arXiv:cond-mat/0406293 · Physical Review Letters

Quantum Critical Properties of the Bose-Fermi Kondo Model in a Large- N Limit

Lijun Zhu, Stefan Kirchner, Qimiao Si A, Antoine Georges

Published Dec 21, 2004• 61 citations

Abstract

Studies of non-Fermi-liquid properties in heavy fermions have led to the current interest in the Bose-Fermi Kondo model. Here we use a dynamical large-$N$ approach to analyze an $\mathrm{S}\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}(\ensuremath{\kappa}N)$ generalization of the model. We establish the existence in this limit of an unstable fixed point when the bosonic bath has a sub-Ohmic spectrum ($|\ensuremath{\omega}{|}^{1\ensuremath{-}ϵ}\mathrm{s}\mathrm{g}\mathrm{n}\ensuremath{\omega}$, with $0<ϵ<1$). At the quantum-critical point, the Kondo scale vanishes and the local spin susceptibility (which is finite on the Kondo side for $\ensuremath{\kappa}<1$) diverges. We also find an $\ensuremath{\omega}/T$ scaling for an extended range (15 decades) of $\ensuremath{\omega}/T$. This scaling violates (for $ϵ\ensuremath{\ge}1/2$) the expectation of a naive mapping to certain classical models in an extra dimension; it reflects the inherent quantum nature of the critical point.

DOI: 10.1103/physrevlett.93.267201

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arXiv:cond-mat/0406293 · Physical Review Letters

Quantum Critical Properties of the Bose-Fermi Kondo Model in a Large- N Limit

Lijun Zhu, Stefan Kirchner, Qimiao Si A, Antoine Georges

Published Dec 21, 2004• 61 citations

arXiv HTML PDF

Abstract

Studies of non-Fermi-liquid properties in heavy fermions have led to the current interest in the Bose-Fermi Kondo model. Here we use a dynamical large-

N

approach to analyze an

\mathrm{S}\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}(\ensuremath{\kappa}N)

generalization of the model. We establish the existence in this limit of an unstable fixed point when the bosonic bath has a sub-Ohmic spectrum (

|\ensuremath{\omega}{|}^{1\ensuremath{-}ϵ}\mathrm{s}\mathrm{g}\mathrm{n}\ensuremath{\omega}

, with

0<ϵ<1

). At the quantum-critical point, the Kondo scale vanishes and the local spin susceptibility (which is finite on the Kondo side for

\ensuremath{\kappa}<1

) diverges. We also find an

\ensuremath{\omega}/T

scaling for an extended range (15 decades) of

\ensuremath{\omega}/T

. This scaling violates (for

ϵ\ensuremath{\ge}1/2

) the expectation of a naive mapping to certain classical models in an extra dimension; it reflects the inherent quantum nature of the critical point.

DOI: 10.1103/physrevlett.93.267201

Related papers

Quantum chaos of the Bose-Fermi Kondo model at intermediate temperature

Xinloong Han, Zuodong Yu

We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large- $N$ limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling ${J}_{K}$ increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as ${J}_{K}$ approaches the multichannel Kondo fixed point ( $MCK$ ). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling $g$ increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large-$N$ limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling ${J}_{K}$ increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as ${J}_{K}$ approaches the multichannel Kondo fixed point ($MCK$). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling $g$ increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.

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Kondo destruction and fixed-point annihilation in a Bose-Fermi Kondo model

Haoyu Hu, Qimiao Si A

Quantum criticality that goes beyond the Landau framework of order-parameter fluctuations is playing a central role in elucidating the behavior of strange metals. A prominent case appears in Kondo lattice systems, which have been extensively analysed in terms of an effective Bose-Fermi Kondo model. Here, a spin is simultaneously coupled to conduction electron bands and gapless vector bosons that represent magnetic fluctuations. The Bose-Fermi Kondo model features interacting fixed points of Kondo destruction with such properties as dynamical Planckian ( $\hbar ω/ k_{\rm B} T$ ) scaling and loss of quasiparticles. Here we carry out a renormalization-group analysis of the model with spin isotropy and identify pair-wise annihilations of the fixed points as the spectrum of the bosonic bath evolves. Our analysis not only provides an essentially complete understanding of the previous numerical results of an SU(2)-symmetric model, but also reveals a surprising feature of sequential fixed-point annihilation. Our results lay the foundation for the understanding of quantum criticality in spin-isotropic heavy-fermion metals as well as in doped Mott-Hubbard systems.Quantum criticality that goes beyond the Landau framework of order-parameter fluctuations is playing a central role in elucidating the behavior of strange metals. A prominent case appears in Kondo lattice systems, which have been extensively analysed in terms of an effective Bose-Fermi Kondo model. Here, a spin is simultaneously coupled to conduction electron bands and gapless vector bosons that represent magnetic fluctuations. The Bose-Fermi Kondo model features interacting fixed points of Kondo destruction with such properties as dynamical Planckian ($\hbar ω/ k_{\rm B} T$) scaling and loss of quasiparticles. Here we carry out a renormalization-group analysis of the model with spin isotropy and identify pair-wise annihilations of the fixed points as the spectrum of the bosonic bath evolves. Our analysis not only provides an essentially complete understanding of the previous numerical results of an SU(2)-symmetric model, but also reveals a surprising feature of sequential fixed-point annihilation. Our results lay the foundation for the understanding of quantum criticality in spin-isotropic heavy-fermion metals as well as in doped Mott-Hubbard systems.

arXiv:2207.08744Jul 18, 20220 citations

Local quantum criticality out of equilibrium : Effective temperatures and scaling in the steady-state regime

Ribeiro, P., Si, Q., Kirchner, S.

