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Using ab initio density functional theory with static mean-fieldcorrelations, we calculate the Heisenberg and Dzyaloshinskii-Moriyainteractions (DMI) for an atomistic spin Hamiltonian for the lacunar spinel,GaV$_4$S$_8$. The parameters describing these interactions are used inatomistic spin dynamics and micromagnetic simulations. The magnetic propertiesof the lacunar spinel GaV$_4$S$_8$, a material well-known from experiment tohost magnetic skyrmions of N\'eel character, are simulated with these ab initiocalculated parameters. The Dzyaloshinskii-Moriya contribution to themicromagnetic energy is a sum of two Lifshitz invariants, supporting theformation of N\'eel skyrmions and its symmetry agrees with what is usuallyexpected for $C_{3\nu}$-symmetric systems. The are several conclusions one maydraw from this work. One concerns the quantum nature of the magnetism, where weshow that the precise magnetic state of the V$_4$ cluster is crucial forunderstanding quantitatively the magnetic phase diagram. In particular wedemonstrate that a distributed-moment state of each V$_4$ cluster explains wella variety of properties of GaV$_4$S$_8$, such as the band gap, observed Curietemperature and especially the stability of N\'eel skyrmions in theexperimentally relevant temperature and magnetic-field range. In addition, wefind that electronic correlations change visibly the calculated value of theDMI.

Dzyaloshinskii-Moriya interactions, N\'eel skyrmions and V$_4$ magnetic clusters in multiferroic lacunar spinel GaV$_4$S$_8$