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Collective behaviors of an electron gas in the mean-field regime

Dong Hao Ou Yang

TL;DR

The paper derives a rigorous mean-field description of the momentum distribution for an electron gas on a 3D torus in the high-density limit, using a trial state $\Psi_N=e^{-\mathcal{K}}\Psi_{FS}$ built on a Slater determinant. It splits the momentum occupation into bosonization and exchange-correlation components, obtaining explicit formulas and sharp error bounds that hold for a broad class of singular interactions with square-summable Fourier modes. A bootstrap argument controls the key auxiliary quantity $\mathcal{Q}$, leading to $\mathcal{Q}=O(k_F^{-1/2+\delta})$, and yielding overall error $\mathcal{O}(k_F^{-3/2+\delta})$ in the momentum observable $n(f)$. The results generalize prior high-density/metallic-density analyses (Daniel–Vosko, Lam) to a wider potential class, connect with recent work on mean-field momentum distribution, and provide a rigorous framework for understanding collective excitations and exchange effects beyond the Hartree–Fock picture in the mean-field regime.

Abstract

In this paper, we study the momentum distribution of an electron gas in a $3$-dimensional torus. The goal is to compute the occupation number of Fourier modes for some trial state obtained through random phase approximation. We obtain the mean-field analogue of momentum distribution formulas for electron gas in [Daniel and Voskov, Phys. Rev. 120, (1960)] in high density limit and [Lam, Phys. Rev. \textbf{3}, (1971)] at metallic density. Our findings are related to recent results obtained independently by Benedikter, Lill and Naidu, and our analysis applies to a general class of singular potentials rather than just the Coulomb case.

Collective behaviors of an electron gas in the mean-field regime

TL;DR

The paper derives a rigorous mean-field description of the momentum distribution for an electron gas on a 3D torus in the high-density limit, using a trial state built on a Slater determinant. It splits the momentum occupation into bosonization and exchange-correlation components, obtaining explicit formulas and sharp error bounds that hold for a broad class of singular interactions with square-summable Fourier modes. A bootstrap argument controls the key auxiliary quantity , leading to , and yielding overall error in the momentum observable . The results generalize prior high-density/metallic-density analyses (Daniel–Vosko, Lam) to a wider potential class, connect with recent work on mean-field momentum distribution, and provide a rigorous framework for understanding collective excitations and exchange effects beyond the Hartree–Fock picture in the mean-field regime.

Abstract

In this paper, we study the momentum distribution of an electron gas in a -dimensional torus. The goal is to compute the occupation number of Fourier modes for some trial state obtained through random phase approximation. We obtain the mean-field analogue of momentum distribution formulas for electron gas in [Daniel and Voskov, Phys. Rev. 120, (1960)] in high density limit and [Lam, Phys. Rev. \textbf{3}, (1971)] at metallic density. Our findings are related to recent results obtained independently by Benedikter, Lill and Naidu, and our analysis applies to a general class of singular potentials rather than just the Coulomb case.
Paper Structure (20 sections, 35 theorems, 285 equations)

This paper contains 20 sections, 35 theorems, 285 equations.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. Trial states
  4. Main result
  5. Preliminaries
  6. Quasi-bosonic operators
  7. Useful identities
  8. Momentum distribution: Bosonization and exchange contributions
  9. Momentum distribution from bosonization
  10. Momentum distribution from exchange correlation
  11. Generic structure of error terms
  12. Analysis on error terms
  13. Preliminary estimates
  14. Analysis on errors in $\mathcal{W}_{s_{1},s_{2}}^{(j_{1},j_{2})}(\varepsilon_{k})$
  15. Analysis on $\mathcal{W}_{s_{1},s_{2}}^{(j_{1},j_{2})}(\mathcal{G}_{k})$ for $(j_{1},j_{2})\in\Sigma_{*}^{1}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume $N=|B_{F}|=4\pi k_{F}^{3}/3$ and suppose $V$ satisfies Then, for each observable $f\in\ell^{\infty}(\mathbb{Z}^{3})$ such that $f(-\xi)=f(\xi)$, the expectation $n(f)$ in the trial state $\Psi_{N}$ constructed in trial-N is given where we have the bosonization contribution the exchange contribution and some error term $\mathcal{E}(\xi)$. In above, $\mathcal{D}_{k,\xi}:=\{\pm\xi,k\pm\xi\

Theorems & Definitions (70)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Lemma 2.1: CHN-23, Lemma 1.3
  • Proposition 2.1: CHN-23, Proposition A.2
  • ...and 60 more