  We show that a one-dimensional topological superconductor can be realized in
carbon nanotubes, using a relatively small magnetic field. Our analysis relies
on the intrinsic curvature-enhanced spin-orbit coupling of the nanotubes, as
well as on the orbital effect of a magnetic flux threaded through the nanotube.
Tuning experimental parameters, we show that a half-metallic state may be
induced in the nanotube. Coupling the system to an Ising superconductor, with
an appreciable spin-triplet component, can then drive the nanotube into a
topological superconducting phase. The proposed scheme is investigated by means
of real-space tight-binding simulations, accompanied by an effective continuum
low-energy theory, which allows us to gain some insight on the roles of
different terms in the Hamiltonian. We calculate the topological phase diagram
and ascertain the existence of localized Majorana zero modes near the edges.
Moreover, we find that in the absence of a magnetic field, a regime exists
where sufficiently strong interactions drive the system into a
time-reversal-invariant topological superconducting phase.
