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 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 | → +∞. 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 finetuning 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. arXiv:1902.05477v1 [nucl-th]

Local energy density functional for superfluid Fermi gases from effective field theory

“…We still lack a general framework that allow for systematically improvable form of such quantities starting from the bare interaction. Among recent attempts to obtain a DFT from first principle, one can refer to the EFT for dilute Fermi gas [62,93,[115][116][117][118][119][120], the bilocal Legendre transforms techniques [90,[121][122][123], the DFT driven by ab initio calculations [69,124] which did not lead to satisfying results [125,126]. On the other hand, the SLDA have proven to provide the formidable precision in the description of strongly correlated Fermi gases, despite its astonishing simplicity.…”

“…the λ-dependence of the associated effective mass, e.g. discussed in [69] related to ab initio adjustments;…”

“…For the ground-state energy or the generalized Bertsch parameter ξ λ , we chose the APS functional [69] displayed in fig. 1(a, d) and the associated inverse effective mass α λ (shown in fig.…”

“…Actually, starting from a microscopic point of view, the knowledge of the quasi-particle properties of the system can be used to obtain a universal systematic expansion of the functional parameters, that is to say, the coupling constant of the effective Hamiltonian. In other words, starting from a single-particle 4): the ground-state energy and the effective mass correspond to the APS functional [69] by setting x −3/4 adjusted to reproduce the result of [71, green asterisks] at unitarity, and the paring gap function is given by eq. ( 17).…”

“…( 17). For reference, we display the second order MBPT results, known as the Lee -Yang [73][74][75] and Galitskii formula [76] for the ground-state energy and the effective mass respectively, and the BCS equation [61] (orange dashed lines) as well as the ladder resummation of the particle-particle [66,69] (green dotted lines), particle-particle and hole-hole [66,69] (blue dotted lines), and particle-hole channels [77,78] picture, the coupling constant for the low-energy degrees of freedom, i.e. the classical density fields, have been obtained in a systematic improvable expansion and explicitly expressed through quasi-particle properties.…”