The localized phase of the Anderson model on the Bethe lattice

TL;DR

This paper presents a highly accurate computational method for analyzing Anderson localization on the Bethe lattice, confirming theoretical predictions and revealing new singular behavior near the transition point.

Contribution

It introduces a cavity approach for calculating inverse participation ratios with unprecedented precision, validating existing theories and uncovering new critical phenomena.

Findings

High-precision validation of the Anderson transition predictions

Observation of a previously unreported singular behavior of IPRs near the critical point

Confirmation of theoretical results using the non-linear sigma model

Abstract

In this paper, we investigate the Anderson model on the Bethe lattice, focusing on the localized regime. Employing the cavity approach, we derive compact expressions for the inverse participation ratios (IPRs) that are equivalent to those obtained using the supersymmetric formalism and naturally facilitate a highly efficient computational scheme. This method yields numerical results with unprecedented accuracy, even very close to the localization threshold. Our approach allows for high-precision validation of all theoretical predictions from the analytical solution, including the finite jump of the IPRs at the transition. Additionally, we reveal a singular behavior of the IPRs near the critical point that has not been previously reported in the literature. This singular behavior is further confirmed by the numerical solution of the non-linear $\sigma$ model on the Bethe lattice, which provides an effective description of Anderson localization.