Electric Dipolar Susceptibility of the Anderson-Holstein Model

The temperature dependence of electric dipolar susceptibility \chi_P is discussed on the basis of the Anderson-Holstein model with the use of a numerical renormalization group (NRG) technique. Note that P is related with phonon Green's function D. In order to obtain correct temperature dependence of P at low temperatures, we propose a method to evaluate P through the Dyson equation from charge susceptibility \chi_c calculated by the NRG, in contrast to the direct NRG calculation of D. We find that the irreducible charge susceptibility estimated from \chi_c agree with the perturbation calculation, suggesting that our method works well.Comment: 4 pages, 4 figure

Electron Mass Enhancement due to Anharmonic Local Phonons

In order to understand how electron effective mass is enhanced by anharmonic local oscillation of an atom in a cage composed of other atoms, i.e., {\it rattling}, we analyze anharmonic Holstein model by using a Green's function method. Due to the evaluation of an electron mass enhancement factor $Z$, we find that $Z$ becomes maximum when zero-point energy is comparable with potential height at which the amplitude of oscillation is rapidly enlarged. Cooperation of such quantum and rattling effects is considered to be a key issue to explain the electron mass enhancement in electron-rattling systems.Comment: 3 pages, 3 figures, to appear in J. Phys. Soc. Jpn. Suppl. (Proceedings for International Conference on Heavy Electrons

Cooperative Effect of Coulomb Interaction and Electron-Phonon Coupling on the Heavy Fermion State in the Two-Orbital Periodic Anderson Model

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 $n_f=1$.Comment: 8 pages, 11 figures, accepted for publication in JPS

Heavy-Electron Formation and Bipolaronic Transition in the Anharmonic Holstein Model

The emergence of the bipolaronic phase and the formation of the heavy-electron state in the anharmonic Holstein model are investigated using the dynamical mean-field theory in combination with the exact diagonalization method. For a weak anharmonicity, it is confirmed that the first-order polaron-bipolaron transition occurs from the observation of a discontinuity in the behavior of several physical quantities. When the anharmonicity is gradually increased, the polaron-bipolaron transition temperature is reduced as well as the critical values of the electron-phonon coupling constant for polaron-bipolaron transition. For a strong anharmonicity, the polaron-bipolaron transition eventually changes to a crossover behavior. The effect of anharmonicity on the formation of the heavy-electron state near the polaron-bipolaron transition and the crossover region is discussed in detail.Comment: 11 pages, 13 figure

Kondo Effect in an Electron System with Dynamical Jahn-Teller Impurity

We investigate how Kondo phenomenon occurs in the Anderson model dynamically coupled with local Jahn-Teller phonons. It is found that the total angular moment composed of electron pseudo-spin and phonon angular moments is screened by conduction electrons. Namely, phonon degrees of freedom essentially contribute to the formation of singlet ground state. A characteristic temperature of the Kondo effect due to dynamical Jahn-Teller phonons is explained by an effective $s$-$d$ Hamiltonian with anisotropic exchange interaction obtained from the Jahn-Teller-Anderson model in a non-adiabatic region.Comment: 5 pages, 3 figure

Strong-Coupling Theory of Rattling-Induced Superconductivity

In order to clarify the mechanism of the enhancement of superconducting transition temperature $T_{\rm c}$ due to anharmonic local oscillation of a guest ion in a cage composed of host atoms, i.e., {\it rattling}, we analyze the anharmonic Holstein model by applying the Migdal-Eliashberg theory. From the evaluation of the normal-state electron-phonon coupling constant, it is found that the strong coupling state is developed, when the bottom of a potential for the guest ion becomes wide and flat. Then, $T_{\rm c}$ is enhanced with the increase of the anharmonicity in the potential, although $T_{\rm c}$ is rather decreased when the potential becomes a double-well type due to very strong anharmonicity. From these results, we propose a scenario of anharmonicity-controlled strong-coupling tendency for superconductivity induced by rattling. We briefly discuss possible relevance of the present scenario with superconductivity in $\beta$-pyrochlore oxides.Comment: 8 pages, 6 figure

