Energetics of a strongly correlated Fermi gas

The energy of the two-component Fermi gas with the s-wave contact interaction is a simple linear functional of its momentum distribution:Einternal=ℏ2ΩC/4πam+∑kσ(ℏ2k2/2m)(nkσ-C/k4)where the external potential energy is not included, a is the scattering length, Ω is the volume, nkσ is the average number of fermions with wave vector k and spin σ, and C≡limk→∞k4nk↑=limk→∞k4nk↓. This result is a universal identity. Its proof is facilitated by a novel mathematical idea, which might be of utility in dealing with ultraviolet divergences in quantum field theories. Other properties of this Fermi system, including pair correlations and the dimer-fermion scattering length, are also studied.