Clearing-out of dipoles for minimisers of 2-dimensional discrete energies with topological singularities

TL;DR

This paper introduces a combinatorial dipole-removal method for discrete models of material defects, establishing conditions under which boundary energy controls interior defect charges, applicable to models like XY systems and dislocations.

Contribution

It develops a novel dipole-removal argument using max-flow min-cut and duality for planar graphs, providing sharp conditions for defect control in discrete energy minimizers.

Findings

Sharp conditions for boundary energy controlling interior defects

Robust dipole-removal method applicable to various models

Use of max-flow min-cut theorem and planar graph duality

Abstract

A key question in the analysis of discrete models for material defects, such as vortices in spin systems and superconductors or isolated dislocations in metals, is whether information on boundary energy for a domain can be sufficient for controlling the number of defects in the interior. We present a general combinatorial dipole-removal argument for a large class of discrete models including XY systems and screw dislocation models, allowing to prove sharp conditions under which controlled flux and boundary energy guarantee to have minimizers with zero or one charges in the interior. The argument uses the max-flow min-cut theorem in combination with an ad-hoc duality for planar graphs, and is robust with respect to changes of the function defining the interaction energies.