Microstructures in a two-dimensional frustrated spin system: Scaling regimes and a discrete-to-continuum limit

We study pattern formation within the $J_1$-$J_3$ - spin model on a two-dimensional square lattice in the case of incompatible (ferromagnetic) boundary conditions on the spin field. We derive the discrete-to-continuum $\mathrm{\Gamma }$-limit at the helimagnetic/ferromagnetic transition point, which turns out to be characterized by a singularly perturbed multiwell energy functional on gradient fields. Furthermore, we study the scaling law of the discrete minimal energy. The constructions used in the upper bound include besides rather uniform or complex branching-type patterns also structures with vortices. Our results show in particular that in certain parameter regimes the formation of vortices is energetically favorable.