Transition between localized and extended states in the hierarchical Anderson model

TL;DR

This paper provides numerical evidence for a localization-delocalization transition in a hierarchical 1-D Anderson model, demonstrating a finite critical disorder strength where eigenstates transition from localized to extended.

Contribution

It introduces the hierarchical Anderson model as a tractable system to study the Anderson transition, showing a finite critical disorder using an exact renormalization group approach.

Findings

Existence of a finite critical disorder strength Wc for localization transition

Delocalized phase at disorder W < Wc

Consistent with Anderson transition predictions in higher-dimensional models

Abstract

We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping between their phases and those of models with short range hopping in higher dimensions, and also because the renormalization group can be applied exactly without the approximations that generally are required in other models. In the hierarchical Anderson model we find a finite critical disorder strength Wc where the average inverse participation ratio goes to zero; at small disorder W < Wc the model lies in a delocalized phase. This result is based on numerical calculation of the inverse participation ratio in the infinite volume limit using an exact renormalization group approach facilitated by the model's hierarchical structure. Our results are consistent with the presence of an Anderson transition in short-range models with D > 2 dimensions, which was predicted using renormalization group arguments. Our finding should stimulate interest in the hierarchical Anderson model as a simplified and tractable model of the Anderson localization transition which occurs in finite-dimensional systems with short-range hopping.