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arXiv:2301.08005 · Annales Henri Poincaré

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

Asbjørn Bækgaard Lauritsen

Published Jul 2, 2024• 4 citations
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Abstract

Abstract We 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 ).

DOI
10.1007/s00023-024-01450-1

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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)].