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

Abstract

AbstractWe prove an upper bound on the energy density of the dilute spin-$$\frac{1}{2}$$ 1 2 Fermi gas capturing the leading correction to the kinetic energy $$8\pi a \rho _\uparrow \rho _\downarrow $$ 8 π a ρ ↑ ρ ↓ with an error of size smaller than $$a\rho ^{2}(a^3\rho )^{1/3-\varepsilon }$$ a ρ 2 ( a 3 ρ ) 1 / 3 - ε for any $$\varepsilon > 0$$ ε > 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).