Topological quantization of the flow of magnetic skyrmions driven by a ratchet-like potential under thermal fluctuations

We consider a magnetic skyrmion adiabatically driven by a spin-polarized electrical current periodic in both space and time and asymmetric in space, and also subject to a random magnetic field representing the thermal fluctuations. We show that when the random magnetic field is low enough, while the time variation of the driving current is slow enough, the skyrmion flow is an integer multiply of the ratio between the space and time periods, the integer being a topological invariant called Chern number. This result is also demonstrated by numerically solving the stochastic Landau-Lifshitz-Gilbert (sLLG) and Langevin equations. Our work suggests a novel method of manipulating skyrmions with topological stability.