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

We investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting carbon nanotubes made of $^{13}$C$~$[1]. Using the Tomonaga-Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in noninteracting nanotubes. The RKKY interaction forms $q=\pm 2 k_{F}$ peaks with the Fermi wave number $k_{F}$, and induces a novel antiferromagnetic nuclear spin helix with a spatial period $\pi/k_{F}$. Due to the feedback effect through the Overhauser field from the ordered nuclear spins, the transition temperature reaches up to several tens of mK. The nuclear spin helix, combining spin and charge degrees of freedom, results in a synthetic spin-orbit interaction, leading to nontrivial topology. In the presence of the proximity-induced superconductivity, this system has a potential to realize Majorana fermions without fine-tuning the external parameters, such as chemical potential and external magnetic field. Ref: [1] C.-H. Hsu, P. Stano, J. Klinovaja, and D. Loss, Phys. Rev. B 92, 235435 (2015).