Magnetic field induced phase branches of the superconducting transition in two-dimensional square \pi-loop arrays

Based on the results of explicit forms of free energy density for each possible arrangement of magnetization fluxes in large-scale two-dimensional (2D) square \pi-loop arrays given by Li et al 2007 Chin. Phys. 16 1450, the field-cooled superconducting phase transition is further investigated by analysing the free energy of the arrays with a simplified symmetrical model. Our analytical result is exactly the same as that obtained in Li's paper by means of numerical calculations. It is shown that the phase transition splits into two branches with either ferromagnetic or anti-ferromagnetic flux ordering, which depends periodically on the strength of external magnetic flux \phi_\rm e through each loop and monotonically on the screen parameter \beta of the loops in the arrays. In principle, the diagram of the phase branches is similar to that of its one-dimensional counterpart. The influence of thermal fluctuation on the flux ordering during the transition from normal to superconducting states of the \pi -loop arrays is also discussed.