Topological Altermagnonics

• Topological altermagnonics is a field studying collective spin excitations in compensated magnets, where momentum-dependent spin splitting meets nontrivial band topology.
• It integrates mechanisms like Berry curvature, Chern bands, and nodal structures to control magnon transport and thermal Hall responses.
• Experimental platforms such as monolayer AgF₂ and synthetic multilayers validate theoretical models and enable control over edge, surface, and higher-order magnon modes.

Topological altermagnonics denotes the conjunction of altermagnetic magnon phenomenology with nontrivial band topology: momentum-dependent spin splitting of collective spin excitations in compensated magnets, together with Berry-curvature effects, Chern bands, nodal structures, higher-order boundary modes, or related transverse transport. In the present literature, the field is structurally asymmetric. One line of work establishes direct magnonic topology in a concrete altermagnet, most notably monolayer AgF$_2$, where a single ferroelastic polar distortion produces both relativistic altermagnetic electronic bands and topological magnons (González et al., 21 Aug 2025). A broader line develops the symmetry, winding, and boundary-mass mechanisms in electronic, superconducting, and synthetic-magnonic settings that are now being used as design principles for bosonic altermagnetic excitations (Das et al., 2024).

1. Concept, definitions, and classification landscape

Altermagnets are commonly defined as compensated magnetic states with zero net magnetization but momentum-dependent spin splitting, so that Kramers degeneracy is lifted in parts of the Brillouin zone. In the formulation emphasized for graphene-based systems, the essential distinction from ordinary ferromagnets and conventional antiferromagnets is that opposite-spin sectors are related by crystallographic rotations rather than by inversion or translation; as a result, the bands can be Kramers non-degenerate even though the total moment vanishes (Das et al., 2024). Closely related classification work instead takes broken $\mathcal{PT}$ symmetry and fully compensated spin angular momenta as the central definition, and separates strong altermagnets, which are spin split already at zero SOC, from weak altermagnets, which require SOC for spin splitting (Cheong et al., 2024).

The classification problem is not settled into a single universal scheme. One symmetry-based framework extends altermagnetism from the original collinear setting to non-collinear orders with multiple local structural variants, and divides the resulting states into type-I, type-II, and type-III according to whether ferromagnetic-like responses are present intrinsically, appear only under $\mathcal{PT}$-preserving perturbations, or exclude odd-order AHE in equilibrium (Cheong et al., 2024). Another framework classifies M-type, S-type, and A-type altermagnets according to whether $T$ is broken with nonzero net moment, broken with zero net moment, or preserved while $P$ is broken, and then overlays the strong/weak distinction through the number of unbroken orthogonal spin-rotation operations $S_n(r)$ (Cheong et al., 2024).

For altermagnonics, these classificatory differences are consequential because they determine which antiunitary and spin-space symmetries can keep bosonic branches degenerate. The direct magnonic literature and the broader symmetry analyses jointly suggest that broken $\mathcal{PT}$, minimal residual spin-rotation symmetry, and crystalline operations relating opposite sublattices are the key ingredients for momentum-dependent magnon spin splitting and, potentially, for topological magnon bands (Cheong et al., 2024).

2. Symmetry, winding, and band-topological mechanisms

A central theoretical mechanism behind altermagnetic structure is the projection of a local order parameter into a low-energy band basis with nontrivial winding. In monolayer, Bernal bilayer, and rhombohedral trilayer graphene, momentum-independent local spin nematic orders generate $p$-, $d$-, and $f$-wave altermagnets because the low-energy band touchings carry winding numbers $\mathcal{PT}$0, respectively (Das et al., 2024). In the band basis, the local order acquires angular dependence $\mathcal{PT}$1 and $\mathcal{PT}$2, so the sign-changing altermagnetic splitting is inherited from the topology of the band wave functions rather than inserted microscopically by hand. This provides a symmetry-and-band-topology route that is directly suggestive for bosonic collective modes.

