# Correlation energy of a weakly interacting Fermi gas

@article{Benedikter2021CorrelationEO, title={Correlation energy of a weakly interacting Fermi gas}, author={Niels Benedikter and Phan Th{\`a}nh Nam and Marcello Porta and Benjamin Schlein and Robert Seiringer}, journal={Inventiones mathematicae}, year={2021}, volume={225}, pages={885-979} }

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 approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a…

## 2 Citations

The Dilute Fermi Gas via Bogoliubov Theory

- Medicine, PhysicsAnnales Henri Poincare
- 2021

A new derivation of this formula for the ground state energy at second order in the particle density is given, inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems.

Bosonization of Fermionic Many-Body Dynamics

- Physics, MathematicsAnnales 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…

## References

SHOWING 1-10 OF 189 REFERENCES

Hartree-Fock dynamics for weakly interacting fermions

- Mathematics, Physics
- 2014

We review recent results concerning the evolution of fermionic systems. We are interested in the mean field regime, where particles experience many weak collisions. For fermions, the mean field…

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

- Medicine, MathematicsCommunications 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 Excitation Spectrum for Weakly Interacting Bosons

- Physics, Mathematics
- 2011

We investigate the low energy excitation spectrum of a Bose gas with weak, long range repulsive interactions. In particular, we prove that the Bogoliubov spectrum of elementary excitations with…

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

- Physics, Mathematics
- 2018

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…

Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional

- Physics, Mathematics
- 2000

The ground-state properties of interacting Host gases in external potentials, as considered in recent exptriments, are usually described by means of the Gross-Pitaevskii energy functional. We present…

Bosonization of the low energy excitations of Fermi liquids.

- Physics, MedicinePhysical review letters
- 1994

The bosonize the low energy excitations of Fermi liquids in any number of dimensions in the limit of long wavelengths and it is shown that the Landau equation for sound waves is exact in the semiclassical approximation for the bosons.

The Excitation Spectrum for Weakly Interacting Bosons in a Trap

- Physics, Mathematics
- 2012

We investigate the low-energy excitation spectrum of a Bose gas confined in a trap, with weak long-range repulsive interactions. In particular, we prove that the spectrum can be described in terms of…

Bosonization of Fermi liquids.

- Physics, MedicinePhysical review. B, Condensed matter
- 1994

A path-integral representation for the generating function is obtained, which gives the Landau equation for sound waves and in the imaginary time gives the correct form of the specific heat for a Fermi liquid even with the corrections due to the interactions between the fermions.

Functional bosonization of interacting fermions in arbitrary dimensions

- Physics
- 1996

We bosonize the long-wavelength excitations of interacting fermions in arbitrary dimension by directly applying a suitable Hubbard-Stratonowich transformation to the Grassmannian generating…

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…