Phase diagram for a superconducting ring obtained from exact analytical results

We present complete phase diagrams for a simple superconducting ring, obtained from exact analytical solutions of the Ginzburg-Landau (GL) equations. The proterties of the phase diagram depend on both the GL parameter and the aspect ratio of the ring, related to the self-induction coefficient. Exact expressions for the order parameter and the free energy involved in the transitions between states with different quantum numbers are also presented. We determine the role played by metastable solutions, which include modulated solutions and states with ``phase slip centers.'' These allow for a change in the phase winding number. Exact relations for all metastability lines in the phase diagram are derived.