Non-Hermitian Delocalization and Eigenfunctions

TL;DR

This paper highlights that delocalization in non-Hermitian systems can be identified through the analysis of the product of left- and right-eigenfunctions, offering a new perspective beyond traditional eigenvalue sensitivity criteria.

Contribution

It introduces the importance of studying the product of left- and right-eigenfunctions to detect delocalization in non-Hermitian systems, expanding the understanding of eigenfunction behavior.

Findings

Eigenfunction products reveal delocalization more clearly.

Ground state eigenfunctions show intermediate behavior.

Eigenvalue sensitivity alone is insufficient for delocalization detection.

Abstract

Recent literature on delocalization in non-Hermitian systems has stressed criteria based on sensitivity of eigenvalues to boundary conditions and the existence of a non-zero current. We emphasize here that delocalization also shows up clearly in eigenfunctions, provided one studies the product of left- and right-eigenfunctions, as required on physical grounds, and not simply the squared modulii of the eigenfunctions themselves. We also discuss the right- and left-eigenfunctions of the ground state in the delocalized regime and suggest that the behavior of these functions, when considered separately, may be viewed as ``intermediate'' between localized and delocalized.