Constructive Fractional-Moment Criteria for Localization in Random Operators

TL;DR

This paper introduces finite-volume criteria based on fractional moments of Green functions that characterize localization in random operators, linking decay properties to various localization phenomena and quantum Hall effects.

Contribution

The paper develops a new set of constructive criteria for localization using fractional moments, applicable to finite volumes and encompassing exponential decay regimes.

Findings

Criteria imply spectral localization and absence of level repulsion

Preclude power-law decay at mobility edges

Connect decay of Green functions to quantum Hall conductance

Abstract

We present a family of finite-volume criteria which cover the regime of exponential decay for the fractional moments of Green functions of operators with random potentials. Such decay is a technically convenient characterization of localization for it is known to imply spectral localization, absence of level repulsion, dynamical localization and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. The constructive criteria also preclude fast power-law decay of the Green functions at mobility edges.