Metallic and insulating behaviour in an exactly solvable random matrix model

Abstract

The authors propose a novel random transfer matrix model for quantum transport in disordered systems. The model is exactly solvable in the sense that arbitrary n-point correlation functions of the eigenvalues can be obtained in terms of known orthogonal polynomials, and the conductance is a simple linear statistic of these eigenvalues. The model exhibits qualitative deviations from the universal properties normally associated with random matrices, and they observe that such deviations may naturally describe the differences in the distribution of conductance in the metallic versus insulating regimes. In particular, by varying a single parameter q, the authors recover the metallic regime with Ohm's law and universal conductance fluctuation in the limit q=1, and the well known log-normal distribution of conductance for one-dimensional insulators in the opposite limit q<<1. They argue that in this model the metal-insulator transition is related to the qualitative change in the eigenvalue density and the associated opening of a gap in the density at the origin.