Theory of AC Anomalous Hall Conductivity in d-electron systems

To elucidate the intrinsic nature of anomalous Hall effect (AHE) in $d$-electron systems, we study the AC anomalous Hall conductivity (AHC) in a tight-binding model with ($d_{xz},d_{yz}$)-orbitals. We drive a general expression for the AC AHC ${\sigma }_{xy}(\omega )$, which is valid for finite quasiparticle damping rate $\gamma$=$\mathrm{\hslash }\mathrm{/}2\tau$, and find that the AC AHC is strongly dependent on $\gamma$. When $\gamma =+0$, the AC AHC shows a spiky peak at finite energy $\mathrm{\Delta }$ that originates from the interband particle-hole excitation, where $\mathrm{\Delta }$ represents the minimum band-splitting measured from the Fermi level. In contrast, we find that this spiky peak is quickly suppressed when $\gamma$ is finite. By using a realistic value of $\gamma (\omega )$ at $\omega =\mathrm{\Delta }\mathrm{/}2$ in $d$-electron systems, the spiky peak is considerably suppressed. In the present model, the obtained results also represents the AC spin Hall conductivity in a paramagnetic state.