Rules for Diagrams

Here we summarize the rules for drawing and evaluating the diagrams for the perturbational contributions to the resolvent Rα(z), defined in equation (7.8), for the degenerate Anderson model. The expansion is in powers of the hybridization parameter V. We give rules for the U = ∞ model and general N.

(1) Each diagram has a base line running from left to right which corresponds to the impurity state. A full line with an arrow directed from left to right and a label m indicates an occupied state f, nf = 1, m. A dashed line represents the unoccupied state nf = 0. The initialand final lines correspond to the state |α〉.

(2) The vertices are associated with the interaction (7.10), at which a single conduction electron (full line) is created or annihilated. When the arrow is directed away from the vertex the electron is created, when directed towards the vertex it is annihilated. Contributions of the order |V|2n correspond to all possible diagrams with 2n vertices consistent with the direction of the arrows.

(3) Quantum numbers k, m, are assigned to each conduction line and have an associated factor, for lines running from left to right, and a factor, for lines running in the reverse direction. The z-component of angular momentum, m, is conserved at each vertex.