A single defect approximation for localized states on random lattices

TL;DR

This paper introduces a single defect approximation (SDA) method to analytically and numerically analyze localized eigenstates caused by geometrical defects in random lattices, relevant for understanding diffusion and vibrational modes.

Contribution

The paper develops the SDA scheme for describing localized states on random lattices and extends it with a diagrammatic expansion for finite-dimensional systems.

Findings

Localization is caused by sites with abnormal connectivities.

SDA accurately describes extended and localized spectral regions.

The method applies to finite-dimensional amorphous media.

Abstract

Geometrical disorder is present in many physical situations giving rise to eigenvalue problems. The simplest case of diffusion on a random lattice with fluctuating site connectivities is studied analytically and by exact numerical diagonalizations. Localization of eigenmodes is shown to be induced by geometrical defects, that is sites with abnormally low or large connectivities. We expose a ``single defect approximation'' (SDA) scheme founded on this mechanism that provides an accurate quantitative description of both extended and localized regions of the spectrum. We then present a systematic diagrammatic expansion allowing to use SDA for finite-dimensional problems, e.g. to determine the localized harmonic modes of amorphous media.