Higher-Order Topological Magnons

Updated 1 August 2025

Higher-order topological magnons are bosonic spin excitations that exhibit robust, localized corner or hinge modes protected by crystalline and magnetic symmetries.
Theoretical models use decorated honeycomb, kagome, and stacked-layer Heisenberg frameworks with anisotropies and Dzyaloshinskii–Moriya interactions to induce these novel boundary states.
These insights enable practical design of low-dissipation magnonic circuits and quantum devices, with tunability via edge termination, lattice distortions, and external controls.

Higher-order topological magnons are bosonic spin excitations in magnetic insulators exhibiting robust boundary-localized modes of codimension greater than one (i.e., corner or hinge states), in analogy to higher-order topological insulators and superconductors. Unlike first-order topological magnons, whose haLLMark is the presence of gapless edge or surface modes protected by bulk invariants (e.g., Chern numbers or $\mathbb{Z}_2$ indices), higher-order phases feature localized excitations at corners (in 2D) or hinges (in 3D), protected by crystalline or magnetic symmetries and characterized by more sophisticated invariants such as bulk polarizations, mirror winding numbers, or nested Wilson loops. These phases provide new platforms for the design of robust, low-dissipation magnonic circuits and quantum devices, with unique responses to symmetry, disorder, and sample termination.

1. Theoretical Framework and Model Systems

Higher-order topological magnon phases have been theoretically established in a variety of two- and three-dimensional lattice models, frequently built from decorated honeycomb, kagome, or stacked-layer systems (Mook et al., 2020, Hua et al., 2022, Bhowmik et al., 2023, Li et al., 2022, Guo et al., 30 Jul 2025). The core Hamiltonians are typically Heisenberg spin models with competing exchanges, single-ion anisotropies, and in relevant cases, Dzyaloshinskii–Moriya interactions. For instance, the decorated honeycomb model (Bhowmik et al., 2023) employs a six-site unit cell with intra- and inter-unit cell exchanges ( $J_0$ , $J_1$ ):

$\mathcal{H} = -\sum_{\langle ij\rangle} J_{ij} \vec S_i \cdot \vec S_j - \beta \sum_i (S_i^z)^2$

Following Holstein–Primakoff linearization, the magnon Hamiltonian naturally incorporates site-dependent onsite terms, a crucial distinction from its fermionic analogues and a key determinant of boundary mode properties (Bhowmik et al., 2023). In stacked honeycomb and AA-type stacked altermagnetic systems, interlayer antiferromagnetic coupling introduces further degrees of hierarchy, promoting the emergence of hinge rather than only corner modes (Mook et al., 2020, Guo et al., 30 Jul 2025).

Symmetry plays a central role. Time-reversal, inversion, and rotational (e.g., $C_6$ or $C_2$ ) symmetries frequently underlie the quantization of polarization or the protection of localized boundary states. Symmetry-indicator topological invariants are constructed from crystalline eigenvalues at high-symmetry points in reciprocal space, e.g., for $C_6$ symmetry:

$\chi^{(6)}= \left([M_1^{(2)}], [K_1^{(3)}]\right)$

where $[M_1^{(2)}]$ counts specific rotational eigenvalues at the M and $\Gamma$ points; $\chi^{(6)}=2$ signals a second-order topological phase (Bhowmik et al., 2023).

2. Classification and Hierarchy of Topological Modes

In higher-order phases, the conventional bulk-boundary correspondence is extended to a dimensional hierarchy:

2D SOTMI: Gapped bulk and edges, with in-gap magnon modes localized at the corners (zero-dimensional).
3D realizations: Propagating chiral magnon hinge states appear along the intersection of two facets, protected by a spectral gap in both the bulk and surface (Mook et al., 2020, Li et al., 2022, Guo et al., 30 Jul 2025).

In altermagnetic honeycomb lattices, AA-type stacking with AFM interlayer coupling leads to anisotropic surface states and robust hinge modes (Guo et al., 30 Jul 2025). The progression—localized 2D corner modes transitioning to propagating 1D hinge modes in 3D stacking—mirrors the general HOTI concept familiar in electron systems (Saha et al., 2021).

