Theory of ac anomalous Hall conductivity ind-electron systems

Abstract

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