Imaginary part of Hall conductivity in a tilted doped Weyl semimetal with both broken time-reversal and inversion symmetry

We consider a Weyl semimetal with finite doping and tilt within a continuum model Hamiltonian with both broken time-reversal and inversion symmetry. We calculate the absorptive part of the anomalous ac Hall conductivity as a function of photon energy Omega for both type-I and type-II Weyl semimetals. For a given Weyl node, changing the sign of its chirality or of its tilt changes the sign of its contribution to the absorptive Hall conductivity with no change in magnitude. For a noncentrosymmetric system we find that there are ranges of photon energies for which only the positive or only the negative-chirality node contributes to the imaginary (absorptive) part of the Hall conductivity. There are also other photon energies where both chiralities contribute, and there can be other ranges of Omega where there is no absorption associated with the ac Hall conductivity in type-I semimetals and regions where it is instead constant for type-II semimetals. We comment on implications for the absorption of circularly polarized light.