Almost Optimal Upper Bound for the Ground State Energy of a Dilute Fermi Gas via Cluster Expansion

We prove an upper bound on the energy density of the dilute spin- \(\frac{1}{2}\) $\frac{1}{2}$ Fermi gas capturing the leading correction to the kinetic energy \(8\pi a \rho _\uparrow \rho _\downarrow \) $8\pi a{\rho }_{\uparrow }{\rho }_{\downarrow }$ with an error of size smaller than \(a\rho ^{2}(a^3\rho )^{1/3-\varepsilon }\) $a{\rho }^2{(a^3\rho )}^{1/3-\epsilon }$ for any \(\varepsilon > 0\) $\epsilon >0$ , where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).