From the Ginzburg-Landau model to vortex lattice problems

We study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting "Coulombian renormalized energy" $W$ is a logarithmic type of interaction, computed by a "renormalization," and we believe it should be rather ubiquitous. We study various of its properties, and show in particular, using results from number theory, that among lattice configurations the hexagonal lattice is the unique minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its minimization in general remains open. The derivation of $W$ uses energy methods: the framework of $\Gamma$-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem.