Renormalized energy between fractional vortices with topologically induced free discontinuities on 2-dimensional Riemannian manifolds

TL;DR

This paper investigates the asymptotic behavior of a combined Ginzburg-Landau and perimeter energy functional on 2D Riemannian manifolds, revealing how geometry influences interactions between fractional vortices and string defects.

Contribution

It establishes a compactness and Γ-convergence result describing the limit energy and the interplay of vortices and string defects influenced by manifold geometry.

Findings

Limit energy depends on manifold geometry.

Fractional vortices and string defects interact in the limit.

The analysis provides a framework for understanding defect interactions on curved surfaces.

Abstract

On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg-Landau and a perimeter term. Our scaling allows low-energy states to be described by an order parameter which can have finitely many point singularities (vortex-like defects) of (possibly) fractional-degree connected by line discontinuities (string defects) of finite length. Our main result is a compactness and $\Gamma$-convergence theorem which shows how the coarse grained limit energy depends on the geometry of the manifold in driving the interaction between vortices and string defects.