Enhanced Kondo Effect in an Electron System Dynamically Coupled with Local Optical Phonon

We discuss Kondo behavior of a conduction electron system coupled with local optical phonon by analyzing the Anderson-Holstein model with the use of a numerical renormalization group (NRG) method. There appear three typical regions due to the balance between Coulomb interaction $U_{\rm ee}$ and phonon-mediated attraction $U_{\rm ph}$. For $U_{\rm ee}>U_{\rm ph}$, we observe the standard Kondo effect concerning spin degree of freedom. Since the Coulomb interaction is effectively reduced as $U_{\rm ee}-U_{\rm ph}$, the Kondo temperature $T_{\rm K}$ is increased when $U_{\rm ph}$ is increased. On the other hand, for $U_{\rm ee}<U_{\rm ph}$, there occurs the Kondo effect concerning charge degree of freedom, since vacant and double occupied states play roles of pseudo-spins. Note that in this case, $T_{\rm K}$ is decreased with the increase of $U_{\rm ph}$. Namely, $T_{\rm K}$ should be maximized for $U_{\rm ee} \approx U_{\rm ph}$. Then, we analyze in detail the Kondo behavior at $U_{\rm ee}=U_{\rm ph}$, which is found to be explained by the polaron Anderson model with reduced hybridization of polaron and residual repulsive interaction among polarons. By comparing the NRG results of the polaron Anderson model with those of the original Anderson-Holstein model, we clarify the Kondo behavior in the competing region of $U_{\rm ee} \approx U_{\rm ph}$.Comment: 8 pages, 8 figure

Local Heavy Quasiparticle in Four-Level Kondo Model

An impurity four-level Kondo model, in which an ion is tunneling among 4-stable points and interacting with surrounding conduction electrons, is investigated using both perturbative and numerical renormalization group methods. The results of numerical renormalization group studies show that it is possible to construct the ground state wavefunction including the excited ion states if we take into account the interaction between the conduction electrons and the ion. The resultant effective mass of quasiparticles is moderately enhanced. This result offers a good explanation for the enhanced and magnetically robust Sommerfeld coefficient observed in SmOs$_4$Sb$_{12}$, some other filled-skutterudites, and clathrate compounds.Comment: 9 pages, 7 figures. Added references and "Note added

First Order Bipolaronic Transition at Finite Temperature in the Holstein Model

We investigate the Holstein model by using the dynamical mean-field theory combined with the exact diagonalization method. Below a critical temperature Tcr, a coexistence of the polaronic and the bipolaronic solutions is found for the same value of the electron-phonon coupling $ in the range gc1(T)<g<gc2(T). In the coexistence region, the system shows a first order phase transition from the bipolaronic to the polaronic states as T decreases at T=Tp(<Tcr), where the double occupancy and the lattice fluctuation together with the anharmonicity of the effective ion potential change discontinuously without any symmetry breaking. The obtained bipolaronic transition seems to be consistent with the rattling transition in the beta-pyrochlore oxide KOs2O6.Comment: 5 pages, 5 figures, J. Phys. Soc. Jpn. 79 (2010) 09370

Kondo Effect of a Magnetic Ion Vibrating in a Harmonic Potential

To discuss Kondo effects of a magnetic ion vibrating in the sea of conduction electrons, a generalized Anderson model is derived. The model includes a new channel of hybridization associated with phonon emission or absorption. In the simplest case of the localized electron orbital with the s-wave symmetry, hybridization with p-waves becomes possible. Interesting interplay among the conventional s-wave Kondo effect and the p-wave one and the Yu-Anderson type Kondo effect is found and the ground state phase diagram is determined by using the numerical renormalization group method. Two different types of stable fixed points are identified and the two-channel Kondo fixed points are generically realized on the boundary.Comment: 15 pages, 17 figures, J. Phys. Soc. Jpn. 80 (2011) No.6 to be publishe