# A limiting interaction energy for Ginzburg-Landau vortices

# A limiting interaction energy for Ginzburg-Landau vortices

This is a joint work with Etienne Sandier where we study 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 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. Among lattice configurations the hexagonal lattice is the unique minimizer of $W$, thus providing a first rigorous hint at the Abrikosov lattice. I will describe briefly how $W$ also appears in the study of the statistical mechanics of Coulomb gases/random matrices.