We study the effects of a finite density of negative-U centers (NUCs) on the surface of a threedimensional 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 creator and pair breaker, and the competition of the two effects results in non-monotonic 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.Introduction: Topological phases of matter have recently attracted much attention in condensed matter physics. One prominent example is the topological insulator (TI), which is insulating in the bulk but possesses metallic surface states with linear dispersion [1,2]. Such novel properties of TIs are protected by time reversal symmetry (TRS) and charge conservation symmetry (CCS). It is of both theoretical and practical importance to find ways to break these symmetries on the surface of TIs. Breaking TRS on the TI surface gives rise to topological magnetoelectric effect [3], and breaking CCS leads to the formation of Majorana zero modes at the superconducting vortices [4]. In practice, there are two ways to break the surface symmetries. One way is to fabricate heterostructures of TIs and other symmetry broken materials, e.g. magnetic insulators that break TRS, superconductors that break CCS. Proximity effect then induces symmetry breaking at the TI surface. Another way is to deposite certain types of impurities on the TI surface, which has the advantage of simple experimental setup and better tunability for both bulk materials and thin films. Depositing magnetic impurities on the TI surface to break TRS has been extensively studied both theoretically and experimentally (see e.g. [5][6][7][8]). Depositing impurities on the surface of TIs, or more generally Dirac materials including graphene, to generate pairing and break CCS was only proposed very recently by the present authors and collaborators [9,10].
