Triple degenerate point in three dimensions: Theory and realization

Abstract

Dirac and Weyl points in condensed matters could resemble the behaviors of their counterparts, Dirac and Weyl fermions, in high-energy physics. However, some fermions have no analog in fundamental fermions. They are expected to render novel physical properties. In this work we study an important fermion in condensed matters which is threefold degenerate. Such a fermion has no counterpart in high-energy physics and is enforced by crystalline symmetry. We perform a systematic search over all 230 space groups with time-reversal symmetry but without spin-orbital coupling (SOC). We find that a triple degenerate point (TDP) intrinsically appears in a three-dimensional irreducible representation. Based on the order of their energy dispersion, TDPs can be classified into two types, namely linear and quadratic TDPs. Furthermore, the linear TDP can be classified into two categories depending on the definition of topological charge. To distinguish topologies of linear and quadratic TDPs, we study their Landau spectrum under a magnetic field. Notably, these TDPs present dramatically different properties. There exists two chiral Landau levels (LLs) for linear TDPs with nonzero topological charge, whereas no chiral LL appears for the quadratic TDPs. Given the novel physical properties of TDPs, we provide concrete materials to support our discoveries. Consequently, our work provides a platform to study the triple degenerate points in condensed matters.