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 $\bar E(k_f)$ are calculated in 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 $\alpha$ 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 $n$-ring diagrams only. The spin factor $S_n(g)$ associated to the $n$-th order ring diagrams is derived for arbitrary spin-degeneracy $g$.