Low energy configurations of topological singularities in two dimensions: A Γ -convergence analysis of dipoles

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 ε the length scale parameter in such models, we focus on the |logε| energy regime. It is well known that, for configurations whose energy is bounded by c|logε|, the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying π|logε| 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 Γ-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale εs, for 0<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 Γ-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order c|logε| with c<π.