The equation of state of the n-vector model: collective variables method

TL;DR

This paper derives the equation of state for the three-dimensional n-vector model using the collective variables method, providing analytical expressions and scaling functions that align qualitatively with Monte Carlo data.

Contribution

It introduces a microscopic, parameter-free analytical approach to the n-vector model's critical behavior using the collective variables method.

Findings

Analytical expression for free energy for T > T_c

Derived equation of state for small and large external fields

Scaling functions for different order parameters

Abstract

The critical behavior of the three-dimensional n-vector model in the presence of an external field is investigated. Mathematical description is performed with the collective variables (CV) method in the framework of the $\rho^4$ model approximation at the microscopic level without any adjustable parameters. The recurrence relations of the renormalization group (RG) as functions of the external field and temperature were found. The analytical expression for the free energy of the system for temperatures $T>T_c$ and different n was obtained. The equation of state of the n-vector model for general case of small and large external fields was written. The explicit form of the correspondent scaling function for different values of the order parameter was derived. The obtained results are in qualitative agreement with the data of Monte Carlo simulations.