Spin-orbit coupling and proximity effects in metallic carbon nanotubes

We study the spin-orbit coupling in metallic carbon nanotubes (CNTs) within the many-body Tomonaga-Luttinger liquid framework. For a well-defined subclass of metallic CNTs, that contains both achiral zigzag as well as a subset of chiral tubes, an effective low-energy field theory description is derived. We aim to describe systems at finite dopings, but close to the charge neutrality point (commensurability). A new regime is identified where the spin-orbit coupling leads to an inverted hierarchy of minigaps of bosonic modes. We then add a proximity coupling to a superconducting (SC) substrate and show that the only order parameter that is supported within the spin-orbit induced phase is a topologically trivial s-SC.