Physical Review Letters

Non-Fermi-Liquid States in a Generalized Two-Channel Kondo Model

Hanbin Pang

Published Nov 14, 1994• 14 citations

Abstract

Using the numerical renormalization group method, we study a generalized two-channel Kondo model where the impurity has a quadrupole moment (pseudospin) and a magnetic moment (flavor). This model may be of relevance to the Ce ions in ${\mathrm{CeB}}_{6}$ and the Cu ions in cuprate superconductors. We find that the nontrivial fixed point of the two-channel Kondo model is unstable against flavor exchange coupling. The system flows to a non-Fermi-liquid fixed line. In the presence of a double tensor interaction, which breaks particle-hole symmetry, the system flows to a Fermi-liquid fixed line.

DOI: 10.1103/physrevlett.73.2736

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Physical Review Letters

Non-Fermi-Liquid States in a Generalized Two-Channel Kondo Model

Hanbin Pang

Published Nov 14, 1994• 14 citations

Publisher

Abstract

{\mathrm{CeB}}_{6}

and the Cu ions in cuprate superconductors. We find that the nontrivial fixed point of the two-channel Kondo model is unstable against flavor exchange coupling. The system flows to a non-Fermi-liquid fixed line. In the presence of a double tensor interaction, which breaks particle-hole symmetry, the system flows to a Fermi-liquid fixed line.

DOI: 10.1103/physrevlett.73.2736

Related papers

Two-channel pseudogap Kondo and Anderson models: Quantum phase transitions and non-Fermi liquids

Imke Schneider, Lars Fritz, Frithjof B. Anders +2 more

We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic $r=0$ case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For $r>{r}_{\mathrm{max}}$ , this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions $r=0$ and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic $r=0$ case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For $r>{r}_{\mathrm{max}}$, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions $r=0$ and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.

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Repulsion between Conduction Electrons as a Relevant Perturbation against the Stability of Non-Fermi-Liquid Fixed Point of Multichannel Kondo Problem

Kazumasa Miyake, Hiroaki Kusunose, Osamu Narikiyo

It is shown that a non-Fermi-liquid fixed point of the multichannel Kondo problem is unstable for repulsive interaction at the impurity site between conduction electrons with different flavors and the same pseudospin. The anomalous fixed point is shown, by the perturbative renormalization group argument, to move into the strong coupling region with increasing repulsion. The low-temperature properties at the so-called “Toulouse limit” are calculated by the bosonization method to show that the Fermi-liquid like behavior is recovered quickly as soon as repulsion is switched on. This result poses a question as to the reality of the non-Fermi-liquid state for the multichannel Kondo model which has been examined extensively so far.It is shown that a non-Fermi-liquid fixed point of the multichannel Kondo problem is unstable for repulsive interaction at the impurity site between conduction electrons with different flavors and the same pseudospin. The anomalous fixed point is shown, by the perturbative renormalization group argument, to move into the strong coupling region with increasing repulsion. The low-temperature properties at the so-called “Toulouse limit” are calculated by the bosonization method to show that the Fermi-liquid like behavior is recovered quickly as soon as repulsion is switched on. This result poses a question as to the reality of the non-Fermi-liquid state for the multichannel Kondo model which has been examined extensively so far.

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Hiroaki Kusunose, K. Miyake, Yukihiro Shimizu +1 more

Particle-hole symmetry breaking perturbation in the two-channel pseudospin Kondo problem is studied by the numerical renormalization-group method. It is shown that the repulsion among conduction electrons at the impurity site and the single particle potential are relevant perturbations against the conventional non-Fermi-liquid fixed point. Although the repulsion (potential) with realistic strength prevents the overscreening of the pseudospin, it induces in turn a real spin, which is also overscreened again. Thus the real spin susceptibility becomes anomalous, contrary to the conventional two-channel Kondo problem.Particle-hole symmetry breaking perturbation in the two-channel pseudospin Kondo problem is studied by the numerical renormalization-group method. It is shown that the repulsion among conduction electrons at the impurity site and the single particle potential are relevant perturbations against the conventional non-Fermi-liquid fixed point. Although the repulsion (potential) with realistic strength prevents the overscreening of the pseudospin, it induces in turn a real spin, which is also overscreened again. Thus the real spin susceptibility becomes anomalous, contrary to the conventional two-channel Kondo problem.

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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.

