Two-body and three-body parquet theory

One of the fundamental approaches to microscopic many-body theory is through the use of perturbation theory. This paper presents a clear derivation of the equations that sum the two-body and three-body reducible diagrams that are generated from some input set of irreducible diagrams (typically the bare interaction) in a crossing-symmetric form. The diagrammatic structure is useful for both bosons and fermions at zero and finite temperature. The essential features of the procedure for the two-body case are the following. (i) All constructions involve only direct diagrams. (ii) Diagrams can be joined together by five different ladder, chain, and vertex correction operations. (iii) The sequences in which these operations are applied satisfy certain associativity relations. (iv) These associativity relations are used to insure that each permitted diagram is generated exactly once. (v) The ladder-and-chain operations are used to construct reducible vertex sums ${\mathrm{\Gamma }}_0$ which have no external vertex corrections. (vi) The full direct vertex Γ consists of the vertex ${\mathrm{\Gamma }}_0$ plus ${\mathrm{\Gamma }}_0$ dressed on either or both sides by the full direct vertex Γ. (vii) Finally, the full vertex is obtained by adding the exchange of the direct vertex. The extension to the three-body formalism proceeds in much the same way.