On the derivation of effective field theories

TL;DR

This paper introduces a self-consistent method using cumulant expansion to derive effective field theories, enabling systematic treatment of fluctuations and improved predictions of critical phenomena in many-body systems.

Contribution

It presents a novel approach extending mean-field theory with cumulant expansion for better fluctuation analysis and phase transition predictions.

Findings

Calculated critical temperature for $\,\phi^4_d$-theory.

Derived Landau parameters for the field theory.

Addressed the critical temperature problem in phase transition theory.

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 effective field 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 $\phi^4_d$-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. A comprehensive description of the ground state properties of many-body systems is also demonstrated.