Physical review. B, Condensed matter

Exact conformal-field-theory results on the multichannel Kondo effect: Single-fermion Green’s function, self-energy, and resistivity

Ian K. Affleck, Andreas W. W. Ludwig

Published Sep 1, 1993• 433 citations

Abstract

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.

DOI: 10.1103/physrevb.48.7297

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Physical Review LettersNov 25, 1991197 citations

Physical review. B, Condensed matter

Exact conformal-field-theory results on the multichannel Kondo effect: Single-fermion Green’s function, self-energy, and resistivity

Ian K. Affleck, Andreas W. W. Ludwig

Published Sep 1, 1993• 433 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.48.7297

Related papers

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

Olivier Parcollet, Antoine Georges, Gabriel Kotliar +1 more

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.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.

arXiv:cond-mat/9711192Physical review. B, Condensed matterAug 15, 1998180 citations

Relevance of anisotropy in the multichannel Kondo effect: Comparison of conformal field theory and numerical renormalization-group results

Ian K. Affleck, Andreas W. W. Ludwig, H.-B. Pang +1 more

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 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.

Physical review. B, Condensed matterApr 1, 1992204 citations

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We compute exactly the low-energy single-electron Green's function, the impurity and electron self-energies, and the resistivity for the two-channel Anderson impurity model. These results are obtained by exploiting the boundary conformal field theory identified from the Bethe ansatz solution of the model. Using that solution we can make contact with the parameters of the original Hamiltonian and provide the detailed crossover between the two integer valence limits. Our results generalize those obtained previously in the context of the two-channel Kondo model.We compute exactly the low-energy single-electron Green's function, the impurity and electron self-energies, and the resistivity for the two-channel Anderson impurity model. These results are obtained by exploiting the boundary conformal field theory identified from the Bethe ansatz solution of the model. Using that solution we can make contact with the parameters of the original Hamiltonian and provide the detailed crossover between the two integer valence limits. Our results generalize those obtained previously in the context of the two-channel Kondo model.

arXiv:cond-mat/0411533Physical Review BMay 18, 200516 citations

Scaling solution of the overscreened multichannel Kondo model

Ming Shaw, Xingwei Zhang, Duo Jin

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.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.

Physical review. B, Condensed matterApr 1, 19985 citations

Conformal-field-theory approach to the two-impurity Kondo problem: Comparison with numerical renormalization-group results

Ian K. Affleck, Andreas W. W. Ludwig, Barbara Jones

Numerical renormalization-group and conformal-field-theory work indicate that the two-impurity Kondo Hamiltonian has a non-Fermi-liquid critical point separating the Kondo-screening phase from the interimpurity singlet phase when particle-hole (P-H) symmetry is maintained. We clarify the circumstances under which this critical point occurs, pointing out that there are two types of P-H symmetry. Only one of them guarantees the occurrence of the critical point. Much of the previous numerical work was done on models with the other type of P-H symmetry. We analyze this critical point using the boundary conformal-field-theory technique. The finite-size spectrum is presented in detail and compared with about 50 energy levels obtained using the numerical renormalization group. Various Green's functions, general renormalization group behavior, and a hidden SO(7) are analyzed.Numerical renormalization-group and conformal-field-theory work indicate that the two-impurity Kondo Hamiltonian has a non-Fermi-liquid critical point separating the Kondo-screening phase from the interimpurity singlet phase when particle-hole (P-H) symmetry is maintained. We clarify the circumstances under which this critical point occurs, pointing out that there are two types of P-H symmetry. Only one of them guarantees the occurrence of the critical point. Much of the previous numerical work was done on models with the other type of P-H symmetry. We analyze this critical point using the boundary conformal-field-theory technique. The finite-size spectrum is presented in detail and compared with about 50 energy levels obtained using the numerical renormalization group. Various Green's functions, general renormalization group behavior, and a hidden SO(7) are analyzed.

arXiv:cond-mat/9409100Physical review. B, Condensed matterOct 1, 1995246 citations

Exact, asymptotic, three-dimensional, space- and time-dependent, Green’s functions in the multichannel Kondo effect

Andreas W. W. Ludwig, Ian K. Affleck

Exact 3D space- and time-dependent Green's functions are computed for the multichannel Kondo effect in the universal large-distance, long-time regime, using conformal field theory. They exhibit crossover from Fermi-liquid form far away, to singular non-Fermi-liquid behavior, near the impurity, where charge, spin, and ``flavor'' degrees of freedom become ``deconfined.'' Explicit expressions are given for the resistivity, \ensuremath{\rho}, for all k and for the spin-density and pair-density Green's functions, for k=2. For k=2, \ensuremath{\rho}(T)-\ensuremath{\rho}(0)\ensuremath{\propto} ${\mathit{T}}^{1/2}$ and the pair susceptibility \ensuremath{\propto}lnT.Exact 3D space- and time-dependent Green's functions are computed for the multichannel Kondo effect in the universal large-distance, long-time regime, using conformal field theory. They exhibit crossover from Fermi-liquid form far away, to singular non-Fermi-liquid behavior, near the impurity, where charge, spin, and ``flavor'' degrees of freedom become ``deconfined.'' Explicit expressions are given for the resistivity, \ensuremath{\rho}, for all k and for the spin-density and pair-density Green's functions, for k=2. For k=2, \ensuremath{\rho}(T)-\ensuremath{\rho}(0)\ensuremath{\propto}${\mathit{T}}^{1/2}$ and the pair susceptibility \ensuremath{\propto}lnT.

Physical Review LettersNov 25, 1991197 citations

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.

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.

{\mathit{T}}^{1/2}

and the pair susceptibility \ensuremath{\propto}lnT.