Parity-induced generalized Brillouin zone without non-Hermitian skin effect

Abstract: Acute spectral sensitivity to boundary conditions and the formation of a generalized Brillouin zone associated with complex quasimomenta are features frequently attributed to systems with non-trivial non-Hermitian topology, showcasing the non-Hermitian skin effect. We show that, away from the thermodynamic limit, these features themselves are not uniquely tied to this phenomenon; they can similarly arise as parity-induced even-odd effects in non-Hermitian systems without skin effect. Despite an underlying generalized Brillouin zone description, wavefunctions remain delocalized. In addition, the effect can arise in skin-effect models as entirely separate distinguishable feature

• The paper reveals that parity-induced finite-size effects generate a complex generalized Brillouin zone without triggering the non-Hermitian skin effect.
• It employs an extended Altland-Zirnbauer classification and numerical simulations on the SSH* model to confirm delocalized bulk eigenstates despite boundary-sensitive spectra.
• The findings refine non-Hermitian topological diagnostics and suggest experimental parity detection in synthetic lattice devices.

Parity-Induced Generalized Brillouin Zone in Non-Hermitian Lattices Without Skin Effect

Overview

This work rigorously analyzes boundary-condition sensitivity and complex quasimomentum structure in finite non-Hermitian lattices, with a focus on distinguishing features traditionally associated with the non-Hermitian skin effect (NHSE) from phenomena that arise purely due to parity-dependent (even-odd) finite-size effects. The author demonstrates that a nontrivial, complex-valued generalized Brillouin zone (GBZ)—typically linked to the presence of the NHSE—can be realized without localization of bulk states, in a regime where the NHSE is absent. This undermines the presumed necessity of the skin effect for GBZ-induced anomalous spectra and emphasizes the subtle role of symmetry and chain parity in finite non-Hermitian models.

Background and Motivation

In non-Hermitian systems, spectral sensitivity to boundary conditions and the emergence of the GBZ are widely understood as signatures of the NHSE, wherein bulk states accumulate at physical boundaries due to asymmetric (nonreciprocal) couplings or gain/loss. Recent advances in non-Bloch band theory have clarified how these features are rooted in complex spectral topology and symmetry class constraints, mediated by the Altland-Zirnbauer ($AZ$ and $AZ^\dagger$) classifications.

However, not all non-Hermitian systems manifest the NHSE. For instance, PT-symmetric and pseudo-Hermitian extensions of the SSH model exhibit delocalized bulk states and a direct correspondence between open boundary condition (OBC) and periodic spectra under unbroken symmetry. Theoretical understanding has so far tied the emergence of a GBZ away from the real axis—and corresponding deviations from the Bloch band spectrum—to the skin effect per se.

Model Construction and Symmetry Analysis

The focal model, termed the SSH*, is a non-Hermitian bipartite tight-binding chain with alternating complex-conjugate bond strengths:

$$H_{\mathrm{SSH}*} = \sum_n [-t + (-1)^n i\delta]\,(c_n^\dagger c_{n+1} + c_{n+1}^\dagger c_n)$$

Unlike the classic SSH model, the dimerization is purely imaginary ($g \to i\delta$), and the symmetry analysis under the extended Altland-Zirnbauer scheme is essential. For this Hamiltonian:

• Both time-reversal and TRS$^\dagger$ symmetries are present at odd chain lengths, placing the system in the BDI$^\dagger$ class, precluding NHSE.
• Even chain lengths break one of these (TRS), reducing the symmetry class to AI$^\dagger$, which also does not host conventional NHSE.

A significant outcome is that despite the nontrivial (complex) spectrum under OBC at finite (even) sizes, the system's symmetry precludes macroscopic localization of the eigenstates. This is rigorously confirmed both analytically and numerically.

Parity-Dependent Boundary Sensitivity and the GBZ

A key result is the emergence of strong parity-induced spectral behavior:

• Odd-length chains: Both OBC and periodic spectra sample real or imaginary axes, with delocalized eigenstates and conventional Brillouin zone quasimomenta. The system's boundary conditions do not induce any anomalous spectral feature.
• Even-length chains: The system exhibits spectral arcs in the complex energy plane under OBC, which deviate substantially from the periodic spectrum. Critically, these energies are reproduced by evaluating the Bloch Hamiltonian on a set of $N$ complex quasimomenta forming a GBZ that is distinct from the real Brillouin zone (as prescribed via transcendental equations analogous to those in NHSE models).

Despite this, the eigenfunctions remain delocalized and (skew-)symmetric across the chain, as revealed by symmetry analysis and scaling of the (inverse) participation ratio, and are not exponentially localized as in prototypical skin-effect models.

Numerical Evidence

The paper provides comprehensive numerical evidence:

• Spectral plots: Energy eigenvalues form boundary-sensitive arcs for even $N$ but remain confined to real or imaginary axes for odd $N$.
• Generalized Brillouin zone reconstructions: Complex quasimomenta governing even-length spectra deviate from the real axis, but this deviation decays as $AZ^\dagger$0.
• Eigenfunction analyses: Participation ratio scaling aligns with delocalized (Bloch-like) states; no enhanced edge accumulation is observed.

Implications and Extensions

Theoretical

This work demonstrates that spectral boundary sensitivity and analytic continuation to a complex GBZ may, in finite non-Hermitian systems, arise purely from parity-induced symmetry breaking rather than the NHSE. Thus, the presence of a nontrivial GBZ does not unambiguously signal boundary localization (skin effect) except in the thermodynamic limit. The findings refine the topological and symmetry-based understanding of non-Hermitian lattice models and stress the importance of finite-size and parity effects.

The paper also clarifies that even in models permitting a bidirectional skin effect (as in certain AI$AZ^\dagger$1-class systems), the SSH* model does not support any non-vanishing TRS$AZ^\dagger$2 winding number, thereby excluding bidirectional skin modes.

Practical

From an implementation standpoint, the SSH* model's coupling structure is experimentally accessible in synthetic lattices using auxiliary site engineering and gain/loss modulation. The study suggests that parity-driven even-odd effects can function as parity indicators—much like conductance oscillations in Hermitian chains—enabling robust parity detection in non-Hermitian devices via chiral probe response.

Future Directions

The results motivate further theoretical investigations into:

• The classification of non-Hermitian even-odd phenomena, extending beyond single-particle tight-binding models.
• The design of multi-terminal devices leveraging nontrivial GBZs for parity-sensitive or chirality-sensitive functionality.
• The impact of interactions and disorder on parity-induced boundary effects in non-Hermitian many-body systems.

Conclusion

The analysis establishes that parity (even-odd) effects in finite, symmetry-protected non-Hermitian chains may yield GBZ structures and pronounced spectral boundary sensitivity even in the absence of bulk-state localization. Distinct from the canonical NHSE, this mechanism demonstrates the rich and intricate landscape of finite-size, symmetry, and topological effects in non-Hermitian quantum systems. The findings refine the diagnostic use of GBZ and spectral sensitivity, emphasizing that these need not imply the presence of the skin effect away from the thermodynamic limit, and illuminate new routes for exploiting parity-induced phenomena in non-Hermitian device engineering.

Reference: "Parity-induced generalized Brillouin zone without non-Hermitian skin effect" (2605.30978)