Approximate self-energy for Fermi systems with large s-wave scattering length: a step towards density functional theory

Abstract

In the present work, we start from a minimal Hamiltonian for Fermi systems where the s-wave scattering is the only low energy constant (LEC) at play. The many-body perturbative approach that is usually valid at rather low density is first discussed. We then use the resummation technique with the ladder approximation to obtain compact expressions for both the energy and/or the on-shell self-energy in infinite spin-degenerated systems. The diagrammatic resummation technique has the advantage in general to be predictive in a region of density larger compared to many-body perturbation theory. It also leads to a non-diverging limit as . Still, the obtained expressions are a rather complex function of the Fermi momentum kF. We introduce the full phase-space or the partial phase-space approximations respectively applied to the energy or to the self-energy to simplify their dependences in terms of (askF) while keeping the correct limit at low density and the non-diverging property at large . Quasi-particle properties of the Fermi system in various regimes of density and scattering length are then illustrated. Our conclusion is that such simplified expressions where the direct link is made with the LEC without fine-tuning can provide a clear guidance to obtain density functional theory (DFT) beyond the perturbative regime. However, quasi-particle properties close or near unitarity cannot be reproduced unless this limit is explicitly used as a constraint. We finally discuss how such approximate treatment of quasi-particles can guide the development of DFT for strongly interacting Fermi systems.