Second-Order Topological Magnon Insulator

Updated 1 August 2025

SOTMIs are bosonic phases that host symmetry-protected in-gap magnon states localized at corners (2D) or hinges (3D).
They rely on advanced lattice constructions and symmetry engineering, such as bond alternation and inversion or rotation invariants, to transition between trivial and topological regimes.
Robust to disorder and externally tunable, SOTMIs promise low-dissipation spin transport applications and novel magnon-based logic devices.

A second-order topological magnon insulator (SOTMI) is a bosonic quantum phase of matter where topological protection manifests not only at boundaries of codimension one (edges in 2D, surfaces in 3D) but at higher codimension—namely, at zero-dimensional corners (in 2D) or one-dimensional hinges (in 3D). Unlike first-order topological magnon insulators that support chiral or helical edge magnon modes, SOTMIs exhibit gapped bulk and edge spectra but feature robust, symmetry-protected magnonic corner or hinge bound states, offering new paradigms for manipulating and transmitting spin information in insulating magnets.

1. Theoretical Framework and Models

The fundamental construction of SOTMIs relies on extending concepts from electronic higher-order topological insulators to bosonic magnon systems. A generic spin Hamiltonian on two- or three-dimensional lattices (e.g., decorated honeycomb, breathing square, altermagnetic honeycomb, or metal–organic framework lattices) is mapped to quadratic bosonic magnon (spin wave) Hamiltonians via Holstein–Primakoff or similar transformations. These yield tight-binding–style models where bond-dependent exchange interactions, single-ion anisotropy, and further symmetry-breaking perturbations enable topologically distinct phases.

Key models and methodologies include:

Bond-Alternating Lattices: Decorated honeycomb or square lattices with intra- and intercell couplings (J₀, J₁) allow for "kekulé" and "anti-kekulé" distortions, analogously to higher-dimensional SSH-type physics (Bhowmik et al., 2023).
BdG Formalism: For systems such as altermagnets or layered van der Waals structures, the bosonic Bogoliubov–de Gennes (BdG) formalism provides the natural home for symmetry and band topology analyses, including calculation of Wannier centers and polarization invariants (Guo et al., 30 Jul 2025).
Symmetry Engineering: Magnetic crystalline symmetries—rotation (C₄, C₆), inversion (P), combined time-reversal (T), and mirror operations—play a vital role in protecting SOTMI phases. Symmetry indicator invariants built from rotation and inversion eigenvalues at high-symmetry momenta diagnose the presence of SOTMI (Zhan et al., 2023, Bhowmik et al., 2023, Wang et al., 21 Feb 2025).
Topological Invariants: Quantities such as Z₂, Z₄, or real Chern numbers, and rotation topological invariants (e.g., for C₃ or C₂ symmetry), determine the existence and quantization of edge/corner/hinge modes (Zhan et al., 2023, Wang et al., 21 Feb 2025).

2. Distinction from First-Order Topological Magnon Insulators

Conventional topological magnon insulators rely on symmetry-breaking interactions (Dzyaloshinskii–Moriya, spin–orbit, or artificial gauge fields) to open gaps in magnon bands, yielding chiral or helical edge modes described by nontrivial Berry curvature and Chern numbers (Zhang et al., 2012, Nakata et al., 2017). SOTMIs, by contrast:

Feature both bulk and edge gaps; the one-dimensional edge states are themselves gapped by boundary (mass) terms.
Localized boundary modes appear at corners (2D) or hinges (3D), protected by spatial symmetries (e.g., C₃, C₄, C₆ rotation) or crystalline inversion, rather than nontrivial Berry curvature.
Corner or hinge magnon states are detected as in-gap modes within the otherwise gapped system. For example, in the breathing square lattice, the Chern number remains zero, but a quantized ℤ₄ Berry phase signals a HOTI phase where corner magnon states appear (Li et al., 2020).

3. Protogenic Lattice Constructions and Symmetry Protection

The generation of SOTMI phases is facilitated by characteristic bonding geometries and symmetry-breaking perturbations:

Decorated Honeycomb Lattices: By tuning the ratio J₀/J₁, one induces a transition between trivial and topological phases, with the kekulé phase (J₀ < J₁) displaying a bulk gap with embedded bosonic corner modes protected by a nonzero C₆ rotation symmetry indicator (Bhowmik et al., 2023).
Breathing Lattices: In breathing square lattices of magnetic vortices, varying the ratio of alternating bond lengths (d₂/d₁) across a critical value triggers a transition to a HOTI/SOTMI phase, marked by the appearance of three robust corner states (Li et al., 2020).
Collinear Altermagnets: AA-stacked honeycomb lattices with bond alternation and AFM interlayer coupling yield second-order phases with corner or hinge magnon modes, as certified by the winding of bulk polarization determined through the Wilson loop of Wannier centers (Guo et al., 30 Jul 2025).
MOF-based SOTMIs: 2D metal-organic frameworks such as Cr(pyz)₂ and Cr(2-pyzol)₂ exhibit fierrimagnetic SOTI phases, with robust corner states in either spin channel protected by PT symmetry and specific rotation invariants (Wang et al., 21 Feb 2025, Gong et al., 27 Feb 2025).

These symmetry constraints underlie the quantization of topological invariants:

For C₃ or C₆ symmetric lattices: fractional quantization of corner charge Q_corner = n/3 |e| or n/6 |e|, etc. (Zhan et al., 2023, Bhowmik et al., 2023).
For reflection or inversion-protected SOTMI, "real" Chern numbers and rotation indicators classify the phase (Wang et al., 21 Feb 2025, Gong et al., 27 Feb 2025).

