Hartree shift and pairing gap in ultracold Fermi gases in the framework of low-momentum interactions

Abstract

In this paper we consider a two-component gas of fermions on the BCS side of the BCS-BEC crossover at zero temperature. We use a momentum dependent interaction that reproduces the s-wave scattering phase shifts of a contact interaction up to a momentum cutoff that is scaled with the Fermi momentum. Using a diagrammatic formulation of Bogoliubov many-body perturbation theory, suitably augmented by self-consistency conditions, we obtain the Hartree shift and the pairing gap to third order. In the weak-coupling regime, our results are not only well-converged but also agree with the well-established Gor'kov-Melik-Barkhudarov corrections for the gap and the Galitskii result for the Hartree shift. Near the unitary regime, our results for the Nambu-Gor'kov self-energy are less converged, but there is still reasonable agreement with experiments as well as with quantum Monte-Carlo results. Perspectives for improvements and applications of this approach to neutron matter are discussed.

Figures (10)

• Figure 1: Elements for Feynman diagrams: (a) $G^{(0)}_{\alpha\beta}(k,\omega)$, (b) $-V(\tfrac{\bm{\mathrm{k}}+\bm{\mathrm{p}}'}{2},\tfrac{\bm{\mathrm{k}}'+\bm{\mathrm{p}}}{2})$, (c) $-H_{\text{mf},\alpha\beta}(k)$.
• Figure 2: (1.1a), (1.1b), and (1.mf) First-order self-energy diagrams. Diagram (1.1a) has been drawn differently in (1.1a$'$) to show that (1.1a) and (1.1b) can be compactly represented by one generic diagram denoted (1.1). (For simplicity, we denote the loop momentum as $k$ instead of $\bm{\mathrm{k}},\omega$ in the figures.)
• Figure 3: (2.1a), (2.1b), and (2.mf) Diagrams for the second-order self-energy $\Sigma^{(2)}$. (2.1b$'$) Alternative way of drawing diagram (2.1b). (2.1) Generic diagram combining (2.1a) and (2.1b), showing the labeling used in the calculation.
• Figure 4: Second-order self-energy diagrams that cancel as a consequence of Eq. \ref{['eq:HFB-Hmf']}.
• Figure 5: Left column, (3.1)-(3.3): general form of the three classes of diagrams contributing to the third-order self-energy $\Sigma^{(3)}$ in perturbation theory on top of the HFB ground state. (3.4)-(3.mf): additional diagrams appearing in the more self-consistent scheme. Right column (3.1a)-(3.4b): corresponding individual diagrams.