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

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.