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Solution of the Two-Impurity, Two-Channel Kondo Model

Antoine Georges, Anirvan M. Sengupta

We solve the two-impurity, two-channel Kondo model using a combination of conformal invariance and bosonization techniques. In the odd-even symmetric case, the RKKY interaction is exactly marginal, resulting in a line of non-Fermi-liquid fixed points. Explicit formulas are given for the critical exponents and for the finite-size spectrum, which depend continuously on a single parameter. The marginal line spans a range of values of the RKKY coupling $I$ which goes from the infinitely strong ferromagnetic point $I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{\infty}$ to a finite antiferromagnetic critical value ${I}_{\mathrm{min}}<0$ beyond which a Fermi liquid is recovered. When the odd-even symmetry is broken, the marginal line is unstable for ferromagnetic $I$ , while for antiferromagnetic $I$ it extends into a manifold of fixed points.We solve the two-impurity, two-channel Kondo model using a combination of conformal invariance and bosonization techniques. In the odd-even symmetric case, the RKKY interaction is exactly marginal, resulting in a line of non-Fermi-liquid fixed points. Explicit formulas are given for the critical exponents and for the finite-size spectrum, which depend continuously on a single parameter. The marginal line spans a range of values of the RKKY coupling $I$ which goes from the infinitely strong ferromagnetic point $I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{\infty}$ to a finite antiferromagnetic critical value ${I}_{\mathrm{min}}<0$ beyond which a Fermi liquid is recovered. When the odd-even symmetry is broken, the marginal line is unstable for ferromagnetic $I$, while for antiferromagnetic $I$ it extends into a manifold of fixed points.

arXiv:cond-mat/9405054Physical Review LettersApr 3, 199533 citations

Crossover Temperature from Non-Fermi Liquid to Fermi Liquid Behavior in Two Types of Impurity Kondo Model

Satoshi Yotsuhashi, Hideaki Maebashi

Numerical renormalization-group results on entropy of the anisotropic two-channel Kondo model with the band-width cutoff ( $D$ ) in the presence of a magnetic field ( $h$ ) are obtained to determine crossover temperature from the non-Fermi liquid to Fermi liquid fixed point. It is found that the crossover temperature ( $T_{\rm x}$ ) is given by $T_{\rm x} \equiv {r} T_{\rm K} \sim D(ΔJ/J_{\rm av})^2 e^{-1/J_{\rm av}}$ when $(h /T_{\rm K})^2 \ll r \ll 1 $, where $T_{\rm K}$ , $J_{\rm av}$ and $ΔJ$ are the Kondo temperature, the average and difference of the exchange coupling constants, respectively. This result indicates that non-Fermi liquid behavior can be seen even if $ΔJ > T_{\rm K}$ . Robust similarities of the crossover behavior in the region around the non-Fermi liquid critical point to that of the two-impurity Kondo model are also discussed.Numerical renormalization-group results on entropy of the anisotropic two-channel Kondo model with the band-width cutoff ($D$) in the presence of a magnetic field ($h$) are obtained to determine crossover temperature from the non-Fermi liquid to Fermi liquid fixed point. It is found that the crossover temperature ($T_{\rm x}$) is given by $T_{\rm x} \equiv {r} T_{\rm K} \sim D(ΔJ/J_{\rm av})^2 e^{-1/J_{\rm av}}$ when $(h /T_{\rm K})^2 \ll r \ll 1 $, where $T_{\rm K}$, $J_{\rm av}$ and $ΔJ$ are the Kondo temperature, the average and difference of the exchange coupling constants, respectively. This result indicates that non-Fermi liquid behavior can be seen even if $ΔJ > T_{\rm K}$. Robust similarities of the crossover behavior in the region around the non-Fermi liquid critical point to that of the two-impurity Kondo model are also discussed.

arXiv:cond-mat/0203506Journal of the Physical Society of JapanJul 15, 200212 citations

{\mathrm{CeB}}_{6}

r=0

case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For

r>{r}_{\mathrm{max}}

, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions

r=0

and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.

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.

I

which goes from the infinitely strong ferromagnetic point $I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{\infty}$ to a finite antiferromagnetic critical value

{I}_{\mathrm{min}}<0

beyond which a Fermi liquid is recovered. When the odd-even symmetry is broken, the marginal line is unstable for ferromagnetic

I

, while for antiferromagnetic

I

it extends into a manifold of fixed points.

Numerical renormalization-group results on entropy of the anisotropic two-channel Kondo model with the band-width cutoff (

D

) in the presence of a magnetic field (

h

) are obtained to determine crossover temperature from the non-Fermi liquid to Fermi liquid fixed point. It is found that the crossover temperature (

T_{\rm x}

) is given by

T_{\rm x} \equiv {r} T_{\rm K} \sim D(ΔJ/J_{\rm av})^2 e^{-1/J_{\rm av}}

when $(h /T_{\rm K})^2 \ll r \ll 1 $, where

T_{\rm K}

J_{\rm av}

and

ΔJ

are the Kondo temperature, the average and difference of the exchange coupling constants, respectively. This result indicates that non-Fermi liquid behavior can be seen even if

ΔJ > T_{\rm K}

. Robust similarities of the crossover behavior in the region around the non-Fermi liquid critical point to that of the two-impurity Kondo model are also discussed.