Particle-hole ring diagrams for fermions in two dimensions

TL;DR

This paper analyzes particle-hole ring diagrams in two-dimensional fermion systems, deriving the polarization function, calculating energy contributions analytically, and studying the effects of resummation and spin factors.

Contribution

It provides a detailed derivation of the polarization function, analytical energy calculations up to twelfth order, and explores the impact of resummation including exchange diagrams for 2D fermions.

Findings

Analytical expressions for polarization function in terms of square-root functions.

Energy per particle calculated up to twelfth order for contact interactions.

Resummation shows strong dependence on coupling parameter and differences from leading ring diagrams.

Abstract

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 $g$.