Residual orbital magnetization governs the anomalous Hall effect in altermagnets

In altermagnets that exhibit anomalous Hall effect, the small remanent magnetization exists but has been treated as too small to be relevant to the Hall response. In this work, we point out that this dismissal is incomplete because the generalized St\v{r}eda relation ties the intrinsic anomalous Hall conductivity ($\sigma_{xy}$) to the orbital magnetization ($M_z$, the topological component from the modern orbital magnetization) by $\sigma_{xy}=-e\frac{\partial M_z}{\partial \mu}$. We reveal a microscopic mechanism to generate net orbital moment from the interplay of local crystal field and spin-orbit coupling for MnTe-type altermagnets, in which the magnetic anisotropy generates weak net magnetization without invoking exchange between neighboring spins (e.g., Dzyaloshinskii-Moriya interaction). Our work indicates that residual orbital and spin magnetization is an intrinsic thermodynamic property that governs anomalous transport in unconventional antiferromagnets, including altermagnets and noncollinear antiferromagnets.