Method of Calculating the Free Energy of Three-Dimensional Ising-Like System in an External Field with the Use of the $\rho^6$ Model

TL;DR

This paper develops an analytical method to calculate the free energy of a three-dimensional Ising-like system in an external field using the $ ho^6$ model, valid near the critical point across the entire field-temperature plane.

Contribution

It introduces a microscopic approach in the higher non-Gaussian approximation that accounts for both temperature and field fluctuations without relying on power series expansions.

Findings

Analytical free energy expression valid near critical point

Inclusion of temperature and field confluent corrections

Applicable across entire field-temperature plane including crossover region

Abstract

The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the $\rho^6$ model) at temperatures above the critical value of $T_c$ ($T_c$ is the phase-transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short- and long-wave spin-density oscillation modes taking into account both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field $\tilde h_c$, which divides external fields into the weak and strong ones (i.e., the crossover region). In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity, and power series are not efficient. The obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.