SOC enriches this picture by promoting the altermagnetic splitting to a multi-component crystal-harmonic field. A symmetry theory of topological Zeeman splitting shows that, once SOC is present, each spin component acquires its own angular dependence, and the zeros of the splitting become symmetry-protected nodal lines on mirror planes (Fernandes et al., 2023). When such nodal lines intersect the Fermi surface they produce pinch points with single or double type-II Weyl character, and a magnetic field perpendicular to the protecting mirror plane moves the nodal lines until a critical field collapses them, yielding a topological transition from nodal to nodeless Zeeman splitting (Fernandes et al., 2023). Although this is an electronic theory, it isolates a mechanism—mirror-protected zeros of a multi-component altermagnetic field followed by field-driven annihilation—that is naturally portable to nodal altermagnon scenarios.

Exchange-scale topology in a real altermagnet is exemplified by CrSb, where high-resolution ARPES and spin-resolved ARPES reveal momentum-dependent spin splitting up to $\mathcal{PT}$3 eV and a Weyl semimetal phase with both opposite-spin and same-spin Weyl nodes (Li et al., 2024). The same-spin nodes carry both chirality and a magnetic quantum number $\mathcal{PT}$4, and their surface projections are connected by spin-polarized Fermi arcs (Li et al., 2024). This does not establish Weyl magnons in CrSb, but it provides a concrete symmetry template for compensated, zero-net-moment systems in which exchange—not weak SOC—is the dominant topological scale.

A complementary microscopic ingredient is quantum geometry. In a two-band altermagnetic instability theory, the generalized magnetic susceptibility decomposes into mass, velocity, and geometric terms, with the geometric part controlled by the quantum metric $\mathcal{PT}$5 (Heinsdorf, 2024). The instability criterion is $\mathcal{PT}$6, and the paper shows that the quantum metric favors altermagnetism by giving a negative curvature contribution near near-degenerate band regions (Heinsdorf, 2024). This suggests that topological altermagnonics is likely to depend not only on static symmetry classification but also on the quantum-geometric structure of the electronic background from which the ordered spin dynamics emerge.

3. Direct magnonic realizations and synthetic altermagnetic magnons

The most explicit current realization of topological altermagnonics is monolayer AgF$\mathcal{PT}$7. In the centrosymmetric flat $\mathcal{PT}$8 reference phase the magnon branches are degenerate and topologically trivial, but the actual polar ferroelastic $\mathcal{PT}$9 ground state breaks inversion, activates strong DMI, and converts the system into a relativistic altermagnet (González et al., 21 Aug 2025). In the electronic sector, SOC gaps altermagnetically split valence-band crossings and yields isolated bands with $\mathcal{PT}$0 and $\mathcal{PT}$1. In the magnon sector, a DFT+$\mathcal{PT}$2-derived spin Hamiltonian

$\mathcal{PT}$3

produces nondegenerate magnon branches with a full topological gap and Chern numbers $\mathcal{PT}$4 and $\mathcal{PT}$5 (González et al., 21 Aug 2025). The dominant DMI reaches $\mathcal{PT}$6–$\mathcal{PT}$7 meV, the relevant electronic gaps near $\mathcal{PT}$8 and $\mathcal{PT}$9 are up to $T$0 meV, and the clearest experimental fingerprint is a finite magnon thermal Hall conductivity below an estimated $T$1–$T$2 K (González et al., 21 Aug 2025). This is the most literal present meaning of topological altermagnonics: a compensated altermagnet hosting chiral topological magnon bands.

A second, foundational route abandons the crystalline setting and engineers altermagnetic magnons synthetically. In a continuum multilayer of antiferromagnetically coupled ferromagnetic films with alternating in-plane exchange anisotropies, the linearized Landau–Lifshitz equation generates an effective $T$3-wave-like splitting term

$T$4

with nominal nodal directions $T$5 in the exchange-only limit (Gallardo et al., 9 Jun 2026). Long-range dipolar interactions then qualitatively reconstruct the spectrum by lifting the nominal nodal degeneracy, hybridizing opposite-chirality modes, and generating a finite, thickness-dependent wave-vector splitting along those directions (Gallardo et al., 9 Jun 2026). The same work shows parity-dependent multilayer reconstruction, separating surface and bulk altermagnetic excitations in even and odd stacks (Gallardo et al., 9 Jun 2026). No Berry curvature or Chern numbers are computed, but the paper supplies precisely the pre-topological band geometry—anisotropic splitting, avoided crossings, surface–bulk separation—from which topological magnon bands can be engineered.

