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Reviews in Mathematical Physics

Correlation energy of weakly interacting Fermi gases

Benjamin Schlein

Published Jun 23, 2023• 0 citations
Publisher

Abstract

In this paper, based on [N. Benedikter, P. T. Nam, M. Porta, B. Schlein and R. Seiringer. Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime, Commun. Math. Phys. 374(3) (2020) 2097–2150; N. Benedikter, P. T. Nam, M. Porta, B. Schlein and R. Seiringer, Correlation energy of a weakly interacting Fermi gas, Invent. Math. 225(3) (2021) 885–979; N. Benedikter, M. Porta, B. Schlein and R. Seiringer, Correlation energy of a weakly interacting Fermi gas with large interaction potential, preprint (2021), arXiv:2106.13185], we present a recent estimate for the correlation energy of a Fermi gas in the so-called mean-field regime and we outline some of the main ideas of its proof.

DOI
10.1142/s0129055x23600012

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In this paper, based on [N. Benedikter, P. T. Nam, M. Porta, B. Schlein and R. Seiringer. Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime, Commun. Math. Phys. 374(3) (2020) 2097–2150; N. Benedikter, P. T. Nam, M. Porta, B. Schlein and R. Seiringer, Correlation energy of a weakly interacting Fermi gas, Invent. Math. 225(3) (2021) 885–979; N. Benedikter, M. Porta, B. Schlein and R. Seiringer, Correlation energy of a weakly interacting Fermi gas with large interaction potential, preprint (2021), arXiv:2106.13185], we present a recent estimate for the correlation energy of a Fermi gas in the so-called mean-field regime and we outline some of the main ideas of its proof.
We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained by Christiansen, Hainzl, and Nam [Comm. Math. Phys. 401 (2023), pp. 1469–1529], our result establishes an analogue of the Gell-Mann–Brueckner formula c 1 ρ log ⁡ ( ρ ) + c 2 ρ c_{1}\rho \log \left (\rho \right )+c_{2}\rho for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.
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 ∣⋅∣V^∈ℓ1(Z3)|\cdot | \hat{V} \in \ell ^1 (\mathbb {Z}^3)∣⋅∣V^∈ℓ1(Z3) |·|V^∈ℓ1(Z3) . 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.
We rigorously establish a formula for the correlation energy of a two-dimensional Fermi gas in the mean-field regime for potentials whose Fourier transform V^\hat{V}V^ satisfies V^(⋅)∣⋅∣∈ℓ1\hat{V}(\cdot) | \cdot | \in \ell^1V^(⋅)∣⋅∣∈ℓ1. Further, we establish the analogous upper bound for V^(⋅)2∣⋅∣1+ε∈ℓ1\hat{V}(\cdot)^2 | \cdot |^{1 + \varepsilon} \in \ell^1V^(⋅)2∣⋅∣1+ε∈ℓ1, which includes the Coulomb potential V^(k)∼∣k∣−2\hat{V}(k) \sim |k|^{-2}V^(k)∼∣k∣−2. The proof is based on an approximate bosonization using slowly growing patches around the Fermi surface. In contrast to recent proofs in the three-dimensional case, we need a refined analysis of low-energy excitations, as they are less numerous, but carry larger contributions.
We consider a system of N≫1N\gg 1N≫1 interacting fermionic particles in three dimensions, confined in a periodic box of volume 111, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their NNN-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of recent methods introduced in the context of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED.
The time evolution of spin-energy correlations in a very weakly interacting Fermi gas of 6{}^{6}6Li produces complex spin-density profiles, which are observed and quantitatively explained by a one-dimensional mean-field model. This system exhibits rich collective dynamics for exploring the interplay between spin, motion, Fermi statistics, and interactions in many-body systems.