Difference between level statistics, ergodicity and localization transitions on the Bethe lattice

TL;DR

This paper reveals a new intermediate phase in disordered electrons on a Bethe lattice, characterized by delocalization without ergodicity, supported by numerical evidence and linked to glass transition phenomena.

Contribution

It identifies and characterizes a non-ergodic delocalized phase on the Bethe lattice, connecting it to glass transition physics and challenging traditional localization paradigms.

Findings

Existence of a non-ergodic delocalized phase with Poisson statistics

Transport occurs over few disorder-dependent paths

Transition to ergodic phase relates to glass transition of directed polymers

Abstract

We show that non-interacting disordered electrons on a Bethe lattice display a new intermediate phase which is delocalized but non-ergodic, i.e. it is characterized by Poisson instead of GOE statistics. The physical signature of this phase is a very heterogenous transport that proceeds over a few disorder dependent paths only. We show that the transition to the usual ergodic delocalized phase, which takes place for a disorder strength smaller than the one leading to the localization transition, is related to the freezing-glass transition of directed polymers in random media. The numerical study of level and eigenstate statistics, and of the singular properties of the probability distribution of the local density of states all support the existence of this new intermediate phase. Our results suggest that the localization transition may change nature in high dimensional systems.