Anomalous Hall effect and spontaneous Chern insulators in the kagome lattice

We consider a two dimensional electron gas  locally coupled with a magnetic background with non-zero scalar chirality on the kagome lattice. In order to control the magnetic texture, we consider two spin models as source of the magnetic background: i) a spin Hamiltonian with spontaneous net chirality and ii) a model with chirality induced by the magnetic field. In the first case  we analyse the phase diagram as a function of the Fermi energy and the temperature of the classical spins by computing the Density-of-States and the Hall conductance. In the second case we show that a Chern number of the filled bands can be "tuned" with the external magnetic field, resulting in a topological insulator in which the direction of the chiral edge mode can be reverted by reversing the orientantion of the magnetic field applied to the classical magnetic system.