Anomalous Hall Effect due to the spin chirality in the Kagom\'{e} lattice

We consider a model for a two dimensional electron gas moving on a kagom'{e} lattice and locally coupled to a chiral magnetic texture. We show that the transverse conductivity $\sigma \mathrm{\_}xy$ does not vanish even if spin-orbit coupling is not present and it may exhibit unusual behavior. Model parameters are the chirality, the number of conduction electrons and the amplitude of the local coupling. Upon varying these parameters, a topological transition characterized by change of the band Chern numbers occur. As a consequence, $\sigma \mathrm{\_}xy$ can be quantized, proportional to the chirality or have a non monotonic behavior upon varying these parameters.