Topological Nature of Anomalous Hall Effect in Ferromagnet

The anomalous Hall effect in two-dimensional ferromagnets is discussed to be the physical realization of the parity anomaly in (2+1)D, and the band crossing points behave as the topological singularity in the Brillouin zone. This appears as the sharp peaks and the sign changes of the transverse conductance $\sigma_{xy}$ as a function of the Fermi energy and/or the magnetization. The relevance to the experiments including the three dimensional systems is also discussed.