Negative- U superconductivity on the surface of topological insulators

We study the effects of a finite density of negative-U centers (NUCs) on the surface of a three-dimensional topological insulator. The surface Dirac fermions mediate a power-law interaction among the local Cooper pairs at the NUCs, and the interaction remains long-ranged for weak disorder. Superconductivity can be generated in the presence of a random distribution of NUCs. The NUCs play dual roles as both pair creators and pair breakers, and the competition of the two effects results in a nonmonotonic dependence of the mean-field superconducting transition temperature on the density of NUCs. Global phase coherence is established through coupling the locally superconducting puddles via Josephson coupling. Rare fluctuations play important roles, and a globally superconducting phase can only be achieved at large enough concentration of NUCs. The p- wave component of the superconducting order parameter gives rise to frustration among the superconducting grains, which is captured by a Potts-XY type model. New phases with chiral order, glass order, and, possibly, topological order can then emerge in the system of superconducting grains.