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 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.

51 pages, 6 figures; v3: 2 new figures, some heuristic discussion added, typos fixed

References in corpus (25)

• Bosonization and Fermion Liquids in Dimensions Greater Than One
• Bosonization of the Low Energy Excitations of Fermi Liquids
• The excitation spectrum for weakly interacting bosons in a trap
• The Second Order Upper Bound for the Ground Energy of a Bose Gas
• The Excitation Spectrum for Weakly Interacting Bosons
• Mean-field Evolution of Fermionic Systems
• Ground State Energy of the Two-Component Charged Bose Gas
• The ground state energy of a low density Bose gas: a second order upper bound
• Bosonization of interacting fermions in arbitrary dimension beyond the Gaussian approximation
• Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime
• Excitation spectrum of interacting bosons in the mean-field infinite-volume limit
• The ground state energy of the weakly interacting Bose gas at high density
• Ground State Energy of the One-Component Charged Bose Gas
• Mean field evolution of fermions with Coulomb interaction
• Bosonization of Fermi Systems in Arbitrary Dimension in Terms of Gauge Forms
• Functional Bosonization of Interacting Fermions in Arbitrary Dimensions
• Mean-Field Dynamics of Fermions with Relativistic Dispersion
• The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
• How to bosonize fermions with non-linear energy dispersion
• Coulomb Interaction and the Fermi Liquid State: Solution by Bosonization
• Lower Bound on the Hartree-Fock Energy of the Electron Gas
• Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet
• One non-relativistic particle coupled to a photon field
• The Second Order Correction to the Ground State Energy of the Dilute Bose Gas
• Interaction Corrections to Spin-Wave Theory in the Large-S Limit of the Quantum Heisenberg Ferromagnet

Cited by in corpus (17)

• Correlation Energy of a Weakly Interacting Fermi Gas
• The dilute Fermi gas via Bogoliubov theory
• Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential
• The Gell-Mann$-$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime
• On the Effective Quasi-Bosonic Hamiltonian of the Electron Gas: Collective Excitations and Plasmon Modes
• Semi-classical limit of large fermionic systems at positive temperature
• The Scott correction in Dirac-Fock theory
• Effective Dynamics of Extended Fermi Gases in the High-Density Regime
• Bosonic Collective Excitations in Fermi Gases
• Lower bounds on the energy of the Bose gas
• Bosonization of Fermionic Many-Body Dynamics
• Bogoliubov Transformations Beyond Shale-Stinespring: Generic $v^{\ast }v$ for bosons
• The Huang-Yang formula for the low-density Fermi gas: upper bound
• Dynamics of mean-field Fermi systems with nonzero pairing
• Bosonized Momentum Distribution of a Fermi Gas via Friedrichs Diagrams
• Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation