Correlation-induced localization

TL;DR

This paper introduces a new class of translation-invariant random Hamiltonians with long-range hopping, establishing their localization-delocalization phase diagram and providing analytical bounds for localized and ergodic states, advancing understanding of Anderson localization.

Contribution

It proposes a novel framework for Anderson localization driven by correlations in long-range hopping, including a matrix inversion trick and a full phase diagram analysis.

Findings

Localization at all power-law exponents confirmed

Analytical bounds for localized and ergodic states established

Symmetry of power-law localized wave functions proved

Abstract

A new paradigm of Anderson localization caused by correlations in the long-range hopping along with uncorrelated on-site disorder is considered which requires a more precise formulation of the basic localization-delocalization principles. A new class of random Hamiltonians with translation-invariant hopping integrals is suggested and the localization properties of such models are established both in the coordinate and in the momentum spaces alongside with the corresponding level statistics. Duality of translation-invariant models in the momentum and coordinate space is uncovered and exploited to find a full localization-delocalization phase diagram for such models. The crucial role of the spectral properties of hopping matrix is established and a new matrix inversion trick is suggested to generate a one-parameter family of equivalent localization/delocalization problems. Optimization over the free parameter in such a transformation together with the localization/delocalization principles allows to establish exact bounds for the localized and ergodic states in long-range hopping models. When applied to the random matrix models with deterministic power-law hopping this transformation allows to confirm localization of states at all values of the exponent in power-law hopping and to prove analytically the symmetry of the exponent in the power-law localized wave functions.