Abstract
We propose a kind of novel topological quantum state of semimetals in a quasi-one-dimensional (1D) crystals BaMX₃ (M = V, Nb or Ta; X = S or Se) family 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 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 two-fold 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 = ±√3k_y on the k_z=π/c plane. These node-lines are robust against SOC and guaranteed by the symmetries of P6₃/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.