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 Tc. By contrast, for a surface Dirac cone of a 3D topological insulator, superconductivity is stabilized by anisotropies in the dispersion. For C₃ᵥ-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 vₓ vy, relevant to side surfaces of layered materials, the Fermi surface splits into two branches, and nesting favors a pairing symmetry Δ~ sgn(kₓ)cos(ky) reminiscent of organic superconductors.