Genuine localisation transition in a long-range hopping model

TL;DR

This paper introduces a new long-range hopping model in one dimension that exhibits a genuine localization transition with well-defined mobility edges, characterized by a transition in the decay behavior of localized states.

Contribution

The study presents a novel class of Banded Random Matrix models and establishes the phase diagram, revealing a genuine localization transition with unique decay properties of localized states.

Findings

Localization transition occurs when hopping decreases slower than  power law.

Localized states decay from exponential to stretched exponential to a new  decay.

The phase diagram includes well-defined mobility edges.

Abstract

We introduce and study a new class of Banded Random Matrix model describing sparse, long range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and mappings to statistical physics models, we establish the phase diagram of the model. A genuine localisation transition, with well defined mobility edges, appears as the hopping rate decreases slower than $\ell^{-2}$, where $\ell$ is the distance. Correspondingly, the decay of the localised states evolves from a standard exponential shape to a stretched exponential and finally to a novel $\exp(-C\ln^\kappa \ell)$ behaviour, with $\kappa > 1$.