Diagrammatic self-consistent theory of Anderson localization for the tight-binding model

TL;DR

This paper develops a diagrammatic, self-consistent theory for Anderson localization in the tight-binding model, accurately predicting localization lengths, conductivities, and phase diagrams across dimensions without adjustable parameters.

Contribution

It introduces a parameter-free, diagrammatic self-consistent approach to analyze Anderson localization, connecting extended and localized state expansions and validating results with numerical data.

Findings

Quantitative localization lengths in 1D, 2D, and 3D.

Frequency-dependent conductivity calculations.

Phase diagram of localization in 3D.

Abstract

A self-consistent theory of the frequency dependent diffusion coefficient for the Anderson localization problem is presented within the tight-binding model of non-interacting electrons on a lattice with randomly distributed on-site energy levels. The theory uses a diagrammatic expansion in terms of (extended) Bloch states and is found to be equivalent to the expansion in terms of (localized) Wannier states which was derived earlier by Kroha, Kopp and W\"olfle. No adjustable parameters enter the theory. The localization length is calculated in 1, 2 and 3 dimensions as well as the frequency dependent conductivity and the phase diagram of localization in 3 dimensions for various types of disorder distributions. The validity of a universal scaling function of the length dependent conductance derived from this theory is discussed in the strong coupling region. Quantitative agreement with results from numerical diagonalization of finite systems demonstrates that the self-consistent treatment of cooperon contributions is sufficient to explain the phase diagram of localization and suggests that the system may be well described by a one-parameter scaling theory in certain regions of the phase diagram, if one is not too close to the transition point.