Renormalization Group Analysis of the Hierarchical Anderson Model

TL;DR

This paper uses renormalization techniques to establish conditions for localization and Poisson eigenvalue statistics in the hierarchical Anderson model, even in regimes previously thought to differ.

Contribution

It introduces a new criterion on the single-site distribution ensuring localization and eigenvalue statistics in the hierarchical Anderson model, applicable beyond spectral dimension two.

Findings

Exponential dynamical localization is proven under the new criterion.

Poisson statistics of eigenvalues are established.

The criterion applies even for spectral dimension greater than two.

Abstract

We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension $d > 2$, which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned.