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arXiv:cond-mat/9412054 · Physical Review Letters

Fermi- and Non-Fermi-Liquid Behavior in the Anisotropic Multichannel Kondo Model: Bethe Ansatz Solution

Natan Andrei, Andrés Jerez

Published May 29, 1995• 95 citations
arXivHTMLPDF

Abstract

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 $S$. We present a detailed discussion for the two channel case and point out some possible non-Fermi-liquid behavior.

DOI
10.1103/physrevlett.74.4507

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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 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.
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.
The low-temperature behavior of a magnetic impurity of spin S interacting with an electron gas via an anisotropic spin exchange is studied via Bethe’s ansatz. The multichannel Kondo model with U(1) invariance is integrable as a function of two continuous (the exchange and the anisotropy) and two discrete parameters, namely the impurity spin S and the number of channels n. As a function of S and n we distinguish: (i) the compensated case with n=2S, (ii) the overcompensated case if n>2S, and (iii) the undercompensated case (n<2S). While in case (i) the ground state is a singlet, the cases (ii), and (iii) yield quantum critical points. The undercompensated one is of a new type with the critical exponents depending on the anisotropy.
We reinvestigate the two channel flavor anisotropic model (2CFAK) and one channel compacitified Kondo model (1CCK). For these two models, all the possible fixed points and their symmetries are identified; the finite size spectra, the electron conductivity and pairing susceptibility are calculated. It is shown that the only non-fermi liquid (NFL) fixed point of the 2CFAK is the NFL of the two channel Kondo model (2CK) with the symmetry O(3)×O(5)O(3) \times O(5)O(3)×O(5). Any flavor anisotropies between the two channels drive the system to the fermi-liquid (FL) fixed point with the symmetry O(4)×O(4)O(4) \times O(4)O(4)×O(4) where one of the two channels suffers the phase shift π/2 and the other remains free. The NFL fixed point of the 1CCK has the symmetry O(3)×O(1)O(3) \times O(1)O(3)×O(1) and has the same thermodynamics as the NFL fixed point of the 2CK. However, in contrast to the 2CK, its conductivity shows T2T^{2}T2 behavior and there is no pairing susceptibility divergence. Any anisotropies between the spin and isospin sectors drive the system to the FL fixed point with the symmetry O(4) where the electrons suffer the phase shift π/2. The connection and differences between the two models are explicitly demonstrated. The recent conjectures and claims on the NFL behaviors of the two models are commented.
We construct and solve numerically the thermodynamic Bethe-ansatz equations for the spin-anisotropic two-channel Kondo model in arbitrary external field h. At high temperatures the specific heat and the susceptibility show power-law dependence. For $\stackrel{\ensuremath{\rightarrow}}{h}0$ and at temperatures below the Kondo temperature TK{T}_{K}TK​ a two-channel Kondo effect develops characterized by a Wilson ratio 8/3,8/3,8/3, and a logarithmic divergence of the susceptibility and the linear specific-heat coefficient. A finite magnetic field, h>0,h>0,h>0, drives the system to a Fermi-liquid fixed point with an unusual Wilson ratio which depends sensitively on h.
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.