Altermagnetism in MnF$_2$: Band Splitting and Its Physical Consequences

Abstract: MnF$_2$ is widely regarded as a candidate altermagnet, but the magnitude and implications of its altermagnetic band splitting remain debated. Using electronic-structure calculations, we construct minimal models that capture the magnetic and electronic properties of MnF$_2$. These models show that the parameters governing the chiral magnon splitting and the spin splitting of the electronic bands are relatively small. Moreover, the electronic system lies in the strong-coupling regime, where most magnetic properties are controlled by the ratio $t/U$ between the characteristic hopping amplitude $t$ and the large on-site Coulomb repulsion $U$. Consequently, all exchange interactions scale as $1/U$, so a small altermagnetic hopping $δt$ produces only a proportionally small exchange term. Upon doping, the altermagnetic contribution to the anomalous Hall effect is likewise suppressed, being smaller than the conventional (non-altermagnetic) contribution by a factor of order $δt/U$. In contrast, the behavior of the conductivity tensor $\hatσ(ω)$ at $\hbar ω\sim U$ differs qualitatively, because $δt$ enters the energies of interband optical transitions \emph{directly} rather than through the reduced ratio $δt/U$. This contribution strongly reshapes $\hatσ(ω)$, leading to a dramatic enhancement of the magneto-optical response.

• The paper demonstrates that MnF₂’s small altermagnetic exchange splitting, derived using first-principles-based Wannier, Hartree–Fock, and spin-model calculations, produces negligible magnon splitting and is suppressed by roughly three orders of magnitude in the anomalous Hall effect.
• The paper separates spin–orbit-driven and altermagnetic mechanisms, showing that band splitting alone cannot produce anomalous Hall or magneto-optical responses but can substantially modify them when spin–orbit coupling is present.
• The paper finds that optical spectroscopy near MnF₂’s approximately 7.2 eV Mott gap is highly sensitive to altermagnetic splitting, with a roughly 0.3 eV excitation bandwidth and strongly enhanced off-diagonal conductivity and Kerr response despite modest absolute rotation.

Context and motivation

MnF$_2$ has become a canonical candidate altermagnet: a centrosymmetric rutile-structure antiferromagnet ($P4_2/mnm$) in which the fourfold screw rotation $\{\mathcal{C}_{4z}|\mathbf{t}\}$ connects opposite-spin sublattices and symmetry permits spin splitting of electronic bands even without relativistic spin-orbit (SO) coupling. Yet neutron measurements report nearly vanishing chiral magnon splitting (2606.01515), prompting the central question addressed in this paper: does altermagnetic band splitting actually control the time-reversal-symmetry-breaking responses of such materials? The author argues that it does not — at least not uniformly — and that judging a material's promise by the magnitude of its band splitting alone is misleading.

A useful framing point is historical. The paper emphasizes that "altermagnetism" as proposed in 2022 is not fundamentally distinct from Turov's decades-old classification of centrosymmetric antiferromagnets on antipolar lattices; most of the 221 candidates in recent reviews fall into that older class. What is genuinely new is the recognition of band splitting itself, an effect outside Landau phenomenology. Symmetry guarantees only that splitting may occur; its magnitude is set by microscopic parameters.

Electronic structure and magnetic interactions

The analysis proceeds from LDA electronic structure (F $2p$ and Mn $3d$ manifolds separated by ~4 eV), Wannier construction of the Mn $3d$ model, constrained-RPA parameters $U = 3.6$ eV and $J_{\rm H} = 0.9$ eV, Hartree–Fock solution, and linear-response extraction of exchange constants for $S = 5/2$. The dominant exchanges are strongly antiferromagnetic, $J_1 = -1.0101$ meV and $P4_2/mnm$0 meV, while the altermagnetic anisotropy of the fourth-neighbor exchange is comparatively small, $P4_2/mnm$1 meV. DM vectors are also small, $P4_2/mnm$2 meV.

Two structural conclusions follow. First, because MnF$P4_2/mnm$3 lies deep in the strong-coupling regime, all exchange interactions scale as $P4_2/mnm$4 with $P4_2/mnm$5; hence a small altermagnetic hopping $P4_2/mnm$6 produces only a proportionally small exchange splitting. The computed magnon dispersion indeed shows chiral branch splittings comparable to the experimental bound $P4_2/mnm$7 meV. Second, the calculated uniaxial anisotropy $P4_2/mnm$8 meV falls well short of the experimental $P4_2/mnm$9 meV, which is attributed to intersite dipole–dipole interactions; the paper concedes that possible ferromagnetic contributions from polarization of F $\{\mathcal{C}_{4z}|\mathbf{t}\}$0 states (invoked to reconcile the theoretical $\{\mathcal{C}_{4z}|\mathbf{t}\}$1, overestimated by a factor of two) remain unresolved and are left open.

