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

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

doi:10.1007/s00220-019-03505-5

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.

Highlights

We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy
While Hartree–Fock theory describes some aspects of interacting fermionic systems very well, it utterly fails at others
In this paper we show that the random-phase approximation (RPA) is mathematically rigorous, insofar as the RPA correlation energy provides an upper bound on the ground state energy of interacting fermions in the mean-field scaling regime

Summary

Introduction

While Hartree–Fock theory describes some aspects of interacting fermionic systems very well, it utterly fails at others. In this paper we show that the RPA is mathematically rigorous, insofar as the RPA correlation energy provides an upper bound on the ground state energy of interacting fermions in the mean-field scaling regime. It turns out that Hartree–Fock theory provides more than an upper bound for the ground state energy: the method developed in [2,3,33] for the jellium model can be applied to show that in the present mean-field scaling the Hartree–Fock minimum agrees with the many-body ground state energy up to an error of size o(1) for N → ∞. In the context of interacting fermions in the mean-field regime, the first rigorous result on the correlation energy has been recently obtained in [37], for small interaction potentials, via upper and lower bounds matching at leading order. The Lee–Huang–Yang formula for the low-density limit has been proven as an upper bound in [22] for small potential and in [67] for general potential, and only very recently as a lower bound [23]

Main Result

Collective Particle–Hole Pairs

Construction of the Trial State

Evaluating the Energy of the Trial State

Counting Particle–Hole Pairs in Patches

The Bosonic Effective Theory