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

Abstract

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 $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian and our classification of emergent particles [Yu et al., Sci. Bull. (2022)], 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 $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonians, and the topological character of these particles, are explicitly presented. Remarkably, combining this work and our homemade $\mathtt{SpaceGroupIrep}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\mathtt{MSGCorep}$ packages will provide an efficient way to search topological magnetic materials with novel quasiparticles.