The Huang-Yang formula for the low-density Fermi gas: upper bound

We study the ground state energy of a gas of spin (Formula presented.) fermions with repulsive short-range interactions. We derive an upper bound that agrees, at low density (Formula presented.), with the Huang–Yang conjecture. The latter captures the first three terms in an asymptotic low-density expansion, and in particular the Huang–Yang correction term of order (Formula presented.). Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation.