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To clarify a key role of $f$ orbitals in the emergence ofantiferro-quadrupole structure in PrPb$_{3}$, we investigate the ground-stateproperty of an orbital-degenerate Kondo lattice model by numericaldiagonalization techniques. In PrPb$_{3}$, Pr$^{3+}$ has a $4f^{2}$configuration and the crystalline-electric-field ground state is a non-Kramersdoublet $\Gamma_{3}$. In a $j$-$j$ coupling scheme, the $\Gamma_{3}$ state isdescribed by two local singlets, each of which consists of two $f$ electronswith one in $\Gamma_{7}$ and another in $\Gamma_{8}$ orbitals. Since in a cubicstructure, $\Gamma_{7}$ has localized nature, while $\Gamma_{8}$ orbitals arerather itinerant, we propose the orbital-degenerate Kondo lattice model for aneffective Hamiltonian of PrPb$_{3}$. We show that an antiferro-orbital state isfavored by the so-called double-exchange mechanism which is characteristic ofmulti-orbital systems.

Antiferro-quadrupole state of orbital-degenerate Kondo lattice model with f^2 configuration