Surprises in the phase diagram of the Anderson model on the Bethe lattice

TL;DR

This paper reviews recent surprising findings in the Anderson model on the Bethe lattice, revealing extended states at weak disorder due to disorder-induced resonances and discussing open questions about eigenfunction structure.

Contribution

It uncovers unexpected extended states at weak disorder levels caused by resonances, challenging previous assumptions about the phase diagram of the Anderson model on the Bethe lattice.

Findings

Extended states exist beyond unperturbed energies into Lifshitz tails at weak disorder.

Disorder-induced resonances are responsible for the appearance of extended states.

Open questions remain about eigenfunction structure and spectral statistics on the Bethe lattice.

Abstract

The Anderson model on the Bethe lattice is historically among the first for which an energy regime of extended states and a separate regime of localized states could be established. In this paper, we review recently discovered surprises in the phase diagram. Among them is that even at weak disorder, the regime of diffusive transport extends well beyond energies of the unperturbed model into the Lifshitz tails. As will be explained, the mechanism for the appearance of extended states in this non-perturbative regime are disorder-induced resonances. We also present remaining questions concerning the structure of the eigenfunctions and the associated spectral statistics problem on the Bethe lattice.