Interactions between topological point singularities

The interaction between point singularities is investigated as a function of dimension d for the isotropic $n=d$ fixed length spin model. For dimensions greater than or equal to 3, the region of d-dimensional space which is perturbed by the pair of singularities collapses into a string joining the two singular cores. Thus the interaction between point singularities in the three-dimensional (3D) classical Heisenberg model ($n=d=3$) is radically different from the interactions between vortices in the 2D XY model ($n=d=2$).