arXiv:cond-mat/9311054

Conformal Field Theory Approach to Quantum Impurity Problems

Ian K. Affleck

Published Nov 23, 1993• 0 citations

Abstract

A brief review is given of a new method for studying the critical behavior of quantum impurity problems, based on conformal field theory techniques, which I developed with Andreas Ludwig. Some results on the overscreened Kondo problem are reviewed. It is shown that the simple open and periodic fixed points, which occur in quantum spin chain impurity models, are related to each other by fusion.

DOI: 10.48550/arxiv.cond-mat/9311054

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arXiv:cond-mat/9311054

Conformal Field Theory Approach to Quantum Impurity Problems

Ian K. Affleck

Published Nov 23, 1993• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.cond-mat/9311054

Related papers

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Satoshi Fujimoto, N. Kawakami, Yang, S. -K.

arXiv:cond-mat/9512059Dec 8, 19950 citations

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Anirvan M. Sengupta, Yong Baek Kim

We consider the single-channel Kondo problem with the Kondo coupling between a spin $S$ impurity and conduction electrons with spin $j$ . These problems arise as multicritical points in the parameter spaces of two-and higher-level tunneling systems and some impurity models of heavy fermion compounds. In contrast to the previous Bethe-ansatz conjectures, it turns out that the dynamics of the spin sector is the same as that of a spin- $S$ impurity coupled to $k(j)$ channels of spin-\textonehalf{} electrons with $k(j)=2\frac{j(j+1)(2j+1)}{3}$ . As a result, for $2S<k(j)$ , the system shows non-Fermi-liquid behavior with the same exponents for the thermodynamic quantities as those of the $k(j)$ -channel Kondo problem. However, both the finite-size spectrum and the operator content are different due to the presence of the other sectors and can be obtained by conformal field theory techniques.We consider the single-channel Kondo problem with the Kondo coupling between a spin $S$ impurity and conduction electrons with spin $j$. These problems arise as multicritical points in the parameter spaces of two-and higher-level tunneling systems and some impurity models of heavy fermion compounds. In contrast to the previous Bethe-ansatz conjectures, it turns out that the dynamics of the spin sector is the same as that of a spin-$S$ impurity coupled to $k(j)$ channels of spin-\textonehalf{} electrons with $k(j)=2\frac{j(j+1)(2j+1)}{3}$. As a result, for $2S<k(j)$, the system shows non-Fermi-liquid behavior with the same exponents for the thermodynamic quantities as those of the $k(j)$-channel Kondo problem. However, both the finite-size spectrum and the operator content are different due to the presence of the other sectors and can be obtained by conformal field theory techniques.

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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

We consider the single-channel Kondo problem with the Kondo coupling between a spin

S

impurity and conduction electrons with spin

j

. These problems arise as multicritical points in the parameter spaces of two-and higher-level tunneling systems and some impurity models of heavy fermion compounds. In contrast to the previous Bethe-ansatz conjectures, it turns out that the dynamics of the spin sector is the same as that of a spin-

S

impurity coupled to

k(j)

channels of spin-\textonehalf{} electrons with

k(j)=2\frac{j(j+1)(2j+1)}{3}

. As a result, for

2S<k(j)

, the system shows non-Fermi-liquid behavior with the same exponents for the thermodynamic quantities as those of the

k(j)

-channel Kondo problem. However, both the finite-size spectrum and the operator content are different due to the presence of the other sectors and can be obtained by conformal field theory techniques.

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.