content_image/423081.jpg content_image/423082.jpg content_image/423096.jpg content_image/423097.jpg content_image/423083.jpg
where ρ ( ϵ ) is the density of states (DOS) for the c - electron, ρ ( ϵ ) = ∑ $_{k}$δ ( ϵ − ϵ$_{k}$ ). In the above equation, ˜ G$_{fl}$ (i ω$_{n}$ ) is the bare Green’s function for the effective impurity Anderson model with U = g = 0 in an effec- tive medium which will be determined self-consistently. The effective impurity Anderson model with finite U and/or g is solved by using the ED method for a finite- size cluster to obtain G$_{flσ}$ (i ω$_{n}$ ) and Σ$_{lσ}$(i ω$_{n}$ ) at T = 0. 26, 27, 30–32, 35–37) In the present study, we use 4 site clus- ter and the cutoff of phonon number is set to be 9 for each Jahn-Teller mode. We assume a semielliptic DOS with the bandwidth W = 1, ρ ( ϵ $_{) = 2}$√$_{1}$ − ϵ $^{2}$/π , and we set V = 0 . 1 and ω$_{0}$ = 0 . 01. We concentrate our attention on the nonmagnetic state with Σ$_{lσ}$(i ω$_{n}$ ) = Σ(i ω$_{n}$ ) at half- filling with n$_{f}$ = 2 and quarter-filling with n$_{f}$ = 1, where n$_{f}$ ≡ ∑ lσ 〈 ˆ n$_{filσ}$ 〉 is the average number of f -electrons per atom. 

## 3. Results 

## 3.1 Renormalization factor 

In Fig.1, we plot the renormalization factor Z = (1 − dΣ( ω ) d ω | ω $_{=0}$) − 1 as a function of g for several values of U at half- and quarter-filling. When g = 0, the ef- fective mass of the quasiparticle m $^{∗}$/m = Z − 1 increases with increasing U and then the heavy fermion state with m $^{∗}$/m ≫ 1 is realized in the strong correlation regime. When the electron-phonon coupling increases, the effec- tive mass increases for all values of U resulting in the heavy fermion state due to the cooperative effect of the Coulomb interaction and the electron-phonon coupling. This is a striking contrast to the case with the single- orbital periodic Anderson-Holstein model where the both effects are compete with each other as mentioned be- fore. 30–32) The effective mass m $^{∗}$/m for half-filling is con- siderably larger than that for quarter-filling as shown in Figs.1 (a) and (b). This is due to the effect of the local or- bital fluctuation which is largely enhanced for half-filling as compared to quarter-filling as mentioned later. 

## 3.2 Local orbital fluctuation 

Fig.2 shows the local orbital fluctuation written by 

with ˆ n$_{fil}$ = ∑ σ ˆ n$_{filσ}$ as a function of g for several values of U , where 〈 τ$_{z}$ 〉 = 〈 τ$_{x}$ 〉 = 0 in the absence of orbital order and 〈 τ 2 z 〉 = 〈 τ 2 x 〉 due to the symmetry of e$_{u}$ repre- sentation. When g = 0, 〈 τ 2 z 〉 increases with increasing U as previously obtained in the two-orbital periodic Ander- son model. 33) When g increases for U ̸ = 0, 〈 τ 2 z 〉 steeply increases at half-filling, while it gradually increases at quarter-filling except for U = 0 as shown in Figs. 2 (a) and (b). Remarkably, the increase in the local orbital fluctuation in the strong coupling regime is largely en- hanced due to the Coulomb interaction for half-filling in contrast to the case for quarter-filling where it is sup- pressed. The difference between half-filling and quarter- filling in the local orbital fluctuation is due to that in the double-occupancy probabilities as shown below. 

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## 3.3 Double occupancy probabilities 

The local orbital fluctuation 〈 τ 2 z 〉 given in eq.(4) can be rewritten by 

<img src='content_image/423083.jpg'>

with ˆ n$_{fi}$ = ∑ l ˆ n$_{fil}$ . To discuss the local orbital fluc- tuation in detail, we also plot the double-occupancy probabilities as functions of g for several values of U at half- and quarter-filling in Fig.3, where 〈 ˆ n$_{filσ}$ ˆ n$_{fil}$$_{¯}$ σ 〉 , 〈 ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ l σ 〉 and 〈 ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ lσ 〉 are on-site correlation func- tions for intra-orbital spin-singlet, inter-orbital spin- singlet and inter-orbital spin-triplet, respectively. When g = 0, the double-occupancy probabilities coincide to each other due to the spin-orbital symmetry with U = U $^{′}$, and then, they are suppressed due to the electron cor- 

