  The present study concerns the nonlocal-to-local convergence of a family of
exchange energy functionals in the limit of very short-range interactions. The
analysis accounts for both symmetric and antisymmetric exchange. Our result is
twofold. First, we extend the Bourgain-Brezis-Mironescu formula to encompass
the scenario where antisymmetric contributions are encoded into the energy.
Second, we prove that, under physically relevant assumptions on the families of
exchange kernels, the family of nonlocal functionals Gamma-converges to their
local counterparts. As a byproduct of our analysis, we obtain a rigorous
justification of Dzyaloshinskii-Moriya interactions in chiral magnets under the
form commonly adopted in the variational theory of micromagnetism when modeling
antisymmetric exchange interactions.
