Reentrant phase transition of granular superconductors

It is shown that the effects of the charging energy on the mean-field transition temperature of a granular superconductor depend in a profound manner on the choice of the interval for the phase $\phi$ of the order parameter. For $-\infty <\phi <\infty$ a well pronounced reentrant behavior is obtained for the diagonal model, in which the Coulomb interaction is neglected. The reentrance disappears in this model when the interval is taken to be $-\pi <\phi <\pi$. Dipolar excitations of the nondiagonal model, considered recently by Efetof are shown to diminish the extent of reentrance for the $-\infty <\phi <\infty$ model and to produce a weak reentrance in the $-\pi <\phi <\pi$ case. The reentrant behavior is shown to be due to the thermally induced phase coherence via the low-lying excited states characterized by the vanishing of the frequency of the phase rotation.