Structure of Superconducting State in Inhomogeneous Non-Fermi-Liquids

Understanding superconductivity in strongly correlated systems remains a central challenge, especially in high-temperature superconductors that exhibit pronounced spatial inhomogeneity of the superconducting gap. One manifestation of this inhomogeneity is the persistence of superconducting “puddles” well above the transition temperature - a phenomenon that has received theoretical attention only recently.

We develop a framework for systematically describing superconducting states of non-Fermi liquids with strong spatial inhomogeneity, extending existing approaches beyond translational invariance settings. Specifically, we investigate a disordered metal coupled to localized fluctuations of a non-conserved order parameter near a disorder-smeared quantum critical point and demonstrate the emergence of a highly inhomogeneous superconducting phase.

In the normal state, electron scattering off disordered order-parameter fluctuations produces marginal Fermi-liquid behavior and localization of these fluctuations. By incorporating pairing mediated by the same modes, we derive a modified Usadel equation for the two-particle propagator, in which localized bosonic modes induce a correlated random potential and random pairing vertex. Analysis across regimes—from disorder-averaged limits to rare superconducting puddles—reveals distinct disorder-driven superconducting behaviors and clarifies their microscopic origins.