Delocalization and Heisenberg's uncertainty relation

TL;DR

This paper compares localization phenomena in the Anderson and Harper models, highlighting how phase space and momentum coupling influence delocalization and localization transitions in quantum systems.

Contribution

It introduces a phase space perspective to understand the differences between Anderson and Harper models, emphasizing the role of momentum coupling at weak potentials.

Findings

Anderson model eigenstates are localized for any disorder level.

Harper model exhibits a transition from extended to localized states.

Phase space analysis reveals the importance of momentum coupling in delocalization.

Abstract

In the one-dimensional Anderson model the eigenstates are localized for arbitrarily small amounts of disorder. In contrast, the Harper model with its quasiperiodic potential shows a transition from extended to localized states. The difference between the two models becomes particularly apparent in phase space where Heisenberg's uncertainty relation imposes a finite resolution. Our analysis points to the relevance of the coupling between momentum eigenstates at weak potential strength for the delocalization of a quantum particle.