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We present a standard field theoretical derivation of the dynamic density andspin linear response functions of a dilute superfluid Fermi gas in the BCS-BECcrossover in both three and two dimensions. The derivation of the responsefunctions is based on the elegant functional path integral approach whichallows us to calculate the density-density and spin-spin correlation functionsby introducing the external sources for the density and the spin density. Sincethe generating functional cannot be evaluated exactly, we consider two gaplessapproximations which ensure a gapless collective mode (Goldstone mode) in thesuperfluid state: the BCS-Leggett mean-field theory and theGaussian-pair-fluctuation (GPF) theory. In the mean-field theory, our resultsof the response functions agree with the known results from the random phaseapproximation. We further consider the pair fluctuation effects and establish atheoretical framework for the dynamic responses within the GPF theory. We showthat the GPF response theory naturally recover three kinds of famousdiagrammatic contributions: the Self-Energy contribution, the Aslamazov-Lakincontribution, and the Maki-Thompson contribution. We also show that unlike theequilibrium state, in evaluating the response functions, the linear(first-order) terms in the external sources as well as the induced orderparameter perturbations should be treated carefully. In the superfluid state,there is an additional order parameter contribution which ensures that in thestatic and long wavelength limit, the density response function recovers theresult of the compressibility (compressibility sum rule). We expect that the$f$-sum rule is manifested by the full number equation which includes thecontribution from the Gaussian pair fluctuations.

Dynamic density and spin responses of a superfluid Fermi gas in the BCS-BEC crossover: Path integral formulation and pair fluctuation theory