Microscopic theory of electric polarization induced by skyrmionic order in GaV₄S₈

The lacunar spinel GaV$_{4}$S$_{8}$ was recently suggested to be a prototype multiferroic material hosting skyrmion lattice states with a sizeable polarization $\boldsymbol{P}$ coupled to magnetic order. We explain this phenomenon on the microscopic level. On the basis of density functional theory, we construct an effective model describing the behavior of magnetically active electrons in a weakly coupled lattice formed by molecular orbitals of the (V$_{4}$S$_{4}$)$^{5+}$ clusters. By applying superexchange theory combined with the Berry-phase theory for $\boldsymbol{P}$, we derive a compass model relating the energy and polarization change with the directions of spins $\boldsymbol{e}_{i}$ in magnetic bonds. We argue that, although each skyrmion layer is mainly formed by superexchange interactions in the same plane, the spin-dependence of $\boldsymbol{P}$ arises from the stacking misalignment of such planes in the perpendicular direction, which is inherent to the lacunar spinel structure. We predict a strong competition of isotropic, $\sim \boldsymbol{e}_{i}\boldsymbol{e}_{j}$, and antisymmetric, $\sim \boldsymbol{e}_{i} \times \boldsymbol{e}_{j}$, contributions to $\boldsymbol{P}$ that explains the experimentally observed effect.