Circular One-Dimensional Model for the Electron-Ion System. II. First-Order Phase Transitions in the Random-Phase Approximation

This paper is the second in a series on a model of "electrons" and "ions" on a circle, with interaction Fourier coefficients $\frac{1}{n^2}$ ($n$ integer). In this sequel to the random-phase-approximation (RPA) analysis of paper I, we show that the model undergoes, in the RPA, a first-order phase transition. This is shown by calculating the free energy $F$ from the partition function and demonstrating that $F$ is unstable with respect to the formation of a second phase. In the mechanism of the phase transition, it is crucial that quantum theory ascribes discrete energy levels to the collective (plasma) degrees of freedom, which are unexcited or "frozen out" if the thermal energy is small with respect to the spacing of these discrete levels, and whose number $s$ furthermore depends on the temperature and density. Thus one obtains a system with an effective number of degrees of freedom, $N-s$, which increases as the temperature increases or the density decreases. This is taken heuristically as the feature underlying the possibility for the transition.