Keisuke Mitsumoto and Yoshiaki ¯ Ono 7 Full Paper J. Phys. Soc. Jpn. 0 0.005 0.01 0 0.01 0.02 0.005 0.01 0 g (a) half−filling U=0 0.2 0.4 0.6 (b) quarter−filling ~ 0 ~ scheme, Hotta has developed the realistic three orbital Anderson model with Γ$_{7}$ doublet and Γ$_{8}$ quartet CEF states simulating filled skutterudite compounds. 34) It is interesting to extend our approach to such realistic mod- els and is now under the way. The authors thank T. Goto, Y. Nemoto and T. Hotta for many useful comments and discussions. This work is supported by a Grant-in-Aid for Scientific Research in a Priority Area ”Skutterudite” (No. 15072202) from the Ministry of Education, Culture, Sports, Science and Technology. spectral function is almost unchanged due to the effect of g except further narrowing of the low energy quasi- particle bands. In the two-orbital periodic Anderson model with the coupling g between the local orbital fluctuation and the Jahn-Teller phonon, the orbital fluctuation is enhanced due to the both effects of U and g , and then the heavy- fermion state is realized due to the cooperative effect. This is a striking contrast to the case with the single- orbital periodic Anderson model with the coupling g be- tween the local charge fluctuation and the local phonon, where the spin (charge) fluctuation is enhanced (sup- pressed) due to the effect of U , while the charge (spin) fluctuation is enhanced (suppressed) due to the effect of g , and then the effects of U and g on the heavy- fermion state compete with each other. 30–32) As the ab- solute value of the local orbital fluctuation for n$_{f}$ = 2 is larger than that for n$_{f}$ = 1, the cooperative effect of U and g for n$_{f}$ = 2 is more pronounced than that for n$_{f}$ = 1. Although the present model is the simplest version, the multi orbital periodic Anderson model coupled with the Jahn-Teller phonons has the potential to account for the experimental observations. The heavy fermion state of nonmagnetic origin due to the cooperative effect of U and g seems to be consistent with the magnetically robust heavy fermion state observed in the filled skutterudite SmOs$_{4}$Sb$_{12}$. The enhancement of the orbital fluctuation and the anomalous phonon softening due to the cooper- ative effect of U and g seemes to be responsible for the extra softening of the low temperature elastic constant which is not reproduced only due to the quadrupolar sus- ceptibility in the CEF state observed in PrOs$_{4}$Sb$_{12}$. To discuss these compounds in detail, however, we need to consider more realistic models including the effects such as the spin-orbit interaction, the Hund’s rule coupling and the crystalline-electric-field. Within the j-j coupling Fig. 11. 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