Altermagnetism and Weak Ferromagnetism

Using a realistic model relevant to La$_2$CuO$_4$ and other altermagnetic perovskite oxides, we study interrelations between weak ferromagnetism (WF), anomalous Hall effect (AHE), and net orbital magnetization (OM). All of them can be linked to the form of Dzyaloshinskii-Moriya (DM) interactions. Nevertheless, while spin WF is induced by the DM vector components having the same sign in all equivalent bonds, AHE and OM are related to alternating-sign components, which do not contribute to any canting of spins. The microscopic model remains invariant under the symmetry operation $\{ \mathcal{S}|{\bf t} \}$, combining the shift ${\bf t}$ of antiferromagnetically coupled sublattices to each other with the spin flip $\mathcal{S}$. Thus, the band structure remains spin-degenerate, but the time-reversal symmetry is broken, providing a possibility to realize AHE in antiferromagnetic substances. The altermagnetic splitting of bands, breaking the $\{ \mathcal{S}|{\bf t}\}$ symmetry, does not play a major role in the problem. More important is the orthorhombic strain, responsible for finite values of AHE and OM.