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 negligible effect on the ground state energy.
We thank Christian Hainzl for helpful discussions and a referee for very careful reading of the paper and many helpful suggestions. NB and RS were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 694227). Part of the research of NB was conducted on the RZD18 Nice–Milan–Vienna–Moscow. NB thanks Elliott H. Lieb and Peter Otte for explanations about the Luttinger model. PTN has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2111-390814868). MP acknowledges financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC StG MaMBoQ, grant agreement No. 802901). BS gratefully acknowledges financial support from the NCCR SwissMAP, from the Swiss National Science Foundation through the Grant “Dynamical and energetic properties of Bose-Einstein condensates” and from the European Research Council through the ERC-AdG CLaQS (grant agreement No. 834782). All authors acknowledge support for workshop participation from Mathematisches Forschungsinstitut Oberwolfach (Leibniz Association). NB, PTN, BS, and RS acknowledge support for workshop participation from Fondation des Treilles.
Benedikter NP, Nam PT, Porta M, Schlein B, Seiringer R. Correlation energy of a weakly interacting Fermi gas. Inventiones Mathematicae. 2021;225:885-979. doi:10.1007/s00222-021-01041-5
Benedikter, N. P., Nam, P. T., Porta, M., Schlein, B., & Seiringer, R. (2021). Correlation energy of a weakly interacting Fermi gas. Inventiones Mathematicae. Springer. https://doi.org/10.1007/s00222-021-01041-5
Benedikter, Niels P, Phan Thành Nam, Marcello Porta, Benjamin Schlein, and Robert Seiringer. “Correlation Energy of a Weakly Interacting Fermi Gas.” Inventiones Mathematicae. Springer, 2021. https://doi.org/10.1007/s00222-021-01041-5.
N. P. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer, “Correlation energy of a weakly interacting Fermi gas,” Inventiones Mathematicae, vol. 225. Springer, pp. 885–979, 2021.
Benedikter NP, Nam PT, Porta M, Schlein B, Seiringer R. 2021. Correlation energy of a weakly interacting Fermi gas. Inventiones Mathematicae. 225, 885–979.
Benedikter, Niels P., et al. “Correlation Energy of a Weakly Interacting Fermi Gas.” Inventiones Mathematicae, vol. 225, Springer, 2021, pp. 885–979, doi:10.1007/s00222-021-01041-5.
All files available under the following license(s):
Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
2021_InventMath_Benedikter.pdf 1.09 MB