Quantum Fluctuations and Multifractally enhanced Superconductivity in Disordered Thin Films

The interplay between electron-electron interactions and weak localization (or anti-localization) phenomena in two-dimensional systems can significantly enhance the superconducting transition temperature. We develop the theory of quantum fluctuations within such multifractally enhanced superconducting states in thin films. In conditions of weak disorder, we employ the Finkel'stein nonlinear sigma model to derive an effective action for the superconducting order parameter and the quasiclassical Green's function, meticulously accounting for the influence of quantum fluctuations. This effective action, applicable for interactions of any strength, reveals the critical role of well-known collective modes in a dirty superconductor, and its saddle-point analysis leads to modified Usadel and gap equations. These equations comprehensively incorporate the renormalizations stemming from the interplay between interactions and disorder, resulting in the non-trivial energy dependence of the gap function. Notably, our analysis establishes a direct relation between the self-consistent gap equation at the superconducting transition temperature and the known renormalization group equations for interaction parameters in the normal state.