Dimension transcendence and anomalous charge transport in magnets with moving multiple- Q spin textures

Multiple-Q spin textures, such as magnetic bubble and skyrmion lattices, have been observed in a large family of magnets. These spin textures can be driven into motion by external stimuli. The motion of spin textures affects the electronic states. Here we show that to correctly describe the electronic dynamics, the momentum space needs to be transcended to higher dimensions by including the ancillary dimensions associated with phason modes of the translational motion of the spin textures. The electronic states have nontrivial topology characterized by the first and second Chern numbers in the high-dimensional hybrid momentum space. This gives rise to an anomalous electric charge transport due to the motion of spin textures. By deforming the spin textures, a nonlinear response associated with the second Chern number can be induced and results in the piezoelectricity. The charge transport is derived from the semiclassical equation of motion of electrons that depends on the Berry curvature in the hybrid momentum space. Our results suggest that the motion of multiple-Q spin textures has significant effects on the electronic dynamics and provides a platform to explore high-dimensional topological physics.