An optimal lower bound for the low density Fermi gas in three dimensions

We consider a dilute Fermi gas in three dimensions interacting through a positive, radially symmetric, compactly supported, and integrable potential in the thermodynamic limit. We establish a second-order lower bound for the ground-state energy density, with an error term that is optimal in the sense that it matches the order of the next correction term conjectured by Huang and Yang [Phys. Rev. 105, 767-775 (1957)]. Although a first rigorous derivation of the Huang-Yang formula has recently been obtained in the combined papers [Giacomelli et al. CPAM, e70040 (2026), Giacomelli et al. arXiv:2505.22340], the present work takes a different approach, inspired by the construction of suitable trial states introduced in Giacomelli et al. [CPAM, e70040 (2026)]. In particular, we provide a simple proof that is effective enough to yield the optimal error bound, by adapting the argument originally used to establish the upper bound on the energy density in Giacomelli et al. [CPAM, e70040 (2026)].