Strong-disorder renormalization group study of the Anderson localization transition in three and higher dimensions

TL;DR

This paper develops an efficient strong-disorder renormalization-group method to study Anderson localization transitions across dimensions, confirming the infinite upper critical dimension and revealing the strong-coupling nature of conductance scaling.

Contribution

It introduces a new SDRG algorithm applicable in any dimension, demonstrating its accuracy and efficiency in analyzing Anderson localization and critical properties.

Findings

Upper critical dimension for Anderson localization is infinite.

Excellent agreement with numerical results at the critical point.

Mirror symmetry of conductance scaling is a strong-coupling effect.

Abstract

We implement an efficient strong-disorder renormalization-group (SDRG) procedure to study disordered tight-binding models in any dimension and on the Erdos-Renyi random graphs, which represent an appropriate infinite dimensional limit. Our SDRG algorithm is based on a judicious elimination of most (irrelevant) new bonds generated under RG. It yields excellent agreement with exact numerical results for universal properties at the critical point without significant increase of computer time, and confirm that, for Anderson localization, the upper critical dimension duc = infinite. We find excellent convergence of the relevant 1/d expansion down to d=2, in contrast to the conventional 2+epsilon expansion, which has little to say about what happens in any d>3. We show that the mysterious mirror symmetry of the conductance scaling function is a genuine strong-coupling effect, as speculated in early work. This opens an efficient avenue to explore the critical properties of Anderson transition in the strong-coupling limit in high dimensions.