Domes of <i>T_c</i> in single and multiband dilute superconductors with extended attractive interactions

One of the common features shared among many families of superconductors is the appearance of a superconducting dome as a function of doping. Although this is often associated with quantum critical points, the connection is not clear for dilute superconductors such as strontium titanate. In an effort to explain experimental signatures such as the weak dependence of T_c on the carrier density, the emergence of multiple domes, and the very dilute regime over which superconductivity is observed in both 2D and 3D, we propose a toy model of electrons coupled via a spatially extended interaction. Using a combination of mean field theory and functional renormalization group to account for competing orders, we show that for a characteristic interaction range L, there exists a peak in T_c around k_FL∼1. Our analysis holds in 2D and 3D systems and predicts the existence of multiple domes in the vicinity of Lifshitz transitions for multiband materials.