Node-surface and node-line fermions from nonsymmorphic lattice symmetries

We propose a kind of topological quantum state of semimetals in the quasi-one-dimensional (1D) crystal family ${\text{Ba}MX}_3$ ($M=\mathrm{V}$, Nb, or Ta; $X=\mathrm{S}$ or Se) by using symmetry analysis and first-principles calculation. We find that in ${\mathrm{BaVS}}_3$ the valence and conduction bands are degenerate in the $k_z=\pi /c$ plane ($c$ is the lattice constant along the $\widehat{z}$ axis) of the Brillouin zone (BZ). These nodal points form a node surface, and they are protected by a nonsymmorphic crystal symmetry consisting of a twofold rotation about the $\widehat{z}$ axis and a half-translation along the same $\widehat{z}$ axis. The band degeneracy in the node surface is lifted in ${\mathrm{BaTaS}}_3$ by including strong spin-orbit coupling (SOC) of Ta. The node surface is reduced into 1D node lines along the high-symmetry paths $k_x=0$ and $k_x=\pm \sqrt{3}{k}_y$ on the $k_z=\pi /c$ plane. These node lines are robust against SOC and guaranteed by the symmetries of the $P{6}_3/mmc$ space group. These node-line states are entirely different from previous proposals which are based on the accidental band touchings. We also propose a useful material design for realizing topological node-surface and node-line semimetals.