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

## TL;DR

This paper develops a resummation-based approximation for the self-energy in Fermi systems with large s-wave scattering length, aiming to improve density functional theory in strongly interacting regimes.

## Contribution

It introduces a simplified, non-diverging expression for the self-energy and energy in Fermi systems with large scattering length, aiding the development of density functional theory beyond perturbation.

## Key findings

- Resummation technique yields compact expressions for energy and self-energy.
- Expressions remain valid at large scattering lengths, including near unitarity.
- Simplified models guide DFT development for strongly interacting Fermi systems.

## 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 at play. 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. 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 non-diverging limit as $|a_s| \rightarrow + \infty$. Still, the obtained expressions are rather complex functional of the Fermi momentum $k_F$. We introduce the full phase-space average or the partial phase-space methods respectively applied to the energy or to the self-energy to simplify their dependences in terms of $(a_s k_F)$ while keeping the correct limit at low density and the non-diverging property at large $|a_s k_F|$. Quasi-particle properties of Fermi system in various regime of density and scattering length are then illustrated. Our conclusion is that such simplified expressions where the direct link is made with the low energy constant without fine-tuning can provide a clear guidance to obtain density functional theory beyond the perturbative regime. However, quasi-particle properties close or near unitary cannot be reproduced unless this limit is explicitly used as a constraint. We finally discuss how such approximate treatment of quasi-particle can guide the development of density functional theory for strongly interacting Fermi systems.