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Particle-Hole Pair Localization on the Fermi Surface and its Impact on the Correlation Energy

Niels Benedikter

Abstract

In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface at bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.

Particle-Hole Pair Localization on the Fermi Surface and its Impact on the Correlation Energy

Abstract

In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface at bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.

Paper Structure

This paper contains 10 sections, 20 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction and Main Result
  2. Particle-Hole Transformation and Hartree-Fock Theory
  3. Almost-Bosonic Collective Operators
  4. Almost-Bosonic Quasifree Trial State
  5. Calculating the Interaction Energy
  6. Estimating the Error Terms.
  7. Calculating the Kinetic Energy
  8. Estimating the Error Terms.
  9. Optimizing the Trial State
  10. Estimating the Error Terms

Key Result

Theorem 1.1

Assume that $\hat{V}(k)$ is such that the quantities $A_1$ through $A_5$ given in Lemma lem:collectederrors are all finite, and also $\sum_{k \in \mathbb{Z}^3} \lvert \hat{V}(k)\rvert$ is finite. (For example, $\hat{V}$ with compact support.) Then we have where $\omega$ is the projection onto the $N$ plane waves $f_k$ with smallest $\lvert k\rvert$ ($k \in \mathbb{Z}^3$), and The infimum in eq:i

Figures (1)

  • Figure 1: The two balls represent the projections $e^{ikx}\omega e^{-ikx}$ and $\omega$, respectively. The normalization constant $n_k^2$ is given by the number of lattice points in the lense marked in gray.

Theorems & Definitions (38)

  • Theorem 1.1: Main Result
  • proof : Proof of Main Result
  • Corollary 1.2: Second Order in the Potential
  • proof
  • Theorem 1.3: Optimal Correlation Energy BNP+20BNP+21BPSS23
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Normalization Constant
  • ...and 28 more