Multipole ordering inf-electron systems on the basis of aj−jcoupling scheme

Abstract

We investigate microscopic aspects of multipole ordering in $f$-electron systems with emphasis on the effect of lattice structure. For this purpose, first we construct $f$-electron models on three kinds of lattices, simple cubic (sc), bcc, and fcc, by including $f$-electron hopping through $(ff\ensuremath{\sigma})$ bonding in a tight-binding approximation on the basis of a $j\text{\ensuremath{-}}j$ 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 $(ff\ensuremath{\sigma})$. By applying mean-field theory to such effective models, we find different types of multipole ordered states, depending on the lattice structure. For the sc lattice, a ${\ensuremath{\Gamma}}_{3g}$ antiferro-quadrupole transition occurs at a finite temperature and, as we further lower the temperature, we find another transition to a ferromagnetic state. For the bcc lattice, a ${\ensuremath{\Gamma}}_{2u}$ 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-$\mathbf{q}\phantom{\rule{0.3em}{0ex}}{\ensuremath{\Gamma}}_{5u}$ octupole ordering.