Nonsymmorphic-symmetry-protected hourglass Dirac loop, nodal line, and Dirac point in bulk and monolayer $X_3$SiTe$_6$ ($X=$ Ta, Nb)

Abstract

Nonsymmorphic space group symmetries can generate exotic band-crossings in topological metals and semimetals. Here, based on symmetry analysis and first-principles calculations, we reveal rich band-crossing features in the existing layered compounds Ta$_3$SiTe$_6$ and Nb$_3$SiTe$_6$, enabled by nonsymmorphic symmetries. We show that in the absence of spin-orbit coupling (SOC), these three-dimensional (3D) bulk materials possess accidental Dirac loops and essential fourfold nodal lines. In the presence of SOC, there emerges an hourglass Dirac loop---a fourfold degenerate nodal loop, on which each point is a neck-point of an hourglass-type dispersion. We show that this interesting type of band-crossing is protected and dictated by the nonsymmorphic space group symmetries, and it gives rise to drumhead-like surface states. Furthermore, we also investigate these materials in the monolayer form. We show that these two-dimensional (2D) monolayers host nodal lines in the absence of SOC, and the nodal lines transform to essential spin-orbit Dirac points when SOC is included. Our work suggests a realistic material platform for exploring the fascinating physics associated with nonsymmorphic band-crossings in both 3D and 2D systems.