Direct non-collinear altermagnonic splitting has also been obtained on the maple leaf lattice. Linear spin-wave theory for several compensated non-collinear altermagnetic-type orders reveals momentum-dependent non-relativistic magnon spin splitting without SOC, with the $T$6 order showing the strongest splitting near $T$7 and the canted-$T$8 order near $T$9 and $P$0 (Ghosh et al., 23 Jan 2026). The spectra have six magnon branches and three zero modes in the studied phases, as expected from exhaustive SU(2) spin-rotation symmetry breaking (Ghosh et al., 23 Jan 2026). The work stops short of Berry-curvature or edge-state analysis, but it establishes frustration-driven non-collinear altermagnonics as a realistic precursor to topological extensions.

4. Boundary, higher-order, and hybrid topological routes

One of the most active directions adjacent to topological altermagnonics is the use of altermagnetic symmetry to generate boundary Dirac masses with edge-dependent signs. In a 2D TI/altermagnet heterostructure, the proximity-induced $P$1-wave altermagnetic splitting yields an edge Hamiltonian

$P$2

with boundary mass

$P$3

so adjacent edges of a square can acquire opposite mass signs and bind Jackiw–Rebbi corner states (Li et al., 2024). Because the mass depends on the Néel-vector direction, the corner states can be moved controllably around the sample by rotating the altermagnetic order (Li et al., 2024). This is an electronic heterostructure result, but the edge-mass mechanism is directly suggestive for higher-order altermagnon constructions.

An even more explicit lattice realization is the Lieb-lattice program. There, systematically constructed $P$4-, $P$5-, and $P$6-wave altermagnetic orders reconstruct strip edge states, generate edge Dirac points through magnetic unit-cell folding, and gap those Dirac points when the AM moments lie in plane (Huo et al., 19 Dec 2025). In open square geometry, the resulting mass-domain walls produce corner-localized states, and the effective edge theory takes the form

$P$7

The same work verifies that the higher-order topology is most pronounced for altermagnetism when compared with ferromagnetism and ferrimagnetism (Huo et al., 19 Dec 2025). Although magnons are not calculated there, the boundary-mass logic is highly transferable.

Boundary-only altermagnetism provides another route. A theory of topologically protected surface altermagnetism shows that a conventional antiferromagnetic bulk can remain fully spin degenerate while the reduced surface symmetry allows a $P$8-wave-like spin spectral density on topological drumhead states or Fermi arcs (Leeb et al., 10 Feb 2026). The protection comes from bulk topology of a Dirac nodal line or Dirac semimetal, not from bulk spin splitting (Leeb et al., 10 Feb 2026). A plausible implication is that surface altermagnonic signatures may likewise exist even when the bulk bosonic spectrum remains more symmetric.

Finally, superconducting boundary modes have become an important comparative template. Topological altermagnetic Josephson junctions use $P$9-wave altermagnetic spin splitting instead of an external Zeeman field, robustly hosting Majorana end modes for $S_n(r)$0-wave but not $S_n(r)$1-wave altermagnets (Yang et al., 27 Feb 2025). The orientation angle $S_n(r)$2 of the altermagnet acts as a topological control parameter, and the distinction reduces to whether the altermagnetic term vanishes at the relevant momenta $S_n(r)$3 (Yang et al., 27 Feb 2025). This orientation-selective topology is likely to remain important in future altermagnonic devices.

5. Real-space textures, transport fingerprints, and control knobs

Topological altermagnonics is not only a momentum-space problem. In MnTe, nanoscale full-vector imaging has resolved $S_n(r)$4, $S_n(r)$5, and $S_n(r)$6 domain walls, vortex-like textures, vortex–antivortex pairs, and microscale single-domain states using XMCD-PEEM and XMLD-PEEM (Amin et al., 2024). The local altermagnetic order parameter is $S_n(r)$7, and the direct identification of a clockwise $S_n(r)$8 winding and its opposite establishes real-space vortex and antivortex defects in the altermagnetic order field (Amin et al., 2024). In patterned filled hexagons, geometry plus field cooling forces a total winding of $S_n(r)$9, so the formation of a vortex pair is required (Amin et al., 2024). This suggests that domain walls and vortices should be regarded as natural real-space scattering, guiding, and localization centers for altermagnonic modes.

