When and why non-Hermitian eigenvalues miss eigenstates in topological physics

TL;DR

This paper reveals that in non-Hermitian topological systems, eigenvalues can fail to detect certain eigenstates, especially in models with skin effects, highlighting limitations of eigenvalue-based analysis and proposing eigenstate-based characterization.

Contribution

It introduces a framework explaining how non-Hermitian Hamiltonians support hidden eigenstates undetectable by eigenvalues, with analytical solutions for the Hatano-Nelson model and implications for topological physics.

Findings

Eigenvalues can miss eigenstates in non-Hermitian systems.

Hidden modes and exceptional points are linked to bulk winding.

Eigenstate analysis improves understanding of non-Hermitian topology.

Abstract

Non-Hermitian systems exhibit a fundamental spectral dichotomy absent in Hermitian physics: the eigenvalue spectrum and the eigenstate spectrum can deviate significantly in the thermodynamic limit. We explain how non-Hermitian Hamiltonians can support eigenstates completely undetected by eigenvalues, with the unidirectional Hatano-Nelson model serving as both a minimal realization and universal paradigm for this phenomenon. Through exact analytical solutions, we show that this model contains not only hidden modes but multiple macroscopic hidden exceptional points that appear more generally in all systems with a non-trivial bulk winding. Our framework explains how the apparent bulk-edge correspondence failures in models like the non-Hermitian SSH chain instead reflect the systematic inability of the eigenvalue spectrum to detect certain eigenstates in systems with a skin-effect. These results establish the limitation of the eigenvalue spectrum and suggest how the eigenstate approach can lead to improved characterization of non-Hermitian topology.