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Momentum Distribution of a Fermi Gas in the Random Phase Approximation

Niels Benedikter, Sascha Lill

TL;DR

The paper rigorously analyzes the momentum distribution of a three-dimensional Fermi gas in the mean-field scaling limit using the random phase approximation implemented via bosonization of particle--hole pairs. A carefully constructed trial state $\psi_N = \mathfrak{R} T \Omega$ combining a particle--hole transformation and an almost-bosonic Bogoliubov transform yields an explicit, near-ground-state energy expansion and a computable bosonized momentum distribution. The authors prove a jump in the momentum distribution at a universal Fermi momentum, provide a precise bound on the quasiparticle weight decline, and establish that RPA-type correlations suffice to capture a nontrivial Fermi liquid phase in this setting. This work provides a rigorous bridge between HF theory and beyond-HF physics in a controlled mean-field regime, validating RPA as a reliable tool for identifying Fermi liquid behavior in fermionic many-body systems.

Abstract

We consider a system of interacting fermions on the three-dimensional torus in a mean-field scaling limit. Our objective is computing the occupation number of the Fourier modes in a trial state obtained through the random phase approximation (in its collective bosonization formulation) for the ground state. We prove that the trial state's momentum distribution has a jump discontinuity, i.e., a well-defined Fermi surface. Moreover the Fermi momentum does not depend on the interaction potential (it is universal). Our result shows that the random phase approximation in the mean-field scaling limit is in principle sufficiently precise to identify a non-trivial Fermi liquid phase.

Momentum Distribution of a Fermi Gas in the Random Phase Approximation

TL;DR

The paper rigorously analyzes the momentum distribution of a three-dimensional Fermi gas in the mean-field scaling limit using the random phase approximation implemented via bosonization of particle--hole pairs. A carefully constructed trial state combining a particle--hole transformation and an almost-bosonic Bogoliubov transform yields an explicit, near-ground-state energy expansion and a computable bosonized momentum distribution. The authors prove a jump in the momentum distribution at a universal Fermi momentum, provide a precise bound on the quasiparticle weight decline, and establish that RPA-type correlations suffice to capture a nontrivial Fermi liquid phase in this setting. This work provides a rigorous bridge between HF theory and beyond-HF physics in a controlled mean-field regime, validating RPA as a reliable tool for identifying Fermi liquid behavior in fermionic many-body systems.

Abstract

We consider a system of interacting fermions on the three-dimensional torus in a mean-field scaling limit. Our objective is computing the occupation number of the Fourier modes in a trial state obtained through the random phase approximation (in its collective bosonization formulation) for the ground state. We prove that the trial state's momentum distribution has a jump discontinuity, i.e., a well-defined Fermi surface. Moreover the Fermi momentum does not depend on the interaction potential (it is universal). Our result shows that the random phase approximation in the mean-field scaling limit is in principle sufficiently precise to identify a non-trivial Fermi liquid phase.
Paper Structure (21 sections, 23 theorems, 257 equations, 3 figures)

This paper contains 21 sections, 23 theorems, 257 equations, 3 figures.

Table of Contents

  1. Introduction and Main Result
  2. Comparison with the Physics Literature
  3. Construction of the Trial State
  4. The trial state
  5. The particle--hole transformation
  6. Particle--hole pair operators
  7. The almost-bosonic Bogoliubov transformation
  8. Optimality of the Main Result
  9. Strategy of Proof of the Main Theorem
  10. The Bootstrap
  11. Post-Processing
  12. Generalized Pair Operators
  13. Momentum Distribution from Bosonization
  14. Controlling the Bosonization Error
  15. Controlling the Bosonized Terms
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that the Fourier transform $\hat{V}$ of the interaction potential is non-negative and compactly supported. Then, there exists a sequence of trial states $\psi_N \in L_{\textnormal{a}}^2(\mathbb{T}^{3N})$ with particle numbers $N$ corresponding to completely filled Fermi balls as in eq:fermiba The error term $\mathcal{E}$ is bounded by In the trial state $\psi_N$ which we construct in sec:t

Figures (3)

  • Figure 1: Patches on the Fermi ball in momentum space, with patch $B_{\alpha_q}$ including $q$.
  • Figure 2: Close-up of a patch: in \ref{['eq:edgeofthepatch']}, $q_1$ is an included momentum, whereas $q_2$ and $q_3$ are excluded.
  • Figure 3: Left: Reflecting part of $\mathcal{C}^q$ renders $\mathcal{D}^q$. Right: The integration range for a fixed $|k|$ in spherical coordinates.

Theorems & Definitions (44)

  • Theorem 1.1: Main Result
  • Theorem 1.2: Jump at the Fermi Surface
  • Proposition 2.1: Optimality
  • Theorem 3.1: Bosonized Momentum Distribution
  • Proposition 3.2
  • Lemma 4.1: Generalized approximate CCR
  • proof
  • Lemma 4.2: Occupation number of a single mode in $c^*_\alpha(k)$
  • proof
  • Lemma 4.3: Approximate CCR
  • ...and 34 more