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We propose that the noncentrosymmetric LiGaGe-type hexagonal $ABC$ crystalSrHgPb realizes a new type of topological semimetal that hosts both Dirac andWeyl points in momentum space. The symmetry-protected Dirac points arise due toa band inversion and are located on the sixfold rotation $z$-axis, whereas thesix pairs of Weyl points related by sixfold symmetry are located on theperpendicular $k_z=0$ plane. By studying the electronic structure as a functionof the buckling of the HgPb layer, which is the origin of inversion symmetrybreaking, we establish that the coexistence of Dirac and Weyl fermions definesa phase separating two topologically distinct Dirac semimetals. These two Diracsemimetals are distinguished by the $\mathbb{Z}_2$ index of the $k_z=0$ planeand the corresponding presence or absence of 2D Dirac fermions on sidesurfaces. We formalize our first-principles calculations by deriving andstudying a low-energy model Hamiltonian describing the Dirac-Weyl semimetalphase. We conclude by proposing several other materials in thenon-centrosymmetric $ABC $ material class, in particular SrHgSn and CaHgSn, ascandidates for realizing the Dirac-Weyl semimetal.

The Dirac-Weyl semimetal: Coexistence of Dirac and Weyl fermions in polar hexagonal $ABC$ crystals