identified with negative couplings

Finite size effects and external field dependencies explicitly calculated

Abstract

Given an arbitrary finite dimensional Hamiltonian H_0, we consider the model H=H_0+\Delta H, where \Delta H is a generic fully connected interaction. By using the strong law of large numbers we easily prove that, for all such models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional mean-field models the term H_0 gives rise to short-range correlations and, furthermore, when H_0 has negative couplings, first-order phase transitions and inverse transition phenomena may take place even when only two-body interactions are present. The dependence from a non uniform external field and finite size effects are also explicitly calculated. Partially, these results were derived long ago by using min-max principles but remained almost unknown.