Systematic analysis for triple points in all magnetic symmorphic systems and symmetry-allowed coexistence of Dirac points and triple points

Similar to Weyl fermions, a recently discovered topological fermion 'triple point' can be generated from the splitting of Dirac fermion in the systems with inversion symmetry (IS) breaking or time-reversal symmetry (TRS) breaking. Inducing triple points in IS breaking symmorphic systems have been well studied, but the same cannot be said for the TRS breaking symmorphic systems. In this work, we extend the theory of searching for triple points to all symmorphic magnetic systems. We list among all symmorphic systems all the k paths which allow the existence of triple points. With this systematic study, we also found that the coexistence of Dirac points and triple points is allowed in some particular symmetric systems. Besides theoretical analysis, we carried out numerical analysis as well. According to our first-principles calculations, $B_3{Re}_7$ and ${As}_2{Ni}_5$ are the candidates for realizing the coexistence of Dirac and triple points. We have not only provided an exhaustive triple point search mechanism for the symmorphic systems, but also identified material systems that host the Dirac and the triple points. (paper)