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

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 manybody 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.

Semi-classical limit of large fermionic systems at positive temperature

“…We will first show that the expression (5) of the minimizers is correct by computing the Euler-Lagrange equation associated with any such minimizer m 0 . This gives automatically the expression of the minimum energy (6). We conclude, in the case w ≥ 0, by showing that the chemical potential µ is given by (8).…”

“…Before [16], this limit has been rigorously considered at T = 0 for atoms by Lieb and Simon in [34,33] and for pseudo-relativistic stars by Lieb, Thirring and Yau in [37,38]. Upper and lower bounds on the next order correction have recently been derived in [22,6], for particles evolving on the torus. For atoms the positive Thomas-Fermi model was derived for the first time in [43].…”

The Dilute Fermi Gas via Bogoliubov Theory

“…The proof is based on rigorous second-order perturbation theory, first developed in [19,20]. The method that we introduce in the present paper is related to the bosonization approach of [5,6]. The method of [5,6] allowed to compute the correlation energy of weakly interacting, mean-field fermionic systems, at all orders in the interaction strength.…”

“…The method that we introduce in the present paper is related to the bosonization approach of [5,6]. The method of [5,6] allowed to compute the correlation energy of weakly interacting, mean-field fermionic systems, at all orders in the interaction strength. The result of [5,6] confirmed the prediction of the random phase approximation, see [3] for a review.…”

“…The method of [5,6] allowed to compute the correlation energy of weakly interacting, mean-field fermionic systems, at all orders in the interaction strength. The result of [5,6] confirmed the prediction of the random phase approximation, see [3] for a review. Both [5,6,22] make use of Fock space methods and fermionic Bogoliubov transformations, extending ideas previously introduced in the context of many-body fermionic dynamics [4,[7][8][9]29].…”