Superconductivity in the ultra-low density limit

According to standard theory, systems with a low concentration of charge carriers, such as semiconductors and semimetals, are not expected to exhibit superconductivity. Their density of states is very small and their Fermi energy is of the same order as the Debye frequency of the crystal. Nonetheless, superconductivity is observed in many low-density systems. In this talk I will give an overview of the theoretical challenges in understanding this phenomena and discuss possible resolutions. I will then focus on a particular scenario, a crystal close to a ferroelectric quantum critical point, in which our theory predicts superconductivity at arbitrary low density.