From the Ginzburg-Landau Model to Vortex Lattice Problems

We introduce a "Coulombian renormalized energy" W which is a logarithmic type of interaction between points in the plane, computed by a "renormalization." We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open. Our motivation is the study of minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields H _c1 and H _c2. In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a Γ-limit of the Ginzburg-Landau energy. More precisely we show that the vortices of minimizers of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W, thus providing a first rigorous hint at the Abrikosov lattice. This is a next order effect compared to the mean-field type results we previously established. The derivation of W uses energy methods: the framework of Γ-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem, that we introduce.