Phase transitions in the mesoscopic superconducting square

We solve the Ginzburg-Landau equation for a mesoscopic thin film of square shape in the magnetic field. In the limit of large screening length we find a series of first- and second-order phase transitions as temperature and/or magnetic field changes. First-order phase transitions between giant-flux states can be described with a simple variational procedure. We discuss the similarity with rotating liquid He-4 and derive a simple formula for H-cl. We identify order parameters based on symmetry arguments and we propose a Landau functional describing the second-order phase transition.