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

@article{Gontier2019LowerBO,
title={Lower bound on the Hartree-Fock energy of the electron gas},
author={David Gontier and Christian Hainzl and Mathieu Lewin},
journal={Physical Review A},
year={2019}
}
• Published 29 November 2018
• Physics
• Physical Review A
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 posteriori the use of the non-interacting Fermi Gas as a reference state in the…

## Figures from this paper

Spin Symmetry Breaking in the Translation-Invariant Hartree-Fock Electron Gas
• Physics
SIAM J. Math. Anal.
• 2019
The breaking of spin symmetry for the nonlinear Hartree-Fock model describing an infinite translation-invariant interacting quantum gas (fluid phase) is studied and it is proved that the equilibrium state is unique and paramagnetic at high temperature or high density.
Spin symmetry breaking in the translation-invariant Hartree-Fock Uniform Electron Gas
• Physics
• 2018
We study the breaking of spin symmetry for the nonlinear HartreeFock model describing an infinite translation-invariant interacting quantum gas (fluid phase). At zero temperature and for the Coulomb
Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime
• Physics
Communications in mathematical physics
• 2020
This paper starts from the Hartree–Fock state given by plane waves and introduces collective particle–hole pair excitations, and uses Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy.
Symmetry Breaking in Density Functional Theory due to Dirac Exchange for a Hydrogen Molecule
• Physics
• 2019
We study symmetry breaking in the mean field solutions to the 2 electron hydrogen molecule within Kohn Sham (KS) local spin density function theory with Dirac exchange (the XLDA model). This
Bosonic collective excitations in Fermi gases
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
On the Correlation Energy of Interacting Fermionic Systems in the Mean-Field Regime
• Physics
Communications in Mathematical Physics
• 2020
We consider a system of $$N\gg 1$$ N ≫ 1 interacting fermionic particles in three dimensions, confined in a periodic box of volume 1, in the mean-field scaling. We assume that the interaction
Localization and IDS Regularity in the Disordered Hubbard Model within Hartree–Fock Theory
• Physics
• 2019
Using the fractional moment method it is shown that, within the Hartree-Fock approximation for the Disordered Hubbard Hamiltonian, weakly interacting Fermions at positive temperature exhibit
Correlation energy of a weakly interacting Fermi gas
• Physics
Inventiones mathematicae
• 2021
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
Large coupling-strength expansion of the Møller-Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms.
• Physics
The Journal of chemical physics
• 2020
The asymptotic H atom solution for the spin-unpolarized case is shown to be variationally optimal for the many-electron spin-restricted closed-shell case, providing expressions for the large coupling-strength density functionals up to the third leading order.
Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory due to Dirac Exchange for a Hydrogen Molecule
• Physics
• 2021
We study symmetry breaking in the mean field solutions to the electronic structure problem for the 2 electron hydrogen molecule within the Kohn Sham (KS) local spin density functional theory with

## References

SHOWING 1-10 OF 86 REFERENCES
Hartree-Fock ground state of the three-dimensional electron gas.
• Physics
Physical review letters
• 2008
Using numerical calculations for finite systems and analytic techniques, this work studies the unrestricted HF ground state of the three-dimensional electron gas and finds broken spin symmetry states with a nearly constant charge density at high density.
Metal-insulator transition in the Hartree-Fock phase diagram of the fully polarized homogeneous electron gas in two dimensions
• Physics
• 2008
We determine numerically the ground state of the two-dimensional fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low
Hartree-Fock ground state phase diagram of jellium.
• Physics
Physical review letters
• 2013
We calculate the ground state phase diagram of the homogeneous electron gas in three dimensions within the Hartree-Fock approximation and show that broken symmetry states are energetically favored at
Upper bounds of spin-density wave energies in the homogeneous electron gas
• Physics
• 2015
Studying the jellium model in the Hartree-Fock approximation, Overhauser has shown that spin density waves (SDW) can lower the energy of the Fermi gas, but it is still unknown if these SDW are
Spin Density Waves in an Electron Gas
It is shown rigorously that the paramagnetic state of an electron gas is never the Hartree-Fock ground state, even in the high-density-- or weak- interaction-limit. The paramagnetic state is always
Properties of Hartree-Fock solutions of the three-dimensional electron gas
• Physics
• 2014
In a previous letter, L. Baguet et al., (Phys. Rev. Lett. {\bf 111}, 166402 (2013)), we presented the ground state phase diagram of the homogeneous electron gas in three dimensions within the
Fermionic path-integral Monte Carlo results for the uniform electron gas at finite temperature.
• Physics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2015
Alternative direct fermionic path integral Monte Carlo (DPIMC) simulations that are independent from RPIMC are presented that take into account quantum effects not only in the electron system but also in their interaction with the uniform positive background.
Hartree-Fock phase diagram of the two-dimensional electron gas
• Physics
• 2011
Univ. Grenoble 1/CNRS, LPMMC UMR 5493, Maison des Magisteres, 38042 Grenoble, France(Dated: August 3, 2011)We calculate the ground state phase diagram of the homogeneous electron gas in two
Energy and Pressure of a Zero-Temperature Plasma
The equation of state is considered for matter consisting of electrons and nuclei of atomic weight A and charge Z, at zero temperature and at densities much larger than that of the solid at zero
Ab Initio Thermodynamic Results for the Degenerate Electron Gas at Finite Temperature.
• Physics
Physical review letters
• 2015
Novel first-principles configuration path integral Monte Carlo results for electrons for r_{s}≤4 are presented and quantum statistical data within the e^{4} approximation are presented that are in good agreement with the simulations at small to moderate r_s.