A new treatment of fluctuation correlations near phase transition points

TL;DR

This paper introduces a self-consistency method using cumulant expansion to improve mean-field theories, enabling systematic analysis of fluctuation effects near phase transition points and providing new insights into critical temperature calculations.

Contribution

It presents a novel self-consistency approach that extends mean-field theory with cumulant expansion, offering a systematic way to account for fluctuations at various scales.

Findings

Calculated critical temperature and Landau parameters for the Ising model.

Provided a new perspective on mean-field theories through cumulant expansion.

Addressed the problem of accurately determining the true critical temperature.

Abstract

A general self-consistency approach allows a thorough treatment of the corrections to the standard mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a new point of view on the MF theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a new coarse graining procedure. We outline and justify our method by some preliminary calculations. Concrete results are given for the critical temperature and the Landau parameters of the theory -- the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions -- the problem for the calculation of the true critical temperature, is considered within the framework of the present approach.