Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry

TL;DR

This paper investigates how non-Hermitian properties like reciprocity and hopping asymmetry influence localization phenomena on directed graphs, revealing new localization states and transitions near exceptional points.

Contribution

It introduces a comprehensive analysis of localization in non-Hermitian systems on directed graphs, highlighting the roles of reciprocity, asymmetry, and disorder in localization behavior.

Findings

Localization states depend on reciprocity and asymmetry parameters.

Biorthogonal localization occurs near exceptional points.

Disorder affects localization and fractal dimensions.

Abstract

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal and right localized states depending on both parameters and graph structure properties such as node degree $d$. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around $W \sim 20$ for a random regular graph $d=4$. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around $W \sim 5$ localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.