Chiral Magnon Splitting in Altermagnets

• Chiral magnon splitting is the non-degeneracy of spin-wave modes with opposite chirality in altermagnets, stemming from unique crystal symmetry operations.
• The phenomenon is modeled via anisotropic Heisenberg exchanges and spin-wave theory, yielding d-wave and g-wave momentum-dependent splitting patterns observed experimentally.
• This splitting underpins robust directional spin transport, topological magnon effects, and tunable spin currents, paving the way for advanced spintronic device applications.

Chiral magnon splitting denotes the momentum-dependent non-degeneracy of spin-wave modes with opposite chirality in certain magnetic systems, particularly altermagnets. Unlike conventional ferromagnets (where only one magnon chirality exists) or antiferromagnets (where opposite-chirality magnons are degenerate throughout the Brillouin zone), altermagnets generically exhibit nontrivial splitting of these modes owing to their unique symmetry properties. This phenomenon has significant implications for high-frequency spintronics, magnonics, and topological transport.

1. Symmetry-Principled Origins of Chiral Magnon Splitting

The essential microscopic mechanism for chiral magnon splitting is encoded in the magnetic point group and space group symmetries of altermagnets. In these materials, the compensated sublattices transform into each other under a crystal rotation or mirror—but not under any simple translation or inversion. This symmetry—manifest in, for example, the [C₂∥C₄] group in RuO₂ or improper sixfold rotations in hexagonal MnTe/CrSb—permits a momentum-dependent splitting of magnon branches associated with handed circular precession of spins. The splitting vanishes along high-symmetry lines—nodal planes where the twofold operation maps sublattices with no phase winding—but is generically finite elsewhere, producing characteristic d-wave (C₄ symmetry, e.g., RuO₂) or g-wave (C₆ symmetry, e.g., MnTe, CrSb) form factors (Šmejkal et al., 2022, Liu et al., 2024, Kravchuk et al., 7 Apr 2025).

Unlike Dzyaloshinskii–Moriya or magnetostatic interactions, the requisite exchange anisotropy that produces chiral splitting is "symmetric exchange"—arising from inequivalent same-sublattice couplings on long-range bonds. In metallic systems, the effect directly reflects bulk electronic band splitting: the sign and magnitude of the electronic ΔE^{↑↓}(k) along specific BZ directions correlate with the magnitude and sign of the magnon splitting Δω(k) (Beida et al., 12 May 2025, Singh et al., 20 Nov 2025). Time-reversal is explicitly broken by the magnetic order, but the compound symmetry operations ensure compensated net moment.

2. Microscopic Theory: Spin Hamiltonians, Linear and Nonlinear Spin-Wave Analysis

Model Hamiltonians and Chiral Splitting

A broad theoretical consensus has emerged: the minimal microscopic Hamiltonian supporting chiral splitting consists of an anisotropic Heisenberg exchange extended to sublattice-inequivalent long-range neighbors. For instance, the minimal Hamiltonian for CrSb is (Singh et al., 20 Nov 2025): $$H = \sum_{⟨ij⟩} J_1\,\mathbf S_i\!\cdot\!\mathbf S_j + \sum_{⟨⟨ij⟩⟩} J_2\,\mathbf S_i\!\cdot\!\mathbf S_j + \sum_{⟨⟨⟨ij⟩⟩⟩} J_3\,\mathbf S_i\!\cdot\!\mathbf S_j + D \sum_i (S_i^c)^2 + \sum_{⟨ij⟩_{11,12}} J_{11,12}\mathbf S_i\!\cdot\!\mathbf S_j$$ where the last term includes two symmetry-inequivalent eleventh- and twelfth-neighbor exchanges crucial for chiral splitting.

Linear spin-wave theory based on Holstein–Primakoff bosonization, Fourier transforms, and paraunitary Bogoliubov diagonalization yields two magnon branches: $$\omega_\pm(\mathbf k) = \sqrt{A_\mathbf k^2 - B_\mathbf k^2} \pm C_\mathbf k$$ with A and B encoding symmetric exchange, and $C_\mathbf k$ the antisymmetric combination of long-range exchanges whose sign and amplitude encode the g- or d-wave symmetry of the splitting (Zhang et al., 17 Mar 2025, Kravchuk et al., 7 Apr 2025).