We study the out-of-equilibrium steady-state properties of the Bose-Fermi-Kondo model, describing a local magnetic moment coupled to two ferromagnetic leads that support bosonic (magnons) and fermionic (Stoner continuum electrons) low-energy excitations. This model describes the destruction of the Kondo effect as the coupling to the bosons is increased. Its phase diagram comprises three non-trivial fixed points. Using a dynamical large-N approach on the Keldysh contour, we study two different non-equilibrium setups: a) a finite bias voltage and b) a finite temperature gradient, imposed across the leads. The scaling behavior of the charge and energy currents is identified and characterized for all fixed points. We report the existence of a fixed-point-dependent effective temperature, defined though the fluctuation-dissipation relations of the local spin-susceptibility in the scaling regime, which permits to recover the equilibrium behavior of both dynamical and static spin-susceptibilities. Copyright (c) EPLA, 2013We study the out-of-equilibrium steady-state properties of the Bose-Fermi-Kondo model, describing a local magnetic moment coupled to two ferromagnetic leads that support bosonic (magnons) and fermionic (Stoner continuum electrons) low-energy excitations. This model describes the destruction of the Kondo effect as the coupling to the bosons is increased. Its phase diagram comprises three non-trivial fixed points. Using a dynamical large-N approach on the Keldysh contour, we study two different non-equilibrium setups: a) a finite bias voltage and b) a finite temperature gradient, imposed across the leads. The scaling behavior of the charge and energy currents is identified and characterized for all fixed points. We report the existence of a fixed-point-dependent effective temperature, defined though the fluctuation-dissipation relations of the local spin-susceptibility in the scaling regime, which permits to recover the equilibrium behavior of both dynamical and static spin-susceptibilities. Copyright (c) EPLA, 2013

Jun 1, 20135 citations

Studies of non-Fermi-liquid properties in heavy fermions have led to the current interest in the Bose-Fermi Kondo model. Here we use a dynamical large-

N

approach to analyze an

\mathrm{S}\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}(\ensuremath{\kappa}N)

generalization of the model. We establish the existence in this limit of an unstable fixed point when the bosonic bath has a sub-Ohmic spectrum (

|\ensuremath{\omega}{|}^{1\ensuremath{-}ϵ}\mathrm{s}\mathrm{g}\mathrm{n}\ensuremath{\omega}

, with

0<ϵ<1

). At the quantum-critical point, the Kondo scale vanishes and the local spin susceptibility (which is finite on the Kondo side for

\ensuremath{\kappa}<1

) diverges. We also find an

\ensuremath{\omega}/T

scaling for an extended range (15 decades) of

\ensuremath{\omega}/T

. This scaling violates (for

ϵ\ensuremath{\ge}1/2

) the expectation of a naive mapping to certain classical models in an extra dimension; it reflects the inherent quantum nature of the critical point.

We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large-

N

limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling

{J}_{K}

increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as

{J}_{K}

approaches the multichannel Kondo fixed point (

MCK

). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling

g

increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.

0<r<\frac{1}{2}

and $\frac{1}{2}\ensuremath{\le}s<1$, and applying the numerical renormalization group to the corresponding Kondo model. At particle-hole symmetry, the models exhibit a quantum-critical point between a Kondo (fermionic strong-coupling) phase and a localized (Kondo-destroyed) phase. The two solution methods, which are in good agreement in their domain of overlap, provide access to the many-body spectrum, as well as to correlation functions including, in particular, the single-particle Green's function and the static and dynamical local spin susceptibilities. The quantum-critical regime exhibits the hyperscaling of critical exponents and $\ensuremath{\omega}/T$ scaling in the dynamics that characterize an interacting critical point. The

(r,s)

plane can be divided into three regions: one each in which the calculated critical properties are dominated by the bosonic bath alone or by the fermionic bath alone, and between these two regions, a third in which the bosonic bath governs the critical spin response but both baths influence the renormalization-group flow near the quantum-critical point.

\mathrm{SU}(2)

-invariant non-Fermi-liquid fixed points studied earlier, and thus the quantum phase transition is usually governed by

\mathrm{XY}

or Ising-type fixed points. We furthermore derive an exact result that relates the anomalous exponent of the Bose field to that of the susceptibility at any finite coupling fixed point. Implications for the dynamical mean-field approach to locally quantum critical phase transitions are also discussed.

\hbar ω/ k_{\rm B} T

) scaling and loss of quasiparticles. Here we carry out a renormalization-group analysis of the model with spin isotropy and identify pair-wise annihilations of the fixed points as the spectrum of the bosonic bath evolves. Our analysis not only provides an essentially complete understanding of the previous numerical results of an SU(2)-symmetric model, but also reveals a surprising feature of sequential fixed-point annihilation. Our results lay the foundation for the understanding of quantum criticality in spin-isotropic heavy-fermion metals as well as in doped Mott-Hubbard systems.