Multipole ordering in f -electron systems

In order to investigate multipole ordering in f -electron systems from a microscopic viewpoint, we study the so-called Γ8 models on three kinds of lattices, simple cubic (sc), bcc, and fcc, based on a j-j coupling scheme with f -electron hopping integrals through (ffσ) bonding. From the Γ8 model, we derive an effective model for each lattice structure by using the second-order perturbation theory with respect to (ffσ). By further applying mean-field theory to the effective model, we find a Γ3g antiferro-quadrupole transition for the sc lattice, a Γ2u antiferro-octupole transition for the bcc lattice, and a longitudinal triple-q Γ5u octupole transition for the fcc lattice.