Systematic investigation of emergent particles in type-III magnetic space groups

In three-dimensional (3D) crystals, emergent particles arise when two or multiple bands contact and form degeneracy (band crossing) in the Brillouin zone. Recently a complete classification of emergent particles in 3D nonmagnetic crystals, described by the type-II magnetic space groups (MSGs), has been established. However, symmetries become complex in magnetically ordered structures. Consequently, we further perform a systematic investigation of emergent particles in magnetic crystals, and here we address this challenging task by exploring the possibilities of the emergent particles in the 674 type-III MSGs. Based on effective Formula Presented Hamiltonian and our classification of emergent particles [Yu , Sci. Bull. (2022)10.1016/j.scib.2021.10.023], we identify all possible emergent particles, including spinful and spinless, essential and accidental particles in the type-III MSGs. We find that all emergent particles in type-III MSGs also exist in type-II MSGs, with only one exception, i.e., the combined quadratic nodal line and nodal surface. Moreover, tabulations of the emergent particles in each of the 674 type-III MSGs, together with the symmetry operations, the small corepresentations, the effective Formula Presented Hamiltonians, and the topological character of these particles, are explicitly presented. Remarkably, combining this work and our homemade Formula Presented packages will provide an efficient way to search topological magnetic materials with novel quasiparticles.