Resonant optical topological Hall conductivity from skyrmions

We study the high-frequency Hall conductivity in a two-dimensional model of conduction electrons coupled to a background magnetic skyrmion texture via an effective Hund's coupling term. For an ordered skyrmion crystal, a Kubo formula calculation using the basis of skyrmion crystal Chern bands reveals a resonant Hall response at a frequency set by the Hund's coupling, ω ~ J_H. A complementary real-space Kubo formula calculation for an isolated skyrmion in a box reveals a similar resonant Hall response. A linear relation between the area under the Hall resonant curve and the skyrmion density is discovered numerically and is further elucidated using a gradient expansion which is valid for smooth textures and a local approximation based on a spin-trimer calculation. We point out the issue of distinguishing this skyrmion contribution from a similar feature arising from spin-orbit interactions, as demonstrated in a model for Rashba spin-orbit coupled electrons in a collinear ferromagnet, which is analogous to the difficulty of unambiguously separating the dc topological Hall effect from the anomalous Hall effect. The resonant feature in the high-frequency topological Hall effect is proposed to provide a potentially useful local optical signature of skyrmions via probes such as scanning magneto-optical Kerr microscopy.