Weyl fermions and the anomalous Hall effect in metallic ferromagnets

We reconsider the problem of the anomalous Hall effect in ferromagnetic SrRuO$_3$, incorporating insights from the recently developed theory of Weyl semimetals. We demonstrate that SrRuO$_3$ possesses a large number of Weyl nodes, separated in momentum space, in its bandstructure. While the nodes normally do not coincide with the Fermi energy, unless the material is doped, we show that even the nodes inside the Fermi sea have a significant effect on the physical properties of the material. In particular, we show that the common belief that (non-quantized part of) the intrinsic anomalous Hall conductivity of a ferromagnetic metal is entirely a Fermi surface property, is incorrect: there generally exists a contribution to the anomalous Hall conductivity that arises from topological Fermi-arc surface states, associated with the Weyl nodes, which is of the same order of magnitude as the Fermi surface contribution.