Physical Review Letters

Perturbative approach to the non-Fermi-liquid fixed point of the overscreened Kondo problem

Junwu Gan, Natan Andrei, Piers Coleman

Published Feb 1, 1993• 51 citations

Abstract

The non-Fermi-liquid fixed point of the overscreened Kondo problem is studied in the limit of a large number of conduction electron channels, k\ensuremath{\gg}1, using perturbation theory in 1/k. By expanding the beta function and self-energy to sub-leading-order in 1/k, we examine the scaling, thermodynamics, and dynamic response functions in detail. Our results compliment and extend previous work based on conformal field theory. We present the dynamical spin susceptibility for the first time and show that the impurity spin is marginally quenched, with an inelastic linewidth of order T.

DOI: 10.1103/physrevlett.70.686

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Physical Review Letters

Perturbative approach to the non-Fermi-liquid fixed point of the overscreened Kondo problem

Junwu Gan, Natan Andrei, Piers Coleman

Published Feb 1, 1993• 51 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.70.686

Related papers

Overscreened single-channel Kondo problem

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.

arXiv:cond-mat/9602100Physical review. B, Condensed matterDec 1, 199619 citations

Overscreened Kondo fixed point in S=1 spin liquid

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On the multichannel Kondo model

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A detailed and comprehensive study of the one-impurity multichannel Kondo model is presented. In the limit of a large number of conduction electron channels k>>1, the low-energy fixed point is accessible to a renormalization group improved perturbative expansion in 1/k. This straightforward approach enables us to examine the scaling, thermodynamics and dynamical response functions in great detail and make clear the following features: (i) the criticality of the fixed point, (ii) the universal non-integer degeneracy, and (iii) that the compensating spin cloud has a spatial extent of the order of one lattice spacing.A detailed and comprehensive study of the one-impurity multichannel Kondo model is presented. In the limit of a large number of conduction electron channels k>>1, the low-energy fixed point is accessible to a renormalization group improved perturbative expansion in 1/k. This straightforward approach enables us to examine the scaling, thermodynamics and dynamical response functions in great detail and make clear the following features: (i) the criticality of the fixed point, (ii) the universal non-integer degeneracy, and (iii) that the compensating spin cloud has a spatial extent of the order of one lattice spacing.

arXiv:cond-mat/9309045Journal of Physics Condensed MatterJun 13, 199434 citations

Non-Fermi-liquid fixed point

Tae‐Suk Kim, L. Nunes de Oliveira, Daniel Lee Cox

We study an exchange interaction in which conduction electrons with pseudospin ${\mathrm{S}}_{\mathrm{c}}$ =3/2 interact with the impurity spin ${\mathrm{S}}_{\mathrm{I}}$ =1/2. Due to the overscreening of the impurity spin by higher conduction electron spin, a nontrivial intermediate-coupling-strength fixed point is realized. Using the numerical renormalization group (NRG), we show that the low-energy spectra are described by a non-Fermi-liquid excitation spectrum. A conformal field theory analysis is compared with NRG results and excellent agreement is obtained. Using the double-fusion rule to generate the operator spectrum with the conformal theory, we find that the specific heat coefficient and magnetic susceptibility will diverge as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}$, that the scaling dimension of an applied magnetic field is 5/6, and that exchange anisotropy is always relevant. We discuss the possible relevance of our work to two-level-system Kondo materials and dilute cerium alloys, and we point out a paradox in understanding the Bethe-ansatz solutions to the multichannel Kondo model.We study an exchange interaction in which conduction electrons with pseudospin ${\mathrm{S}}_{\mathrm{c}}$=3/2 interact with the impurity spin ${\mathrm{S}}_{\mathrm{I}}$=1/2. Due to the overscreening of the impurity spin by higher conduction electron spin, a nontrivial intermediate-coupling-strength fixed point is realized. Using the numerical renormalization group (NRG), we show that the low-energy spectra are described by a non-Fermi-liquid excitation spectrum. A conformal field theory analysis is compared with NRG results and excellent agreement is obtained. Using the double-fusion rule to generate the operator spectrum with the conformal theory, we find that the specific heat coefficient and magnetic susceptibility will diverge as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}$, that the scaling dimension of an applied magnetic field is 5/6, and that exchange anisotropy is always relevant. We discuss the possible relevance of our work to two-level-system Kondo materials and dilute cerium alloys, and we point out a paradox in understanding the Bethe-ansatz solutions to the multichannel Kondo model.

arXiv:cond-mat/9606095Physical review. B, Condensed matterMay 1, 199713 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.

We propose a possible realization of the overscreened Kondo impurity problem by a magnetic

s=1/2

impurity embedded in a two-dimensional

S=1

U(1)

spin liquid with a Fermi surface. This problem contains an interesting interplay between non-Fermi-liquid behavior induced by a

U(1)

gauge field coupled to fermions and a non-Fermi-liquid fixed point in the overscreened Kondo problem. Using a large-

N

expansion together with an expansion in the dynamical exponent of the gauge field, we find that the coupling to the gauge field leads to weak but observable changes in the physical properties of the system at the overscreened Kondo fixed point. We discuss the extrapolation of this result to a physical case and argue that the realization of overscreened Kondo physics could lead to observations of effects due to gauge fields.

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 Kondo problem, which describes the interaction of a spin

s

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/s

with the renormalization group. We study both the case of an impurity interacting with a single channel

K=1

of fermions in the

s \rightarrow \infty

limit and the case with

K

channels in the double-scaling limit

K \rightarrow \infty

s \to \infty

K/s

fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large

s

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 > 2s

we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.

We study an exchange interaction in which conduction electrons with pseudospin

{\mathrm{S}}_{\mathrm{c}}

=3/2 interact with the impurity spin

{\mathrm{S}}_{\mathrm{I}}

=1/2. Due to the overscreening of the impurity spin by higher conduction electron spin, a nontrivial intermediate-coupling-strength fixed point is realized. Using the numerical renormalization group (NRG), we show that the low-energy spectra are described by a non-Fermi-liquid excitation spectrum. A conformal field theory analysis is compared with NRG results and excellent agreement is obtained. Using the double-fusion rule to generate the operator spectrum with the conformal theory, we find that the specific heat coefficient and magnetic susceptibility will diverge as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}$, that the scaling dimension of an applied magnetic field is 5/6, and that exchange anisotropy is always relevant. We discuss the possible relevance of our work to two-level-system Kondo materials and dilute cerium alloys, and we point out a paradox in understanding the Bethe-ansatz solutions to the multichannel Kondo model.