Spin crystalline group in magnetic materials

The physical properties in magnetic-ordered materials were ultimately believed to rely on the symmetry theory of magnetic groups. Recently, it has come to light that a more comprehensive group, known as spin group, which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic-ordered materials such as altermagnets. In this talk, we introduce the recent developments of spin group theory, including extensive enumeration of over 100000 spin space groups, identification of spin groups for collinear, coplanar, and noncoplanar configurations, and derivation of irreducible co-representations in momentum space leading to more energy degeneracies that are disallowed by magnetic groups. We further show some theoretical and experimental results of spin group applications in understanding physical effects beyond the framework of magnetic groups.