Extensivity of boundary-state spectral accumulation

Determine whether the number of eigenvalues accumulating in a nontrivial point gap is extensive, so that the associated boundary states contribute nonzero weight to the normalized thermodynamic-limit spectral density.

Background

The paper uses Hermitization to identify boundary zero modes associated with nontrivial point-gap topology. Such modes locate regions where open-boundary eigenvalues may accumulate, but the existence of boundary states alone does not establish that their multiplicity scales with the system volume.

The distinction is important because extensive multiplicity produces a finite contribution to the normalized limiting density of states, whereas boundary states whose multiplicity scales only with boundary area remain visible in the spectrum as a set but disappear from the normalized density. The authors explicitly state that their approach does not resolve this scaling question in general.

References

The present approach alone, however, does not determine whether the number of accumulating levels is extensive. Consequently, topology by itself does not imply that the limiting spectral density is nonzero there, or that $G_{\Omega}\neq G_{\mathrm B}$ throughout this component.

— Universal aspects of bulk density of states in non-Hermitian lattices  (2608.14508 - Pavliuk et al., 14 Aug 2026) in Caption of Fig. 6, Section 3.3, “Universal aspects of the energy level accumulation”

Furthermore, noting that boundary spectral accumulation tied to nontrivial stable point-gap topology is, by the presented analysis, related to the appearance of zero-energy modes in the Hermitian doubled Hamiltonian, it will be interesting to consider whether analogous collapse occurs for points gaps that are nontrivial in refined classifications. Notable examples to consider include higher-order topology protected by crystalline symmetry and weak topology protected by translation symmetry.

— Universal aspects of bulk density of states in non-Hermitian lattices  (2608.14508 - Pavliuk et al., 14 Aug 2026) in Section 4, “Conclusion”