Highly Accurate Critical Exponents from Self-Similar Variational Perturbation Theory

TL;DR

This paper introduces an enhanced variational perturbation theory using self-similar approximation, enabling highly accurate calculation of critical exponents in O(N)-symmetric $ ext{phi}^4$ theory from limited perturbation data.

Contribution

The authors develop a self-similar variational perturbation approach that accelerates convergence and yields analytic critical exponents with high precision from low-order perturbation expansions.

Findings

Accurate analytic critical exponents from three-loop expansions.

Reproduces large-N behavior of exponents with high precision.

Explains the specific heat exponent in superfluid helium experiments.

Abstract

We extend field theoretic variational perturbation theory by self-similar approximation theory, which greatly accelerates convergence. This is illustrated by re-calculating the critical exponents of O(N)-symmetric $\vp^4$ theory. From only three-loop perturbation expansions in $4- \epsilon $ dimensions we obtain {\em analytic results for the exponents, with practically the same accuracy as those derived recently from ordinary field-theoretic variational perturbational theory to seventh order. In particular, the theory explains the best-measured exponent $\al\approx-0.0127$ of the specific heat peak in superfluid helium, found in a satellite experiment with a temperature resolution of nanoKelvin. In addition, our analytic expressions reproduce also the exactly known large-N behaviour of the exponents $ \nu $ and $ \gamma= \nu (2- \eta) $ with high precision.