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

TL;DR

This paper systematically analyzes the presence of triple points in all symmorphic magnetic systems, revealing conditions for their coexistence with Dirac points and expanding understanding of topological fermions.

Contribution

It extends the theory of triple points to all symmorphic magnetic systems and identifies symmetry conditions for their coexistence with Dirac points.

Findings

All $k$ paths allowing triple points are listed.

Coexistence of Dirac and triple points is symmetry-allowed in certain systems.

Provides a comprehensive framework for searching topological fermions.

Abstract

Similar to Weyl fermions, a recently discovered topological fermion "triple point" can be generated from the splitting of Dirac fermion while the system has inversion symmetry (IS) breaking or time reversal symmetry (TRS) breaking. Inducing triple points in IS breaking symmorphic systems have been well studied, whereas in TRS breaking symmorphic systems have not yet. In this work, we extend the theory of searching triple points to all symmorphic magnetic systems. We list all $k$ paths of all symmorphic systems which allow the existence of triple points. With this systematic study, we also find out that the coexistence of Dirac points and triple points is symmetrically allowed in some particular symmetric systems. Our works will not only be helpful for searching triple points but also extend the knowledge of such a new topological fermion.