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Interactions of composite magnetic skyrmion-superconducting vortex pairs in ferromagnetic superconductors

We study composite topological excitations in ferromagnetic superconductors consisting of bound states of magnetic spin textures (skyrmions) and superconducting vortices. Using a Ginzburg--Landau framework with Zeeman coupling between the magnetization and superconducting magnetic field, we demonstrate that skyrmion-vortex pairs (SVPs) form energetically stable bound states. By analyzing their asymptotic interactions, we identify regimes in which SVPs exhibit both short-range repulsion and long-range attraction, leading to clustering phenomena. Our results provide a field-theoretical basis for understanding suggest pathways for controlling hybrid topological matter through long-range interactions.