Advanced Interaction Effects

Beyond harmonic approximation, chiral splitting can be further renormalized by nonlinear magnon–magnon interactions. Notably, three-magnon (three-wave mixing) processes—absent in fermionic systems—produce a 1/S self-energy giving an additional, symmetry-protected chiral splitting which can dominate at finite temperature and for low-spin systems (Jin et al., 29 Jul 2025, Yan et al., 5 Nov 2025). The resulting analytic structure respects the same symmetry form factors (e.g., d_{{x^2}-y^2} for C₄T-invariant systems, g-wave for D₆h-invariant altermagnets).

3. Experimental Evidence and Measurement

Comprehensive inelastic neutron scattering (INS), circular dichroism resonant inelastic X-ray scattering (CD-RIXS), and microwave/microwave-optical spectroscopies have substantiated chiral magnon splitting in both insulating and metallic altermagnets:

• In MnTe, both INS and CD-RIXS resolve clear g-wave chiral splitting, with maximum Δω ≈ 2 meV observed off the nodal planes (Liu et al., 2024, Jost et al., 29 Jan 2025).
• In CrSb, polarized neutron diffraction and high-resolution INS detect chiral splitting up to ≈20–30 meV along low-symmetry directions, with splitting tied directly to symmetry-inequivalent long-range exchanges (Singh et al., 20 Nov 2025, Zhang et al., 17 Mar 2025).
• In RuO₂ and hematite (α-Fe₂O₃), first-principles calculations and experimental spectra confirm d- or g-wave magnon splitting proportional to the difference of long-range exchange paths (Šmejkal et al., 2022, Hoyer et al., 14 Mar 2025).
• Absence of observable splitting in MnF₂, despite nominal altermagnetic symmetry, is traced to negligible scale of the requisite anisotropic exchanges (Δω < 120 μeV), demonstrating that symmetry is necessary but not sufficient—energetic magnitudes are critical (Morano et al., 2024).

A recurrent theme is the experimental necessity of matching spectrometer resolution to the predicted splitting amplitude and discriminating chiral effects from possibly larger dipolar-induced splittings (e.g., in FeF₂) (Sears et al., 7 Jan 2026).

4. Functional Consequences: Spin Transport, Topological Effects, and Control

Chiral magnon splitting produces robust directional and nonreciprocal spin transport phenomena. The essential mechanism is that a thermal gradient or optical/microwave drive can selectively excite one chirality, generating net spin, heat, or angular-momentum currents even in the absence of net magnetization or stray fields.

Thermal spin transport calculations, often via the Kubo formalism, predict substantial spin Seebeck and spin Nernst effects due to the non-cancellation of left- and right-chiral magnon contributions when Δω(k) is finite and alternates in sign as a function of direction. Such effects are seen and predicted in MnTe, RuO₂, rare-earth melilite compounds, and d-wave bilayer altermagnets (Siam et al., 11 Jul 2025, Šmejkal et al., 2022, Yuan et al., 29 Jan 2026).

Further, topological magnonic phenomena (quantum spin Hall effect with helical edge states, magnonic Chern phases) can emerge in altermagnets with nontrivial chiral magnon splitting and Berry curvature, as realized in d-wave bilayer V₂WS₄ (Yuan et al., 29 Jan 2026).

Crucially, electric fields can tune, induce, or even reverse the splitting in select symmetry classes, enabling efficient, ultrafast control of magnon spin signals, as recently demonstrated in PT-symmetric AFIs such as Cr₂CBr₂ and classified in large-scale symmetry taxonomies (Ni et al., 22 Jan 2026, Xie et al., 21 Jan 2026).

5. Thermodynamic and Dynamic Tuning: Temperature, Interactions, Domain Walls

The amplitude and even sign of chiral magnon splitting is strongly modulated by temperature due to the competition between isotropic symmetric exchange (ISE) and anisotropic spin exchange (ASE) channels. At low T the splitting is ISE dominated; increasing T populates magnon–magnon interactions that enhance the ASE component, potentially reversing the sign and causing a spin-current reversal at a critical temperature $T_c$ (Yan et al., 5 Nov 2025). The scaling of splitting with temperature, spin value S, and exchange coefficients is computationally accessible and verified in several fluoride antiferromagnets.

In real devices, domain wall (DW) engineering offers further degrees of freedom. Altermagnetic domain walls confine gapless, chirality-split magnon modes, with their spectrum tunable by wall orientation and spin–orbit torque (Zeng et al., 4 Jan 2026). Dzyaloshinskii–Moriya interactions in these DWs cause hybridization and unidirectional magnon-magnon coupling, forming the basis for programmable, electrically reconfigurable magnonic nanocircuits.

