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arXiv:cond-mat/9803137 · Physical review. B, Condensed matter

Solution of the multichannel Coqblin-Schrieffer impurity model and application to multilevel systems

Andrés Jerez, Natan Andrei, Gergely Zaránd

Published Aug 15, 1998• 64 citations
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Abstract

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 $N$ and $f:$ for $N<~f,$ the specific heat and the susceptibility are dominated by the NFL contributions; for $N>f$ the leading contributions are Fermi-liquid-like, and the NFL behavior can be seen only to subleading order; and for $N=f$ the behavior is marginal. We also analyze the possibility of physical realizations. We show by a detailed renormalization-group and $1/f$ analysis that the tunneling $N$-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.

DOI
10.1103/physrevb.58.3814

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A complete Bethe ansatz solution of the SU(N)\ifmmode×\else\texttimes\fiSU(f)\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)SU(N)\ifmmode×\else\texttimes\fiSU(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 \ensuremathμ=f,\ensuremath{\mu}=f,\ensuremathμ=f, we find a Fermi-liquid fixed point, with the impurity spin completely screened. For f>\ensuremathμ,f>\ensuremath{\mu},f>\ensuremathμ, 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 SU(N)\ifmmode×\else\texttimes\fiSU(f)\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)SU(N)\ifmmode×\else\texttimes\fiSU(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 SU(3)\ifmmode×\else\texttimes\fiSU(2)\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)SU(3)\ifmmode×\else\texttimes\fiSU(2) model if the parameters of the dots are tuned appropriately.
The multichannel Kondo model with SU(N)\mathrm{SU}(N)SU(N) spin symmetry and SU(K)\mathrm{SU}(K)SU(K) channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of SU(N),\mathrm{SU}(N),SU(N), corresponding to a fixed number of Abrikosov fermions ${\ensuremath{\sum}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{\ensuremath{\alpha}}=Q.$ For more than one channel (K>1),(K>1),(K>1), and all values of NNN and Q,Q,Q, the model displays non-Fermi behavior associated with the overscreening of the impurity spin. Universal low-temperature thermodynamic and transport properties of this non-Fermi-liquid state are computed using conformal field theory methods. A large-NNN limit of the model is then considered, in which $K/N\ensuremath{\equiv}\ensuremath{\gamma}$ and $Q/N\ensuremath{\equiv}{q}_{0}$ are held fixed. Spectral densities satisfy coupled integral equations in this limit, corresponding to a (time-dependent) saddle point. A low-frequency, low-temperature analysis of these equations reveals universal scaling properties in the variable $\ensuremath{\omega}/T,$ in agreement with conformal invariance. The universal scaling form is obtained analytically and used to compute the low-temperature universal properties of the model in the large-NNN limit, such as the T=0T=0T=0 residual entropy and residual resistivity, and the critical exponents associated with the specific heat and susceptibility. The connections with the ``noncrossing approximation'' and the previous work of Cox and Ruckenstein are discussed.
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.
I construct non-Fermi liquid (NFL) quantum impurity models that are similar to the overscreened multi-channel Kondo models with the difference that an odd number of electron species screen the impurity. The simplest of them, named sesqui-channel (i.e. one-and-a-half-channel) Kondo (1.5CK) model, has less degrees of freedom and is simpler than the two-channel Kondo model, and yet it exhibits NFL physics. Using representation theory I derive the 1.5CK model for a spin-half impurity surrounded with electrons in cubic crystal field and solve it with the numerical renormalization group.
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.
We solve the multichannel Kondo model with channel anisotropy using the Bethe ansatz method. The model generates energy scales, characterizing the neighborhoods of the various infrared fixed points, reflecting the structure of the symmetry breaking in the channel sector. The nature of these fixed points also depends on the magnitude of the impurity spin SSS. We present a detailed discussion for the two channel case and point out some possible non-Fermi-liquid behavior.
The Kondo problem, which describes the interaction of a spin sss magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in 1/s1/s1/s with the renormalization group. We study both the case of an impurity interacting with a single channel K=1K=1K=1 of fermions in the s→∞s \rightarrow \inftys→∞ limit and the case with KKK channels in the double-scaling limit K→∞K \rightarrow \inftyK→∞, s→∞s \to \inftys→∞, K/sK/sK/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large sss and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime K>2sK > 2sK>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.