A classical \(\mathbb{S}^2\) spin system with discrete out-of-plane anisotropy: variational analysis at surface and vortex scalings. (English) Zbl 1512.49016
Summary: We consider a classical Heisenberg system of \(\mathbb{S}^2\) spins on a square lattice of spacing \(\varepsilon\). We introduce a magnetic anisotropy by constraining the out-of-plane component of each spin to take only finitely many values. Computing the \(\varGamma\)-limit of a suitable scaling of the energy functional as \(\varepsilon \to 0\) we prove that, in the continuum description, the system concentrates energy at the boundary of sets in which the out-of-plane component of the spin is constant. In a second step we analyze a different scaling of the energy and we prove that, in each of such phases, the energy can further concentrate on finitely many points corresponding to vortex-like singularities of the in-plane components of the spins.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49M25 | Discrete approximations in optimal control |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82D40 | Statistical mechanics of magnetic materials |
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