Ab initio self-consistent Gorkov-Green's function calculations of semimagic nuclei: Formalism at second order with a two-nucleon interaction

An ab initio calculation scheme for finite nuclei based on self-consistent Green's functions in the Gorkov formalism is developed. It aims at describing properties of doubly magic and semimagic nuclei employing state-of-the-art microscopic nuclear interactions and explicitly treating pairing correlations through the breaking of U(1) symmetry associated with particle number conservation. The present paper introduces the formalism necessary to undertake applications at (self-consistent) second order using two-nucleon interactions in a detailed and self-contained fashion. First applications of such a scheme will be reported soon in a forthcoming publication. Future works will extend the present scheme to include three-nucleon interactions and implement more advanced truncation schemes.

Figure 1

First-order normal ${\Sigma }^{11\left(1\right)}$ (left) and anomalous ${\Sigma }^{21\left(1\right)}$ (right) self-energy diagrams. Double lines denote self-consistent normal (two arrows in the same direction) and anomalous (two arrows in opposite directions) propagators while dashed lines embody antisymmetrized matrix elements of the $NN$ interaction.

Figure 2

Second-order normal self-energies ${\Sigma }^{11\left(2^{\prime }\right)}$ (left) and ${\Sigma }^{11\left(2^{\prime \prime }\right)}$ (right). See Fig. 1 for conventions.

Figure 3

Second-order anomalous self-energies ${\Sigma }^{21\left(2^{\prime }\right)}$ (left) and ${\Sigma }^{21\left(2^{\prime \prime }\right)}$ (right). See Fig. 1 for conventions.

Figure 4

Diagrams contributing to ${\Phi }^{\left(1\right)}$.

Figure 5

Diagrams contributing to ${\Phi }^{\left(2\right)}$.

Figure 6

Allowed (left) and forbidden (right) interaction lines.

Figure 7

Example of topologically equivalent second-order diagrams.

Figure 8

Example of second-order diagrams topologically differing from the ones in Fig. 7. Right diagram is the exchange of left diagram.

Figure 9

Example of connected (left) and disconnected (right) first-order diagrams.

Figure 10

Example of second-order diagram with no equivalent lines.

Figure 11

Example of first-order diagram (left) and corresponding self-energy contribution (right).

Figure 12

Example of skeleton (upper) and composed (lower) second-order self-energy diagrams.