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Quadrupole partial orders and triple-$q$ states on the face-centered cubic lattice

Kazumasa HattoriTakayuki IshitobiHirokazu Tsunetsugu
Dec 2022
We study $\Gamma_3$ quadrupole orders in a face-centered cubic lattice. The$\Gamma_3$ quadrupole moments under cubic symmetry possess a unique cubicinvariant in their free energy in the uniform ($q=0$) sector and the triple-qsector for the X points $q=(2\pi,0,0),(0,2\pi,0)$, and $(0,0,2\pi)$.Competition between this cubic anisotropy and anisotropic quadrupole-quadrupoleinteractions causes a drastic impact on the phase diagram both in the groundstate and at finite temperatures. We show details about the model constructionand its properties, the phase diagram, and the mechanism of the varioustriple-$q$ quadrupole orders reported in our preceding letter [J. Phys. Soc.Jpn. 90, 43701 (2021), arXiv:2102.06346]. By using a mean-field approach, weanalyze a quadrupole exchange model that consists of a crystalline-electricfield scheme with the ground-state $\Gamma_3$ non-Kramers doublet and theexcited singlet $\Gamma_1$ state. We find various triple-$q$ orders in thefour-sublattice mean-field approximation. A few partial orders of quadrupolesare stabilized in a wide range of parameter space at a higher transitiontemperature than single-$q$ orders. With lowering the temperature, thesepartial orders undergo phase transitions into further symmetry broken phases inwhich nonvanishing quadrupole moments emerge at previously disordered sites.The obtained phases in the mean-field approximation are investigated by aphenomenological Landau theory, which clearly shows that the cubic invariantplays an important role for stabilizing the triple-$q$ states. We also discussits implications for recent experiments in a few f- and d-electron compounds.