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

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