3. Role of Symmetry, Edge Termination, and Disorder

The spatial profile and existence of magnonic corner or hinge states are finely sensitive to boundary terminations and sample symmetry:

Edge and Corner Sensitivity: In the decorated honeycomb SOTMI, two edge terminations delineate “intrinsic” and “pseudo” SOTMI phases. For Type I (whole cell preserved), robust intrinsic corner modes arise; for Type II (incomplete cells at the edge), modes are only partially localized and may hybridize with bulk states (Bhowmik et al., 2023).
Symmetry Indicators: The presence or absence of topological corner modes correlates with integer-valued symmetry indicators, e.g., a nonzero $[M_1^{(2)}]$ (Bhowmik et al., 2023), bulk ( ${C}_2$ -graded) polarization (Li et al., 2022), or mirror winding number (Hua et al., 2022).
Disorder Robustness: Random out-of-plane exchange anisotropy disorder acts as a site-dependent potential (through variations in effective on-site magnon energy). Intrinsic SOTMI corner modes remain robust under moderate disorder provided inter-cell couplings are strong; in pseudo phases, corner states can be destroyed for sufficient disorder strength (Bhowmik et al., 2023). Hinge magnon localization length is inversely related to the surface gap, setting a limit for design tolerances in engineered devices (Mook et al., 2020).

4. Kekulé and Anti-Kekulé Distortions, Engineering of Topological Phases

Tuning exchange interactions across the lattice allows one to switch between SOTMI and trivial magnonic phases:

Kekulé Distortion: For $J_0 < J_1$ , inter-cell coupling dominates, opening a bulk magnon gap $\Delta = 2S|J_0 - J_1|$ and yielding a second-order topological phase characterized by robust corner modes (Bhowmik et al., 2023).
Anti-Kekulé Distortion: For $J_0 > J_1$ , the system becomes topologically trivial. Boundary modes degenerates into Tamm-Shockley type bond-localized modes, which are not protected by topology but depend on edge termination and can be considered accidental (Bhowmik et al., 2023).

This offers a highly flexible mechanism for engineering topological magnonic devices, as exchange couplings can be tuned by external strain, chemical substitution, or nanofabrication, enabling on-demand control of the topological phase.

5. Distinctions and Comparisons with Fermionic Analogues

Bosonic magnonic SOTMIs exhibit several unique features compared to fermionic HOTIs:

Onsite Energy Effects: Magnonic Hamiltonians include site-dependent onsite energies arising from both exchange and anisotropy, linked to coordination number and bond environment. This leads to a “bond-dependent onsite energy difference” not present in standard fermionic tight-binding models, causing shifts of in-gap states away from exact midgap and affecting their localization (Bhowmik et al., 2023).
Particle–Hole Asymmetry: Magnon in-gap modes are not necessarily centered in the spectral gap, and generic mid-gap degeneracies found in fermionic models due to particle–hole symmetry are absent.
Topology Detection: Symmetry indicator invariants and polarization measures adapt to the bosonic case but must account for altered commutation relations and possible absence of fermionic time-reversal analogues.

6. Prospective Applications and Experimental Outlook

Higher-order topological magnons provide low-dissipation, robust channels for magnonic information transfer, supporting energy-efficient logic and quantum information applications:

Device Prospects:
- Robust magnonic circuits based on corner or hinge modes, immune to backscattering and tolerant of defect perturbations (Mook et al., 2020, Guo et al., 30 Jul 2025).
- Vertical (3D) integration via hinge modes, allowing high-density design.
- Tunability via stacking, edge engineering, and external fields or strain (Li et al., 2022, Bhowmik et al., 2023).
Detection and Characterization: Corner and hinge states can potentially be probed by highly resolved techniques such as nitrogen-vacancy center magnetometry, inelastic neutron scattering, or near-field Brillouin light scattering.
Material Platforms: Twisted bilayer honeycomb magnets (notably twisted bilayer CrI $_3$ ) offer practical realizations where SOTMI phases and their topological invariants (mirror winding, bulk polarization) are accessible using current thin-film technologies (Hua et al., 2022, Kim et al., 2022). Engineering interlayer exchange (FM vs AFM) and stacking orientation (AA-type) is critical.

7. Future Directions and Open Challenges

Higher-order topology in magnonic systems is undergoing rapid development. Open areas of research include:

Exploration of new lattice geometries and stacking patterns to stabilize more exotic higher-order phases (including third-order or “network” higher-order magnonics) (Kim et al., 2022).
Investigation of disorder-induced transitions, thermal fluctuations, and interactions beyond linear spin-wave theory.
Systematic search for material candidates, aided by symmetry-based indicators and group-theoretical filters, to expand the database of real higher-order topological magnonic insulators (Karaki et al., 2022).
Design of reconfigurable or programmable magnonic devices, exploiting the sensitivity of higher-order topological modes to edge and boundary manipulation (Guo et al., 30 Jul 2025).

In summary, higher-order topological magnons constitute a distinctly bosonic manifestation of bulk-boundary correspondence, enabling robust and highly tunable corner and hinge magnon states. Their realization leverages subtle interplay of lattice geometry, exchange anisotropy, edge termination, and crystalline symmetry, opening compelling pathways for both fundamental exploration and functional magnonic device engineering.