Calculating critical temperature and critical exponents by self-similar approximants

TL;DR

This paper introduces a simplified self-similar approximant method for accurately calculating critical temperatures and exponents in the $O(N)$-symmetric $^4$ field theory, avoiding complex numerical procedures.

Contribution

The paper demonstrates that self-similar factor approximants provide an easier yet accurate alternative to traditional summation methods for critical phenomena calculations.

Findings

Effective sums of asymptotic series are achieved with simpler calculations.

The method yields results comparable to more complex techniques.

Application to the $O(N)$-symmetric $^4$ theory confirms its accuracy.

Abstract

Self-similar approximation theory allows for defining effective sums of asymptotic series. The method of self-similar factor approximants is applied for calculating the critical temperature and critical exponents of the $O(N)$-symmetric $\varphi^4$ field theory in three dimensions by summing asymptotic $\varepsilon$ expansions. This method is shown to be essentially simpler than other summation techniques involving complicated numerical calculations, while enjoying comparable accuracy.