Inhomogeneous phase stiffness in two-dimensional s-wave disordered superconductors

We study the effect of white-noise disorder on the local phase stiffness and thermodynamic properties of a two-dimensional s-wave superconductor. Starting from a local attractive model within the path-integral formalism, we derive a gauge-invariant effective action by separating the superconducting order parameter into amplitude and phase components. Perturbative treatment of the phase fluctuation sector leads to an effective phase-only XY model for disordered superconducting systems. Using Bogoliubov-de Gennes theory, we compute the distributions of nearest-neighbor couplings for various disorder strengths: a Gaussian-like single peak at weak disorder evolves into a bimodal structure with negative couplings at strong disorder. The local phase stiffness exhibits random spatial variations uncorrelated with pairing amplitudes. Monte Carlo simulations show that at strong disorder and low temperatures, the superfluid stiffness J_s increases anomalously with temperature, signaling possible glassy dynamics. The corresponding Edwards-Anderson order parameter further supports the emergence of a low-temperature phase-glass state.