Differential–geometric method of integrating the two-dimensional Heisenberg model

We have integrated the two-dimensional Heisenberg model using classical differential geometry methods. Following a hodograph transformation, the model equations have been stated in terms of a metric tensor and its derivatives in a curvilinear coordinate system. It has been shown that their general solution describes all previously known exact solutions except for a plane vortex. We have predicted and analyzed a new type of vortex structure, a “vortex ring”, in a two-dimensional ferromagnet. Among the latter’s distinctive properties are the limited dimensions of the definition interval, the finiteness of its full energy, and the lack of the vortex core upon the existence of a vortex structure. The present paper covers a two-vortex solution.