Low energy configurations of topological singularities in two dimensions: A $\Gamma$-convergence analysis of dipoles

Abstract

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg-Landau energy. Denoting by $\epsilon$ the length scale parameter in such models, we focus on the $|\log \epsilon |$ energy regime. It is well known that, for configurations whose energy is bounded by $c|\log \epsilon |$, the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying $\pi |\log \epsilon |$ energy, plus a measure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses. Here we perform a compactness and $\Gamma$-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale ${\epsilon }^s$, for $0<s<1$), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical $\Gamma$-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order $c|\log \epsilon |$ with $c<\pi$\,.