A circular model for an interacting Fermi gas

Abstract

In this work we study a circular one dimensional model for an interacting Fermi gas. We consider an interparticle potential having the properties that its Fourier transform has a functional form (-)k/k2, and."whose spatial form is such that it can be solved exactly in the thermodynamic limit using the Green's function technique. The results obtained from such an exact solution are compared with the conclusions reached, by studying the problem using three dimentional systems. It turns out however that the RPA brakes down for our model system. We therfore make a further approximation (which we call the Modified Random Phase Approximation (MRPA) ). _ l. __The study of the system in the MRPA shows that a :first order phase transition is possible. The thermodynamic properties of the system and the question of the possibility of a first order phase transition is also studied in the exact solution of our .model system. __Upon further study of our circular one dimensional model within the MRPA and using the BCS simplification and the BCS ansatz for the trial function for the system, we show that our system exhibits superconductive behavior in the sense that we show the existence of an energy gap in the single particle energy spectrum. We show the existence of a critical magnetic field and a discontinuity in the specific heat for the system. i.e. we show the existence of a second order phase transition.