Spin anisotropy and quantum Hall effect in the kagomé lattice: Chiral spin state based on a ferromagnet

A ferromagnet with spin anisotropies on the two-dimensional (2D) kagomé lattice is theoretically studied. This is a typical example of the flat-band ferromagnet. The Berry phase induced by the tilting of the spins opens the band gap and quantized Hall conductance ${\sigma }_{\mathrm{xy}}=\pm e^2/h$ is realized. This is the most realistic chiral spin state based on the ferromagnetism. We also discuss the implication of our results to the anomalous Hall effect observed in the metallic pyrochlore ferromagnets $R_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ $(R=\mathrm{Nd},$ Sm, Gd).