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The lacunar spinel GaV$_{4}$S$_{8}$ was recently suggested to be a prototypemultiferroic material hosting skyrmion lattice states with a sizeablepolarization $\boldsymbol{P}$ coupled to magnetic order. We explain thisphenomenon on the microscopic level. On the basis of density functional theory,we construct an effective model describing the behavior of magnetically activeelectrons in a weakly coupled lattice formed by molecular orbitals of the(V$_{4}$S$_{4}$)$^{5+}$ clusters. By applying superexchange theory combinedwith the Berry-phase theory for $\boldsymbol{P}$, we derive a compass modelrelating the energy and polarization change with the directions of spins$\boldsymbol{e}_{i}$ in magnetic bonds. We argue that, although each skyrmionlayer is mainly formed by superexchange interactions in the same plane, thespin-dependence of $\boldsymbol{P}$ arises from the stacking misalignment ofsuch planes in the perpendicular direction, which is inherent to the lacunarspinel 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.

Microscopic theory of electric polarization induced by skyrmionic order in GaV$_{4}$S$_{8}$