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Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential

Niels Benedikter, Marcello Porta, Benjamin Schlein, Robert Seiringer

Abstract

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only $|\cdot| \hat{V} \in \ell^1 (\mathbb{Z}^3)$. Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.

Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential

Abstract

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only . Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.

Paper Structure

This paper contains 16 sections, 27 theorems, 324 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Correlation Hamiltonian
  3. Strategy of the Proof: Approximate Bosonization
  4. A--Priori Estimates on Excitations of the Fermi Ball
  5. The case $p^2 - (p-k)^2 \geq 4 N^{1/3}$.
  6. The case $p^2 - (p-k)^2 < 4 N^{1/3}$.
  7. Patch Decomposition and Almost Bosonic Operators
  8. Reduction to an Almost Bosonic Quadratic Hamiltonian
  9. Approximate Bogoliubov Transformations
  10. Bound for $\| \widetilde{O}-1 \|_{\textnormal{HS}}$.
  11. Bound for ${\left| \left|O-1\right| \right|}_{\textnormal{HS}}$.
  12. Linearization of the Kinetic Energy
  13. Proof of \ref{['thm:main']}
  14. Lower bound.
  15. Upper bound.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose $V \in L^1 (\mathbb{T}^3)$ with $\hat{V} \geq 0$ and For $k_\textnormal{F} > 0$ let $N := | B_\textnormal{F} | = |\{ k \in \mathbb{Z}^3 : |k| \leq k_\textnormal{F} \}|$. Then there exists $\alpha > 0$ such that where the RPA energy formula is

Figures (2)

  • Figure 1: Decomposition of (a shell around) the Fermi surface into patches. The vectors $\hat{\omega}_\alpha$ (marked with dots) are the patch centers. The decomposition of the southern half sphere is obtained through reflection by the origin. See BNPSS0 for the details of the construction.
  • Figure 2: Illustration for the condition $N^{2\delta}R^2 \ll M$ of \ref{['lem:counting']}. The angle between patch center and patch boundary is $\theta_1 \sim 1/\sqrt{M}$. The angle between the tangent at the center and at the boundary is $\theta_2 = \theta_1$ by elementary geometry. We know $k\cdot\hat{\omega}_\alpha \geq N^{-\delta}$ by definition of $\mathcal{I}_k$. This means that the angle between $k$ and the tangent at the center (being perpendicular to $\omega_\alpha$) is at least of order $\sim N^{-\delta}/R$. To have $k$ pointing from the inside to the outside of the Fermi ball even at the boundary we need $N^{-\delta}/R \gg 1/\sqrt{M}$.

Theorems & Definitions (51)

  • Theorem 1.1: Main result: RPA correlation energy
  • Lemma 4.1: A--priori bound on kinetic energy
  • Remark
  • proof : Proof of \ref{['lm:H0-apri']}
  • Corollary 4.2: A--priori bounds on particle number
  • proof
  • Lemma 4.3: Kinetic bound on particle--hole pairs
  • Proposition 4.4: Lattice points in convex bodies, Hux
  • Remark
  • proof : Proof of \ref{['lm:kinetic']}
  • ...and 41 more