Emergence of the Abrikosov lattice in several models with two dimensional Coulomb interaction

Abstract

We consider three different models coming from physics and involving the Coulomb interaction of points in the plane. The first is the classical Coulomb gas in an external potential, the second is the Ginzburg–Landau model of superconductivity, the third is the Ohta–Kawasaki model of polymers. In superconductivity, one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattices named "Abrikosov lattices" in physics. We show how these Abrikosov lattices are expected to emerge in the three systems, via the minimization of a "Coulombian renormalized energy" that we defined with Etienne Sandier. We also present applications to the statistical mechanics of Coulomb gases, which are related to some random matrix models, and show how the previous results lead to expecting crystallisation in the low temperature limit.