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arXiv:2505.23136

The second order Huang-Yang approximation to the Fermi thermodynamic pressure

Xuwen Chen, Jiahao Wu, Zhifei Zhang

Published May 29, 2025• 0 citations

Abstract

We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length $\mathfrak{a}_0$ at temperature $T>0$. We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature $T<ρ^{\frac{2}{3}+\frac{1}{6}}$, which is, by scaling, also necessary for the Huang-Yang formula to hold. Here, $T_F\simρ^{\frac{2}{3}}$ is the Fermi temperature. We also establish during the course of the proof, a conjecture regarding the second order approximation of density $ρ$ by R. Seiringer \cite{FermithermoTpositive}. Our proof uses frequency localization techniques from the analysis of nonlinear PDEs and does not involve spatial localization or Bosonization. In particular, our method covers the classical Huang-Yang formula at zero temperature.

DOI: 10.48550/arxiv.2505.23136

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We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length

\mathfrak{a}_0

at temperature

T>0

. We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature

T<ρ^{\frac{2}{3}+\frac{1}{6}}

, which is, by scaling, also necessary for the Huang-Yang formula to hold. Here,

T_F\simρ^{\frac{2}{3}}

is the Fermi temperature. We also establish during the course of the proof, a conjecture regarding the second order approximation of density

ρ

by R. Seiringer \cite{FermithermoTpositive}. Our proof uses frequency localization techniques from the analysis of nonlinear PDEs and does not involve spatial localization or Bosonization. In particular, our method covers the classical Huang-Yang formula at zero temperature.

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.

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)].

We consider a dilute gas in 3D composed of two species of bosons interacting through positive inter-species and intra-species pairwise potentials. We prove a second order expansion for the energy density in the thermodynamic limit. For the case of compactly supported, integrable potentials, we derive the correct second order of the expansion. If we make the further assumption of having soft potentials, we also derive the correct coefficient of the second order and the resulting formula is coherent with the physics literature. If we let the density and scattering length of one of the species go to zero, we obtain the Lee-Huang-Yang formula for one species of bosons. The paper also contains a proof of BEC for a mixture of bosons in a box with length scale larger than the Gross-Pitaevskii one.

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.