Altermagnetic splitting of magnons in hematite ($\alpha$-Fe$_2$O$_3$)

We develop a four-sublattice spin-wave theory for the $g$-wave altermagnet candidate hematite ($\alpha$-Fe$_2$O$_3$), 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 band width of about 100 meV. 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. 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. This result suggests that the $g$-wave magnon spin splitting in hematite enables transverse heat-to-spin conversion without requiring an external magnetic field.