  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.
