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 BaMX3 (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 kz=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 kx=0 and kx=±3ky on the kz=π/c plane. These node lines are robust against SOC and guaranteed by the symmetries of the 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.