We present ground state calculations for low-density Fermi gases described by two model interactions, an attractive square-well potential and a Lennard-Jones potential, of varying strength. We use the optimized Fermi-Hypernetted Chain integral equati
on method which has been proved to provide, in the density regimes of interest here, an accuracy better than one percent. We first examine the low-density expansion of the energy and compare with the exact answer by Huang and Yang (H. Huang and C. N. Yang, {em Phys. Rev./} {bf 105}, 767 (1957)). It is shown that a locally correlated wave function of the Jastrow-Feenberg type does not recover the quadratic term in the expansion of the energy in powers of $a0KF$, where $a0$ is the vacuum $s$-wave scattering length and $KF$ the Fermi wave number. The problem is cured by adding second-order perturbation corrections in a correlated basis. Going to higher densities and/or more strongly coupled systems, we encounter an instability of the normal state of the system which is characterized by a divergence of the {em in-medium/} scattering length. We interpret this divergence as a phonon-exchange driven dimerization of the system, similar to what one has at zero density when the vacuum scattering length $a0$ diverges. We then study, in the stable regime, the superfluid gap and its dependence on the density and the interaction strength. We identify two different corrections to low-density expansions: One is medium corrections to the pairing interaction, and the other one finite-range corrections. We show that the most important finite-range corrections are a direct manifestation of the many-body nature of the system.
We present a theoretical study of the dynamic structure function of a resonantly interacting two-component Fermi gas at zero temperature. Our approach is based on dynamic many-body theory able to describe excitations in strongly correlated Fermi syst
ems. The fixed-node diffusion Monte Carlo method is used to produce the ground-state correlation functions which are used as an input for the excitation theory. Our approach reproduces recent Bragg scattering data in both the density and the spin channel. In the BCS regime, the response is close to that of the ideal Fermi gas. On the BEC side, the Bose peak associated with the formation of dimers dominates the density channel of the dynamic response. When the fraction of dimers is large our theory departs from the experimental data, mainly in the spin channel.
