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Physical review. B, Condensed matter

Scaling solution of the overscreened multichannel Kondo model

Ming Shaw, Xingwei Zhang, Duo Jin

Published Apr 1, 1998• 5 citations
Publisher

Abstract

In the Kondo problem the overscreened multichannel model is the only one that can be assessed perturbatively. However, until now there are still problems remaining in studies by means of a perturbative renormalization group due to the absence of a proper treatment on the constraint of the pseudofermion. In this paper we try to settle these problems. We studied the general overscreened multichannel Kondo problem by means of a field-theory renormalization group with the technique of treating the pseudofermion constraint as developed recently. We calculated the critical exponent, specific heat, magnetic susceptibility, residue entropy, and Wilson ratio for the overscreened multichannel Kondo model. From these results we conclude that all overscreened multichannel Kondo models with the same channel number $K$ but different $S$ have the same fixed point properties. Therefore, they belong to the same universal class, which agrees with the result obtained by Bethe ansatz and conformal field theory.

DOI
10.1103/physrevb.57.8381

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In the Kondo problem the overscreened multichannel model is the only one that can be assessed perturbatively. However, until now there are still problems remaining in studies by means of a perturbative renormalization group due to the absence of a proper treatment on the constraint of the pseudofermion. In this paper we try to settle these problems. We studied the general overscreened multichannel Kondo problem by means of a field-theory renormalization group with the technique of treating the pseudofermion constraint as developed recently. We calculated the critical exponent, specific heat, magnetic susceptibility, residue entropy, and Wilson ratio for the overscreened multichannel Kondo model. From these results we conclude that all overscreened multichannel Kondo models with the same channel number KKK but different SSS have the same fixed point properties. Therefore, they belong to the same universal class, which agrees with the result obtained by Bethe ansatz and conformal field theory.
We study the residue entropy of overscreen multi-channel Kondo model by means of perturbathe renormalization group. In this field theory framework one must use pseudofermion representation for local spin operator, and the constraint of pseudofermion must be treated properly. We treat the constraint by a method developed recently. In this method the partition function of the Kondo model is separated into proper weighted partition functions of subsystems. We study the overscreen multi-channel Kondo model through M -subsystem. From calculating the residue entropy of M -subsystem, we obtain the residue entropy of original system which agrees with the result of conformal field theory.
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.
A conformal-field-theory approach has recently been developed for quantum-impurity problems including the ``overscreened'' multichannel Kondo model. We present the details of our calculation of the single-fermion Green's function. The universal zero-temperature resistivity and leading temperature-dependent term are derived. Effects of particle-hole symmetry breaking are included. We also give our perturbative proof of the ``g theorem'' governing the change in residual entropy under a renormalization-group flow between two boundary fixed points.
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.
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.
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.