Realization of Strong Coupling Fixed Point in Multilevel Kondo Models

Abstract

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 ...

Heavy-Electron Formation and Polaron–Bipolaron Transition in the Anharmonic Holstein Model

“…In fact, fourand six-level Kondo systems have been analyzed and the magnetically robust heavy-electron state has been actually obtained. [26][27][28] The periodic Anderson-Holstein model has been analyzed using the dynamical mean-field theory, and the mass enhancement due to large lattice fluctuations and phonon softening towards a double-well potential have been addressed. [29][30][31] Kondo phenomena in the conduction electron system coupled with local Jahn-Teller phonons 32,33) and Holstein phonons 34,35) have been discussed for the realization of a nonmagnetic Kondo effect.…”

Heavy Fermions due to Cooperative Effects of Coulomb and Electron–Phonon Interactions in the Two-Orbital Periodic Anderson Model

“…SmOs 4 Sb 12 shows a large specific heat coefficient γ = 820 mJ/(K 2 · mol) which is almost independent of applied magnetic fields, 1) suggesting a heavy fermion state of nonmagnetic origin such as charge, valence and phonon degrees of freedom. [2][3][4][5][6][7][8][9] PrOs 4 Sb 12 shows a large specific heat coefficient γ = 750 mJ/(K 2 · mol) together with a large jump in the specific heat ∆C/T c > ∼ 500 mJ/(K 2 · mol) at the superconducting transition temperature T c = 1.85 K. 10) In the ultrasonic measurements, remarkable frequency dependence of the elastic constant (ultrasonic dispersion) around 30 K 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.…”

Electric dipolar susceptibility of the Anderson-Holstein model

“…Such oscillation with large amplitude is frequently called rattling and it is considered to play crucial roles for the formation of magnetically-robust heavy electron state in SmOs 4 Sb 12 [2]. This peculiar heavyelectron state has been theoretically investigated from various aspects by several groups [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].Recently, the Kondo effect of a vibrating magnetic ion in a cage has been theoretically discussed on the basis of a two-channel conduction electron system hybridized with a vibrating magnetic ion [18][19][20]. Note here that a vibrating ion inevitably induces electric dipole moment.…”