Symmetry-adapted models for multifold fermions with spin-orbit coupling

Multifold fermions, quasiparticles with multiple degeneracy protected by crystalline symmetries, exhibit a variety of intriguing phenomena stemming from their large topological charges and unique band structures. A comprehensive understanding of their response to external stimuli remains challenging, especially for types protected by nonsymmorphic symmetries where degrees of freedom, such as orbital, spin, sublattice, and crystal momentum, are intricately coupled. Here, we systematically construct effective models for multifold fermions that incorporate external fields based on crystalline symmetry. Specifically, we develop a $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model for the threefold fermion protected by space group $I{2}_{1}{31}^{\ensuremath{'}}$ (No. 199) in the presence of spin-orbit coupling and derive the terms for external fields. By complementing this with a tight-binding model, we investigate the magnetic field response and reveal the pair annihilation of magnetic monopoles. Furthermore, we construct a $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model for the eightfold fermion in space group $P\overline{4}3n{1}^{\ensuremath{'}}$ (No. 218), including its coupling to external fields. This work provides a robust theoretical foundation for advancing the study of external field responses and transport phenomena in multifold fermions, opening new avenues to explore their rich physics.