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

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 $t\mathrm{/}U$ between the characteristic hopping amplitude $t$ and the large on-site Coulomb repulsion $U$. Consequently, all exchange interactions scale as $1\mathrm{/}U$, so a small altermagnetic hopping $\delta t$ 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 $\delta t\mathrm{/}U$. In contrast, the behavior of the conductivity tensor $\widehat{\sigma }(\omega )$ at $\mathrm{\hslash }\omega \sim U$ differs qualitatively, because $\delta t$ enters the energies of interband optical transitions \emph{directly} rather than through the reduced ratio $\delta t\mathrm{/}U$. This contribution strongly reshapes $\widehat{\sigma }(\omega )$, leading to a dramatic enhancement of the magneto-optical response.