Topological phase transition in metallic single-wall carbon nanotube induced by magnetic field

The single-wall carbon nanotube (SWNT) can be regarded as a one-dimensional topological insulator owing to the sublattice symmetry for $A$ and $B$ lattice sites [1]. It is characterized by a $\mathbb{Z}$ topological invariant, winding number, in both the absence (class BDI) and presence (AIII) of magnetic field. We theoretically study the topological phase transition in a metallic SWNT, in which a small energy gap is opened by the mixing between $\sigma$ and $\pi$ orbitals owing to a finite curvature of the tube surface and closed by applying a magnetic field $B=B^*$ parallel to the tube axis [2]. We demonstrate discontinuous changes in the winding number at $B^*$, which can be observed as a change in the number of edge states owing to the bulk-edge correspondence. This is confirmed by numerical calculations for finite SWNTs of length $\sim$ 1 $\mu$m, using a 1D lattice model to effectively describe the mixing between $\sigma$ and $\pi$ orbitals and spin-orbit interaction [3]. --- [1] W. Izumida, R. Okuyama et al., Phys. Rev. B 93, 195442 (2016). [2] R. Okuyama, W. Izumida, and M. Eto, arXiv:1610.05034. [3] W. Izumida et al., J. Phys. Soc. Jpn 78, 074707 (2009).