# Topological classification of the single-wall carbon nanotube

## TL;DR

This paper classifies the topological phases of single-wall carbon nanotubes using winding numbers, revealing most are nontrivial insulators and identifying phase transitions induced by magnetic fields, with implications for experimental detection.

## Contribution

It provides an analytical classification of SWNT topologies across all chiralities and magnetic conditions, linking winding numbers to edge states and experimental observables.

## Key findings

- Most SWNTs are nontrivial topological insulators without magnetic field.
- Magnetic fields induce topological phase transitions in SWNTs.
- Winding number correlates with the number of edge states at nanotube ends.

## Abstract

The single-wall carbon nanotube (SWNT) can be a one-dimensional topological insulator, which is characterized by a $\mathbb{Z}$-topological invariant, winding number. Using the analytical expression for the winding number, we classify the topology for all possible chiralities of SWNTs in the absence and presence of a magnetic field, which belongs to the topological categories of BDI and AIII, respectively. We find that the majority of SWNTs are nontrivial topological insulators in the absence of a magnetic field. In addition, the topological phase transition takes place when the band gap is closed by applying a magnetic field along the tube axis, in all the SWNTs except armchair nanotubes. The winding number determines the number of edge states localized at the tube ends by the bulk-edge correspondence, the proof of which is given for SWNTs in general. This enables the identification of the topology in experiments.