Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates in topological physics

The gist The eigenvalue spectrum can miss most of the eigenstate spectra in non-Hermitian systems, which establishes a limitation for eigenvalue-based diagnostics and suggests that eigenstate approaches can lead to improved characterization of non-Hermitian topology.

Non-Hermitian Spectral Dichotomy

Non-Hermitian systems exhibit a fundamental spectral dichotomy absent in Hermitian physics, where the eigenvalue spectrum and the eigenstate spectrum can deviate significantly in the thermodynamic limit. This is because non-Hermitian Hamiltonians can support eigenstates completely undetected by eigenvalues, as demonstrated by the unidirectional Hatano-Nelson model serving as a minimal realization and universal paradigm for this phenomenon. The paper explains how the apparent bulk-edge correspondence failures in models like the non-Hermitian SSH chain reflect the systematic inability of the eigenvalue spectrum to detect certain eigenstates in systems with a skin-effect.

Formal Definitions and Spectrum Mismatch

The definitions of eigenvalue spectrum and eigenstate spectrum coincide for finite-dimensional matrices when excluding degeneracies. However, in the thermodynamic limit for a system of size L with Hamiltonian HL, two different ways to generalize the notion of spectrum exist: the eigenvalue spectrum is defined as the limit of roots of the characteristic polynomial, while the eigenstate spectrum is defined as states ψL⟩ such that HL ψL⟩ −→ 0. The fact that these two definitions no longer coincide for some non-Hermitian systems is an important result discussed in the rest of the paper.

The Hatano-Nelson Model as a Minimal Realization

The simplest model for which the eigenstate and eigenvalue spectrum differ is the leftward-hopping Hatano-Nelson model. This model is both a physical system and a mathematical archetype: an order-L Jordan block exceptional point. While its eigenvalue spectrum is a single point (σeig(HL) = 0 for all L), it supports almost eigenstates for any E < 1, as shown by the scaling of the normalized state error which grows linearly with system size. In the thermodynamic limit (L → ∞), σstates(H∞) therefore contains the entire unit disk E < 1, which is much larger than the corresponding thermodynamic limit of σeig(HL) L→∞ −−−−→ 0.

Relationship with Alternative Spectrum Notions

The eigenstate spectrum can be directly connected to two alternative notions of spectrum: the singular value spectrum, derived from H = USV †, and the notion of spectrum used in mathematics literature defined as the set of energies E for which (H − E) is not invertible. These three approaches are in fact equivalent, meaning a complex energy E lies outside the spectrum if and only if there exists a spectral gap ∆ which can be defined in three equivalent ways. The eigenstate spectrum offers much stronger theoretical stability guarantees because the eigenstate gap can be expressed as an Hermitian gap of (H − E)†(H − E) or (H − E)(H − E)†, which is stable, in stark contrast to the unstable eigenvalue gap Dist(E, σeig(H)).

Topological Implications and Hidden Exceptional Points

Using topological arguments based on the index theorem, a non zero winding number W(Hbulk − E) ≠ 0 forces a proliferation of generalized eigenmodes, which verifies that H - E acts as a Jordan block of size n like in the minimal model. This leads to the conclusion that whenever H − E has nontrivial winding, the Hamiltonian must host exceptional points of macroscopic order which are typically hidden from the eigenvalue spectrum. The non-Hermitian skin effect is directly related to this proliferation, stating that a macroscopic number of boundary states exist as long as the index of the Hamiltonian is nontrivial.

Case Study: Non-Hermitian SSH Model

In the non-Hermitian SSH model, the eigenstate spectrum differs significantly from its eigenvalues, showing that while a bulk gap closes for values t1 ± γ/2 = t2, this does not correspond to the appearance or disappearance of a vanishing eigenvalue. The analysis demonstrates that the closing of the bulk gap corresponds instead to the appearance or disappearance of an exponentially precise edge eigenstate but not to a vanishing eigenvalue. This discrepancy is interpreted as the eigenvalue spectrum’s inability to detect eigenstates, which is already shown in the simpler Hatano-Nelson model. The non-Hermitian SSH model hosts infinite families of states ψk⟩ forming quasi-Jordan blocks with H ψk⟩ = ψk−1>, illustrating the general proximity of non-Hermitian Hamiltonians to macroscopic-order Jordan blocks. The universal presence of spectral singularities demands a paradigm shift towards state-centric analysis for non-Hermitian topology.