6. Consequences for Magnon–Phonon Coupling and Device Engineering

Spin-lattice coupling in altermagnets transmits the chiral magnon splitting into the phononic sector, imbuing phonons with a d- or g-wave angular momentum texture and enabling phonon angular momentum splitter effects—bosonic analogues to electronic spin conductance in spin–split bands (Bendin et al., 11 Nov 2025). These findings suggest possibilities for chiral magnon–phonon hybrid devices capable of transverse phonon angular momentum currents controllable by temperature gradients or electric fields.

Control of magnon splitting and directionality by chemical composition (pnictogen substitution: As, Sb, Bi) and lattice strain has been demonstrated in CrSb and related systems, providing routes to tune not only Δω but also magnon lifetimes and damping, offering design strategies for nonreciprocal, high-frequency spin transport (Beida et al., 12 May 2025, Singh et al., 20 Nov 2025).

7. Summary Table: Key Systems and Features

Material / Model	Splitting Symmetry	Max Δω (experiment)	Tunability/control	Notable features
CrSb (metallic, hexag.)	g-wave	~20–30 meV	Chemical, electric, strain	THz bandwidth, nonreciprocity, large Δω
MnTe (insulating, hexag.)	g-wave	~2 meV	Magnetic domain, geometry	Linear dispersion, independent chiral channels
RuO₂ (metallic, rutile)	d-wave	~15–20 meV (DFT)	Symmetry, lattice strain	Suppressed damping, THz linear dispersions
Hematite (α-Fe₂O₃)	g-wave	~2 meV	Anisotropy, neighbor exchange	Four-sublattice, nonlinear spin currents
FeF₂ (rutile)	None (masked by dipolar)	<0.1 meV	Select momenta, domain	Dipolar coupling overwhelms altermagnetic
MnF₂ (rutile)	None (degenerate)	<0.12 meV	—	Splitting negligibly small
V₂WS₄ bilayer (2D)	d-wave (bilayer)	~0.2–0.3 meV	Electric, layer orientation	Quantum spin Hall effect, edge states

A plausible implication is that device-focused research should prioritize materials and architectures with strong, symmetry-allowed long-range exchange anisotropy, high moment per site (moderate S to retain strong splitting without damping losses), and amenability to electric-field or domain wall manipulation for on-chip spin current control. Systems where dipolar or conventional anisotropy terms overwhelm the intrinsic altermagnetic exchange (as in MnF₂, FeF₂) are unlikely to show useful chiral magnonic transport unless ultrahigh precision can resolve μeV-scale spectral features.

• (Liu et al., 2024) Chiral-Split Magnon in Altermagnetic MnTe
• (Morano et al., 2024) Absence of altermagnetic magnon band splitting in MnF₂
• (Hoyer et al., 14 Mar 2025) Altermagnetic splitting of magnons in hematite
• (Zhang et al., 17 Mar 2025) Chiral magnon splitting in altermagnetic CrSb from first principles
• (Kravchuk et al., 7 Apr 2025) Chiral magnetic excitations and domain textures of g-wave altermagnets
• (Beida et al., 12 May 2025) Chiral split magnons in metallic g-wave altermagnets
• (Jin et al., 29 Jul 2025) Interaction-Driven Altermagnetic Magnon Chiral Splitting
• (Siam et al., 11 Jul 2025) Chiral-split magnons in the S = 1 Shastry-Sutherland model
• (Yan et al., 5 Nov 2025) Competitive Orders in Altermagnetic Chiral Magnons
• (Singh et al., 20 Nov 2025) Chiral Spin-Split Magnons in the Metallic Altermagnet CrSb
• (Šmejkal et al., 2022) Chiral magnons in altermagnetic RuO₂
• (Bendin et al., 11 Nov 2025) D-Wave Phonon Angular Momentum Texture in Altermagnets by Magnon-Phonon-Hybridization
• (Zeng et al., 4 Jan 2026) Alignment-Dependent Gapless Chiral Split Magnons in Altermagnetic Domain Walls
• (Sears et al., 7 Jan 2026) Altermagnetic and dipolar splitting of magnons in FeF₂
• (Xie et al., 21 Jan 2026) A General Theory of Chiral Splitting of Magnons in Two-Dimensional Magnets
• (Ni et al., 22 Jan 2026) Electric-Switchable Chiral Magnons in PT-Symmetric Antiferromagnets
• (Yuan et al., 29 Jan 2026) Magnonic Quantum Spin Hall Effect with Chiral Magnon Transport in Bilayer Altermagnets