Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method

TL;DR

This paper proves exponential decay of fractional moments of the Green function for a class of Anderson models with sign-indefinite potentials, establishing localization under certain conditions using the fractional moment method.

Contribution

It introduces a finite-volume criterion for fractional moment decay in models with sign-indefinite potentials, enabling proof of localization in these complex cases.

Findings

Fractional moment decay holds under typical perturbative regimes.

The criterion applies at spectral boundaries with Lifshitz tail estimates.

Exponential spectral localization is established for the considered models.

Abstract

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators we establish a finite-volume criterion which implies that above mentioned the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e. g. at spectral boundaries which satisfy 'Lifshitz tail estimates' on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.