Generalized Fermi sea for plane-wave Hartree-Fock theory: One dimensional model calculation

The Hartree-Fock theory for many fermions with plane-wave orbitals but with abnormal occupation is studied. A one-dimensional problem with an interparticle interaction of finite range which saturates the $N$ body system at a fixed density and finite binding energy is solved explicitly. A direct variation of the resulting Hartree-Fock energy with respect to the additional variational parameters introduced is carried out numerically and results reminiscent of a gas-to-liquid phase transition are found.