Equal-spin and opposite-spin density-density correlations in the BCS-BEC crossover: Gauge Symmetry, Pauli Exclusion Principle, Wick's Theorem and Experiments

We develop a general theory of spin-dependent density-density correlations, that is valid for any temperature, interactions, dimensions and mass or population status of Fermi gases with two internal states. We use gauge invariance and the Pauli principle to establish constraints on the spin-dependent density-density correlations that are consistent with the fluctuation-dissipation and Wick's theorem. As an example, we study the spin-dependent density-density correlations from the BCS to the Bose regime in two dimensions at zero temperature, inspired by experiments in 6Li. We show that two-particle irreducible contributions involving collective excitations, many-particle scattering and vertex corrections, are essential to describe experiments. In particular they turn out to be responsible for the emergence of an experimentally observed minimum in the opposite-spin density-density correlations.