Constructions and applications of irreducible representations of spin-space groups

Spin-space groups (SSGs), including the traditional space groups (SGs) and magnetic space groups (MSGs) as subsets, describe the complete symmetries of magnetic materials with weak spin-orbit coupling (SOC). In the present work, we systematically study the irreducible representations (irreps) of SSGs by focusing on the projective irreps of the little cogroup $L(k)$ of any momentum point $\mathbit{k}$. We analyze the factor systems of $L(k)$, and then reduce the projective regular representation of $L(k)$ into a direct sum of irreps using the Hamiltonian approach. Especially, for collinear SSGs which contain continuous spin rotation operations, we adopt discrete subgroups to effectively capture their characteristics. Furthermore, we apply the representation theory of SSGs to study the band structure of electrons and magnons in magnetic materials. After identifying the SSG symmetry group, we extract relevant irreps and determine the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ models. As an example, we illustrate how our approach works for the material ${\mathrm{Mn}}_{3}\mathrm{Sn}$. Degeneracies facilitated by SSG symmetry are observed, underscoring the effectiveness of application in material analysis. The SSG recognition and representation code is uploaded to GitHub, and the information of irreps of all SSGs is also available in the online Database. Our work provides a practical toolkit for exploring the intricate symmetries of magnetic materials and paves the way for future advances in materials science.