Symmetry classification of magnetic orders using oriented spin space groups.

Magnetism has seen substantial progress in recent decades, driven largely by its potential for next-generation storage devices. However, the classification of magnetic orders, even for fundamental concepts such as ferromagnetism (FM) and antiferromagnetism (AFM), remains a topic of active evolution, particularly with the discovery of unconventional magnetic materials and advances in antiferromagnetic spintronics1-4. Here we present a classification of magnetic order using the state-of-the-art spin space group (SSG) theory5-11. On the basis of whether the net spin magnetization is constrained to zero by the SSG framework, we systematically categorize magnetic orders into FM (including ferrimagnetism) and AFM. We further introduce an 'oriented spin space group' (OSSG) description, that is, a SSG with a fixed magnetic orientation, thereby unifying the SSG and magnetic space group (MSG)12-14 frameworks. This approach clearly reveals the symmetry-breaking pathway induced by spin-orbit coupling (SOC). On the basis of the proposed group framework, we identify a distinct magnetic phase, termed spin-orbit magnetism (SOM), in which the net spin magnetization is induced by SOC. Our work provides a comprehensive symmetry-based perspective for classifying magnetic order, offering fresh insights into unconventional magnets and broad applicability in spintronics and quantum materials design.