Vortex patterns in Ginzburg-Landau minimizers

We present a survey of results obtained with Etienne Sandier on vortices in the minimiz- ers of the 2D Ginzburg-Landau energy of superconductivity with an applied magnetic field, in the asymptotic regime of large kappa where vortices become point-like. We de- scribe results which characterize the critical values of the applied field for which vortices appear, their numbers and locations. If the applied field is large enough, it is observed in experiments that vortices are densely packed and form triangular (hexagonal) lattices named Abrikosov lattices. Part of our results is the rigorous derivation of a mean field model describing the optimal density of vortices at leading order in the energy, and then the derivation of a next order limiting energy which governs the positions of the vortices after blow-up at their inter-distance scale. This limiting energy is a logarithmic- type interaction between points in the plane. Among lattice configurations it is uniquely minimized by the hexagonal lattice, thus providing a first justification of the Abrikosov lattice in this regime.