Superconductivity in a Dilute Two Dimensional Electron System

We develop a finite temperature mean field theory in the path integral picture for an extremely dilute system of interacting Fermions in a plane. In the limit of short ranged interaction, the system is shown to undergo a phase transition to a superconducting regime. Unlike the well known BCS transition for metals, this phase transition is found to be very sensitive to the two dimensional nature of the problem. A quantitative estimate of this sensitivity is carried out by repeating the analysis at non - zero thickness. The validity of the mean field results are also proved using renormalization group techniques.