Keisuke Mitsumoto and Yoshiaki ¯ Ono Full Paper 6 J. Phys. Soc. Jpn. 0 0.01 0.02 0.03 g 0.008 0.006 0.004 0.002 0 g 0.004 0.003 0.002 0.001 0 (a) U=0.2 (b) U=0.6 phonon spectral function 0 0.2 0.4 0.6 0 10 20 30 0 0.02 0.04 0 20 40 60 0 0.2 0.4 0 40 80 −0.5 0 0.5 0 0.1 0.2 0 40 80 (a) U=0 g=0 (b) U=0 g=0.021 (c) U=0.6 g=0 (d) U=0.6 g=0.008 f−electron spectral function DOS Fig. 9. The phonon spectral function as a function of ω at half- filling for several values of g with U = 0 . 2 (a) and U = 0 . 6 (b). Fig. 8. (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 quarter-filling, where the Fermi level is set to zero. 0 0.01 0.02 0.03 g 0.020 0.014 0.010 0.006 0 g 0.008 0.006 0.004 0.002 0 (a) U=0.2 (b) U=0.6 phonon spectral function a large value of U = 0 . 6, the spectral weight of the lowest excited state is relatively small and a consider- able amount of the spectral weight remains around the original phonon level. This result is consistent with that obtained for the Hubbard-Holstein model in the strong coupling regime, where the sharp soft phonon mode with a large spectral weight is observed for small- U , while the broad soft phonon mode with a small spectral weight is observed for large- U . 26) To see the energy shift in more detail, we show the lowest excited energy ˜ ω$_{0}$ in the phonon spectral function as a function of g in Fig.11. Remarkably, the softening, i. e., the decrease in ˜ ω$_{0}$ , observed in the strong coupling regime is largely enhanced due to the Coulomb inter- action especially for half-filling in contrast to the case with the single-orbital periodic Anderson-Holstein model where the softening is suppressed due to the Coulomb in- teraction. 30–32) In Summary, we have investigated the two-orbital pe- riodic Anderson model coupled with the two-fold degen- erate Jahn-Teller phonon by using the dynamical mean- field theory, and have found that the heavy fermion state of nonmagnetic origin is realized due to the cooperative effect of the Coulomb interaction U and the electron- phonon coupling g . The specific features of the heavy fermion state for large U and g are as follows: (1) The local orbital and lattice fluctuations are enhanced, while the local charge (valence) and spin fluctuations are sup- pressed. (2) The sharp soft phonon mode with a large spectral weight is observed for small U , while the broad soft phonon mode with a small spectral weight is ob- served for large U . (3) The cooperative effect for half- filling with n$_{f}$ = 2 is more pronounced than that for quarter-filling with n$_{f}$ = 1. In the absence of g , when U increases, the effective mass increases with increasing spin and orbital fluctua- tions together with decreasing charge fluctuation. Then, the heavy fermion state of magnetic origin with large (small) spin and orbital (charge) fluctuations is realized for large U due to the strong correlation effect. When we increase g for a fixed value of U , the effective mass further increases with increasing orbital and lattice fluctuations together with decreasing spin and charge fluctuations. Then, the heavy fermion state of nonmagnetic origin with large (small) orbital and lattice (spin and charge) fluctu- ations is realized for large U and g due to the cooperative effect of the strong correlation and the strong coupling. In this heavy fermion state, the phonon spectral function shows the broad soft phonon mode while the f electron 4. Summary and Discussions Fig. 10. The phonon spectral function as a function of ω at quarter-filling for several values of g with U = 0 . 2 (a) and U = 0 . 6 (b).