Keisuke Mitsumoto and Yoshiaki ¯ Ono 5 Full Paper J. Phys. Soc. Jpn. Fig. 6. The normalized local lattice fluctuation 〈 Q $^{2}$〉 / 〈 Q $^{2}$〉 0 , as a function of g for several values of U at half-filling (a) and quarter-filling (b). Fig. 7. (Color online) The f-electron spectral function (vertical bars) and the DOS (dotted lines) as functions of ω for several values of U and g at half-filling, where the Fermi level is set to zero. 3.6 Local lattice fluctuation The local lattice fluctuation is defined by 〈 Q 2 ν 〉 = 〈 ( ˆ Q$_{iν}$ − 〈 ˆ Q$_{iν}$ 〉 ) $^{2}$〉 and 〈 Q 2 1 〉 = 〈 Q 2 〉 ( ≡ 〈 Q $^{2}$〉 ) due to the symmetry of E$_{g}$ modes. In Fig.6, we plot the normal- ized local lattice fluctuation, 〈 Q $^{2}$〉 / 〈 Q $^{2}$〉 $_{0}$, where 〈 Q $^{2}$〉 0 = 1 / 2 ω$_{0}$ is the value for the zero-point oscillation with g = 0. When g increases, 〈 Q $^{2}$〉 monotonically increases for all values of U . Remarkably, the increase in 〈 Q $^{2}$〉 observed in the strong coupling regime is largely en- hanced due to the Coulomb interaction especially for half-filling, where the orbital fluctuation coupled with the local phonons is largely enhanced. This is a strik- ing contrast to the case with the single-orbital periodic Anderson model where 〈 Q $^{2}$〉 is suppressed due to the Coulomb interaction as well as the valence fluctuation as mentioned before. 30–32) 3.7 F -electron spectral function Finally, we calculate the local phonon Green’s func- tion defined by D$_{ν}$ ( τ ) = −〈 T$_{τ}$b$_{iν}$ ( τ ) b + $_{iν}$(0) 〉 , whose spec- tral function is plotted as a function of ω for half-filling in Fig.9 and for quarter-filling in Fig.10, respectively. When g increases, the lowest excited energy shifts to low en- ergy as previously observed for the single-orbital peri- odic Anderson-Holstein model. 30–32) For a small value of U = 0 . 2, the lowest excited state has a dominant contri- bution to the spectral weight. On the other hand, for 3.8 Phonon spectral function In Fig.8, we plot the f -electron spectral function as a function of ω at quarter half-filling for U = g = 0 ( Z = 1), U = 0 and g = 0 . 021 ( Z ∼ 0 . 20), U = 0 . 6 and g = 0 ( Z ∼ 0 . 11), and U = 0 . 6 and g = 0 . 008 ( Z ∼ 0 . 066). The specific features of the f -electron spec- tral function and the DOS for quarter-filling shown in Fig.8 are almost the same as those for half-filling shown in Fig.7 except that the DOS is asymmetric in contrast to the case with half-filling where the DOS is symmet- ric. Then, the system becomes heavy fermion metal with large effective mass m $^{∗}$/m = Z − $^{1}$, instead of the Kondo insulator with narrow c - f hybridization gap ∼ Z Δ$_{cf}$ re- alized for half-filling. W , however, the clear lower and upper Hubbard bands are not observed, instead incoherent sub-bands are ob- served together with the quasiparticle bands. For U = 0 . 6 and g = 0 . 004 shown in Fig.7 (d), the c - f hybridiza- tion gap becomes narrower ( ∼ Z Δ$_{cf}$ with Z ∼ 0 . 065) than that for U = 0 . 6 and g = 0 (see Fig.7 (c)) due to the cooperative effect of the Coulomb interaction and the electron-phonon coupling. In this case, the high en- ergy structure of the lower and upper Hubbard bands observed for U = 0 . 6 and g = 0 is almost unchanged by the effect of g . 0 5 10 15 0 0.01 0.02 5 Q 2 g (a) half−filling U=0 0.2 0.4 0.6 (b) quarter−filling Q$^{2}$ $_{0}$ Q 2 Q$^{2}$ $_{0}$ 0 0.2 0.4 0 10 20 30 0 0.1 0 20 40 0 0.1 0 50 100 −0.5 0 0.5 0 0.1 0 50 (a) U=0 g=0 (b) U=0 g=0.018 (c) U=0.6 g=0 (d) U=0.6 g=0.004 f−electron spectral function DOS Fig.7 shows the f -electron spectral function as a func- tion of ω at half-filling, where we also plot the analytic DOS for U = g = 0 in Fig.7 (a) and the DOS eval- uated by a Lorentzian broadening with width of 0 . 1 | ω | for finite U and/or g in Figs.7 (b)-(d). For U = g = 0, the Fermi level is in the c - f hybridization gap Δ$_{cf}$ as shown in Fig.7 (a). For U = 0 . 6 and g = 0 shown in Fig.7 (c), the c - f hybridization gap is renormalized as ∼ Z Δ$_{cf}$ with the renormalization factor Z ∼ 0 . 099 due to the strong electron correlation effect, i. e., the system becomes Kondo insulator. In this case, we also observe the lower and upper Hubbard bands around ω = ± U/ 2. For U = 0 and g = 0 . 018 shown in Fig.7 (b), the c - f hybridization gap is renormalized as ∼ Z Δ$_{cf}$ with Z ∼ 0 . 048 due to the effect of strong electron-phonon coupling. We note that, in the case with large phonon frequency, ω$_{0}$ ≫ W , the system can be described by the effective negative- U$_{eff}$ Anderson model with | U$_{eff}$ | = 2 g $^{2}$/ω$_{0}$ , where the lower and upper bands are formed around U$_{eff}$ / 2. In the present case with small ω$_{0}$