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arXiv:cond-mat/9711192 · Physical review. B, Condensed matter

Overscreened multichannel SU(N) Kondo model: Large- N solution and conformal field theory

Olivier Parcollet, Antoine Georges, Gabriel Kotliar, Anirvan M. Sengupta

Published Aug 15, 1998• 180 citations
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Abstract

The multichannel Kondo model with $\mathrm{SU}(N)$ spin symmetry and $\mathrm{SU}(K)$ channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of $\mathrm{SU}(N),$ 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 $(K>1),$ and all values of $N$ and $Q,$ 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-$N$ 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-$N$ limit, such as the $T=0$ 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.

DOI
10.1103/physrevb.58.3794

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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∑\ensuremathαf\ensuremathα\ifmmode†\else\textdagger\fif\ensuremathα=Q.{\ensuremath{\sum}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{\ensuremath{\alpha}}=Q.\ensuremath∑\ensuremathα​f\ensuremathα\ifmmode†\else\textdagger\fi​f\ensuremathα​=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≡\ensuremathγK/N\ensuremath{\equiv}\ensuremath{\gamma}K/N\ensuremath≡\ensuremathγ and Q/N\ensuremath≡q0Q/N\ensuremath{\equiv}{q}_{0}Q/N\ensuremath≡q0​ 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 \ensuremathω/T,\ensuremath{\omega}/T,\ensuremathω/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 novel large- NNN limit of the multichannel Kondo model is introduced, for representations of the impurity spin described by Schwinger bosons. Three cases are found, associated with underscreening, overscreening, and exact Kondo screening of the impurity. The saddle-point equations derived in this limit are reminiscent of the ``noncrossing approximation,'' but preserve the Fermi-liquid nature of the model in the exactly screened case. Several physical quantities are computed, both numerically and analytically in the low- $\ensuremath{\omega},T$ limit, and compared to other approaches.
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 discuss a multichannel SU(N) Kondo model which displays nontrivial zero-temperature phase transitions due to a conduction electron density of states vanishing with a power law at the Fermi level. In a particular large- N limit, the system is described by coupled integral equations corresponding to a dynamic saddle point. We exactly determine the universal low-energy behavior of spectral densities at the scale-invariant fixed points, obtain anomalous exponents, and compute scaling functions describing the crossover near the quantum-critical points. We argue that our findings are relevant to recent experiments on impurity-doped d-wave superconductors.
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.
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 develop a conformal field theory approach for investigation of the quantum charge, heat, and thermoelectric transport through a quantum impurity fine-tuned to a non-Fermi liquid regime. The non-Fermi-liquid operational mode is associated with the overscreened spin Kondo effect and controlled by the number of orbital channels. The universal low-temperature scaling and critical exponents for Seebeck and Peltier coefficients are investigated for the multichannel geometry. We discuss the universality of Lorenz ratio and power factor beyond the Fermi-liquid paradigm. Different methods of verifying our findings based on the recent experiments are proposed.