Self-consistent Green's functions formalism with three-body interactions

We extend the self-consistent Green's functions formalism to take into account three-body interactions. We analyze the perturbative expansion in terms of Feynman diagrams and define effective one- and two-body interactions, which allows for a substantial reduction of the number of diagrams. The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fully correlated density matrices. We give examples up to third order in perturbation theory. To define nonperturbative approximations, we extend the equation-of-motion method in the presence of three-body interactions. We propose schemes that can provide nonperturbative resummation of three-body interactions. We also discuss two different extensions of the Koltun sum rule to compute the ground state of a many-body system.

Figure 1

Diagrammatic representation of the effective 1B interaction of Eq. (10). This is given by the sum of the original 1B potential (dotted line), the 2B interaction (dashed line) contracted with a dressed SP propagator $G$ (double line with arrow), and the 3B interaction (long-dashed line) contracted with a dressed 2B propagator $G^{II}$. The correct symmetry factor of 1/4 in the last term is also shown explicitly.

Figure 2

Diagrammatic representation of the effective 2B interaction of Eq. (11). This is given by the sum of the original 2B interaction (dashed line) and the 3B interaction (long-dashed line) contracted with a dressed SP propagator $G$.

Figure 3

1PI, skeleton and interaction-irreducible self-energy diagrams appearing at second order in the perturbative expansion of Eq. (7), using the effective Hamiltonian of Eq. (9).

Figure 4

These four diagrams are contained in diagram Fig. 3a. They correspond to one 2B interaction-irreducible diagram (a), and three interaction-reducible diagrams (b)–(d).

Figure 5

1PI, skeleton and interaction-irreducible self-energy diagrams appearing at third order in the perturbative expansion of Eq. (7) using the effective Hamiltonian of Eq. (9).

Figure 6

Diagrammatic representation of the EOM, Eq. (22), for the dressed 1B propagator $G$. The first term, given by a single line, defines the free 1B propagator $G^{\left(0\right)}$. The second term denotes the interaction with a bare 1B potential, whereas the third and the fourth terms describe interactions involving the intermediate propagation of two- and three-particle configurations.

Figure 7

Exact separation of the four-point Green's function, $G^{4-\mathrm{pt}}$, in terms of noninteracting lines and a vertex function, as given in Eq. (23). The first two terms are the direct and exchange propagation of two noninteracting and fully dressed particles. The last term defines the four-point vertex function, ${\Gamma }^{4-\mathrm{pt}}$, involving the sum of all 1PI diagrams.

Figure 8

Exact separation of the six-point Green's function, $G^{6-\mathrm{pt}}$, in terms of noninteracting dressed fermion lines and vertex functions, as given in Eq. (24). The first two terms gather noninteracting dressed lines and subgroups of interacting particles that are fully connected to each other. Round brackets with numbers above (below) these diagrams indicate the numbers of permutations of outgoing (incoming) legs needed to generate all possible diagrams. The last term defines the six-point 1PI vertex function ${\Gamma }^{6-\mathrm{pt}}$.

Figure 9

Diagrammatic representation of the irreducible self-energy ${\Sigma }^{☆}$ by means of effective 1B and 2B potentials and 1PI vertex functions, as given in Eq. (25). The first term is the energy-independent part of ${\Sigma }^{☆}$ and contains all diagrams depicted in Fig. 1. The second and third terms are dynamical terms consisting of excited configurations generated through 2B and 3BFs. This is an exact equation for Hamiltonians including 3BFs and it is not derived from perturbation theory.

Figure 10

Diagrammatic representation of the EOM for the four-point propagator, $G^{4-\mathrm{pt}}$, given in Eq. (26). The last term, involving an eight-point GF, arises due to the presence of 3B interactions.

