Berry Phase Theory of the Anomalous Hall Effect: Application to Colossal Magnetoresistance Manganites

We show that the anomalous Hall effect (AHE) observed in colossal magnetoresistance manganites is a manifestation of Berry phase effects caused by carrier hopping in a nontrivial spin background. We determine the magnitude and temperature dependence of the Berry phase contribution to the AHE, finding that it increases rapidly in magnitude as the temperature is raised from zero through the magnetic transition temperature $T_c$, peaks at a temperature $T_{\max }>T_c$, and decays as a power of $T$, in agreement with experimental data. We suggest that our theory may be relevant to the anomalous Hall effect in conventional ferromagnets as well.