Pairing around a Single Dirac Point: A Unifying View of Kohn-Luttinger Superconductivity in Chern Bands, Quarter Metals, and Topological Surface States

Abstract

Superconductivity of a single two-dimensional Dirac fermion 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 \emph{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 $U^2$. 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 \ce{Bi2Te3}, pairing is strongest when the Fermi surface becomes hexagonal, leading to order in the $(d\pm id)\times (p+ip)$ channel with accidental near-nodes. In the highly anisotropic limit $v_x\gg v_y$, relevant to side surfaces of layered materials, the Fermi surface splits into two branches, and nesting favors a pairing symmetry $\Delta \sim \mathrm{sgn}(k_x)\cos (k_y)$ reminiscent of organic superconductors.

17 pages, 9 figures

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