Singular Structure and Enhanced Friedel Oscillations in the Two-Dimensional Electron Gas

We calculate the leading order corrections (in $r_s$) to the static polarization $\Pi^{*}(q,0,)$, with dynamically screened interactions, for the two-dimensional electron gas. The corresponding diagrams all exhibit singular logarithmic behavior in their derivatives at $q=2 k_F$ and provide significant enhancement to the proper polarization particularly at low densities. At a density of $r_s=3$, the contribution from the leading order {\em fluctuational} diagrams exceeds both the zeroth order (Lindhard) response and the self-energy and exchange contributions. We comment on the importance of these diagrams in two-dimensions and make comparisons to an equivalent three-dimensional electron gas; we also consider the impact these finding have on $\Pi^{*}(q,0)$ computed to all orders in perturbation theory.