Low energy configurations of topological singularities in two dimensions: A Γ-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 [Formula: see text] the length scale parameter in such models, we focus on the [Formula: see text] energy regime. It is well known that, for configurations whose energy is bounded by [Formula: see text], the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying [Formula: see text] 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 [Formula: see text]-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale [Formula: see text], for [Formula: see text]), 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 [Formula: see text]-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order [Formula: see text] with [Formula: see text].