Quadrupole Orders on the Fcc Lattice

We theoretically study electric quadrupole orders in f-electron systems on the fcc lattice. Quadrupole degrees of freedom O-20 and O-22 originate in the non-Kramers doublet ground state Gamma(3) of ion with f(2) electron configuration. For discussing quadrupole orders, we use a minimal model with isotropic (J) and anisotropic (K) nearest-neighbor interactions, and determine the phase diagram using a four-site mean-field approximation at zero and finite temperatures. Quadrupoles couple the Gamma(3) doublet to the singlet excited state Gamma(1), and its effects on canting antiferro orders are examined in detail. We found that this coupling leads to a rich phase diagram including two- and four-sublattice antiferro phases, and that two phases show a partial order of quadrupoles at finite temperatures.