Fig. 1: Crystal structure, magnetic structure and basic properties of MnGe thin film.

a The crystal structure of MnGe. b The schematic illustration of the three-dimensional spin hedgehog lattice. c The positions of the emergent monopoles (green dots) and anti-monopoles (yellow dots) at zero magnetic field in the green box of panel b²¹. The red and blue colors represent the positive and negative emergent magnetic fields, respectively. d The magnetic-field (H) dependence of the Hall conductivity at each temperature (solid curves). The data are shifted vertically for clarity. The dashed curves (σ_xy^A) represent the estimated magnetic-field dependence of the anomalous Hall effect estimated from the magnetization curves (see also the discussion in Supplementary Note 1). The shaded areas, which represents the deviation from σ_xy^A, are ascribed mainly to topological Hall effect (THE). e The schematic illustration of the experimental setup. The direction of the magnetic field is parallel to the light propagation direction (Faraday configuration). WG1-4 are wire-grid polarizers. f The longitudinal terahertz conductivity spectra (σ_xx(ω)). Filled circles at zero frequency are the DC values obtained from the transport measurement. Dashed lines are the fits with Drude formula (see the text). g The Faraday rotation spectra. Filled circles at zero frequency indicate the values expected from the DC transport properties (see Methods). The error bars indicate the standard deviation.