Vortex lattices in two-dimensional ferromagnets

We apply methods of classical differential geometry to integrate the two-dimensional Heisenberg model. We formulate the model equations using the metric tensor and its derivatives, associated with a curvilinear coordinate system obtained via the hodograph transformation. We find a solution for the metric tensor depending on two variables. Based on this solution, we predict and analyze a new type of magnetic structure, the “vortex ring” in a two-dimensional ferromagnet. Its distinctive features include a finite region of existence, finite total energy, and the absence of a vortex core despite the presence of a vortex structure. Finally, we derive various types of vortex lattices through conformal transformations of the vortex ring using Jacobi and Weierstrass elliptic functions.