Concurrent Skin-scale-free Localization and Criticality under M\"obius Boundary Conditions in a Non-Hermitian Ladder

TL;DR

This paper explores how non-Hermitian ladder systems under M"obius boundary conditions exhibit coexisting skin effect and scale-free localization, revealing energy-dependent localization exchanges driven by critical behaviors from weak interchain coupling.

Contribution

It uncovers the coexistence and energy-dependent exchange of two localization phenomena in non-Hermitian ladders with M"obius boundary conditions, highlighting the role of criticality and boundary effects.

Findings

Eigenstates show coexistence of skin effect and scale-free localization.

Localization characteristics can exchange between chains depending on eigenenergy.

M"obius boundary conditions enhance critical behaviors and localization effects.

Abstract

Non-Hermitian systems possess exotic localization phenomena beyond their Hermitian counterparts, exhibiting massive accumulation of eigenstates at the system boundaries with different scaling behaviors. In this study, we investigate two weakly coupled non-Hermitian Hatano-Nelson chains under M\"obius boundary conditions (MBCs), and reveal the coexistence of two distinct localization behaviors for eigenstates. Namely, eigenstates exhibit non-Hermitian skin effect in one chain and scale-free localization in the other. Notably, the localization characteristics of eigenstates can exchange between the two chains depending on their eigenenergies. This phenomenon is found to emerge from the critical behaviors induced by the weak interchain coupling, which can even be enhanced by MBCs in comparison \red{to} the system under other boundary conditions. Our findings deepen the understanding of non-Hermitian localization and criticality, and offer new insights into engineering tunable edge-localized states in synthetic quantum systems.