Theory of superconductivity in an ultra-dilute fermionic liquid in the proximity of a quantum phase transition.

We study the problem of superconductivity in an extremely weakly doped narrow gap semiconductor, driven by electron-phonon coupling in a non-adiabatic regime. We derive an analytic theory analogue to Eliashberg equations that shall allow us to determine the critical temperature. In the first stage, based on our recent correlated electron-ion dynamics study, we obtain the effective non-adiabatic electron-phonon coupling function. We then add long-range Coulomb type interactions to derive propagators for the emergent collective state. Since the electron-phonon coupling is highly anisotropic, in the second step we test whether coupling to cooperons is able to induce a thermodynamic phase transition. Finally, we discuss transport properties of the emergent superconducting condensate, in particular (magneto-)thermoelectricity driven by vortex states. Our study is applicable in various classes of novel superconducting phases in materials with extremely small carrier concentrations such as doped SrTiO, but also a mysterious case of elementary bismuth.