Spin amplitude wave due to dipole-quadrupole hybridization in spin-1 pyrochlore magnets

We explore the quantum pseudospin-1 pyrochlore magnet featuring Fe$^{2+}$-based spinel oxides that addresses the formation of amplitude-modulated spin-density waves. We propose that the relatively small spin-orbit coupling and the small extra crystal field splitting in these materials create anisotropic exchange interactions and strong single-ion anisotropy, respectively, whose interplay becomes the source of quadrupolar moments selectively appearing on certain sublattices, leading to a spatially modulated hybrid of dipolar and quadrupolar moments. This mechanism represents the possibility of insulating magnets to form an exotic phase with coexisting liquid-solid properties.