Anomalous Hall Effect in type-I Weyl metals

We study the ac anomalous Hall conductivity $\sigma_{xy}(\omega)$ of a Weyl semimetal with broken time-reversal symmetry. Even in the absence of free carriers these materials exhibit a "universal" anomalous Hall response determined solely by the locations of the Weyl nodes. We show that the free carriers, which are generically present in an undoped Weyl semimetal, give an additional contribution to the ac Hall conductivity. We elucidate the physical mechanism of the effect and develop a microscopic theory of the free carrier contribution to $\sigma_{xy}(\omega)$. The latter can be expressed in terms of a small number of parameters (the electron velocity matrix, the Fermi energy $\mu$, and the "tilt" of the Weyl cone). The resulting $\sigma_{xy}(\omega)$ has resonant features at $\omega \sim 2 \mu$ which may be used to separate the free carrier response from the filled-band response using, for example, Kerr effect measurements. This may serve as diagnostic tool to characterize the doping of individual valleys.