Figure 11

Exact separation of the eight-point Green's function, $G^{8-\mathrm{pt}}$, in terms of noninteracting lines and vertex functions. The first four terms gather noninteracting dressed lines and subgroups of interacting particles that are fully connected to each other. Round brackets with numbers above (below) these diagrams indicate the numbers of permutations of outgoing (incoming) legs needed to generate all possible diagrams. The last term defines the eight-point 1PI vertex function ${\Gamma }^{8-\mathrm{pt}}$.

Figure 12

Self-consistent expression for the ${\Gamma }^{4-\mathrm{pt}}$ vertex function derived from the EOM for $G^{4-\mathrm{pt}}$. The round brackets underneath some of the diagrams indicate that the term obtained by exchanging the $\left\{\beta {\omega }_{\beta }\right\}$ and $\left\{\delta {\omega }_{\delta }\right\}$ arguments must also be included. Diagrams (a), (b), (c), and (f) are the only ones present for 2B Hamiltonians, although (f) also contains some intrinsic 3BF contributions such as the $\left\{\alpha {\omega }_{\alpha }\right\}\leftrightarrow \left\{\gamma {\omega }_{\gamma }\right\}$ exchange of (e). All other diagrams arise from the inclusion of 3B interactions. Diagram (b) is responsible for generating the ladder summation, the direct part of (c) generates the series of antisymmetrized rings, and the three sets together [(b), (c), and the exchange of (c)] would give rise to a Parquet-type resummation.

Figure 13

Skeleton and interaction-irreducible diagrams contributing to the ${\Gamma }^{4-\mathrm{pt}}$ vertex function up to second order. The round brackets above (below) some diagrams indicate that the exchange diagram between the $\left\{\alpha {\omega }_{\alpha }\right\}$ and $\left\{\gamma {\omega }_{\gamma }\right\}$ ($\left\{\beta {\omega }_{\beta }\right\}$ and $\left\{\delta {\omega }_{\delta }\right\}$) arguments must also be included.

Figure 14

The same as Fig. 13 for the ${\Gamma }^{6-\mathrm{pt}}$ vertex function. The round brackets above (below) some diagrams indicate that cyclic permutations of the $\left\{\alpha {\omega }_{\alpha }\right\}$, $\left\{\gamma {\omega }_{\gamma }\right\}$ and $\left\{\epsilon {\omega }_{\epsilon }\right\}$ ($\left\{\beta {\omega }_{\beta }\right\}$, $\left\{\delta {\omega }_{\delta }\right\}$ and $\left\{\eta {\omega }_{\eta }\right\}$) arguments must also be included.

Figure 15

Diagrammatic representation of the self-energy correction $\Delta {\Sigma }^{{☆}_{\Gamma W\Gamma }}$ given in Eq. (31).

Figure 16

Examples of diagrams containing symmetric and interacting lines, with explicit symmetry factors. Diagrams (b) to (d) are obtained by expanding the effective interaction of diagram (a) according to Eq (11). Swapping the 3B and 2B internal vertices in (c) gives a distinct, but topologically equivalent, contribution.

Figure 17

Examples of diagrams entering the static part of the self-energy. Applying rule 9-ii, diagrams (a) and (b) take a factor $S_{si}=\frac{1}{2}$ from the symmetry between the two bubbles attached to the upper three body vertex. The symmetry is broken in diagram (c), where the overall factor is $S_{si}=1$

Figure 18

Diagrams entering the effective one-body interaction, Eq. (10), obtained by substituting the right-hand side of Fig. 13 into Eq. (23). The two bubble terms correctly reproduce the symmetric factor inferred by applying rules 9-i and 9-ii.

Figure 19

Examples of a diagram where equivalent group of lines are present and one where rule 9-iii does not apply. Swapping the two chains of bubbles in (a), one finds an identical diagram. This is precisely the case of rule 9-iii, which brings in a factor $S_{egl}$ $=$ $\frac{1}{2}$. Performing the same exchange in diagram (b) generates a graph where the direction of the internal loop is reversed. No symmetry rule applies here and $S_{egl}$ $=$ 1