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.

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We investigate antiferro quadrupole orders in systems with non-Kramers doublet ground state with total angular momentum J = 4 in T d point group symmetry. We demonstrate that a pure O 2 2 antiferro quadrupole order is impossible in general crystalline-electric field potential and should be accompanied by ferro O 0 2 quadrupole moment. The temperature and magnetic-field phase diagram is obtained by mean-field approximation of intersite quadrupole interactions and the excitation spectrum is analyzed by "spin"-wave approximations. Gapless excitations emerge at the boader of antiferro O 2 2 quadrupole phases under magnetic field. Quadrupole susceptibilities in the antiferro quadrupole ordered state exhibit unusual singularity and especially the uniform quadrupole susceptibility diverges in addition to the staggered ones. These unusual singularities are also realized at the critical field along [111] direction. We also discuss recent experimental results in PrT 2X20 (T =Ir, Rh, Ti, V, and X=Zn, Al). IntroductionOrbital order appears in varieties of systems in condensed matter physics such as in d-and f -electron strongly correlated systems with orbital degrees of freedom.1, 2) Quadrupole orders are typical and most intensively studied cases. Such orbital orders can be in principle described by similar theoretical approaches as in spin systems. As is evident from the nature of orbital degrees of freedom, interactions have spatial anisotropies depending on what kinds of orbital is considered and the symmetry is in general not fully isotropic in the orbital space. These differ from the typical case of spin systems, where the spin anisotropy is zero or not so strong unless the spin-orbit interaction is very strong. Each orbital order possesses a unique property and exploring such uniqueness is an important issue of the condensed matter theories and experiments.Recently, Pr-based compounds PrT 2 X 20 (T =Ir,Rh,Ti,V, and X=Zn,Al), so-called 1-2-20 compounds, have attracted great attention.3-12) In these compounds, each Pr ion has (4f) 2 electron configuration. Its ground state under the crystalline electric field (CEF) is a non-Kramers doublet, 13) and this doublet couples with conduction electrons. This is a typical situation that the two-channel Kondo effects take place. Thus, it is expected that these compounds show exotic properties due to the two-channel Kondo effects.

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. An interesting interplay among the conventional s-and p-wave Kondo effects 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 at the boundary.