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We investigate the topological defect populations for superconductingvortices and magnetic skyrmions on random pinning substrates under drivingamplitudes that are swept at different rates or suddenly quenched. When thesubstrate pinning is sufficiently strong, the system exhibits a nonequilibriumphase transition at a critical drive into a more topologically ordered state.We examine the number of topological defects that remain as we cross theordering transition at different rates. In the vortex case, the systemdynamically orders into a moving smectic, and the Kibble-Zurek scalinghypothesis gives exponents consistent with directed percolation. Due to theirstrong Magnus force, the skyrmions dynamically order into an isotropic crystal,producing different Kibble-Zurek scaling exponents that are more consistentwith coarsening. We argue that in the skyrmion crystal, the topological defectscan both climb and glide, facilitating coarsening, whereas in the vortexsmectic state, the defects cannot climb and coarsening is suppressed. We alsoexamine pulsed driving across the ordering transition and find that the defectpopulation on the ordered side of the transition decreases with time as a powerlaw, indicating that coarsening can occur across nonequilibrium phasetransitions. Our results should be general to a wide class of nonequilibriumsystems driven over random disorder where there are well-defined topologicaldefects.

The Kibble-Zurek Scenario and Coarsening Across Nonequilibrium Phase Transitions in Driven Vortices and Skyrmions