# Genuine localisation transition in a long-range hopping model

### Xiangyu Cao 1 Alberto Rosso 1 Jean-Philippe Bouchaud 2 Pierre Le Doussal 3

#### Physical Review E , American Physical Society (APS), 2017, 95 (6), pp.062118

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 $\kappa > 1$.

• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
• 2. CFM - Capital Fund Management
• 3. LPTENS - Laboratoire de Physique Théorique de l'ENS