Phase degree of freedom and topological properties of multiple-Q spin textures*

Topological spin textures, such as skyrmion lattices (SkLs) and hedgehog lattices (HLs) are approximately represented by superpositions of multiple spin density waves, and hence, called multiple-Q spin structures. In such structures, the phase degree of freedom of the superposed waves plays an important role in the topological properties as well as the symmetry of the magnetic textures [1], but the systematic investigation has not been performed thus far. In this study, we theoretically investigate the evolution of the two-dimensional SkL (3Q-SkL) and the three-dimensional HL (4Q-HL) while changing the phases as well as the magnetization. For the 3Q-SkL, we elucidate the topological phase diagram including the SkLs with the skyrmion number -2, -1, 1, and 2. In the case of the 4Q-HL, we find a variety of HLs with different numbers of topological objects called the magnetic hedgehogs and antihedgehogs connected by the Dirac strings. Analyzing the spin textures obtained for the specific Hamiltonians, we find that an external magnetic field can cause topological transitions with phase shifts. Our findings clarify that the phase is a key parameter to realize unprecedented magnetic and topological phases, and the emergent electromagnetic phenomena associated with the noncoplanar spin textures.

[1] T. Kurumaji et al., Science 365, 914 (2019); S. Hayami, T. Okubo, and Y. Motome, Nat. Commun. 12, 6927 (2021).