Gorkov algebraic diagrammatic construction for infinite nuclear matter

Abstract

We propose a novel many-body truncation for Gorkov self-consistent Green's function (SCGF) theory where pairing correlations are handled at first order, while dynamical correlations are described using the particle-number-conserving Dyson-SCGF scheme up to third order in the algebraic diagrammatic construction. The new method is enabled by the introduction of a scheme that allows to approximate the Gorkov propagator in terms of a particle-number-conserving optimized reference state. The approach provides state-of-the-art predictions of the equation of state and spectral properties of infinite nuclear matter at zero temperature and in the presence of pairing. We find satisfactory results using modern saturating Hamiltonians at next-to-next-to-leading order in chiral effective field theory.

Figures (16)

• Figure 1: Third-order non-skeleton diagrams that contribute to the dynamical self-energy. These complement the ADC(3) diagrams of Fig. \ref{['fig: adc3 diagrams']}, and should be included whenever calculations are performed with non-self-consistent propagators. Wiggly lines denote the effective 2B interaction \ref{['eq: eff interaction 2B']} and the effective 1B interaction $\widetilde{U}^{(1)}$\ref{['eq:Utilde1']}. Fermion lines refer to the OpRS propagator $g^{OpRS}$.
• Figure 2: Diagrammatic representation of the ADC(3)-D truncation scheme. Diagram (a) shows a representative contribution to the $M_{r \alpha}$ 2p1h vertex in standard ADC(3) that incorporates perturbative 2p2h excitations through the amplitude $t^{(0)}$. See e.g. Ref. Raimondi2017, Fig. 7(a). All fermion lines are explicitly labeled. The substitution in diagram (b) of $(t^{(0)})^{n_1\,n_2}_{k_3\,k_4}$ with the converged 2p2h amplitude $t^{n_1\,n_2}_{k_3\,k_4}$ from the solution of the CCD equations, represented as a thick horizontal line, amounts to incorporating additional infinite classes of diagrams within the ADC self-energy, some of which are shown in (c). Diagrams contributing to the CCD amplitudes can be found e.g. in Refs. ShavittBartlettMarino2024.
• Figure 3: MBPT(2) energies per nucleon as a function of the cutoff $N_{max}$ of the s.p. basis. Calculations are performed with the NNLO$_{ \rm{sat} }\,(450)$ (squares) and $\Delta$NNLO$_{\rm{go}}\,(394)$ (circles) interactions in both PNM (upper panels) and SNM (lower panels), at densities $\rho=$ 0.16 fm$^{-3}$ (left) and 0.32 fm$^{-3}$ (right). Lines are a guide to the eye.
• Figure 4: Schematic representation of the momentum space. The Fermi sphere is represented as a circle of radius $k_F$. The model space is defined by the region within the dashed circle of radius $k_{max}$. A representative 'hole' state lying on the Fermi surface with momentum $\mathbf{k}_{i}$ is shown. The red arrow represents an excitation process in which $\bf{k}_{i}$ is scattered into a particle state $\bf{k}_{a}$ by the 2B interaction. We indicate the momentum transfer $\bf{q} = \bf{k}_{a} - \bf{k}_{i}$ with a red arrow.
• Figure 5: Energy per nucleon as a function of $N_{max}$ for SNM, $A=132$ and density $\rho=$ 0.16 fm$^{-3}$. Calculations are performed at the level of Gorkov-ADC(3) with the Cen-$k_F$ prescription (see text). Inset: difference between the energy per particle at a given cutoff and the energy for $N_{max} =28$. The NNLO$_{ \rm{sat} }\,(450)$ interaction is employed.