Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

TL;DR

This paper introduces a new analytical approach using the inverted order thermodynamic limit and RSB formalism to identify a non-ergodic delocalized phase in Anderson models on Bethe lattices and regular graphs, revealing a broad non-ergodic phase region.

Contribution

The paper develops an innovative analytical method to characterize the non-ergodic delocalized phase in infinite-dimensional Anderson models, providing explicit expressions for fractal dimensions and phase diagrams.

Findings

Existence of a broad non-ergodic delocalized phase.

Analytical expressions for fractal dimension D_{1} and self-energy.

Phase diagrams as functions of disorder and energy.

Abstract

We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graphs. The key ingredient of the approach is the notion of the inverted order thermodynamic limit (IOTL) in which the coupling to the environment goes to zero before the system size goes to infinity. Using IOTL and Replica Symmetry Breaking (RSB) formalism we derive analytical expressions for the fractal dimension D_{1} that distinguishes between the extended ergodic, D_{1}=1, and extended non-ergodic (multifractal), 0<D_{1}<1 states on the Bethe lattice and random regular graphs with the branching number K. We also employ RSB formalism to derive the analytical expression ln(1/S_{typ})~(W_{c}-W)^{-1} for the typical imaginary part of self-energy S_{typ} in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit. We prove the existence of an extended non-ergodic phase in a broad range of disorder strength and energy and establish the phase diagrams of the models as a function of disorder and energy. The results of the analytical theory are compared with large-scale population dynamics and with the exact diagonalization of Anderson model on random regular graphs. We discuss the consequences of these results for the many body localization.