The second order Huang-Yang formula to the 3D Fermi gas: the Gross-Pitaevskii regime

Abstract: For a system of $N$ Fermions of spin $1/2$, with its interaction potential of scattering length $a$, the classical Huang-Yang formula states that the energy density $e(\rho)$ is of the form \begin{equation*} e(\rho)=\frac{3}{5}(3\pi^2)^{\frac{2}{3}}\rho^{\frac{5}{3}}+2\pi a\rho^2 +\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}\pi^{\frac{2}{3}}a^2\rho^{\frac{7}{3}} +o(\rho^{\frac{7}{3}}). \end{equation*} We consider a general system of $N$ Fermions of spin $1/\mathbf{q}$ with the scattering length of the interaction potential at the scale $a\propto N^{-1}$, that is, in the Gross-Pitaevskii regime. We prove the 2nd order ground state energy approximation corresponding to the Huang-Yang formula. The thermodynamic limit case shares a similar logic, and could be dealt with in a separate paper.