Correlations in the low-density Fermi gas: Fermi-liquid state, dimerization, and Bardeen-Cooper-Schrieffer pairing

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 equation 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, Phys. Rev. 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 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