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

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

TL;DR

This work rigorously confirms the Huang–Yang three-term energy expansion for a dilute, spin-$\tfrac{1}{2}$ Fermi gas with short-range repulsive interactions by deriving a matching lower bound to a previously established upper bound. Central to the proof is a refined completion-of-the-square strategy applied to a correlation Hamiltonian obtained after a particle-hole transformation, with careful renormalizations that isolate the HY constant and the universal third-order term depending on the scattering length $a$. The method splits the interaction into $Vf$ and $V\varphi$ pieces and introduces renormalized operators $T,S,\mathcal{T},\mathcal{S}$ so that non-negative square terms can be dropped to yield a valid lower bound, while detailed a priori estimates control all error terms. The result demonstrates universality of the first three terms in the low-density limit and advances mathematical understanding of fermionic many-body energies in the dilute regime, with potential impact on rigorous analyses of related quantum many-body systems.

Abstract

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.

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

TL;DR

This work rigorously confirms the Huang–Yang three-term energy expansion for a dilute, spin- Fermi gas with short-range repulsive interactions by deriving a matching lower bound to a previously established upper bound. Central to the proof is a refined completion-of-the-square strategy applied to a correlation Hamiltonian obtained after a particle-hole transformation, with careful renormalizations that isolate the HY constant and the universal third-order term depending on the scattering length . The method splits the interaction into and pieces and introduces renormalized operators so that non-negative square terms can be dropped to yield a valid lower bound, while detailed a priori estimates control all error terms. The result demonstrates universality of the first three terms in the low-density limit and advances mathematical understanding of fermionic many-body energies in the dilute regime, with potential impact on rigorous analyses of related quantum many-body systems.

Abstract

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

Paper Structure

This paper contains 19 sections, 22 theorems, 446 equations.

Table of Contents

  1. Introduction and main result
  2. Key ingredients in the proof
  3. The correlation Hamiltonian
  4. Heuristic ideas
  5. Main propositions and the proof of Theorem \ref{['thm: main lw bd']}
  6. Preliminaries
  7. Scattering solutions
  8. Configuration space representation and $t$-integral bounds
  9. Some operator bounds
  10. Completion of the square for $V\varphi$
  11. Completion of the square for $Vf$
  12. Proof outline of Proposition \ref{['pro: completion square Vf']}
  13. Proof of Lemma \ref{['lem: Ta']}
  14. Proof of Lemma \ref{['lem: Sa']}
  15. Proof of Lemma \ref{['lem: T']}
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $V_\infty$ be as in Assumption asu: potential V, and let $a>0$ be its scattering length. In the low-density limit $\varrho_\uparrow+ \varrho_\downarrow = \varrho \rightarrow 0$, the ground state energy density defined in eq: def gs energy density satisfies where $F$ is defined in eq:def-F. In particular, for $\varrho_\uparrow = \varrho_\downarrow = \varrho/2$, eq:HY holds true.

Theorems & Definitions (44)

  • Theorem 1.2: Huang--Yang formula
  • Definition 2.1: Particle-hole transformation
  • Proposition 2.2: Conjugation of $\mathcal{H}$ by $R$
  • Lemma 2.3: Basic a priori estimates for $\mathbb{H}_0$, $\mathbb{Q}_4$, $\mathcal{N}$
  • Proposition 2.4: Improved a priori bounds
  • Definition 2.5: Periodic scattering solution
  • Remark 2.6
  • Definition 2.7: Renormalization terms for $\mathbb{Q}_2$
  • Definition 2.8: Renormalization terms for $\mathbb{Q}_3$
  • Proposition 2.9: Completion of the square for $V\varphi$
  • ...and 34 more