In systems with large local Hilbert space, multipolar degrees of freedom naturally emerge and enrich the landscape of ordered phases [1]. Recent studies have highlighted nonuniform multipolar orders, where competition between dipolar and quadrupolar moments leads to spatially modulated structures. An illustrative case is the amplitude-modulated spin-density-wave (SDW) state observed experimentally in the spinel magnet GeFe2O4 [2], whose microscopic origin remains unclear. The localized moments of Fe2+ ions in this compound can be described as spin-orbit-coupled pseudospin-1 states, where spin-orbit coupling and the trigonal crystal field give rise to exchange anisotropy and single-ion anisotropy (SIA), respectively. To investigate the implications of these features, we analyze a pseudospin-1 pyrochlore system using SU(3) coherent states to explore amplitude-modulated magnetic phases involving both dipolar and quadrupolar components. By focusing on the interplay between anisotropic dipolar exchange and the SIA acting on quadrupoles, we show that their cooperation provides a microscopic route to stabilizing nonuniform multipolar orders [3].
[1] L. V. Pourovskii et al., Nat. Rev. Mater. 10, 674 (2025).
[2] G. Perversi et al., Commun. Phys. 1, 69 (2018).
[3] H. Nakai and C. Hotta, Phys. Rev. B 112, 054401 (2025).