Antiferromagnetic nuclear spin helix and topological superconductivity in $^{13}$C nanotubes

TL;DR

This paper explores how hyperfine interactions in $^{13}$C nanotubes induce an antiferromagnetic nuclear spin helix that can lead to topological superconductivity with Majorana fermions, observable through conductance changes.

Contribution

It demonstrates the formation of a nuclear spin helix in carbon nanotubes and its role in inducing topological phases without fine-tuning, linking nuclear spin order to Majorana fermions.

Findings

Nuclear spin helix forms at tens of mK transition temperature.

Helical order causes conductance reduction by a factor of 2.

System can host topological superconductivity with Majorana bound states.

Abstract

We investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting $^{13}$C carbon nanotubes. Using the Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in noninteracting carbon nanotubes, and it leads to an antiferromagnetic nuclear spin helix in finite-size systems. The transition temperature reaches up to tens of mK, due to a strong boost by a positive feedback through the Overhauser field from ordered nuclear spins. Similar to GaAs nanowires, the formation of the helical nuclear spin order gaps out half of the conduction electrons, and is therefore observable as a reduction of conductance by a factor of 2 in a transport experiment. The nuclear spin helix leads to a density wave combining spin and charge degrees of freedom in the electron subsystem, resulting in synthetic spin-orbit interaction, which induces non-trivial topological phases. As a result, topological superconductivity with Majorana fermion bound states can be realized in the system in the presence of proximity-induced superconductivity without the need of fine tuning the chemical potential. We present the phase diagram as a function of system parameters, including the pairing gaps, the gap due to the nuclear spin helix, and the Zeeman field perpendicular to the helical plane.