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

Topological spin textures, such as skyrmion lattices (SkLs) and hedgehog lattices (HLs), have attracted a lot of interest due to the robustness protected by the topology and the intriguing transport and optical responses associated with the emergent electromagnetic field through the spin Berry phase mechanism [1,2]. Such spin textures are approximately represented by superpositions of multiple spin waves, and hence, called multiple-$Q$ spin structures. Thus, not only the magnetic structures but also the topological properties are modulated by various parameters in the superpositions, e.g., the number, periods, amplitudes, and propagation directions of the constituent helices [3,4,5]. Among them, the phase degree of freedom of the superposed waves may play an important role in topological transitions between different multiple-$Q$ spin structures [6,7], but the systematic investigation has not been performed thus far.
In this study, we theoretically investigate the evolution of the two-dimensional SkL composed of three spin waves (3$Q$-SkL) and the three-dimensional HL composed of four spin waves (4$Q$-HL) while changing the phases of the superposed waves [8]. For a systematic analysis of the phase degree of freedom, we construct a hyperspace representation of multiple-$Q$ spin structures by introducing additional dimensions, where the phase degree of freedom in the original physical space can be regarded as a spatial translation in the hyperspace. By using this framework, we elucidate the topological phase diagram for the 3$Q$-SkL while changing the sum of the phases of three waves, $\tilde{\varphi}$, and the uniform magnetization $m$; the skyrmion number $N_{\rm sk}$ changes among -2, -1, 0, 1, and 2 in a different way depending on the type of the constituent waves. In the case of the 4$Q$-HL, we obtain richer phase diagrams where the density of topological objects called hedgehogs and antihedgehogs changes in a wide range. We elucidate that the topological number and the positions of topological objects depend on the sum of phases of four waves, $\tilde{\varphi}$, and accordingly, the emergent magnetic field, which is directly related with the topological Hall effect, changes while varying $\tilde{\varphi}$. Our results of the complete phase diagrams for the phase shift provide good references to discuss how the actual systems experience the emergent electromagnetic phenomena, and the magnetic and topological phase transitions in an applied magnetic field.
This research was supported by Grant-in-Aid for Scientific Research Grants (Nos. JP18K03447, JP19H05822, JP19H05825, and JP21J20812), JST CREST (No. JP-MJCR18T2), and the Chirality Research Center in Hiroshima University and JSPS Core-to-Core Program, Advanced Research Networks. K.S. was supported by the Program for Leading Graduate Schools (MERIT-WINGS). Parts of the numerical calculations were performed in the supercomputing systems in ISSP, the University of Tokyo.

[1] G. E. Volovik, J. Phys. C $\mathbf{20}$, L87 (1987).
[2] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. $\mathbf{8}$, 899 (2013).
[3] B. Binz, A. Vishwanath, and V. Aji, Phys. Rev. Lett. $\mathbf{96}$, 207202 (2006); B. Binz and A. Vishwanath, Phys. Rev. B $\mathbf{74}$, 214408 (2006).
[4] J. H. Park and J. H. Han, Phys. Rev. B $\mathbf{83}$, 184406 (2011).
[5] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome, Phys. Rev. B $\mathbf{103}$, 054427 (2021); ibid, $\mathbf{103}$, 184421 (2021).
[6] T. Kurumaji $et~al$., Science $\mathbf{365}$, 914 (2019).
[7] S. Hayami, T. Okubo, and Y. Motome, preprint (arXiv:2005.03168).
[8] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome, in preparation.