Finite-size scaling from self-consistent theory of localization

TL;DR

This paper derives finite-size scaling functions for localization transitions in various dimensions based on the self-consistent theory, aligning well with numerical data and clarifying critical exponents and scaling behaviors.

Contribution

It provides a derivation of finite-size scaling procedures from the self-consistent localization theory and clarifies the interpretation of critical exponents in different dimensions.

Findings

Scaling functions for d=2 and d=3 match numerical results.

Critical exponent =1.3-1.6 for d=3 explained by L+L_0 dependence.

Modified scaling relations for der 4 are derived and discussed.

Abstract

Accepting validity of self-consistent theory of localization by Vollhardt and Woelfle, we derive the finite-size scaling procedure used for studies of the critical behavior in d-dimensional case and based on the use of auxiliary quasi-1D systems. The obtained scaling functions for d=2 and d=3 are in good agreement with numerical results: it signifies the absence of essential contradictions with the Vollhardt and Woelfle theory on the level of raw data. The results \nu=1.3-1.6, usually obtained at d=3 for the critical exponent of the correlation length, are explained by the fact that dependence L+L_0 with L_0>0 (L is the transversal size of the system) is interpreted as L^{1/\nu} with \nu>1. For dimensions d\ge 4, the modified scaling relations are derived; it demonstrates incorrectness of the conventional treatment of data for d=4 and d=5, but establishes the constructive procedure for such a treatment. Consequences for other variants of finite-size scaling are discussed.