Phase transitions in the mesoscopic superconducting square

We solve the Ginzburg-Landau equation (GLE) for the mesoscopic thin film of the square shape in the magnetic field. In the limit of Ginzburg-Landau parameter $\kappa \to \infty$ 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 $H_{c1}$. We identify order parameters based on symmetry arguments and we propose Landau functional describing the second order phase transition.