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Quenched disorder and the BCS-BEC crossover in the Hubbard model

We study the impact of weak quenched disorder on the BCS-BEC crossover in the Hubbard model within a functional-integral approach. By deriving the thermodynamic potential up to second order in both the disorder potential and pairing fluctuations, we obtain self-consistent expressions for the number equation, condensate fraction, superfluid fraction and sound speed at zero temperature. In the dilute BEC limit, our results analytically reproduce the known continuum limits of weakly-interacting bosons, where weak disorder depletes the superfluid more strongly than the condensate due to broken translational symmetry, and enhances the sound speed through the overcompensation of the static compressibility. These findings establish a unified and controlled approach for describing the BCS-BEC crossover in disordered lattice models, and they provide a foundation for future extensions to finite temperatures and multiband Hubbard models.