Conclusion

The instabilities inherent to non-Hermitian systems are simply a consequence of the eigenvalue spectrum missing (nearly all) eigenstates of operators in the thermodynamic limit. The eigenstate spectrum is generally stable and well-defined, reproducing well-known results due to its connection to singular value analysis. The instabilities are understood as a consequence of the proximity of hidden macroscopic Jordan blocks that proliferate in the thermodynamic limit. The universal presence of spectral singularities demands a paradigm shift: state-centric analysis is indispensable for non-Hermitian topology, many-body physics, and beyond.

Appendix B on Boundary Conditions

For any local non-Hermitian Hamiltonian, the eigenstate spectrum share the property of the Hermitian spectrum that the open boundary condition spectrum includes, in the thermodynamic limit, the bulk bands of the periodic boundary condition spectrum. A state ψ′⟩ can be constructed from a Bloch state ψ⟩ by applying a smooth envelope function g(x/L) that vanishes at the boundaries, ensuring that its amplitude is negligible near the boundary which suppresses boundary-related errors by a factor O(1/L). This proves that all periodic bulk eigenstates have open-boundary counterparts in the eigenstate spectrum for large enough systems.

Appendix A on Spectrum Equivalence

The appendix demonstrates that the eigenstate approach, the singular value spectrum, and the bounded inverse spectrum provide equivalent characterizations of a spectral gap. Specifically, (A1) Pseudospectrum criteria is equivalent to (A2) Singular value criteria because squaring the inequality in (A1) yields an inequality for A†A which implies σmin(A†A). Furthermore, the equivalence between (A1) and (A3) shows that if the bounded inverse criterion holds, it implies the pseudospectrum criterion.

Figure Interpretation

Figure 2a shows that for the non-Hermitian SSH model, eigenstates with zero energy are present despite the absence of topological modes in the eigenvalue spectrum. The eigenvalue spectrum is deficient in the red regions with either a gap while there is one mode at zero energy or a single zero eigenvalue where there are two modes at zero energy. Singular value has however no problem detecting those modes at zero energy. The eigenstate spectrum is generally stable and well-defined, and reproduce well-known results due to its connection to singular value analysis. The instabilities are simply a consequence of the eigenvalue spectrum missing (nearly all) eigenstates of operators in the thermodynamic limit. The instabilities can also be understood as a consequence of the proximity of hidden macroscopic Jordan blocks that proliferate in the thermodynamic limit.

1. textbfState-Centric Topological Characterization of Non-Hermitian Systems: The Improved AI System Can perform robust topological phase classification by analyzing eigenstate stability rather than eigenvalue locations, as the eigenstate spectrum offers much stronger theoretical stability guarantees.

2. textbfHandling Spectral Singularities and Exceptional Points: The improved system can detect hidden macroscopic exceptional points that control the spectral behavior across all skin-effect systems by leveraging the connection between topological winding numbers and the proliferation of generalized eigenstates, which are otherwise hidden from the eigenvalue spectrum.

3. textbf Enhanced Numerical Stability for Large Systems: The AI system will employ eigenstate analysis to bypass the instability of numerical diagonalization near exceptional points, as this approach is preferred over eigenvalue-based diagnostics when rounding errors due to the floating point precision play the role of a random perturbation.

4. textbf Robust Bulk-Edge Correspondence Verification: The system can reliably distinguish between bulk and boundary phenomena in open systems by analyzing states, utilizing the proof that all periodic bulk eigenstates have open-boundary counterparts in the eigenstate spectrum for large enough systems, contrasting this with the failure of eigenvalue correspondence due to the skin effect.

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 extrapolating finite-size eigenvalues can miss important asymptotic eigenstates, with the unidirectional Hatano-Nelson model serving as a minimal realization of this phenomenon. Through exact analytical solutions, we show that this model contains not only hidden asymptotic modes but also quasi-Jordan structures that become macroscopic in the thermodynamic limit and appear more generally in systems with nontrivial bulk winding. Our framework explains how the apparent failure of the bulk-edge correspondence in models such as the non-Hermitian SSH chain can reflect the failure of finite-size eigenvalue extrapolation to detect asymptotic edge states in systems with a skin effect. These results clarify the limitations of using finite-size eigenvalues alone to infer thermodynamic-limit topology.