Singular Behavior of Anderson Localized Wavefunctions for a Two-Site Model

TL;DR

This paper analytically demonstrates non-analytic behavior in the inverse participation ratio and density of states in a simple two-site Anderson model, revealing insights into localization phenomena and their evolution to larger systems.

Contribution

It shows that non-analyticities observed in complex models also occur in a simple two-site model, providing a clearer understanding of Anderson localization.

Findings

Non-analyticity in IPR and DOS at specific energies.

Evolution of non-analyticity from two sites to larger systems.

Existence of higher derivative non-analyticities for all bounded disorder distributions.

Abstract

We show analytically that the apparent non-analyticity discovered recently in the inverse participation ratio (IPR) of the eigenstates in Anderson's model of localization is also present in a simple two-site model, along with a concurrent non-analyticity in the density of states (DOS) at the same energy. We demonstrate its evolution from two sites to the thermodynamic limit by numerical methods. For the two site model, non-analyticity in higher derivatives of the DOS and IPR is also proven to exist for all bounded distributions of disorder.