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Ground state energy of a dilute inhomogeneous Fermi gas

Thomas Gamet

TL;DR

The paper analyzes the ground-state energy of a dilute, two-spin Fermi gas in a fixed inhomogeneous trap under a short-range interaction with a diluteness scaling. It combines a semiclassical Thomas–Fermi framework with a careful partitioning of space and a Dyson lemma-based regularization to derive matching upper and lower bounds, yielding E(N) = N E^{\mathrm{TF}} + 2\pi a_w N^{1/3-\beta} \int (\rho^{\mathrm{TF}})^2 + o(N^{4/3-\beta}) for 1/3 < \beta < 34/81. The analysis hinges on proving convergence of the many-body densities to the TF minimizer, including a detailed treatment of the interaction term via semiclassical approximations and density regularization. The results extend the understanding of how Thomas–Fermi theory emerges in inhomogeneous, dilute Fermi systems and quantify the leading-order interaction correction through the scattering length $a_w$, with potential extensions to other scalings by replacing $a_w$ with the interaction range. The work has implications for predicting ground-state energies in trapped ultracold Fermi gases and clarifies the role of short-range correlations in the dilute regime.

Abstract

We study the ground state energy of a system of N fermions with two spin states in the large N limit. The particles are placed in an inhomogeneous trapping potential and interact via scaled interactions. We study a dilute limit where the range of the interaction potential is much smaller than the typical inter-particle distance. We show that the energy per particle converges to the Thomas-Fermi energy of the system, with a perturbative term corresponding tot he interaction and exhibiting the scattering length of the potential. The proof is decomposed into two bounds. First, we construct an appropriate test-state to prove the upper bound. Then, we prove the lower bound by the Dyson lemma, which allows us to regularize the interaction potential, and several semi-classical tools.

Ground state energy of a dilute inhomogeneous Fermi gas

TL;DR

The paper analyzes the ground-state energy of a dilute, two-spin Fermi gas in a fixed inhomogeneous trap under a short-range interaction with a diluteness scaling. It combines a semiclassical Thomas–Fermi framework with a careful partitioning of space and a Dyson lemma-based regularization to derive matching upper and lower bounds, yielding E(N) = N E^{\mathrm{TF}} + 2\pi a_w N^{1/3-\beta} \int (\rho^{\mathrm{TF}})^2 + o(N^{4/3-\beta}) for 1/3 < \beta < 34/81. The analysis hinges on proving convergence of the many-body densities to the TF minimizer, including a detailed treatment of the interaction term via semiclassical approximations and density regularization. The results extend the understanding of how Thomas–Fermi theory emerges in inhomogeneous, dilute Fermi systems and quantify the leading-order interaction correction through the scattering length , with potential extensions to other scalings by replacing with the interaction range. The work has implications for predicting ground-state energies in trapped ultracold Fermi gases and clarifies the role of short-range correlations in the dilute regime.

Abstract

We study the ground state energy of a system of N fermions with two spin states in the large N limit. The particles are placed in an inhomogeneous trapping potential and interact via scaled interactions. We study a dilute limit where the range of the interaction potential is much smaller than the typical inter-particle distance. We show that the energy per particle converges to the Thomas-Fermi energy of the system, with a perturbative term corresponding tot he interaction and exhibiting the scattering length of the potential. The proof is decomposed into two bounds. First, we construct an appropriate test-state to prove the upper bound. Then, we prove the lower bound by the Dyson lemma, which allows us to regularize the interaction potential, and several semi-classical tools.
Paper Structure (22 sections, 24 theorems, 291 equations, 1 figure)

This paper contains 22 sections, 24 theorems, 291 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Main result
  4. Plan of the paper
  5. Preliminaries
  6. Thomas Fermi functional and density
  7. Scattering length
  8. Some general notation
  9. Upper bound
  10. Decomposition into boxes
  11. Upper bound in the boxes
  12. Conclusion
  13. Lower bound
  14. Main term - low frequencies and potential energy
  15. Dyson lemma: high frequencies and interactions
  16. ...and 7 more sections

Key Result

Theorem 1.2

For $1/3 < \beta < 34/81$, and under (H_1) and (H_2), we have

Figures (1)

  • Figure 1: Representation of the double decomposition into boxes: for the second decomposition see the proof of Lemma \ref{['lemma_energy_bound_small_box']}. The grayed region is the support of $\rho^{\mathrm{TF}}$.

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2: Main result: convergence of the ground state energy
  • Remark 1.3: Comments on Theorem \ref{['theorem']}
  • Proposition 2.1: Existence and properties of the minimizing density
  • proof
  • Lemma 2.2: Strong convergence of a TF-minimizing sequence
  • proof
  • Proposition 2.3: Equality between Thomas-Fermi energies
  • proof
  • Proposition 2.4: Properties of $f_v$
  • ...and 41 more