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

TL;DR

The paper addresses the fundamental mismatch between the eigenvalue spectrum $\sigma_{\text{eig}}(H)$ and the eigenstate spectrum $\sigma_{\text{states}}(H)$ in non-Hermitian systems, a discrepancy that becomes pronounced in the thermodynamic limit and is linked to the non-Hermitian skin effect. Using the Hatano-Nelson model as a minimal solvable system and the non-Hermitian SSH chain as a case study, it demonstrates that many eigenstates can exist without being detected by the eigenvalue spectrum, with almost-eigenstates occupying the unit disk $\{|E|<1\}$ even when $\sigma_{\text{eig}}(H_L)$ collapses to a single point as $L$ grows. The work develops a framework anchored in topological winding $W$ to show that nonzero $W(H-E)$ enforces generalized eigenstates that form macroscopic Jordan blocks, revealing hidden exceptional points that are often invisible to $\sigma_{\text{eig}}$. Through case studies and analysis, it shows that bulk-edge correspondence failures in models like the non-Hermitian SSH chain arise from the eigenvalue spectrum’s failure to detect certain edge/localized states, while the eigenstate/singular-value/pseudospectrum perspective remains robust. The results argue for adopting eigenstate-centric diagnostics, establish equivalences among eigenstate, singular-value, and bounded-inverse criteria for spectral gaps, and suggest extensions to many-body and topological classifications in non-Hermitian systems.

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.

Figures (2)

• Figure 1: a) Bulk spectrum of the leftward-hopping Hatano-Nelson model \ref{['eq:Jordanblock']} for $L=30$ sites, showing periodic (red) and open boundary (blue) eigenvalue spectra. The yellow background denotes the $\epsilon$-pseudospectrum of the open system with $\epsilon=10^{-3}$. b) Temporal stability of the normalized state error \ref{['eq:error']} for the almost eigenstate $\ket{\psi_E}$\ref{['eq:JordanState']} for $E$ at the green marker in a) showing how the state lifetime (vertical lines), until which the error remain below 1%, scales with system size.
• Figure 2: a) Spectrum of the non-Hermitian SSH model defined in Eq. \ref{['eq:NHSSHchain']} for $(t_1, t_2,\gamma)=(0.2, 1, 4/3)$ and $L = 100$. The dots correspond to the eigenvalue spectrum for periodic and open boundary conditions, while the yellow background is the eigenstate spectrum of the open system. It is the full area enclosed by the periodic spectrum. b) Similar spectrum for $(t_1, t_2,\gamma)=(1.4, 1, 4/3)$. Note that now eigenstates with zero energy are present despite the absence of topological modes in the eigenvalue spectrum. Plot of c) eigenvalue spectrum and d) singular value spectrum for $(t_2,\gamma)=(1,4/3)$ and $L=20$ sites with open boundary conditions, reproducing the configuration of Yao2018. Parameters for which topological edge modes at zero energy exist are denoted in green. The eigenvalue spectrum is deficient in the red regions with either a gap while there is one mode at zero energy ($1.2\lesssim|t_1|\leq 5/3$) or a single zero eigenvalue where there is two modes at zero energy ($|t_1|\leq 1/3$). Singular value has however no problem detecting those modes at zero energy.