The Entropy Influence Conjecture

The antisymmetric exchange arises when relativistic effects, i.e the spin-orbit coupling, are taken into account. The coupling between spins ${𝐒}_A$ and ${𝐒}_B$ on sites A and B respectively takes the form $H_{DM}={𝐃}_{AB}\cdot ({𝐒}_A\times {𝐒}_B)$ where the coupling constant ${𝐃}_{AB}\propto (𝐱\times {𝐫}_{AB})$ depends on the vector ${𝐫}_{AB}$ connecting the site A to the site B, and on the displacement $𝐱$ of the ligand ion from the AB line.

Visualization of the antisymmetric exchange mechanism

It is clear that the Dzyaloshinskii-Moriya Interaction is non-zero only for non-collinear spin configurations, and when the ligand atom doesn't belong to the AB line. The magnetoelectric effect arises when, upon the application of an electric field, the ligand ion moves inducing as a consequence spin canting.