Anderson localization transition and eigenfunction multifractality in ensemble of ultrametric random matrices

TL;DR

This paper investigates Anderson localization and eigenfunction multifractality in ultrametric random matrices, revealing a metal-insulator transition with analytical and numerical agreement on critical parameters and multifractal properties.

Contribution

It introduces a new class of models on ultrametric spaces exhibiting Anderson transition, with analytical derivation and numerical validation of critical points and multifractal exponents.

Findings

Identification of Anderson transition in ultrametric random matrices

Analytical calculation of critical parameters and multifractal exponents

Numerical simulations confirm analytical predictions

Abstract

We demonstrate that by considering disordered single-particle Hamiltonians (or their random matrix versions) on ultrametric spaces one can generate an interesting class of models exhibiting Anderson metal-insulator transition. We use the weak disorder virial expansion to determine the critical value of the parameters and to calculate the values of the multifractal exponents for inverse participation ratios. Direct numerical simulations agree favourably with the analytical predictions.