# The Gell-Mann$-$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime

@inproceedings{Christiansen2022TheGF, title={The Gell-Mann\$-\$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime}, author={Martin Ravn Christiansen and Christian Hainzl and Phan Th{\`a}nh Nam}, year={2022} }

We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c1ρ log (ρ)+c2ρ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.

## One Citation

### On the effective quasi-bosonic Hamiltonian of the electron gas: collective excitations and plasmon modes

- PhysicsLetters in Mathematical Physics
- 2022

We consider an effective quasi-bosonic Hamiltonian of the electron gas which emerges naturally from the random phase approximation and describes the collective excitations of the gas. By a rigorous…

## References

SHOWING 1-10 OF 22 REFERENCES

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

- PhysicsCommunications 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.

### The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime

- Physics
- 2021

We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three-dimensions in the mean-field scaling regime. We consider a system of $N$ fermions on a…

### On the Correlation Energy of Interacting Fermionic Systems in the Mean-Field Regime

- PhysicsCommunications 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…

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

- PhysicsPhysical Review A
- 2019

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…

### Correlation energy of a weakly interacting Fermi gas

- PhysicsInventiones 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…

### A CORRELATION ESTIMATE WITH APPLICATIONS TO QUANTUM SYSTEMS WITH COULOMB INTERACTIONS

- Physics
- 1994

We consider some two-body operators acting on a Fock space with either fermionic or no statistics. We prove that they are bounded below by one-body operators which mimic exchange effects. This allows…

### On the energy of a large atom

- Mathematics, Physics
- 1990

We announce a proof of an asymptotic formula for the groundstate energy of a large atom. The early work of Thomas-Fermi, Hartree-Fock, Dirac, and Scott predicted that for an atomic number Z , the…

### Bosonization of Fermionic Many-Body Dynamics

- PhysicsAnnales Henri Poincaré
- 2021

We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing…

### Error bound for the Hartree-Fock energy of atoms and molecules

- Physics
- 1992

We estimate the error of the Hartree-Fock energy of atoms and molecules in terms of the one-particle density matrix corresponding to the exact ground state. As an application we show this error to be…

### A COLLECTIVE DESCRIPTION OF ELECTRON INTERACTIONS: IV. ELECTRON INTERACTION IN METALS

- Physics
- 1953

The effects of the Coulomb interaction between free electrons in an electron gas are considered for a variety of phenomena. The analysis is based on the collective description, which describes the…