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arXiv:cond-mat/9606095 · Physical review. B, Condensed matter

Non-Fermi-liquid fixed point

Tae‐Suk Kim, L. Nunes de Oliveira, Daniel Lee Cox

Published May 1, 1997• 13 citations
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Abstract

We study an exchange interaction in which conduction electrons with pseudospin ${\mathrm{S}}_{\mathrm{c}}$=3/2 interact with the impurity spin ${\mathrm{S}}_{\mathrm{I}}$=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.

DOI
10.1103/physrevb.55.12460

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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 T\ensuremath−2/3{\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}T\ensuremath−2/3, 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.
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.
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We study a generalized multichannel single-impurity Kondo model, in which the impurity spin is described by a representation of the SU(N)\mathrm{SU}(N)SU(N) group that combines bosonic and fermionic degrees of freedom. The impurity spin states are described by Abrikosov pseudofermions, and we make use of a method initiated by Popov and Fedotov that allows a proper handling of the fermionic constraint. The partition function is derived within a path integral approach. We use renormalization group techniques to calculate the $\ensuremath{\beta}$ scaling function perturbatively in powers of the Kondo coupling constant, which is justified in the weak coupling limit. The truncated expansion is valid in the overscreened (Nozi\`eres-Blandin) regime, for an arbitrary SU(N)\mathrm{SU}(N)SU(N) group and any value of the parameters characterizing the impurity spin representation. The intermediate coupling fixed point is identified. We derive the temperature dependence of various physical quantities at low TTT, controlled by a unique critical exponent, and show that the physics of the system in the overscreened regime governed by the intermediate coupling fixed point is characterized by a non-Fermi liquid behavior. Our results are in accordance with those obtained by other methods, as Bethe ansatz and boundary conformal field theory, in the case of various impurity spin symmetries. We establish in a unified way that the Kondo models in which the impurity spin is described successively by a fundamental, symmetric, antisymmetric, and mixed symmetry representation yield all the same low-energy physics in the overscreened regime. Possible generalizations of the analysis we present to the case of arbitrary impurity spin representations of SU(N)\mathrm{SU}(N)SU(N) are also discussed.
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A complete Bethe ansatz solution of the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)$ Coqblin-Schrieffer model and a detailed analysis of some physical applications of the model are given. As in the usual multichannel Kondo model, a variety of Fermi-liquid and non-Fermi-liquid (NFL) fixed points is found, whose nature depends on the impurity representation \ensuremath{\mu}. For $\ensuremath{\mu}=f,$ we find a Fermi-liquid fixed point, with the impurity spin completely screened. For $f>\ensuremath{\mu},$ the impurity is overscreened and the model has NFL properties. The form the NFL behavior takes depends on the NNN and f:f:f: for N< f,N<~f,N< f, the specific heat and the susceptibility are dominated by the NFL contributions; for N>fN>fN>f the leading contributions are Fermi-liquid-like, and the NFL behavior can be seen only to subleading order; and for N=fN=fN=f the behavior is marginal. We also analyze the possibility of physical realizations. We show by a detailed renormalization-group and 1/f1/f1/f analysis that the tunneling NNN-state problem can be mapped into the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)$ exchange model, and discuss the subtle differences between the two models. As another physical realization we suggest a double quantum dot structure that can be described by means of an $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ model if the parameters of the dots are tuned appropriately.
Hund metals have attracted attention in recent years due to their unconventional superconductivity, which supposedly originates from non-Fermi-liquid (NFL) properties of the normal state.When studying Hund metals using dynamical mean-field theory, one arrives at a self-consistent "Hund impurity problem" involving a multiorbital quantum impurity with nonzero Hund coupling interacting with a metallic bath.If its spin and orbital degrees of freedom are screened at different energy scales, T sp < T orb , the intermediate energy window is governed by a novel NFL fixed point, whose nature had not yet been clarified.We resolve this problem by providing an analytical solution of a paradigmatic example of a Hund impurity problem, involving two spin and three orbital degrees of freedom.To this end, we combine a state-ofthe-art implementation of the numerical renormalization group, capable of exploiting non-Abelian symmetries, with a generalization of Affleck and Ludwig's conformal field theory (CFT) approach for multichannel Kondo models.We characterize the NFL fixed point of Hund metals in detail for a Kondo model with an impurity forming an SUð2Þ × SUð3Þ spin-orbital multiplet, tuned such that the NFL energy window is very wide.The impurity's spin and orbital susceptibilities then exhibit striking power-law behavior, which we explain using CFT arguments.We find excellent agreement between CFT predictions and numerical renormalization group results.Our main physical conclusion is that the regime of spinorbital separation, where orbital degrees of freedom have been screened but spin degrees of freedom have not, features anomalously strong local spin fluctuations: the impurity susceptibility increases as χ imp sp ∼ ω -γ , with γ > 1.