A Bourgain–Brezis–Mironescu Formula Accounting for Nonlocal Antisymmetric Exchange Interactions

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 by-product 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.