Altermagnetic splitting of magnons in hematite α-Fe₂O₃

We develop a four-sublattice spin-wave theory for the g-wave altermagnet candidate hematite (alpha-Fe2O3), considering both its easy-axis phase below and its weak ferromagnetic phase above the Morin temperature. A key question is whether the defining altermagnetic feature-magnon spin splitting (also called chirality or polarization splitting) due to nonrelativistic time-reversal symmetry breaking-remains intact when relativistic corrections, which contribute to hematite's magnetic order, are included. Using a detailed symmetry analysis supported by density functional theory, we show that capturing the magnon splitting within a Heisenberg model requires exchange interactions extending at least to the 13th neighbor. We find an altermagnetic band splitting of approximately 2 meV, which contrasts with the total bandwidth of about 100 meV. The splitting scales as k4 in the long-wavelength limit (k is the crystal momentum) and exhibits complex direction dependence. While three of the four expected altermagnetic nodal surfaces align with crystallographic mirror planes, the fourth deviates from planarity, leading to twelve instead of six nodes in the kz = 0 plane. To evaluate the experimental observability of this splitting, we analyze relativistic corrections to the magnon spectrum in both magnetic phases. We show that spin-orbit coupling-manifesting as magnetocrystalline anisotropies and the Dzyaloshinskii-Moriya interaction (DMI)-does not obscure the key altermagnetic features. In the easy-axis phase, DMI introduces small spectral corrections on the order of 100 mu eV. In the easy-plane phase, while DMI induces the well-documented weak ferromagnetic moment due to spin canting, its effect on the magnon spectrum is negligibly small, on the order of 25 mu eV. The dominant relativistic effect arises from easy-plane anisotropy, which splits magnon modes at the Brillouin-zone center and suppresses their spin expectation value. However, this effect remains weaker than the altermagnetic splitting and rapidly diminishes away from the zone center. Our analysis suggests that nonrelativistic altermagnetic splitting dominates at energies above approximate to 30 meV, where the magnon spin polarization nearly recovers its quantized value. These findings indicate that inelastic neutron scattering can directly probe altermagnetic magnon splitting in hematite. We also discuss implications for magnon transport, particularly magnonic contributions to the thermal Hall effect (which requires spin-orbit coupling) and to spin splitter effects (which do not). Notably, we predict a third-order nonlinear magnon spin splitter effect: when a temperature gradient is applied along a direction in the ab plane that does not coincide with a mirror plane, a spin current emerges along the c axis. This result suggests that the g-wave magnon spin splitting in hematite enables transverse heat-to-spin conversion without requiring an external magnetic field.