We investigate the role of four-spin interactions in stabilizing multiple-$Q$ topological spin textures and demonstrate their ability to realize a skyrmion crystal. While such higher-order interactions are known to be important, their intricate nature makes systematic model construction challenging. To address this issue, we develop a theoretical framework that connects microscopic real-space four-spin couplings to their effective interactions in momentum space, providing a clear route to engineer target magnetic phases. Applying this framework to a frustrated Heisenberg model with designed four-spin interactions, we identify the stabilization of the zero-field skyrmion crystal with a skyrmion number of two per magnetic unit cell via simulated annealing. Furthermore, our momentum-space analysis reveals the intrinsic mechanism by which the well-known ring-exchange interaction also favors the skyrmion crystal. Our findings not only present a concrete model for a higher-order skyrmion crystal but also offer a general methodology for understanding and designing a wide range of complex multiple-$Q$ magnetic orders driven by multispin interactions.

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• Exchange interaction
• Frustrated magnetism
• Magnetic texture
• Skyrmions
• Heisenberg model
• Methods in magnetism