Effective one-orbital model and symmetry analysis

Mapping the superexchange expressions onto an effective one-orbital model yields transfer integrals $\{\mathcal{C}_{4z}|\mathbf{t}\}$2 meV, $\{\mathcal{C}_{4z}|\mathbf{t}\}$3 meV, and $\{\mathcal{C}_{4z}|\mathbf{t}\}$4 meV, plus bond SO amplitudes $\{\mathcal{C}_{4z}|\mathbf{t}\}$5 meV. The sublattice-pseudospin Hamiltonian separates cleanly into two distinct sources of time-reversal breaking:

• SO-driven breaking: with $\{\mathcal{C}_{4z}|\mathbf{t}\}$6, the combined operation $\{\mathcal{C}_{4z}|\mathbf{t}\}$7 remains a symmetry, so by the generalized Bloch theorem the system maps onto an effective ferromagnet on a reduced unit cell. This explains why AHE, orbital magnetization, and magneto-optical effects — all odd in SO coupling — exist even without any altermagnetic splitting.
• Altermagnetic splitting: with finite $\{\mathcal{C}_{4z}|\mathbf{t}\}$8 but zero SO term, the Hamiltonian is real and $\{\mathcal{C}_{4z}|\mathbf{t}\}$9-invariant; the state can be described as ferroically ordered spin octupoles, but AHE and net orbital magnetization vanish.

The key implication is stated directly: band splitting alone cannot generate AHE or magneto-optical response; it can only modulate their magnitude when combined with SO coupling.

Anomalous Hall effect under doping

Assuming hole doping within the rigid-band approximation, the Berry curvature decomposes into an odd-in-$2p$0 term proportional to $2p$1 (independent of $2p$2) and an even-in-$2p$3 correction proportional to $2p$4. Their ratio scales as $2p$5, so the altermagnetic contribution to the AHE is suppressed by roughly three orders of magnitude relative to the conventional SO-driven part. Within this model, altermagnetism is essentially irrelevant to the doped Hall response.

Optical conductivity and magneto-optics

The situation reverses qualitatively at photon energies near the Mott gap, $2p$6 eV. Here $2p$7 enters interband transition energies directly, whereas $2p$8 contributes only at order $2p$9 meV — so despite being thirteen times smaller than $3d$0, $3d$1 dominates the optical lineshape. Three consequences are identified: the optical excitation bandwidth is of order $3d$2 eV and set primarily by $3d$3; the spectral weight distribution around $3d$4 changes qualitatively between $3d$5 and $3d$6; and while the diagonal conductivity obeys the f-sum rule (only reshaping, no weight change), the off-diagonal conductivity is strongly enhanced by $3d$7.

The complex Kerr effect inherits this sensitivity. Although $3d$8 originates from SO coupling and remains finite at $3d$9, its magnitude is dramatically boosted by the altermagnetic splitting. Quantitatively, the predicted Kerr rotation in AFM MnF$3d$0 is still about an order of magnitude smaller than in bcc Fe or fcc Ni — a sobering absolute scale — but the demonstration that a modest $3d$1 meV can strongly reshape $3d$2 establishes optical spectroscopy as a sensitive probe of altermagnetic splitting where transport probes are blind to it.

Limitations and open questions

Several caveats are explicit in the paper. The AHE analysis assumes rigid-band hole doping of a material that is experimentally insulating. The underestimate of $3d$3 and the factor-of-two overestimate of $3d$4 point to missing physics, plausibly magnetic polarization of F $3d$5 states, whose treatment is deferred. The Kerr formula used is exact only for cubic materials with magnetization along $3d$6, an approximation for tetragonal MnF$3d$7. More broadly, the conclusion that "large" versus "small" splitting is property-dependent rests on the strong-coupling scaling argument specific to Mott insulators like MnF$3d$8; whether the same hierarchy holds in weakly correlated or metallic altermagnets is not addressed.

Conclusion

Using first-principles-derived minimal models, this work shows that in MnF$3d$9 the altermagnetic band splitting is genuinely small on the scale relevant to magnons and the doped anomalous Hall effect, where all interactions scale as $U = 3.6$0. At optical frequencies near the charge gap, however, the same splitting enters transition energies unsuppressed and strongly enhances the off-diagonal conductivity and Kerr response. The broader lesson is methodological: the significance of altermagnetic splitting cannot be assessed from band-structure magnitude alone, but must be evaluated property by property against the appropriate energy scales of each response.