BdG equations within lattice Hubbard model for the description of $\pi$-states in nanoscale S-FF-S junctions

We calculate the persistent currents in $S-F_{\uparrow }{F}_{\uparrow [\downarrow ]}-S$ structures (S denotes the superconductor in a closed ring geometry, F the ferromagnet in a two dimensional geometry, and the arrows the alignment of exchange field) as a function of the applied magnetic flux, self consistently by solving the Bogoliubov-de Gennes equations within the two dimensional Hubbard model. The local current shows a sign change i.e. a 0 to $\pi$ transition in the parallel alignment of the magnetizations and it does not change sign in antiparallel alignment.