Spin-Point-Group Classification of Multipoles for Nonrelativistic Collinear Magnets
Multipole moments provide a unified symmetry language for describing electronic degrees of freedom and their associated physical responses. Although active multipoles have been systematically classified for crystallographic and magnetic point groups, their classification in spin point groups, which naturally describe magnetic systems in the absence of spin-orbit coupling, has remained unexplored. In this work, we present a complete classification of active orbital and spin multipoles for all 32 nonmagnetic and 122 collinear spin point groups. Treating orbital and spin degrees of freedom independently, we identify the symmetry-allowed multipoles associated with nonmagnetic and collinear magnetic orderings and clarify their hierarchy through comparisons among the corresponding spin point groups. The resulting classification provides a symmetry-based database that distinguishes structural and magnetic contributions to active multipoles and directly identifies microscopic order parameters, including those responsible for $d$-, $g$-, and $i$-wave altermagnetism. Furthermore, by classifying response tensors within the same framework, we establish a correspondence between active multipoles and symmetry-allowed physical responses. This correspondence systematically distinguishes nonrelativistic responses that survive without spin-orbit coupling from those requiring relativistic effects, providing a unified framework for understanding and predicting spin-dependent electromagnetic and transport phenomena in magnetic materials.