Multipole ordering in f-electron systems on the basis of a j-j coupling   scheme
  We investigate microscopic aspects of multipole ordering in f-electron systems with emphasis on the effect of lattice structure. For the purpose, first we construct f-electron models on three kinds of lattices, simple cubic (sc), bcc, and fcc, by including f-electron hopping through (ff\sigma) bonding in a tight-binding approximation on the basis of a j-j coupling scheme. Then, an effective model is derived in the strong-coupling limit for each lattice structure with the use of second-order perturbation theory with respect to (ff\sigma). By applying mean-field theory to such effective models, we find different types of multipole ordered state depending on the lattice structure. For the sc lattice, a \Gamma_{3g} antiferro-quadrupole transition occurs at a finite temperature and as further lowering temperature, we find another transition to a ferromagnetic state. For the bcc lattice, a \Gamma_{2u} antiferro-octupole ordering occurs first, and then, a ferromagnetic phase transition follows it. Finally, for the fcc lattice, we find a single phase transition to the longitudinal triple-q \Gamma_{5u} octupole ordering. 
Katsunori Kubo and Takashi Hotta