Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

An optimal lower bound for the low density Fermi gas

Emanuela L. Giacomelli

Published Apr 1, 2026• 0 citations
Publisher

Abstract

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)].

DOI
10.1063/5.0309369

Related papers

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

Emanuela L. Giacomelli

We consider the 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 which is optimal in the sense that it matches the order of the next correction term conjectured by Huang-Yang in 1957.

arXiv:2410.08904•Oct 11, 2024•0 citations

The Huang-Yang conjecture for the low-density Fermi gas

Emanuela L. Giacomelli, Christian Hainzl, Phan Thành Nam +1 more

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.

arXiv:2505.22340•May 28, 2025•0 citations

The free energy of dilute Bose gases at low temperatures

Florian Haberberger, Christian Hainzl, Phan Thành Nam +2 more

We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.

arXiv:2304.02405•Apr 5, 2023•3 citations

A new second-order upper bound for the ground state energy of dilute Bose gases

Giulia Basti, Serena Cenatiempo, Benjamin Schlein

Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb {R}^3)$ , gives better rates and, in our opinion, is substantially simpler.

arXiv:2101.06222•Forum of Mathematics Sigma•Jan 1, 2021•44 citations

The Huang--Yang formula for a two-dimensional Fermi gas: upper bound

Christian Hainzl, Fabian Saxler, Robert Seiringer

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 $\varrho$ denotes the density and $a$ the scattering length of the interaction potential.

arXiv:2606.03356•Jun 2, 2026•0 citations

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

Emanuela L. Giacomelli, Christian Hainzl, Phan Thành Nam +1 more

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.

arXiv:2409.17914•Communications on Pure and Applied Mathematics•Mar 13, 2026•3 citations
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)].
Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin 1/21/21/2 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.
Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported V∈L3(R3)V \in L^3 (\mathbb {R}^3)V∈L3(R3) , gives better rates and, in our opinion, is substantially simpler.
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 ϱa2\varrho a^2ϱa2, where ϱ\varrhoϱ denotes the density and aaa the scattering length of the interaction potential.
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.