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

TL;DR

This paper studies how superconducting pairing affects the electronic spectrum of finite carbon nanotubes, revealing conditions for zero energy edge states linked to topological invariants and boundary shapes.

Contribution

It develops a topological framework to identify zero energy edge states in superconducting carbon nanotubes, connecting boundary conditions with topological invariants.

Findings

Zero energy edge states emerge in zigzag nanotubes due to curvature and superconductivity.

A topological phase diagram is constructed showing regions with nontrivial winding number.

Edge states depend on boundary shape in armchair nanotubes, with a bulk-edge correspondence established.

Abstract

We investigate the spectrum of finite-length carbon nanotubes in the presence of onsite and nearest-neighbor superconducting pairing terms. A one-dimensional ladder-type lattice model is developed to explore the low-energy spectrum and the nature of the electronic states. We find that zero energy edge states can emerge in zigzag class carbon nanotubes as a combined effect of curvature-induced Dirac point shift and strong superconducting coupling between nearest-neighbor sites. The chiral symmetry of the system is exploited to define a winding number topological invariant. The associated topological phase diagram shows regions with nontrivial winding number in the plane of chemical potential and superconducting nearest-neighbor pair potential (relative to the onsite pair potential). A one-dimensional continuum model reveals the topological origin of the zero energy edge states: A bulk-edge correspondence is proven, which shows that the condition for nontrivial winding number and that for the emergence of edge states are identical. For armchair class nanotubes, the presence of edge states in the superconducting gap depends on the nanotube's boundary shape. For the minimal boundary condition, the emergence of the subgap states can also be deduced from the winding number.