Transport studies supply another set of control principles. In layered altermagnets such as $\mathcal{PT}$0, RuO$\mathcal{PT}$1, MnF$\mathcal{PT}$2, CrSb, and MnTe, crystal Hall, Nernst, and thermal Hall responses are governed by SOC-gapped altermagnetic pseudonodal surfaces, with Berry-curvature hotspots concentrated near those avoided crossings (Yang et al., 23 Feb 2025). The response moments

$\mathcal{PT}$3

generate finite-temperature crystal Hall, Nernst, and thermal Hall coefficients, and spin canting is shown to be an especially effective control knob because it preserves the alternating spin character while breaking the symmetries that enforce cancellation of the Berry curvature (Yang et al., 23 Feb 2025). This is an electronic transport theory, but it offers an immediately relevant blueprint for field-tunable magnon Hall or spin-Nernst analogs in altermagnetic backgrounds.

A useful caution is that nonzero local Berry curvature need not imply a net transverse response. In a symmetry-based comparison of altermagnets and antiferromagnets, local Berry curvature appears in compensated altermagnetic analogs, yet the anomalous Hall conductivity vanishes in the specific collinear example because the integrand is odd over the Brillouin zone (Mineev, 21 Jan 2026). For topological altermagnonics, the corresponding lesson is that band splitting, nodal structure, and local geometric curvature must be distinguished from a nonzero integrated thermal Hall coefficient.

6. Present status and open problems

The present field contains one direct magnon-topological material platform, several direct magnonic but not yet topological models, and a larger set of electronic and BdG theories that function as symmetry blueprints. Monolayer AgF$\mathcal{PT}$4 already realizes chiral topological magnons in an altermagnetic setting (González et al., 21 Aug 2025). Synthetic dipole–exchange multilayers and frustrated non-collinear lattices already realize altermagnetic magnon splitting, nodal directions, chirality mixing, or parity-dependent surface–bulk reconstruction, but have not yet been assigned bosonic Chern numbers or edge-state topology (Gallardo et al., 9 Jun 2026, Ghosh et al., 23 Jan 2026).

The principal open problem is therefore not whether altermagnetic symmetry can structure bosonic spectra, but how often that structure survives as a well-defined topological magnon phase in realistic materials. Many highly relevant works explicitly do not calculate magnon Berry curvature, magnon Chern numbers, or magnon edge states, even when they identify the underlying nodal, Weyl, higher-order, or boundary-mass mechanisms (Das et al., 2024, Li et al., 2024, Fernandes et al., 2023, Li et al., 2024). A plausible implication is that the next stage of the field will consist of translating these now well-developed fermionic symmetry mechanisms—band-winding inheritance, mirror-protected nodal lines, surface-only altermagnetic polarization, Dirac mass sign reversal, and orientation-controlled topology—into bosonic Bogoliubov frameworks.

A second open problem concerns hierarchy of energy scales. CrSb shows that altermagnetic topology can be exchange dominated and reach energy scales far above ordinary SOC (Li et al., 2024), whereas AgF$\mathcal{PT}$5 demonstrates that strong DMI activated by inversion breaking can open a full topological magnon gap in a compensated collinear magnet (González et al., 21 Aug 2025). Topological altermagnonics will likely develop at the intersection of these two regimes: exchange-structured altermagnetic splitting furnishing wide bosonic bandwidths, and relativistic anisotropies or dipolar couplings selecting the final topological gap.

A third frontier is the unification of momentum-space and real-space topology. The direct control of vortices, antivortices, and domain walls in MnTe (Amin et al., 2024), the higher-order boundary-mass mechanisms of TI/altermagnet and Lieb-lattice models (Li et al., 2024, Huo et al., 19 Dec 2025), and the surface-selective topological altermagnetism of topological AFMs (Leeb et al., 10 Feb 2026) collectively suggest that future topological altermagnonics will not be limited to bulk magnon Chern bands. It is likely to include surface altermagnons, corner altermagnons, domain-wall-guided altermagnons, and texture-bound modes whose existence is fixed jointly by crystalline symmetry, magnetic compensation, and boundary or defect geometry.