Interaction energy of domain walls in a nonlocal Ginzburg-Landau type model from micromagnetics

We study a variational model from micromagnetics involving a nonlocal Ginzburg-Landau type energy for S^1-valued vector fields. These vector fields form domain walls, called Neel walls, that correspond to one-dimensional transitions between two directions within the unit circle S^1. Due to the nonlocality of the energy, a Neel wall is a two length scale object, comprising a core and two logarithmically decaying tails. Our aim is to determine the energy differences leading to repulsion or attraction between Neel walls. In contrast to the usual Ginzburg-Landau vortices, we obtain a renormalised energy for Neel walls that shows both a tail-tail interaction and a core-tail interaction. This is a novel feature for Ginzburg-Landau type energies that entails attraction between Neel walls of the same sign and repulsion between Neel walls of opposite signs.