Small two-component Fermi gases in a cubic box with periodic boundary conditions

The properties of two-component Fermi gases become universal if the interspecies s-wave scattering length $a_s$ and the average interparticle spacing are much larger than the range of the underlying two-body potential. Using an explicitly correlated Gaussian basis set expansion approach, we determine the eigen energies of two-component Fermi gases in a cubic box with periodic boundary conditions as functions of the interspecies s-wave scattering length and the effective range of the two-body potential. The universal properties of systems consisting of up to four particles are determined by extrapolating the finite-range energies to the zero-range limit. We determine the eigen energies of states with vanishing and finite momentum. In the weakly-attractive BCS regime, we analyze the energy spectra and degeneracies using first-order degenerate perturbation theory. Excellent agreement between the perturbative energy shifts and the numerically determined energies is obtained. For the infinitely large scattering length case, we compare our results - where available - with those presented in the literature.