Dzyaloshinski-Moriya interactions in the kagome lattice

The kagom\'e lattice exhibits peculiar magnetic properties due to its strongly frustated cristallographic structure, based on corner sharing triangles. For nearest neighbour antiferromagnetic Heisenberg interactions there is no Neel ordering at zero temperature both for quantum and classical s pins. We show that, due to the peculiar structure, antisymmetric Dzyaloshinsky-Moriya interactions (${\bf D} . ({\bf S}_i \times {\bf S}_j)$) are present in this latt ice. In order to derive microscopically this interaction we consider a set of localized d-electronic states. For classical spins systems, we then study the phase diagram (T, D/J) through mean field approximation and Monte-Carlo simulations and show that the antisymmetric interaction drives this system to ordered states as soon as this interaction is non zero. This mechanism could be involved to explain the magnetic structure of Fe-jarosites.