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

Abstract

Toward the understanding of incommensurate ${\Gamma }_3$ 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 ${\Gamma }_7$-${\Gamma }_8$ Hubbard model on a simple cubic lattice with the effective interactions that induce local ${\Gamma }_3$ states. By evaluating multipole susceptibility in a random phase approximation, we find that the hybridization between ${\Gamma }_7$ and ${\Gamma }_8$ orbitals plays a key role in the emergence of ${\Gamma }_3$ quadrupole ordering. We also emphasize that ${\Gamma }_3$ quadrupole ordering can be understood from the concept of {\it multipole nesting}, in which the Fermi surface region with large ${\Gamma }_8$ orbital density should be nested on the area with a significant ${\Gamma }_7$ 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 ${\Gamma }_3$ doublets are mainly composed of two singlets between ${\Gamma }_8$ and ${\Gamma }_7$ 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.

arXiv comment11 pages, 7 Postscript figures, accepted for publication in J. Phys. Soc. Jpn

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