Circular One-Dimensional Model for the Electron-Ion System. I. Random-Phase Approximation

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Abstract
This paper, which is first of a series about a system of "electrons" and "ions" constrained to move on a circle with an interaction potential whose Fourier transform is the inverse square of the momentum variable, applies the conventional tools of the equation of motion method in the random-phase approximation and Bardeen and Pines's canonical transformations, as well as the well-known Cooper-pair considerations. It is found that the model is in all ways similar to the three-dimensional electron-ion system: It displays collective modes (plasmons and phonons), screening, etc., with the familiar renormalization factors. The existence of a phonon-mediated attraction and the Cooper-pair analysis lead to the existence of bound pairs and an energy gap.