Particle–hole ring diagrams for fermions in two dimensions

The set of particle–hole ring diagrams for a many-fermion system in two dimensions is studied. The complex-valued polarization function is derived in detail and shown to be expressible in terms of square-root functions. For a contact-interaction the perturbative contributions to the energy per particle Ē(kf) are calculated in a closed analytical form from third up to twelfth order. The resummation of the particle–hole ring diagrams to all orders is studied and a pronounced dependence on the dimensionless coupling parameter αα is found. There is a substantial difference between the complete ring-sum with all exchange-type diagrams included and the standard resummation of the leading nn-ring diagrams only. The spin factor Sn(g)Sn(g) associated to the nnth order ring diagrams is derived for arbitrary spin-degeneracy gg.