In this talk, I will present our recent studies on zero energy edge states in finite-length carbon nanotubes from topological viewpoint.
An effective one-dimensional (1D) lattice model for a single-wall carbon nanotube (SWNT) is introduced to analyze the quantum system with boundary . Open-ended SWNTs commonly host edge states whose energies lie in the bulk band gap. By analyzing the 1D lattice model a bulk-edge correspondence is proven, which shows that the number of edge states and a winding number topological invariant defined in the corresponding bulk system are identical . Manipulation of the edge states by a magnetic field is suggested in terms of the topological phase transition .
We extend these studies to investigate the nature of edge states which arise when the SWNT is proximity coupled to a superconductor . The zero energy edge states emerge as a combined effect of curvature-induced Dirac point shift and strong superconducting coupling between nearest-neighbor sites. A 1D continuum model reveals the topological origin of the zero energy edge states.
 W. Izumida, R. Okuyama, R. Saito, Phys. Rev. B 91, 235442 (2015)
 W. Izumida, R. Okuyama, A. Yamakage, R. Saito, Phys. Rev. B 93, 195442 (2016)
 R. Okuyama, W. Izumida, M. Eto, J. Phys. Soc. Jpn. 86, 013702 (2017)
 W. Izumida, L. Milz, M. Marganska, M. Grifoni, arXiv:1707.02934