$\zeta$-Pade SRWS theory with lowest order approximation

TL;DR

This paper refines the SRWS-$$ theory for Anderson transition by replacing summation with integral and applying Pade approximants, leading to improved estimates of the critical exponent rom the upper critical dimension.

Contribution

It introduces a lowest order approximation in the SRWS-$ theory by replacing summation with integral and using Pade approximants, enhancing the estimate of the critical exponent or Anderson transition.

Findings

The dimensional dependence of ollows the Riemann unction.

The estimate of the critical exponent  aligns with previous fitting curve estimates.

The approach improves the lower critical dimension estimate.

Abstract

In my previous paper about SRWS-$\zeta$ theory[Y.Ueoka,viXra:2205.014,2022], I proposed an approximation of rough averaged summation of typical critical Green function for the Anderson transition in the Orthogonal class. In this paper, I remove a rough approximate summation for the series of the typical critical Green function by replacing summation with integral. Pade approximant is used to take a summation. The perturbation series of the critical exponent $\nu$ of localization length from upper critical dimension is obtained. The dimensional dependence of the critical exponent is again directly related with Riemann $\zeta$ function. Degree of freedom about lower critical exponent improve estimate compared with previous studies. When I fix lower critical dimension equal to two, I obtained similar estimate of the critical exponent compared with fitting curve estimate of the critical exponent[E.Tarquini et al.,PhysRevB.95(2017)094204]. 1