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 $\rho_{\bf r}$ and its conjugate variable, the phase $\theta_{\bf r}$ of the pairing order parameter $\Delta_{\bf r}$. We recover the standard action of a Bose superfluid of density $\rho_{\bf r}/2$, where the bosons have a mass $m_B=2m$ and interact {\it via} a repulsive contact potential with amplitude $g_B=4\pi a_B/m_B$, $a_B=2a$ ($a$ 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 $J=4t^2/U$ ($t$ and $-U$ are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model, $z$ the number of nearest-neighbor sites).