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We carry out density functional theory calculations which demonstrate thatthe electron dynamics in the skyrmion phase of Fe-rich Mn$_{1-x}$Fe$_x$Gealloys is governed by Berry phase physics. We observe that the magnitude of theDzyaloshinskii-Moriya interaction, directly related to the mixed space-momentumBerry phases, changes sign and magnitude with concentration $x$ in directcorrelation with the data of Shibata {\it et al.}, Nature Nanotech. {\bf 8},723 (2013). The computed anomalous and topological Hall effects in FeGe arealso in good agreement with available experiments. We further develop a simpletight-binding model able to explain these findings. Finally, we show that theadiabatic Berry phase picture is violated in the Mn-rich limit of the alloys.

Dzyaloshinskii-Moriya interaction and Hall effects in the skyrmion phase of MnFeGe alloys