Borel-Pad\'e re-summation of the $\beta$-functions describing Anderson localisation in the Wigner-Dyson symmetry classes

TL;DR

This paper introduces a Borel-Padé re-summation method for the beta-function in Wigner-Dyson classes, enabling analysis of critical exponents and lower critical dimensions in Anderson localization.

Contribution

It presents a novel re-summation technique for beta-functions in symmetry classes, improving the understanding of localization transitions and critical dimensions.

Findings

Estimated the critical exponents across dimensions

Determined the lower critical dimension for the symplectic class

Compared re-summation results with numerical data

Abstract

We describe a Borel-Pad\'e re-summation of the $\beta$-function in the three Wigner-Dyson symmetry classes. Using this approximate $\beta$-function we discuss the dimensional dependence of the critical exponent and compare with numerical estimates. We also estimate the lower critical dimension of the symplectic symmetry class.