A nonsymmorphic-symmetry-protected hourglass Weyl node, hybrid Weyl node, nodal surface, and Dirac nodal line in Pd4X (X = S, Se) compounds

Abstract

Nonsymmorphic symmetry has been proved to protect band crossings in topological semimetals/metals. In this work, based on the symmetry analysis and first-principles calculations, we reveal rich topological phases in compounds Pd₄X (X = S, Se), which are protected by nonsymmorphic symmetry. In the absence of spin–orbit coupling (SOC), it shows the coexistence of the type-I Weyl point and type-II Weyl point. Here, due to the screw rotation, the type-I Weyl point takes an hourglass form. However, this hourglass Weyl point can be gapped in the presence of SOC. Furthermore, a combination of nonsymmorphic twofold screw-rotational symmetry and time-reversal symmetry protects a nodal surface. Particularly, this nodal surface is robust against SOC. In addition, a combination of the glide mirror and time-reversal symmetry contributes a nodal line of double degeneracy. In the presence of SOC, there emerges hybridization of type-I and type-II Weyl points. Meanwhile, there also appears a Dirac nodal line—a fourfold degenerate nodal line under SOC, which is protected by nonsymmorphic symmetries. Our works suggest realistic materials to study Weyl nodes of type-I and type-II, and their hybridization, as well as symmetry-protected nodal surfaces and Dirac nodal lines.