Typeset with jpsj3.cls < ver.1.0 > Full Paper arXiv:1001.2065v2 [cond-mat.str-el] 1 Mar 2010 Cooperative Effect of Coulomb Interaction and Electron-Phonon Coupling on the Heavy Fermion State in the Two-Orbital Periodic Anderson Model Keisuke Mitsumoto 1 ∗ and Yoshiaki ¯ Ono 2 , 3 1 Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan 2 Department of Physics, Niigata University, Niigata 950-2181, Japan 3 Center for Transdisciplinary Research, Niigata University, Niigata 950-2181, Japan We investigate the two-orbital periodic Anderson model, where the local orbital fluctuations of f -electrons couple with a two-fold degenerate Jahn-Teller phonon, by using the dynamical mean-field theory. It is found that the heavy fermion state caused by the Coulomb interaction between f -electrons U is largely enhanced due to the electron-phonon coupling g , in contrast to the case with the single-orbital periodic Anderson model where the effects of U and g compete to each other. In the heavy fermion state for large U and g , both the orbital and lattice fluctuations are enhanced, while the charge (valence) and spin fluctuations are suppressed. In the strong coupling regime, a sharp soft phonon mode with a large spectral weight is observed for small U , while a broad soft phonon mode with a small spectral weight is observed for large U . The cooperative effect of U and g for half-filling with two f -electrons per atom n$_{f}$ = 2 is more pronounced than that for quarter-filling with n$_{f}$ = 1. KEYWORDS: heavy fermion, periodic Anderson model, orbital degeneracy, Coulomb interaction, electron- phonon coupling, Jahn-Teller phonon, dynamical mean-field theory heat measurement claimed the Γ$_{5}$ doublet CEF ground state and there is still controversy about the CEF state. 6) Recently, another class of heavy fermion systems has been observed in filled skutterudite compounds RT$_{4}$X$_{12}$ with R=rare earth, T=Fe, Ru, Os, and X=pnictogen, where the R ion is surrounded by an oversized cage made of X and T atoms. SmOs$_{4}$Sb$_{12}$ shows a large spe- cific heat coefficient γ = 820mJ / K $^{2}$mol which is al- most independent of applied magnetic fields, 1) suggest- ing a heavy fermion state of nonmagnetic origin such as charge, valence and phonon degrees of freedom. Sm ion in SmOs$_{4}$Sb$_{12}$ shows mixed valence state of Sm 3+ and Sm 2+ as observed in X-ray spectroscopy experiments. 2,3) In the Sb-NQR experiment, a peak in 1 /T$_{2}$ has been at- tributed to some electrical fluctuations, probably local charge fluctuations. 4) The specific heat and the ultra- sonic measurements have revealed that the crystalline- electric-field (CEF) ground state of Sm 3+ ion is Γ$_{67}$ quar- tet in T$_{h}$ site symmetry. 1,5) However, another specific Heavy fermion systems have been intensively investi- gated for the last three decades. The effective mass of quasiparticles for the systems, estimated from the T - linear coefficient of specific heat, the Pauli paramagnetic susceptibility and the T 2 coefficient of resistivity, is much larger than that for simple metals. The strong correla- tion effects between f -electrons are considered to play crucial roles, where the large entropy due to the local magnetic moment of the f -electrons remains down to a low temperature T$_{0}$ and results in a large effective mass in proportion to 1 /T$_{0}$ at low temperature below T$_{0}$ . When the degeneracy is resolved by external magnetic fields, the effective mass is expected to be suppressed. Such a magnetic suppression of the effective mass is widely ob- served in heavy fermion compounds, justifying the heavy fermion state of magnetic origin. PrOs$_{4}$Sb$_{12}$ with Γ$_{1}$ singlet ground state also shows a large specific heat coefficient γ = 750mJ / K $^{2}$mol to- gether with a large jump in the specific heat Δ C/T$_{c}$ > ∼ 500mJ / K $^{2}$mol at the superconducting transition temper- ature T$_{c}$ = 1 . 85K. 7) In the ultrasonic measurements, re- markable frequency dependence of the elastic constant (ultrasonic dispersion) around 30K has been observed in PrOs$_{4}$Sb$_{12}$ and has been attributed to large amplitude local vibrations (rattling) of the Pr ion in the cage. 8) In addition, PrOs$_{4}$Sb$_{12}$ shows an anomalous softening of the elastic constant below 10K down to T$_{c}$ . 8) The soft- ening is well accounted for by quadrupolar susceptibility due to f -electrons in the Γ$_{1}$ − Γ (2) 4 CEF state except for an extra softening below 3K, 9) where the coupling be- tween the quadrupolar fluctuations and the rattling may play important roles for the extra softening and also for the heavy fermion behavior down to T$_{c}$ . Similar proper- ties of the ultrasonic dispersion and/or the anomalous softening have been observed also in other filled skut- terudite compounds such as ROs$_{4}$Sb$_{12}$ (R=La, Sm, Nd) and RFe$_{4}$Sb$_{12}$ (R=La, Pr). 5,10–13) After the pioneering works of Holstein, 14) the prob- lem of coupling between local lattice vibrations (Einstein phonons) and local charge fluctuations of conduction electrons has been investigated by many authors. Yu and Anderson originally predicted that the strong electron- phonon coupling causes an effective double-well potential for oscillating ions which are responsible for the anoma- lously large resistivity and Debye-Waller factor observed in A15 compounds such as V$_{3}$Si and Nb$_{3}$Ge. 15,16) As a strong coupling fixed point, the two-level Kondo systems were investigated to describe a heavy-fermion like be- havior observed in such compounds. 17,18) Recently, the 1. Introduction