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arXiv:1809.01902 · Communications in Mathematical Physics

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

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

Published Jul 13, 2019• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s00220-019-03505-5

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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.
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 prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c1ρlog⁡(ρ)+c2ρc_{1}\rho \log \left( \rho \right) +c_{2}\rho c1​ρlog(ρ)+c2​ρ c 1 ρ log ρ + c 2 ρ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.
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 as 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.
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.
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.
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.