Vortex textures for doubly periodic planar nanomagnet with inclusions

We consider periodic states inside a thin film ferromagnetic element with periodic nano-scale array of non-magnetic holes. The starting point is the model [1], which is based on the existence of a well-defined energy hierarchy within a nano-structured magnet with Heisenberg exchange interaction being dominant. From a mathematical point of view the metastable states are the harmonic maps of the domain to the sphere. For the case of a continuous thin film A.A. Belavin and A.M. Polyakov in 1975 and D. Gross in 1978 proposed a series of local solutions with singularities modeling the magnetic vortexes. In the multiply-connected case the solutions are similar, but there are additional constraints on the vortex positions [2,3]. Here we present the full set of constraints for the infinitely-connected case (thin film with periodic array of inclusions). We parametrize all the metastable states in terms of genus two Riemann surfaces. For 6-parametric shapes of inclusions the exact solutions are given. (Joint study with K.L. Metlov.)
[1] K. L. Metlov, 'Magnetization patterns in ferromagnetic nano-elements as functions of complex variable'// Phys. Rev. Lett. 105, 107201 (2010)
[2] Bogatyrev A.B., Metlov K.L. 'Topological constraints on positions of magnetic solitons in multiply-connected planar magnetic nano-elements' Phys. Rev. B 95:2, 024403 (2017) arXiv:1609.02509
[3] A. B. Bogatyrev, “Real meromorphic differentials: A language for describing meron configurations in planar magnetic nanoelements”, Theoret. and Math. Phys., 193:1 (2017), 1547–1559; arXiv: 1610.04984.