Symmetry Classification of Magnetic Orders using Oriented Spin Space Groups

Magnetism has witnessed remarkable progress in recent decades, largely driven by its potential for next-generation storage devices. However, the classification of magnetic orders, even for fundamental concepts such as ferromagnetism and antiferromagnetism, remains a topic of active evolution, particularly with the discovery of unconventional magnetic materials and advances in antiferromagnetic spintronics. Here, we present a unified classification of magnetic order utilizing the state-of-the-art spin space group (SSG) theory. Based on whether the net spin magnetization is constrained to zero by SSG, we systematically categorize magnetic orders into ferromagnetism (including ferrimagnetism) and antiferromagnetism. We further introduce an oriented SSG description, i.e., an SSG with a fixed magnetic orientation, thereby unifying the SSG and magnetic space group frameworks. This approach clearly reveals the symmetry-breaking pathway induced by spin-orbit coupling. The proposed group framework completes the intrinsic logic of magnetic symmetry and identifies a distinct magnetic phase, termed spin-orbit magnetism, in which the net spin magnetization is induced by spin-orbit coupling. 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 material design.