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arXiv:cond-mat/9506092 · Physical Review Letters

Numerical Renormalization-Group Study of Particle-Hole Symmetry Breaking in Two-Channel Kondo Problem: Effect of Repulsion among Conduction Electrons and Potential Scattering

Hiroaki Kusunose, K. Miyake, Yukihiro Shimizu, Osamu Sakai

Published Jan 8, 1996• 41 citations
arXivHTMLPDF

Abstract

Particle-hole symmetry breaking perturbation in the two-channel pseudospin Kondo problem is studied by the numerical renormalization-group method. It is shown that the repulsion among conduction electrons at the impurity site and the single particle potential are relevant perturbations against the conventional non-Fermi-liquid fixed point. Although the repulsion (potential) with realistic strength prevents the overscreening of the pseudospin, it induces in turn a real spin, which is also overscreened again. Thus the real spin susceptibility becomes anomalous, contrary to the conventional two-channel Kondo problem.

DOI
10.1103/physrevlett.76.271

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Particle-hole symmetry breaking perturbation in the two-channel pseudospin Kondo problem is studied by the numerical renormalization-group method. It is shown that the repulsion among conduction electrons at the impurity site and the single particle potential are relevant perturbations against the conventional non-Fermi-liquid fixed point. Although the repulsion (potential) with realistic strength prevents the overscreening of the pseudospin, it induces in turn a real spin, which is also overscreened again. Thus the real spin susceptibility becomes anomalous, contrary to the conventional two-channel Kondo problem.
It is shown that a non-Fermi-liquid fixed point of the multichannel Kondo problem is unstable for repulsive interaction at the impurity site between conduction electrons with different flavors and the same pseudospin. The anomalous fixed point is shown, by the perturbative renormalization group argument, to move into the strong coupling region with increasing repulsion. The low-temperature properties at the so-called “Toulouse limit” are calculated by the bosonization method to show that the Fermi-liquid like behavior is recovered quickly as soon as repulsion is switched on. This result poses a question as to the reality of the non-Fermi-liquid state for the multichannel Kondo model which has been examined extensively so far.
Using the numerical renormalization group method, we study a generalized two-channel Kondo model where the impurity has a quadrupole moment (pseudospin) and a magnetic moment (flavor). This model may be of relevance to the Ce ions in CeB6{\mathrm{CeB}}_{6}CeB6​ and the Cu ions in cuprate superconductors. We find that the nontrivial fixed point of the two-channel Kondo model is unstable against flavor exchange coupling. The system flows to a non-Fermi-liquid fixed line. In the presence of a double tensor interaction, which breaks particle-hole symmetry, the system flows to a Fermi-liquid fixed line.
We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r=0r=0r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For r>rmaxr>{r}_{\mathrm{max}}r>rmax​, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions r=0r=0r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.
The non-Fermi-liquid fixed point of the overscreened Kondo problem is studied in the limit of a large number of conduction electron channels, k\ensuremath{\gg}1, using perturbation theory in 1/k. By expanding the beta function and self-energy to sub-leading-order in 1/k, we examine the scaling, thermodynamics, and dynamic response functions in detail. Our results compliment and extend previous work based on conformal field theory. We present the dynamical spin susceptibility for the first time and show that the impurity spin is marginally quenched, with an inelastic linewidth of order T.
The multichannel Kondo model exhibits non-Fermi-liquid behavior in the overscreened case, when the number of channels, k, is greater than twice the size of the impurity spin, s. We show that, for overscreening, exchange anisotropy is irrelevant at the low-temperature fixed point for s=1/2 or s=k/2-1/2, but relevant for all other values of s. However, an external field or channel asymmetry is relevant, producing crossover to Fermi-liquid fixed points with different phase shifts for each spin in the first case and for each channel in the second. These results are elucidated by explicit comparison of analytic finite-size spectra derived from conformal field theory with those obtained from numerical renormalization-group calculations. The relevance of the results (for k=2 and s=1/2) to the quadrupolar Kondo Hamiltonian (which has been proposed as a model for many uranium-based heavy-fermion materials) will be briefly discussed. Also, a larger, a/Isymplectic symmetry Sp(2k) is shown to be present in the k-channel Kondo model.
We study an exchange interaction in which conduction electrons with pseudospin Sc{\mathrm{S}}_{\mathrm{c}}Sc​=3/2 interact with the impurity spin SI{\mathrm{S}}_{\mathrm{I}}SI​=1/2. Due to the overscreening of the impurity spin by higher conduction electron spin, a nontrivial intermediate-coupling-strength fixed point is realized. Using the numerical renormalization group (NRG), we show that the low-energy spectra are described by a non-Fermi-liquid excitation spectrum. A conformal field theory analysis is compared with NRG results and excellent agreement is obtained. Using the double-fusion rule to generate the operator spectrum with the conformal theory, we find that the specific heat coefficient and magnetic susceptibility will diverge as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}$, that the scaling dimension of an applied magnetic field is 5/6, and that exchange anisotropy is always relevant. We discuss the possible relevance of our work to two-level-system Kondo materials and dilute cerium alloys, and we point out a paradox in understanding the Bethe-ansatz solutions to the multichannel Kondo model.