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 ${\text{Ba}MX}_{3}$ ($M\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\mathrm{V}$, Nb, or Ta; $X\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\mathrm{S}$ 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 ${k}_{z}=\ensuremath{\pi}/c$ plane ($c$ is the lattice constant along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ 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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ axis and a half-translation along the same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ 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 ${k}_{x}=\ifmmode\pm\else\textpm\fi{}\sqrt{3}{k}_{y}$ on the ${k}_{z}=\ensuremath{\pi}/c$ 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.