Lower bound on the Hartree-Fock energy of the electron gas

Abstract

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies a posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.The Homogeneous Electron Gas (HEG), where electrons are placed in a positively-charged uniform background, is a fundamental system in quantum physics and chemistry [1, 2]. In spite of its simplicity, it provides a good description of valence electrons in alcaline metals (e.g. in solid sodium [3]) and of the deep interior of white dwarfs [4,5]. It also plays a central role in the Local Density Approximation of Density Functional Theory [1], where it is used for deriving empirical functionals [6][7][8][9].The ground state of the HEG is highly correlated at low and intermediate densities. It was first predicted by Wigner that the particles form a BCC ferromagnetic crystal at small densities [10,11]. But correlation also plays an important role at high densities: The exact largedensity expansion of the correlation energy has a peculiar logarithm due to the long range of the Coulomb potential, which cannot be obtained from regular second-order perturbation theory [12][13][14][15].In principle, the correlation energy of the HEG is defined as the difference between the Hartree-Fock (HF) ground state energy and the true energy. However, many authors use instead the (paramagnetic) non-interacting Fermi Gas as a reference. This state indeed provides the first two terms of the large-density expansion of the HEG total energy [16]. But it is not the absolute ground state of the Hartree-Fock HEG. This was first suggested by Wigner [11] at high densities and then proved by Overhauser [17][18][19] who showed that the free Fermi Gas is unstable under the formation of spin or charge density waves. Recently, the phase diagram of the Hartree-Fock HEG has been studied numerically in great details [20][21][22][23]. It was discovered that the system is crystallized at all densities and that, at high densities, the electrons form an incommensurate lattice having more crystal sites than electrons [21,23]. Similar conclusions were reached in two space dimensions [23][24][25].These works naturally raise the question of determining the energy gain of the true HF ground state, compared to the free Fermi Gas. A too large deviation could affect the large-density expansion of the exact correlation en...

Spin Symmetry Breaking in the Translation-Invariant Hartree--Fock Electron Gas

“…We need an estimate on this eigenvalue. This is what has been accomplished in [HS10] for regular potentials and in [GHL19] in the critical case d = 2, 3 and s = 1. Following the exact same method, we can prove the Lemma 26 (Estimate on the first eigenvalue of |∆ + 1| − ε|x| −s ).…”

“…Proof of Lemma 26. The cases d = 2, 3 and s = 1 have been handled in [GHL19]. The proof is exactly the same in higher dimensions.…”

“…We then show that g min = g max in the whole region, by studying the region of small densities, where we can prove the uniqueness of critical points. At high densities, our arguments are inspired by our recent work [GHL19] which is itself based on spectral techniques recently developed in the context of Bardeen-Cooper-Schrieffer theory in [FHNS07, HHSS08, HS08, FHS12, HS16, HL17]. For Theorem 10, we directly prove the uniqueness of minimisers, using some estimates derived in the proof of Theorem 9.…”

“…We now prove that this operator is (strictly) positive, which leads to the equality g σ = g 0 . To this end we use an argument from our recent work [GHL19] with Christian Hainzl. We first remark that it is enough to study the case where w = κ| · | s−d .…”

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

“…Some authors define the correlation energy with respect to the minimum of the Hartree–Fock functional instead. For the present translation invariant setting, it was recently proved [ 31 ] that the energy of the plane wave state and the minimal Hartree–Fock energy differ only by an exponentially small amount as . However, for systems that are not translation invariant, the ground state of non-interacting fermions will not even be a stationary point of the interacting Hartree–Fock functional.…”