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 ${\ensuremath{\rho}}_{\mathbf{r}}$ and its conjugate variable, the phase ${\ensuremath{\theta}}_{\mathbf{r}}$ of the pairing order parameter ${\ensuremath{\Delta}}_{\mathbf{r}}$. We recover the standard action of a Bose superfluid of density ${\ensuremath{\rho}}_{\mathbf{r}}∕2$, where the bosons have a mass
mB=2m and interact via a repulsive contact potential with amplitude ${g}_{B}=4\ensuremath{\pi}{a}_{B}∕{m}_{B},{a}_{B}=2a$ (
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
tB=J∕2 and an on-site repulsive interaction
UB=2Jz, where
J=4t2∕U (
t and $\ensuremath{-}U$ are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model,
z the number of nearest-neighbor sites).
We investigate the quantum and thermal phase diagram of fermionic polar molecules loaded in a bilayer trapping potential with perpendicular dipole moment. We use both a BCS-theory approach that is most reliable at weak coupling and a strong-coupling approach that considers the two-body bound dimer states with one molecule in each layer as the relevant degree of freedom. The system ground state is a Bose-Einstein condensate (BEC) of dimer bound states in the low-density limit and a paired superfluid (BCS) state in the high-density limit. At zero temperature, the intralayer repulsion is found to broaden the regime of BCS-BEC crossover and can potentially induce system collapse through the softening of roton excitations. The BCS theory and the strongly coupled dimer picture yield similar predictions for the parameters of the crossover regime. The Berezinskii-Kosterlitz-Thouless transition temperature of the dimer superfluid is also calculated. The crossover can be driven by many-body effects and is strongly affected by the intralayer interaction which was ignored in previous studies.