Rigorous mean field model for CPA: Anderson model with free random variables

TL;DR

This paper introduces a rigorous mean field model for the Anderson CPA using free random variables, extending Wegner's model to include correlated site energies and providing exact Green function calculations.

Contribution

It develops a novel mean field approach for the Anderson model with free random variables, generalizing previous models to include correlated site energies and exact Green function solutions.

Findings

Eigenstates are extended with non-vanishing conductivity.

Green functions are calculated exactly and are universal.

The model rigorously solves the CPA equations for Green functions.

Abstract

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner's $n$-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites but free. Freeness of random variables is an analogue of the concept of independence for non-commuting random operators. A possible realization is the ensemble of at different lattice-sites randomly rotated matrices. The one- and two-particle Green functions of the proposed hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is nonvanishing everywhere inside the band. The long-range behaviour and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA-equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multi-site) model is a rigorous mean field model for the (single-site) CPA. We show how the Llyod model is included in our model and treat various kinds of noises.