When Does a Single Repulsive Dirac Cone Superconduct?

Superconductivity in a single two-dimensional Dirac cone offers a natural route to topological superconductivity. While usually considered extrinsic, arising from proximity to a conventional superconductor, we investigate when a doped Dirac cone can spontaneously develop superconductivity from a short-range repulsive interaction U via the Kohn-Luttinger mechanism.

We show that an ideal, linear Dirac cone is immune to pairing at leading order in U². Superconductivity instead emerges only through higher-order in k corrections to the dispersion, which are unavoidable in any lattice realization and crucially dictate the pairing symmetry. The form of the pairing thus reflects how the well-known obstruction to realizing a single Dirac cone on a lattice is circumvented.

When a Dirac cone arises from broken time-reversal symmetry, for instance, at a transition between Chern insulators or in a valley-polarized phase, we find a topological p - ip state whose chirality is opposite to that of the parent chiral metal above T_c. By contrast, for a surface Dirac cone of a 3D topological insulator, superconductivity is stabilized by anisotropies in the dispersion. For C_3v-symmetric warping, as in Bi2Te3, pairing is strongest when the Fermi surface becomes hexagonal, leading to order in the (d±id)×(p+ip) channel with accidental near-nodes. In the highly anisotropic limit v_x >> v_y, relevant to side surfaces of layered materials, the Fermi surface splits into two branches, and nesting favors a pairing symmetry Δ ~ sgn(k_x)cos(k_y) reminiscent of organic superconductors.