Topological nature of anomalous Hall effect in ferromagnets

The anomalous Hall effect in two-dimensional (2D) ferromagnets is considered 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. These band crossing points appear as sharp peaks and 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 three-dimensional (3D) systems is also discussed.