Vortices and the lower critical field for a Ginzburg–Landau model of superconductors with ferromagnetic interactions

We study a Ginzburg–Landau functional for a vector-valued order parameter that carries a spin and couples directly to the magnetic field. We verify that spin coupling reduces the lower critical field H c1 , the smallest value of the applied field strength at which vortices penetrate the superconductor, and discuss the related phenomena of a ‘spontaneous vortex phase’ for sufficiently strong spin coupling, predicted by physicists. The method uses energy methods and a priori bounds to confine the vortices and spin transitions to very small balls