Momentum Distribution of a Fermi Gas with Coulomb Interaction in the Random Phase Approximation
Niels Benedikter, Sascha Lill, Diwakar Naidu
TL;DR
The paper analyzes the momentum distribution of a three-dimensional Fermi gas in the mean-field scaling regime with Coulomb-type interactions by constructing a patchless, bosonization-based trial state $\Psi_N = R e^{-S} \Omega$ in Fock space. Through a Duhamel expansion, it separates the leading random phase approximation (RPA) contribution $n^{\mathrm{RPA}}(q)$ from exchange and higher-order errors, and proves sharp bounds showing the distribution deviates from the non-interacting step by a controlled RPA correction with error $\mathcal E(q)$ that scales as $k_\mathrm{F}^{-1-1/6+\varepsilon} e(q)^{-1}$ (or $k_\mathrm{F}^{-2+\varepsilon} e(q)^{-1}$ under summability of $\hat V$). The analysis refines prior patch-based approaches, extends to physically relevant Coulomb potentials, and provides precise control near the Fermi surface via a bootstrap on the number-operator growth under the Bogoliubov transformation. Overall, the work demonstrates that the ground-state energy and the momentum distribution of the interacting gas are captured to the leading RPA level with quantifiable, subleading corrections in the mean-field limit.
Abstract
We analyse the momentum distribution of a three-dimensional Fermi gas in the mean-field scaling regime in a trial state that was recently proven to reproduce the Gell-Mann-Brueckner correlation energy for Coulomb potentials. For a class of potentials including the Coulomb potential we show that the momentum distribution is given by a step profile corrected by a random phase approximation contribution as predicted by Daniel and Vosko. Moreover, for potentials with summable Fourier transform we provide optimal error bounds for the deviation from the random phase approximation. This refines a recent analysis by two of the authors to the physically most relevant potentials and to momenta closer to the Fermi surface.