Spin-orbit magnetism in altermagnets

The mechanism enabling antiferromagnets, including altermagnets, to exhibit a prominent anomalous Hall effect despite a vanishingly small net magnetization has long remained elusive. Here, by employing oriented spin group theory and spin-orbit-coupling tensor expansion, we systematically disentangle the perturbative behaviors of orbital and spin magnetizations with respect to spin-orbit coupling. Remarkably, we find that only if the opposite-spin sublattices are connected through a fourfold rotation, the orbital and spin magnetizations exhibit distinct perturbative orders. In these altermagnets, we further discover a coaxial Hall effect characterized by the induced spin and orbital magnetizations aligning parallel to the N\'eel vector, which we further demonstrate by first-principles calculations in the altermagnet KV$_{2}$Se$_{2}$O. This effect holds great promise for achieving deterministic switching of the N\'eel order under weak external fields. Our work provides a systematic symmetry approach to identify potential altermagnetic candidates combining a large anomalous Hall effect with minimal net magnetization, paving the way for high-performance, stray-field-free spintronic applications.