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arXiv:2409.17914 · Communications on Pure and Applied Mathematics

The Huang–Yang Formula for the Low‐Density Fermi Gas: Upper Bound

Emanuela L. Giacomelli, Christian Hainzl, Phan Thành Nam, Robert Seiringer

Published Mar 13, 2026• 3 citations

Abstract

ABSTRACT We study the ground state energy of a gas of spin fermions with repulsive short‐range interactions. We derive an upper bound that agrees, at low density , with the Huang–Yang conjecture. The latter captures the first three terms in an asymptotic low‐density expansion, and in particular the Huang–Yang correction term of order . Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi‐bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero‐energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation.

DOI: 10.1002/cpa.70040

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ABSTRACT We study the ground state energy of a gas of spin fermions with repulsive short‐range interactions. We derive an upper bound that agrees, at low density , with the Huang–Yang conjecture. The latter captures the first three terms in an asymptotic low‐density expansion, and in particular the Huang–Yang correction term of order . Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi‐bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero‐energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation.

Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin

1/2

fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.

We compute an upper bound on the ground state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. Our bound can be viewed as the two-dimensional analogue of a formula derived by Huang and Yang in the three-dimensional case. It captures the first three terms in an asymptotic expansion for small

\varrho a^2

, where

1/2

denotes the density and

a

the scattering length of the interaction potential.

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)].

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(ρ)

is of the form \begin{equation*} e(ρ)=\frac{3}{5}(3π^2)^{\frac{2}{3}}ρ^{\frac{5}{3}}+2πaρ^2 +\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}π^{\frac{2}{3}}a^2ρ^{\frac{7}{3}} +o(ρ^{\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.