Stabilization mechanisms of magnetic skyrmion crystal and multiple-Q states based on momentum-resolved spin interactions

Multiple-$Q$ states as represented by a magnetic skyrmion crystal and hedgehog crystal have been extensively studied in recent years owing to their unconventional physical properties. The materials hosting multiple-$Q$ states have been so far observed in a variety of lattice structures and chemical compositions, which indicates rich stabilization mechanisms inducing the multiple-$Q$ states. We review recent developments in the research of the stabilization mechanisms of such multiple-$Q$ states with an emphasis on the microscopic spin interactions in momentum space. We show that an effective momentum-resolved spin model is a canonical model for not only understanding the microscopic origin of various multiple-$Q$ states but also exploring further exotic multiple-$Q$ states with topological properties. We introduce several key ingredients to realize the magnetic skyrmion crystal with the skyrmion numbers of one and two, hedgehog crystal, meron–antimeron crystal, bubble crystal, and other multiple-$Q$ states. We also review that the effective spin model can be used to reproduce the magnetic phase diagram in experiments efficiently.