Microscopic Theory of ${\mathrm{\Gamma }}_3$ Quadrupole Ordering in Pr Compounds on the Basis of a $j$-$j$ Coupling Scheme

Toward the understanding of incommensurate ${\mathrm{\Gamma }}_3$ quadrupole ordering in PrPb${}_3$, we develop a microscopic theory of multipole ordering in $f^2$-electron systems from an itinerant picture on the basis of a $j$-$j$ coupling scheme. For this purpose, we introduce the ${\mathrm{\Gamma }}_7$-${\mathrm{\Gamma }}_8$ Hubbard model on a simple cubic lattice with the effective interactions that induce local ${\mathrm{\Gamma }}_3$ states. By evaluating multipole susceptibility in a random phase approximation, we find that the hybridization between ${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_8$ orbitals plays a key role in the emergence of ${\mathrm{\Gamma }}_3$ quadrupole ordering. We also emphasize that ${\mathrm{\Gamma }}_3$ quadrupole ordering can be understood from the concept of {\it multipole nesting}, in which the Fermi surface region with large ${\mathrm{\Gamma }}_8$ orbital density should be nested on the area with a significant ${\mathrm{\Gamma }}_7$ component when the positions of the Fermi surfaces are shifted by the ordering vector. This concept cab be intuitively understood from the fact that local ${\mathrm{\Gamma }}_3$ doublets are mainly composed of two singlets between ${\mathrm{\Gamma }}_8$ and ${\mathrm{\Gamma }}_7$ orbitals. Finally, we discuss the possible relevance of the present theory to the experimental results of PrPb${}_3$ and point out some future problems in this direction of research.