Effective action for superfluid Fermi systems in the strong-coupling limit

We derive the low-energy effective action for three-dimensional superfluid Fermi systems in the strong-coupling limit, where superfluidity originates from Bose-Einstein condensation of composite bosons. Taking into account density and pairing fluctuations on the same footing, we show that the effective action involves only the fermion density ρ_r and its conjugate variable, the phase θ_r of the pairing order parameter Δ_r. We recover the standard action of a Bose superfluid of density ρ_r∕2, where the bosons have a mass m_B=2m and interact via a repulsive contact potential with amplitude g_B=4πa_B∕m_B,a_B=2a (a the s-wave scattering length associated to the fermion-fermion interaction in vacuum). For lattice models, the derivation of the effective action is based on the mapping of the attractive Hubbard model onto the Heisenberg model in a uniform magnetic field, and a coherent state path integral representation of the partition function. The effective description of the Fermi superfluid in the strong-coupling limit is a Bose-Hubbard model with an intersite hopping amplitude t_B=J∕2 and an on-site repulsive interaction U_B=2Jz, where J=4t²∕U (t and −U are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model, z the number of nearest-neighbor sites).