Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. Orlando - M. Ruf

A classical $\mathbb{S}^2$ spin system with discrete out-of-plane anisotropy: variational analysis at surface and vortex scalings

created by cicalese on 21 Dec 2021
modified on 07 May 2022


Accepted Paper

Inserted: 21 dec 2021
Last Updated: 7 may 2022

Journal: Nonlinear Analysis
Year: 2021


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 $\Gamma$-limit 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 and 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.