Keisuke Mitsumoto and Yoshiaki ¯ Ono Full Paper 4 J. Phys. Soc. Jpn. 0 0.2 0.4 0 0.01 0.02 0.1 U=0 0.6 <n$_{fil}$ n$_{fil}$ > − <n$_{fil}$ n$_{fil}$ > − − <n$_{fil}$ n$_{fil}$ > − (a) half−filling (b) quarter−filling double−occupancy probabilities g ^ ^ ^ ^ ^ ^ 0 1 0 0.01 0.02 1 <(n fi −<n fi >) 2 > g (a) half−filling U=0 0.2 0.4 0.6 (b) quarter−filling ^ ^ <(n fi −<n fi >) 2 > ^ ^ Fig. 4. The valence fluctuation 〈 (ˆ n$_{fi}$ − 〈 ˆ n$_{fi}$ 〉 ) $^{2}$〉 as a function of g for several values of U at half-filling (a) and quarter-filling (b). Fig. 3. (Color online) The double-occupancy probabilities, 〈 ˆ n$_{filσ}$ ˆ n$_{fil}$ ¯ $_{σ}$〉 (solid lines), 〈 ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ l $_{σ}$〉 (dashed lines) and 〈 ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ $_{lσ}$〉 (dotted lines), as functions of g for U = 0 and 0 . 6 at half-filling (a) and quarter-filling (b), respectively. 0 0.5 1 0 0.01 0.02 0.5 1 <S$^{2}$> g (a) half−filling U=0 0.2 0.4 0.6 (b) quarter−filling <S$^{2}$> relation effect, especially in the case with quarter-filling as shown in Fig.3 (b). On the other hand, when g = 0, the degeneracy is resolved, and then, the intra-orbital spin-singlet correlation 〈 ˆ n$_{filσ}$ ˆ n$_{fil}$$_{¯}$ σ 〉 is enhanced, while the inter-orbital spin-triplet correlation 〈 ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ lσ 〉 and the inter-orbital spin-singlet correlation ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ l σ are suppressed due to the effect of g . As the results, 〈 τ 2 z 〉 given in eq.(5) is largely enhanced for half-filling with large values of the double-occupancy probabilities, while it is slightly enhanced for quarter-filling with small values of those as shown in Fig.2. 3.4 Local charge fluctuation The local charge fluctuation of f -electrons, i. e., the valence fluctuation is written by Fig. 5. The local moment 〈 S $^{2}$〉 as a function of electron-phonon coupling constant g for several values of U at half-filling (a) and quarter-filling (b). 3.5 Local moment 〈 (ˆ n$_{fi}$ − 〈 ˆ n$_{fi}$ 〉 ) $^{2}$〉 = 〈 ˆ n$_{fi}$ 〉 − 〈 ˆ n$_{fi}$ 〉 2 + 2 ∑ l 〈 ˆ n$_{fil}$$_{↑}$ ˆ n$_{fil}$$_{↓}$ 〉 + 2 ∑ σσ ′ 〈 ˆ n$_{fi}$$_{1}$$_{σ}$ ˆ n$_{fi}$$_{2}$$_{σ}$ ′ 〉 , (6) 〈 S $^{2}$〉 = 3 4 ( 〈 ˆ n$_{fi}$ 〉 + 2 ∑ σ 〈 ˆ n$_{fi}$$_{1}$$_{σ}$ ˆ n$_{fi}$$_{2}$$_{σ}$ 〉 − 2 ∑ ll ′ 〈 ˆ n$_{fil}$$_{↑}$ ˆ n$_{fil}$ ′ ↓ 〉 ) and is plotted as a function of g for several values of U at half- and quarter-filling in Fig.4. When g = 0, the valence fluctuation decreases with increasing U to- gether with decreasing double-occupancy probabilities for both half- and quarter-filling due to the electron cor- relation effect. When g = 0, the valence fluctuation is suppressed or almost constant due to the effect of g in contrast to the case with the single-orbital periodic An- derson model where the valence fluctuation is coupled with local phonons and is enhanced due to the effect of g . 30–32) We note that the sign of the final term (the inter- orbital correlation) in eq.(6) is opposite to that in eq.(5), and then, the changes in the intra- and inter- orbital cor- relations cancel with each other (see Fig.3) resulting in the decrease in the valence fluctuation with g in contrast to the increase in the local orbital fluctuation. is plotted as a function of g for several values of U at half- and quarter-filling in Fig.5. When g = 0, 〈 S $^{2}$〉 is enhanced due to the effect of U . When g increases, 〈 S decreases with decreasing (increasing) spin-triplet (singlet) corre- lation as shown in Fig.3. The decrease in 〈 S $^{2}$〉 with g for half-filling is larger than that for quarter-filling, as the double-occupancy probabilities for half-filling are larger than those for quarter-filling (see Fig.3). The local moment, 〈 S $^{2}$〉 = 〈 ˆ S 2 i 〉 = 〈 ˆ S 2 ix + ˆ S 2 iy + ˆ S 2 iz 〉 , written by