Genuine localisation transition in a long-range hopping model

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 $\kappa >1$.

updated version; 14 pages, 11 figures

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• Diluted banded random matrices: Scaling behavior of eigenfunction and spectral properties
• Power-Law Random Banded Matrix Ensemble as the Effective Model for Many-Body Localization Transition
• More current with less particles due to power-law hopping
• Clusters in an epidemic model with long-range dispersal
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization