Non-Bloch band theory of boundary-controlled magnon edge modes in an antiferromagnetic chain

Abstract: We define a winding number within the Non-Bloch band theory framework that captures the emergence of magnon edge modes in a one-dimensional antiferromagnetic spin chain, even when the conventional Bloch winding number is trivial. Within linear spin-wave theory, magnon excitations are governed by a non-Hermitian dynamic matrix, despite the underlying Hamiltonian being Hermitian. The symmetry classification of this matrix yields a trivial bulk invariant, however, finite systems exhibit boundary-localized modes, signaling a breakdown of the conventional bulk-boundary correspondence. We further show that these edge modes can be controlled via boundary perturbations. By tuning the boundary potential, the modes can be driven into or out of the bulk spectrum. To resolve the bulk-boundary mismatch, we develop a non-Bloch framework based on a generalized Brillouin zone and a winding number that correctly predicts the presence of edge states. Our results establish boundary-controlled topological transitions that are experimentally accessible through local Zeeman fields or modified edge anisotropy in antiferromagnetic van der Waals nanostructures.

• The paper introduces a non-Bloch band theory that predicts boundary-induced magnon edge modes in dimerized antiferromagnetic chains.
• It employs a bosonic BdG formulation to reveal a breakdown of conventional bulk-boundary correspondence due to non-Hermiticity.
• Boundary perturbations enable tunable topological transitions, offering practical routes for experimental realizations in magnonics.

Non-Bloch Band Theory and Boundary-Induced Magnon Edge Modes in Antiferromagnetic Chains

Introduction

The work develops a rigorous non-Bloch band theory to explain and quantify the emergence and control of sublattice-polarized magnon edge modes in a one-dimensional (1D) dimerized antiferromagnetic chain with Dzyaloshinskii–Moriya interaction (DMI) and easy-axis anisotropy. The central finding is a robust breakdown of conventional bulk–boundary correspondence in bosonic Bogoliubov-de Gennes (BdG) systems, due to the non-Hermiticity of the magnon dynamic matrix even when the underlying Hamiltonian is strictly Hermitian. This leads to the existence of boundary-localized modes not predicted by typical bulk topological invariants. The theory is substantiated by analytic and numerical investigations and provides routes for experimental realization and control of edge magnon modes via boundary engineering in van der Waals (vdW) antiferromagnetic nanostructures.

Model and Bosonic BdG Formulation

The system under consideration is a 1D antiferromagnetic spin chain with alternating nearest-neighbor Heisenberg exchanges $J_1$, $J_2$, next-nearest-neighbor DMI of alternating sign, and onsite easy-axis anisotropy $\kappa$. The model incorporates inversion symmetry about bond centers, restricting certain DMI terms. Linear spin-wave theory, performed around a collinear Néel state, leads to a bosonic BdG Hamiltonian with a dynamic matrix $D_{\text{BdG}}(k)$ that is generically non-Hermitian but pseudo-Hermitian, with real or conjugate-pair eigenvalues. For $\kappa > 0$, the spectrum exhibits a gap and stability up to a critical DMI $D_c$ where the magnon gap closes.

The dynamic matrix possesses particle-hole and—in the absence of DMI—time-reversal symmetry, and in symmetry classification corresponds to class CI$(\eta_{+-})$, changing to C$(\eta_{-})$ for finite DMI. Importantly, these classes with real line gaps admit no nontrivial bulk topological invariant in 1D [25].

Edge Modes in Finite Chains: Failure of Conventional Bulk Invariants

Open boundary conditions (OBC) reveal edge-localized magnon states, evidenced by strong inverse participation ratios (IPR) and sublattice polarization: the edge mode on one end is supported on the A sublattice in particle component and B in hole component, and vice versa on the other edge. Unlike the Su-Schrieffer-Heeger (SSH) model, these are finite-energy edge modes arising from the boundary-induced modification in the local spin environment, not protected by chiral or sublattice symmetries.

The conventional winding number computed from the Bloch Hamiltonian is shown to be trivial ($w = 0$), as is the relevant $\mathbb{Z}_2$ invariant, precisely as required by the non-Hermitian symmetry classification of the bosonic BdG model. Nevertheless, edge modes robustly persist over an extensive parameter regime, manifesting a clear breakdown of standard bulk–boundary correspondence [26–29].

Boundary Control and Tunable Topological Transitions

Motivated by the physical nature of edge terminations as effective onsite potential shifts, the study introduces a boundary perturbation (on-site energy $J_2$0) at the chain ends. This boundary tuning, implementable by local Zeeman fields or edge-selective anisotropy modifications, drives edge states into or out of the bulk continuum as $J_2$1 is varied. Numerical and analytic results reveal:

• For varying $J_2$2, edge modes merge into the bulk at critical potentials, then re-emerge from the opposite side of the spectrum as $J_2$3 increases.
• The behavior persists for various dimerization strengths $J_2$4 and small DMI.

These results provide an experimentally accessible control handle for topological transitions of magnon edge states in realistic antiferromagnetic nanostructures.

Non-Bloch Band Theory: Generalized Brillouin Zone and Winding Number

To resolve the bulk–boundary correspondence failure, the study constructs the generalized Brillouin zone (GBZ) by finding complex momenta that solve the BdG equations under OBC. The analytic treatment (explicit for $J_2$5) confirms that:

• The non-Bloch (GBZ-based) winding number is quantized to 2 when edge modes are present, and zero otherwise.
• The transitions of the non-Bloch invariant precisely match the emergent/disappearing edge modes induced by boundary tuning.
• For $J_2$6, the GBZ construction and corresponding non-Bloch invariants remain accessible but require numerical solution of higher order in the characteristic equation.

The non-Bloch framework thus restores a generalized bulk–boundary correspondence, accurately predicting the existence of edge-localized magnon modes in the finite chain.

Implications and Perspectives

Practical Implications

This work demonstrates that topological magnonics in insulating antiferromagnets is far richer than allowed by standard Hermitian band theory. The existence and boundary control of edge magnon modes, governed by non-Bloch topology, offers avenues for:

• Nonlocal signal routing and logic architectures in magnetic van der Waals heterostructures.
• Probing and manipulating edge modes with local fields, strain, or spatially resolved magnetometry (e.g., nitrogen-vacancy centers).
• Spectroscopic detection with Brillouin light scattering or magnon transport, leveraging strong spatial localization and sublattice polarization.

Theoretical Implications and Extensions

The identification of boundary-controlled nontrivial topological modes in bosonic BdG chains with trivial bulk invariants underlines the necessity of non-Bloch band theory in non-Hermitian/lossless systems. The construction is generic and extendable to:

• Ferromagnetic chains with quadratic spin-wave Hamiltonians exhibiting anomalous terms [48].
• Generalizations with more complex symmetry and interaction profiles, and higher-dimensional lattices where higher-order boundary phenomena are expected.

The interplay between symmetry class, real vs. complex line gaps, and GBZ topology remains a fertile area for analysis, especially for the classification of edge and corner magnon modes in higher dimensions.

Conclusion

This work establishes the relevance and power of non-Bloch band theory for topological magnon edge modes in dimerized antiferromagnetic chains. The introduction of a boundary-tunable non-Bloch winding invariant enables the prediction and control of edge modes invisible to bulk Bloch invariants, fundamentally extending the understanding of topological band theory for bosonic systems. These results have direct implications for experimentally accessible topological magnonics and suggest a wide landscape of observable boundary phenomena in vdW magnetic nanostructures and beyond.

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References:

• "Non-Bloch band theory of boundary-controlled magnon edge modes in an antiferromagnetic chain" (2606.09267)
• K. Kawabata et al., Phys. Rev. X 9, 041015 (2019) [25]