4. Boundary Modes: Corner and Hinge Magnons

A haLLMark of SOTMIs is the existence of robust magnonic boundary states at higher codimension:

In 2D, the corner states are zero-dimensional and reside in the bulk (and edge) gap. These are robust against moderate disorder provided the relevant crystalline symmetry is retained.
In 3D, system termination and stacking (e.g., honeycomb magnetic slabs with alternating interlayer coupling) engineer chiral hinge magnon modes that propagate unidirectionally along sample hinges, immune to backscattering—even in the presence of lattice defects (Mook et al., 2020, Li et al., 2022, Guo et al., 30 Jul 2025).
In models with bond-dependent onsite energies (a structural feature of magnon systems), the existence and localization profile of in-gap states are strongly influenced by the termination geometry; this may yield "intrinsic" SOTMI phases (full unit cell termination) or "pseudo"-SOTMI (partial termination), in contrast to strictly fermionic models (Bhowmik et al., 2023).
Trivial "anti-kekulé" phases or edge modifications can produce Tamm/Shockley-type bond-localized modes, though these lack topological protection and show sensitivity to disorder or boundary perturbations.

5. Robustness and Tuning Mechanisms

SOTMI phases offer a range of tunability and remarkable robustness:

Disorder: SOTMI corner or hinge states persist under moderate random exchange, out-of-plane anisotropy, or small symmetry-breaking deformations, as numerically demonstrated in decorated honeycomb and MOF-based models (Bhowmik et al., 2023, Wang et al., 21 Feb 2025).
External Perturbations: Strain, ligand rotations, electric fields, or defect engineering do not close the bulk/edge gaps or remove the protected 0D or 1D magnon modes in magnetic MOF platforms (Wang et al., 21 Feb 2025, Gong et al., 27 Feb 2025).
Ferroelectric Chirality Coupling: In 2D MOFs with broken inversion symmetry, the direction and spin polarization of corner states can be continuously tuned by switching ferroelectric chirality, simultaneously rotating the corner localization and inverting the spin channel—a first demonstration of such electrical/structural control in a SOTI/SOTMI platform (Gong et al., 27 Feb 2025).
Symmetry Protection and Higher-Order Invariants: Provided crystalline or magnetic rotation/inversion symmetries are not lost, the higher-order invariant (e.g., half-quantized bulk polarization or Z₄ Berry phase) pins the existence of the corner/hinge state regardless of weak perturbations (Li et al., 2020, Li et al., 2022, Guo et al., 30 Jul 2025).

6. Experimental Realizations and Applications

Recent theoretical and experimental advances enable the exploration and application of SOTMI phases:

Material Platforms: Proposals and predictions span a wide range of systems: 2D van der Waals magnets with AFM interlayer coupling (Li et al., 2022), twisted bilayer honeycomb magnets with FM interlayer coupling (Hua et al., 2022), (MnBi₂Te₄)(Bi₂Te₃)ₘ heterostructures with AFM order (Zhan et al., 2023), MOF-based frameworks with ligand substitution (Wang et al., 21 Feb 2025, Gong et al., 27 Feb 2025), and synthetic magnonic crystals and artificial vortex arrays (Li et al., 2020).
Detection: The robust corner/hinge modes can be probed via inelastic light or neutron scattering (for magnonic excitations), spin-resolved tunneling, or microwave spectroscopy. Devices exploiting these features include vortex-based imaging elements or magnonic logic circuits (Li et al., 2020, Wang et al., 21 Feb 2025).
Spintronic Device Potential: Electrically or structurally tunable SOTMI platforms with low-dissipation, symmetry-protected transport, and high resilience to disorder offer avenues for magnon-based qubits, neuromorphic systems, energy-efficient logic, or programmable magnon waveguides (Gong et al., 27 Feb 2025, Guo et al., 30 Jul 2025).

7. Unique Bosonic Features and Future Directions

SOTMIs exhibit features and challenges not present in fermionic higher-order topological systems:

Magnonic Onsite Energies: The intrinsic bond-dependent onsite energy—arising from the spin–magnon mapping—shifts the energies and spatial profiles of the in-gap states, affects disorder response, and distinguishes bosonic SOTMIs from fermionic HOTIs (Bhowmik et al., 2023).
Dimensional Hierarchy: SOTMIs manifest a dimensional hierarchy: transitions from bulk to edge to hinge/corner as dimensionality and symmetry are engineered. Adiabatic evolution of Wannier centers and bulk polarization via Wilson loop analysis provide practical diagnostics of SOTMI phases (Guo et al., 30 Jul 2025).
Hierarchy of Topological Protection: The combination of crystalline, magnetic, and (in some cases) ferroelectric symmetries enables multiple, overlapping mechanisms of topological protection—with prospects for multifunctional, dynamically reconfigurable devices at the intersection of magnonics, spintronics, and topological quantum information science (Gong et al., 27 Feb 2025, Guo et al., 30 Jul 2025).

Summary Table: Core Characteristics of SOTMIs in Contemporary Models

Feature	SOTMI Realization(s)	Topological Indicator(s)
Bulk and edge gapped	Decorated/stacked honeycomb, MOF, vortices	Z₂/Z₄ index; real Chern; rotation
Corner/hinge magnon	Kekulé, breathing lattice, altermagnetics	Berry/Z₄ phase, inversion, parity
Robust to disorder	All symmetry-protected phases	Symmetry indicator nonvanishing
Tunability	Ferroelectric chirality, stacking, strain	Direction, spin, frequency domains

Emerging SOTMI platforms are defined by their higher-order boundary-bound bosonic modes, nontrivial symmetry indicators, and the potential for robust, energy-efficient quantum information transport and processing. Continuing advances in magnonic crystal synthesis, layered 2D/3D materials, and synthetic quantum systems are expected to further broaden exploration and application of SOTMI physics.