Abstract
The study gives a decisive answer to the recently risen question about the type and origin of interaction between spin vortices and spin vacancies in two-dimensional (2D) spin models. The approach is based on the low-temperature approximation of the 2D $XY$ model known as the Villain model and does not involve any additional approximations, thus preserving the lattice structure. The exact form of the Hamiltonian describing a system of topological charges and a vacant site supports the attractive type of interaction between the vacancy and the charges. The quantitative difference between the characteristics of the vortex behavior in the 2D $XY$ and Villain models due to the different energy of the vortex cores in the two models is pointed out. This leads to a conclusion that the interaction between a vortex and a spin vacancy and between a vortex and the antivortex differs quantitatively for small separations in the two mentioned models.
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