Many-body perturbation methods in a soluble model

Abstract

It is pointed out that because statistics should not be obeyed in Goldstone's perturbation expansion there exist two-body contributions from diagrams other than ladders. The role of diagrams with “wrong” statistics is investigated in a model in which the perturbation is a single particle potential. A summation of all diagrams is carried out in this model and compared to the sum of all ladders. It is found that the “wrong statistics” contributions are important in deciding the region of convergence of the series and the location and nature of the singularities which limit it. While the region of convergence of the series of ladders is limited by a pole on the real axis, the radius of convergence of the complete series (which is smaller) is marked off by two branching points off the real axis. The formal sum of all diagrams displays a finite discontinuity at that point of the real axis where the ladders have a pole, thus changing from one branch to another. Each branch can be continued analytically through this discontinuity which is not a singularity of either branch. The ladders approximate only one branch, turning the finite discontinuity into a pole and thus one branch corresponding to the Cooper state, is lost. The remaining branch is described quite well by Brueckner's ladders even outside the region of convergence of the power series. It corresponds to the “normal” state. In the case of two-body interactions with short range and low density the most important contributions with wrong statistics are the two-body contributions of “many-body diagrams”. Their inclusion is equivalent to using the Cooper type solution of a Bethe-Goldstone equation.