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The midinfrared anomalous Hall effect (AHE) can provide critical new information for resolving the controversial origins of the dc AHE in Ca${}_{x}$Sr${}_{1\ensuremath{-}x}$RuO${}_{3}$. The complex Faraday and Kerr angles, as well as the complex Hall conductivity ${\ensuremath{\sigma}}_{xy}$, are measured in Ca${}_{x}$Sr${}_{1\ensuremath{-}x}$RuO${}_{3}$ films as a function of mid- and near-infrared energy $E$ from 0.1 eV to 1.4 eV, magnetic field $H$, temperature $T$, and Ca concentration $x$. For the ferromagnetic state from $x=0$ to 0.4, the $({d}_{xz},{d}_{yz})$-orbital tight-binding model is employed to investigate the quasiparticle role in the low energy response of the AHE ${\ensuremath{\sigma}}_{xy}(E)$ since the Berry curvature term becomes weak at low energies. The infrared Hall sign reversals with $T$ are observed only at $x=0$ and 0.13, which is narrower than the Ca concentration range in which the dc Hall sign reversal appears. The similarity of the infrared Hall angles between paramagnetic and ferromagnetic Ca${}_{x}$Sr${}_{1\ensuremath{-}x}$RuO${}_{3}$ compounds demonstrates the symmetric nature of the Hall response around the quantum phase transition at $x=0.7$.

The origin of the Hall effect in many itinerant ferromagnets is still not resolved with an anomalous contribution from the sample magnetization that can exhibit extrinsic or intrinsic behavior. We report the midinfrared (MIR) measurements of the complex Hall $({\ensuremath{\theta}}_{H})$, Faraday $({\ensuremath{\theta}}_{F})$, and Kerr $({\ensuremath{\theta}}_{K})$ angles, as well as the Hall conductivity $({\ensuremath{\sigma}}_{xy})$ in a ${\text{SrRuO}}_{3}$ film in the 115--1400 meV energy range. The magnetic field, temperature, and frequency dependence of the Hall effect is explored. The MIR magneto-optical response shows very strong frequency dependence including sign changes. Below 200 meV, the MIR ${\ensuremath{\theta}}_{H}(T)$ changes sign between 120 and 150 K, as is observed in dc Hall measurements. Above 200 meV, the temperature dependence of ${\ensuremath{\theta}}_{H}$ is similar to that of the dc magnetization and the measurements are in good agreement with predictions from a band calculation for the intrinsic anomalous Hall effect (AHE). The temperature and frequency dependence of the measured Hall effect suggests that whereas the behavior above 200 meV is consistent with an intrinsic AHE, the extrinsic AHE may play an important role in the lower-energy response.

The electron-doped cuprate ${\mathrm{Pr}}_{2\ensuremath{-}x}{\mathrm{Ce}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ is investigated using infrared magneto-optical measurements. The optical Hall conductivity ${\ensuremath{\sigma}}_{xy}(\ensuremath{\omega})$ shows a strong doping, frequency, and temperature dependence consistent with the presence of a temperature- and doping-dependent coherent backscattering amplitude which doubles the electronic unit cell and produces a spin density wave state. At low temperatures, the data suggest that the coherent backscattering vanishes at a quantum critical point inside the superconducting dome and is associated with the commensurate antiferromagnetic order observed by other workers. Using a spectral weight analysis, we have further investigated the Fermi-liquid-like behavior of the overdoped sample. The observed Hall-conductance spectral weight is about ten times less than that predicted by band theory, raising the fundamental question of the effect of Mott and antiferromagnetic correlations on the Hall conductance of strongly correlated materials.