Random-matrix modeling of semi-linear response, the generalized variable range hopping picture, and the conductance of mesoscopic rings

TL;DR

This paper develops a random-matrix based semi-linear response theory to analyze absorption and conductance in mesoscopic rings, revealing significant deviations from traditional linear response predictions and revisiting Mott's diffusion and localization concepts.

Contribution

It introduces a generalized variable range hopping approach to model semi-linear response and applies it to mesoscopic ring conductance, challenging conventional theories.

Findings

Absorption coefficients can differ greatly from Kubo predictions with broad transition rate distributions.

A practical approximation for mesoscopic ring conductance is developed.

The approach revisits Mott's diffusion and localization in this context.

Abstract

Semi-linear response theory determines the absorption coefficient of a driven system using a resistor network calculation: Each unperturbed energy level of a particle in a vibrating trap, or of an electron in a mesoscopic ring, is regarded as a node ($n$) of the network; The transition rates ($w_{mn}$) between the nodes are regarded as the elements of a random matrix that describes the network. If the size-distribution of the connecting elements is wide (e.g. log-normal-like rather than Gaussian-like) the result for the absorption coefficient differs enormously from the conventional Kubo prediction of linear response theory. We use a generalized variable range hopping scheme for the analysis. In particular we apply this approach to obtain practical approximations for the conductance of mesoscopic rings. In this context Mott's picture of diffusion and localization is revisited.