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

Abstract

We study ${\mathrm{\ensuremath{\Gamma}}}_{3}$ quadrupole orders in a face-centered cubic lattice. The ${\mathrm{\ensuremath{\Gamma}}}_{3}$ quadrupole moments under cubic symmetry possess a unique cubic invariant in their free energy in the uniform ($\mathbf{q}=\mathbit{0}$) sector and the triple-$\mathbf{q}$ sector for the $X$ points $\mathbf{q}=(2\ensuremath{\pi},0,0),(0,2\ensuremath{\pi},0)$, and $(0,0,2\ensuremath{\pi})$. Competition between this cubic anisotropy and anisotropic quadrupole-quadrupole interactions causes a drastic impact on the phase diagram both in the ground state and at finite temperatures. We show details about the model construction and its properties, the phase diagram, and the mechanism of the various triple-$\mathbf{q}$ quadrupole orders reported in our preceding letter [J. Phys. Soc. Jpn. 90, 043701 (2021)]. By using a mean-field approach, we analyze a quadrupole exchange model that consists of a crystalline-electric field scheme with the ground-state ${\mathrm{\ensuremath{\Gamma}}}_{3}$ non-Kramers doublet and the excited singlet ${\mathrm{\ensuremath{\Gamma}}}_{1}$ state. We have found various triple-$\mathbf{q}$ orders in the four-sublattice mean-field approximation. A few partially ordered phases are stabilized in a wide range of parameter space and they have a higher transition temperature than single-$\mathbf{q}$ orders. With lowering temperature, there occur transitions from these partially ordered phases into further symmetry broken phases in which previously disordered sites acquire nonvanishing quadrupole moments. The identified phases in the mean-field approximation are further analyzed by a phenomenological Landau theory. This analysis reveals results qualitatively consistent with the mean-field results and also shows that the cubic invariant plays an important role for stabilizing the triple-$\mathbf{q}$ states. The present mechanism for the triple-$\mathbf{q}$ states also takes effect in systems with different types of quadrupoles, and we discuss its implications for recent experiments in a few $f$- and $d$-electron compounds.