Conductance of Finite Systems and Scaling in Localization Theory

TL;DR

This paper investigates the conductance of finite systems within localization theory, addressing key questions about its definition, relation to internal properties, and scaling behavior, using two complementary theoretical approaches.

Contribution

It introduces a unified definition of finite-system conductance, derives finite-size scaling relations, and calculates the eta(g) functions for dimensions 1, 2, and 3, considering both metallic and localized phases.

Findings

Finite-system conductance relates closely to the Thouless definition.

Derived eta(g) functions for 1D, 2D, and 3D systems.

In 2D, eta(g) expansion matches two-loop pproach results.

Abstract

The conductance of finite systems plays a central role in the scaling theory of localization (Abrahams et al, 1979). Usually it is defined by the Landauer-type formulas, which remain open the following questions: (a) exclusion of the contact resistance in the many-channel case; (b) correspondence of the Landauer conductance with internal properties of the system; (c) relation with the diffusion coefficient D(\omega,q) of an infinite system. The answers to these questions are obtained below in the framework of two approaches: (1) self-consistent theory of localization by Vollhardt and Woelfle, and (2) quantum mechanical analysis based on the shell model. Both approaches lead to the same definition for the conductance of a finite system, closely related to the Thouless definition. In the framework of the self-consistent theory, the relations of finite-size scaling are derived and the Gell-Mann - Low functions \beta(g) for space dimensions d=1,2,3 are calculated. In contrast to the previous attempt by Vollhardt and Woelfle (1982), the metallic and localized phase are considered from the same standpoint, and the conductance of a finite system has no singularity at the critical point. In the 2D case, the expansion of \beta(g) in 1/g coincides with results of the \sigma-model approach on the two-loop level and depends on the renormalization scheme in higher loops; the use of dimensional regularization for transition to dimension d=2+\epsilon looks incompatible with the physical essence of the problem. The obtained results are compared with numerical and physical experiments. A situation in higher dimensions and the conditions for observation of the localization law \sigma\propto -i\omega for conductivity are discussed.