Topological Phase Transition in Metallic Single-Wall Carbon Nanotube

Abstract

The topological phase transition is theoretically studied in a metallic single-wall carbon nanotube (SWNT) by applying a magnetic field $B$ parallel to the tube. The $\mathbb{Z}$ topological invariant, winding number, is changed discontinuously when a small band gap is closed at a critical value of $B$, 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 $\sim$ 1 $\mu$m length, using a one-dimensional lattice model to effectively describe the mixing between $\sigma$ and $\pi$ orbitals and spin-orbit interaction, which are relevant to the formation of the band gap in metallic SWNTs.Comment: 8 pages, 4 figure

Topological classification of the single-wall carbon nanotube

“…This explains the topological phase transition at |φ| 1/3, which was demonstrated in Ref. [12], for the following reason. ∆k φ is proportional to B along the tube axis, ∆k φ = −eBd t /(4 ), in Eq.…”

“…We generalize Eq. (12) [and Eq. (29)] for both semiconducting and metallic SWNTs in a magnetic field in Appendix B.…”

“…Then positive and negative E µ,s (k)'s form the conduction and valence bands, respectively. It should be mentioned that φ * 1/3 in metallic SWNTs, which corresponds to the magnetic field of a few Tesla [12].…”

“…We construct an effective 1D lattice model for a metallic SWNT, starting from the Hamiltonian of k · p theory [26], as discussed in Ref. [12]. In a magnetic field B in the axial direction, the Hamiltonian in the vicinity of K and K points reads…”

Topology and zero energy edge states in carbon nanotubes with superconducting pairing

“…It extends the 1D lattice model of Refs. [14,18,19] to the superconducting case. On the other hand, the chiral symmetry of the bulk Hamiltonian allows us to introduce a winding number as a topological invariant.…”