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Journal of Mathematical Physics

Bosonization of Fermi gases

Phan Thành Nam

Published Mar 1, 2026• 0 citations
Publisher

Abstract

We review the bosonization method for Fermi gases, initiated by Sawada, Bohm–Pines, and Gell-Mann–Brueckner in the 1950s, and present recent rigorous results that justify this approach in the mean-field regime with regular and Coulomb interactions, as well as in the low-density regime leading to the Huang–Yang formula. We also discuss some open questions on related topics.

DOI
10.1063/5.0313178

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Hartree–Fock theory has been justified as a mean-field approximation for fermionic systems. However, it suffers from some defects in predicting physical properties, making necessary a theory of quantum correlations. Recently, bosonization of many-body correlations has been rigorously justified as an upper bound on the correlation energy at high density with weak interactions. We review the bosonic approximation, deriving an effective Hamiltonian. We then show that for systems with Coulomb interaction this effective theory predicts collective excitations (plasmons) in accordance with the random phase approximation of Bohm and Pines, and with experimental observation.
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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 c1ρlog⁡(ρ)+c2ρc_{1}\rho \log \left( \rho \right) +c_{2}\rho c1​ρlog(ρ)+c2​ρ 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.