Magnetic field induced phase branches of the superconducting transition in two-dimensional square π-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 π-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 φe through each loop and monotonically on the screen parameter β 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 π-loop arrays is also discussed.

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

Based on the results of explicit forms of free energy density foreach possible arrangement of magnetization fluxes in large-scaletwo-dimensional (2D) square $pi $-loop arrays given by Litextit{et al} [2007 {em Chin. Phys.} {bf 16} 1450], thefield-cooled superconducting phase transition is furtherinvestigated by analysing the free energy of the arrays with asimplified symmetrical model. Our analytical result is exactly thesame as that obtained in Li's paper by means of numericalcalculations. It is shown that the phase transition splits into twobranches with either ferromagnetic or anti-ferromagnetic fluxordering, which depends periodically on the strength of externalmagnetic flux $phi_{e}$ through each loop and monotonically on thescreen parameter $beta $ of the loops in the arrays. In principle,the diagram of the phase branches is similar to that of itsone-dimensional counterpart. The influence of thermal fluctuation onthe flux ordering during the transition from normal tosuperconducting states of the $pi $-loop arrays is also discussed.