  We study the high frequency Hall conductivity in a two-dimensional (2D) 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:
$\hbar\omega_{\text{res}} \approx J_H$. 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 d.c. 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.
