Genuine localization transition in a long-range hopping model.

We introduce and study a banded random matrix model describing sparse, long-range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and a mapping to a long-range epidemics model, we establish the phase diagram of the model. A genuine localization transition, with well defined mobility edges, appears as the hopping rate decreases slower than ℓ^{-2}, where ℓ is the distance. Correspondingly, the decay of the localized states evolves from a standard exponential shape to a stretched exponential and finally to a exp(-Cln^{κ}ℓ) behavior, with κ>1.