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 BaM_{X 3} (M =V , Nb, or Ta; X =S or Se) by using symmetry analysis and first-principles calculation. We find that in BaVS₃ the valence and conduction bands are degenerate in the k_z=π /c plane (c is the lattice constant along the 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 z ̂ axis and a half-translation along the same z ̂ axis. The band degeneracy in the node surface is lifted in BaTaS₃ 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=±√{3 }k_y on the k_z=π /c plane. These node lines are robust against SOC and guaranteed by the symmetries of the P 6₃/m m c 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.