Keisuke Mitsumoto and Yoshiaki ¯ Ono Full Paper 2 J. Phys. Soc. Jpn. precise studies for the electron-phonon impurity mod- els 19–22) and the multi-level impurity models 23, 24) have been done by using the numerical renormalization group (NRG) method and have confirmed the previous re- sults including the double-well potential formation in the strong coupling regime. The similar approaches have also applied to solve the lattice models such as the Holstein model 25) and the Hubbard-Holstein model 26) in infinite dimensions on the basis of the dynamical mean-field the- ory 27) (DMFT) and have revealed that the system be- comes bipolaronic insulator when the electron-phonon coupling is larger than a critical value, g > g$_{c}$ , where the effective potential becomes double-well type. More recently, Hattori and Miyake have discussed the ultra- sonic dispersion in the Holstein model by using the self- consistent ladder approximation. 28) To discuss the effect of local lattice vibrations on the rare-earth and actinide compounds, the periodic Anderson-Holstein model, where local phonons couple with valence fluctuations of f -electrons which interact with each other via the Coulomb interaction and hy- bridize with conduction electrons, was extensively stud- ied in early theories for mixed-valence systems such as Sm chalcogenides. 29) As the strong correlation effect due to both the Coulomb interaction and the electron-phonon coupling is crucial for describing the heavy-fermion state in these systems, we need reliable and nonperturbative approaches such as the DMFT. Recently, we have stud- ied the periodic Anderson-Holstein model by using the DMFT. 30–32) What we found are as follows: (1) In the strong electron-phonon coupling regime, g > ∼ g$_{c}$ , the sys- tem shows an anomalous heavy fermion behavior which is accompanied by a large lattice fluctuation and an ex- treme phonon softening. (2) A simple harmonic potential for ions for g < ∼ g$_{c}$ changes into an effective double-well potential for g > ∼ g$_{c}$ . (3) The pairing interaction between the conduction electrons has a maximum at g ≈ g$_{c}$ . (4) The heavy fermion state due to the electron-phonon cou- pling is realized in the wide range of the f-electron num- ber n$_{f}$ , while that due to the Coulomb interaction is real- ized in the narrow range near the half-filling n$_{f}$ ∼ 1. (5) The effect of the electron-phonon coupling on the heavy fermion state and that of the Coulomb interaction are compete with each other. In the rare-earth and actinide compounds, orbital fluc- tuations of f -electrons are considered to play crucial roles in determining the heavy-fermion behavior. 33) Re- cently, Hotta studied the multiorbital impurity Ander- son model in which the orbital fluctuations of f -electrons couple with Jahn-Teller (JT) phonons by using the NRG method and observed the remarkable quasi-Kondo be- havior due to the dynamical JT effect. 34) As for the lattice model, we have studied the two-orbital periodic Anderson model coupled with JT phonons by using the DMFT and have found that the heavy fermion state is realized due to the cooperative effect of the Coulomb in- teraction and the electron-phonon coupling. 35) This is a striking contrast to the case with the periodic Anderson- Holstein model where the both effects are compete with each other as mentioned above. 30–32) However, the effects of the degeneracy of JT phonon mode and the f -electron where c † ilσ ( f † $_{ilσ}$) is a creation operator for a conduction (c-)electron ( f -electron) with orbital l (= 1 , 2) and spin σ (= ↑ , ↓ ) at site i , and ˆ n$_{filσ}$ = f † $_{ilσ}$f$_{ilσ}$ . b † iν is a creation operator for a Jahn-Teller phonon with mode ν (= 1 , 2) at site i , where the normal coordinates for the Jahn-Teller phonons are given by ˆ Q$_{iν}$ = 1 $_{/}$√$_{2}$$_{ω}$$_{0}$$_{ν}$$_{(}$$_{b}$$_{iν}$ + b † $_{iν}$). ϵ$_{k}$ , ϵ$_{f}$ , and V are the dispersion of c-electron, the atomic level of f -electron, and the c- f hybridization, respectively. In the model eq.(1), the Jahn-Teller phonons with the fre- quency ω$_{0}$$_{ν}$ couple with the local orbital fluctuations of f -electrons, ˆ τ$_{ix}$ and ˆ τ$_{iz}$ , via the electron-phonon coupling g$_{ν}$ . 34) The model eq.(1) also includes the Coulomb inter- action between f -electrons: the intra and inter orbital di- rect Coulomb U and U ′ and the exchange coupling J . For simplicity, we assume ω$_{01}$ = ω$_{02}$ ( ≡ ω$_{0}$ ), g$_{1}$ = g$_{2}$ ( ≡ g ), U = U ′ and J = 0 in the present paper. To solve the model eq.(1), we use the DMFT 27) in which the model is mapped onto an effective single im- purity two-orbital Anderson model coupled with Jahn- Teller phonons. 26) The local Green’s function G$_{flσ}$ (i ω$_{n}$ ) and the local self-energy Σ$_{lσ}$(i ω$_{n}$ ) for the f -electron sat- isfy the following self-consistency conditions: G$_{flσ}$ (i ω$_{n}$ ) = ∫ dϵ ρ ( ϵ ) i ω$_{n}$ − ϵ$_{f}$ − Σ$_{lσ}$(i ω$_{n}$ ) − V 2 i ω n − ϵ H = ∑ k lσ ϵ$_{k}$c † k $_{lσ}$c$_{k}$$_{lσ}$ + ϵ$_{f}$ ∑ ilσ f † $_{ilσ}$f$_{ilσ}$ + V ∑ ilσ ( f † $_{ilσ}$c$_{ilσ}$ + h.c. ) + U ∑ il ˆ n$_{fil}$$_{↑}$ ˆ n$_{fil}$$_{↓}$ + U ′ ∑ iσσ ′ ˆ n$_{fi}$$_{1}$$_{σ}$ ˆ n$_{fi}$$_{2}$$_{σ}$ ′ + J ∑ iσσ ′ f † i 1 $_{σ}$f † i 2 σ ′ f$_{i}$$_{1}$$_{σ}$ ′ f$_{i}$$_{2}$$_{σ}$ + g$_{1}$ ∑ i ( b † i 1 + b$_{i}$$_{1}$ )ˆ τ$_{ix}$ + g$_{2}$ ∑ i ( b † i 2 + b$_{i}$$_{2}$ )ˆ τ$_{iz}$ + ∑ iν ω$_{0}$$_{ν}$b † $_{iν}$b$_{iν}$ (1) ˆ τ$_{ix}$ = ∑ σ ( f † i 1 $_{σ}$f$_{i}$$_{2}$$_{σ}$ + f † i 2 $_{σ}$f$_{i}$$_{1}$$_{σ}$ ) (2) ˆ τ$_{iz}$ = ∑ σ ( f † i 1 $_{σ}$f$_{i}$$_{1}$$_{σ}$ − f † i 2 $_{σ}$f$_{i}$$_{2}$$_{σ}$ ) , (3) with Our model Hamiltonian is given by, In the present paper, we investigate the two-orbital periodic Anderson model, where JT phonons with a two-fold degenerate mode couple with orbital fluctua- tions of f -electrons which interact with each other via the Coulomb interaction and hybridize with conduction electrons, by using the DMFT to elucidate the coopera- tive effect of the Coulomb interaction and the electron- phonon coupling on the heavy-fermion state in the cases with half-filling with two f -electrons per atom n$_{f}$ = 2 and quarter-filling with n$_{f}$ = 1. filling have not been discussed there. 2. Formulation