Study of the Nonlinear Dependence of Anomalous Hall Conductivity on Magnetization in Weak Itinerant Ferromagnet ZrZn2

As opposed to the ordinary Hall effect, the anomalous Hall effect (AHE) remained unexplained for decades, and, amazingly, some misconceptions have survived even now, in particular, the claim that AHE is linearly related to the net magnetization. Karplus and Luttinger provided a quantum-mechanical explanation of AHE by explicitly including the SOC and the Berry curvature of electronic bands. They did address the question of linearity, but only in the relatively uncommon limit of the exchange coupling smaller than SOC. Now the linear relation in traditional ferromagnets is understood as a domain population effect: both AHE and magnetization are independently proportional to the domain disbalance. In this connection, it is interesting to check to what extent this relation will hold in {\em single-domain} itinerant ferromagnet, the closest case to that analyzed by Karplus and Luttinger? We answer this question by direct calculations, using the Karplus-Luttinger formula, of AHE in a prototypical itinerant ferromagnet, ZrZn$_2$. We show that in the zero-magnetization limit, $M\rightarrow 0$, the linear relation hold, but at rather small moments of $\sim 0.4\ \mu_B$/Zr breaks down completely and even flips the sign.