The spherical model and Bose-Einstein condensation

In 1968 Gunton and Buckingham pointed out that there is a close relationship between the critical behaviour of the spherical model and that of the ideal Bose gas. In this paper we concentrate on this similarity between the two systems. We show that in the spherical model there is in fact Bose-Einstein condensation into the spin modes with low energy. As in the Bose gas we distinguish between generalised Bose-Einstein condensation and Bose-Einstein condensation. We find that the spherical model in certain cases has two critical temperatures: one temperature T c corresponding to the onset of generalised Bose-Einstein condensation (and of spontaneous magnetisation) and a lower temperature T m at which generalised condensation becomes condensation into the spin mode with the lowest energy