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.

Impurity four-and six-level Kondo model, in which an ion is tunneling among four-and sixstable points and interacting with surrounding conduction electrons, are investigated by using the perturbative and numerical renormalization group methods. It is shown that purely orbital Kondo effects occur at low temperatures in these systems which are direct generalizations of the Kondo effect in the so-called two-level system. This result offers a good explanation for the enhanced and magnetically robust Sommerfeld coefficient observed in SmOs4Sb12 and some other filled-skutterudites. KEYWORDS: Kondo effect, rattling, multi-level systemSince the two-level Kondo effect was studied about three decades ago, 1, 2) the physics arising from the offcenter degrees of freedom of ions have attracted much attention. Recently, off-center motions of ions in clathrate compounds 3) and filled skutterudites 4) have been identified by ultrasonics. One of the recent remarkable experimental results is a realization of magnetically robust heavy fermion in SmOs 4 Sb 12 , in which the linear temperature coefficient of the specific heat at low temperatures amounts to 820mJ/(mol·K 2 ) even up to 8-15 T. 6, 7) Since the Sm ion is considered to be in a mixed valence state, it is mysterious why such a heavy effective mass is realized.Characteristic properties of these materials are the existence of thermally excited rattling modes if an ion in the "cage". In the case of PrOs 4 Sb 12 , it is claimed that there exist degenerate Γ + 3 (cubic O h ) modes of the offcenter motion of the ion (Pr 3+ ) to explain the ultrasonic dispersion in C 11 − C 12 . (This point has been still controversial. There are two different results in C 44 4, 5) ). However, textbooks for quantum mechanics tell us that a ground state is a singlet with a symmetric wavefunction and there exists no node in a spinless one-body problem. From this view point, the interpretation above is not easily accepted.In this paper, we investigate an impurity four-and sixlevel Kondo model, in which an ion is tunneling among four-and six-stable points. We take into account interactions between the ion and the surrounding conduction electrons, and use the perturbative renormalization group (PRG) method to see low energy properties of the system. Since we want to investigate the effects arising from the non-magnetic origin, we assume the conduction electron to be spinless. We restrict ourselves mainly to the six-level model (see ref. 8 for the detailed PRG and numerical renormalization group (NRG) discussions of four-level model). We will make a brief comment about the results of the four-level model in the last part.In order to make the model simple, we consider a Gaussian wavefunction of the ion φ 0 i (x) at each off-center position i,where σ and x i are the width of the Gaussian and coordinate of off-center positions x i = (±a, 0, 0), (0, ±a, 0) and (0, 0, ±a), respectively (a and σ are measured in the unit of inverse Fermi wavenumber k, we can evaluate the transfer integral S directly (we keep ...