Topology in single-wall carbon nanotube of zigzag and armchair type

Single-wall carbon nanotubes can be one-dimensional (1D) topological insulators except for armchair nanotubes. The electronic states are characterized by a non-zero topological invariant, winding number, which is related to the number of 0D edge states via the bulk-edge correspondence. In the present study, we theoretically examine zigzag and armchair nanotubes to elucidate the emergence and absence of edge states. The effective 1D lattice model is employed in order to describe the fine structures due to the finite curvature of tube surface and spin-orbit interaction. We show that the lattice model for a zigzag nanotube is equivalent to the Su-Schrieffer-Heeger model, by which the formation of edge states is explained. An armchair nanotube is described by a ladder model, on the other hand, which does not host any edge states owing to the σ_h symmetry.