We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained by Christiansen, Hainzl, and Nam [Comm. Math. Phys. 401 (2023), pp. 1469–1529], our result establishes an analogue of the Gell-Mann–Brueckner formula c 1 ρ log ⁡ ( ρ ) + c 2 ρ c_{1}\rho \log \left (\rho \right )+c_{2}\rho for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained by Christiansen, Hainzl, and Nam [Comm. Math. Phys. 401 (2023), pp. 1469–1529], our result establishes an analogue of the Gell-Mann–Brueckner formula c 1 ρ log ⁡ ( ρ ) + c 2 ρ c_{1}\rho \log \left (\rho \right )+c_{2}\rho for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.

arXiv:2405.01386Journal of the American Mathematical SocietyMay 27, 20261 citation

Mean-Field Regime for Fermionic Systems

Niels Benedikter, Marcello Porta, Benjamin Schlein

SpringerBriefs in mathematical physicsOct 6, 20153 citations

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

Martin Ravn Christiansen, Christian Hainzl, Phan Thành Nam

Abstract 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 torus, interacting via a two-body repulsive potential proportional to $N^{-\frac {1}{3}}$ . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.Abstract 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 torus, interacting via a two-body repulsive potential proportional to $N^{-\frac {1}{3}}$ . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.

arXiv:2106.11161Forum of Mathematics PiJan 1, 202317 citations

Pairing Correlations in Finite Fermionic Systems

H. Flocard

Springer eBooksJul 3, 20070 citations

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

Martin Ravn Christiansen, Christian Hainzl, Phan Thành Nam

Abstract 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 $c_{1}\rho \log \left( \rho \right) +c_{2}\rho$ c 1 ρ log ρ + c 2 ρ 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.Abstract 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 $$c_{1}\rho \log \left( \rho \right) +c_{2}\rho $$ c 1 ρ log ρ + c 2 ρ 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.

arXiv:2208.01581Communications in Mathematical PhysicsFeb 21, 202316 citations

N^{-\frac {1}{3}}

. In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.

c_{1}\rho \log \left( \rho \right) +c_{2}\rho

c 1 ρ log ρ + c 2 ρ 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.