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arXiv:2005.08933 · Inventiones mathematicae

Correlation energy of a weakly interacting Fermi gas

Niels Benedikter, Phan Thành Nam, Marcello Porta, Benjamin Schlein, Robert Seiringer

Published May 3, 2021• 30 citations
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Abstract

Abstract We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.

DOI
10.1007/s00222-021-01041-5

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Hartree–Fock theory has been justified as a mean-field approximation for fermionic systems. However, it suffers from some defects in predicting physical properties, making necessary a theory of quantum correlations. Recently, bosonization of many-body correlations has been rigorously justified as an upper bound on the correlation energy at high density with weak interactions. We review the bosonic approximation, deriving an effective Hamiltonian. We then show that for systems with Coulomb interaction this effective theory predicts collective excitations (plasmons) in accordance with the random phase approximation of Bohm and Pines, and with experimental observation.
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We prove that the random phase approximation provides an upper bound on the correlation energy of a three-dimensional, weakly interacting, Fermi gas in the thermodynamic limit at high density. More precisely, we consider a mean-field scaling regime where the kinetic and interaction Hamiltonian are scaled by ρ−2/3ρ^{-2/3}ρ−2/3 and ρ−1ρ^{-1}ρ−1, respectively. We use a correlated trial state constructed through a quasi-bosonic Bogoliubov transformation of collective particle--hole excitations associated to patches on the Fermi surface. Key of the proof is to control the position-space localization of the collective excitations. Combined with an iterated Duhamel expansion of the expectation values of (localized) particle number operators we obtain extensive estimates, which justify the reduction to an effective quadratic Hamiltonian and its approximate diagonalization.
Abstract We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to N−13N^{-\frac {1}{3}}N−31​ . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.
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.
While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.