Angular momentum and topology in semiconducting single-wall carbon nanotubes

Semiconducting single-wall carbon nanotubes are classified into two types by means of the orbital angular momentum of the valley state, which is useful to study their low-energy electronic properties in finite length. The classification is given by an integer d, which is the greatest common divisor of two integers n and m specifying the chirality of nanotubes, by analyzing cutting lines. For the case that d is greater than or equal to four, the two lowest subbands from two valleys have different angular momenta with respect to the nanotube axis. Reflecting the decoupling of two valleys, discrete energy levels in finite-length nanotubes exhibit fourfold degeneracy and small lift of fourfold degeneracy by the spin-orbit interaction. For the case that d is less than or equal to two, in which the two lowest subbands from two valleys have the same angular momentum, discrete levels exhibit a lift of fourfold degeneracy reflecting the coupling of two valleys. Especially, two valleys are strongly coupled when the chirality is close to the armchair chirality. An effective one-dimensional lattice model is derived by extracting states with relevant angular momentum, which reveals the valley coupling in the eigenstates. A bulk-edge correspondence, which is a relationship between the number of edge states and the winding number calculated in the corresponding bulk system, is analytically shown by using the argument principle, and this enables us to estimate the number of edge states from the bulk property. The number of edge states depends not only on the chirality but also on the shape of boundary.