Paper detail

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

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