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 \Gamma_8 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 (ff\sigma) bonding. From the \Gamma_8 model, we derive an effective model for each lattice structure by using the second-order perturbation theory with respect to (ff\sigma). By further applying mean-field theory to the effective model, we find a \Gamma_{3g} antiferro-quadrupole transition for the sc lattice, a \Gamma_{2u} antiferro-octupole transition for the bcc lattice, and a longitudinal triple-q \Gamma_{5u} octupole transition for the fcc lattice. 
Katsunori Kubo and Takashi Hotta