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Communications in Theoretical Physics

The Kondo Problem as the Quantum Sine-Gordon Model

Guang-Ming Zhang, Hong Chen, Xiang Xia Wu

Published Jun 1, 1991• 0 citations
Publisher

Abstract

The ground-state properties of the Kondo Hamiltonian is analyzed in detail and we find the problem to be exactly equivalent to a modified quantum sine-Gordon model. Paralleling to the renormalization-group theory of the sine-Gordon model, we easily derive the correct next-to-leading order flow equations and obtain a universal quantity in a linear combination of the coefficients of the higher-order terms.

DOI
10.1088/0253-6102/15/4/497

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The ground-state properties of the Kondo Hamiltonian is analyzed in detail and we find the problem to be exactly equivalent to a modified quantum sine-Gordon model. Paralleling to the renormalization-group theory of the sine-Gordon model, we easily derive the correct next-to-leading order flow equations and obtain a universal quantity in a linear combination of the coefficients of the higher-order terms.
The bosonized Kondo Hamiltonian is obtained by comparing the long-time limit of the reduced grand partition function for the impurity spin. We analyze this bosonized Kondo Hamiltonian at zero temperature and derive an effective Hamiltonian describing the effective interaction between the conduction electrons via an impurity-spin scattering. The infrared divergences encountered in the conventional perturbation theory are caused by this growth effective coupling at the low-energy limit. With the Bogoliubov transformation, we have developed a perturbation approach for this effective Hamiltonian and investigated the nontrivial ferromagnetic-antiferromagnetic crossover of the impurity spin. The critical condition derived here is in agreement with the renormalization-group numerical result. In particular, the ground-state wave function and its excitation spectrum of the conduction electrons are also obtained. Moreover, this effective Hamiltonian may be mapped onto a modified quantum sine-Gordon model. Paralleling the renormalization-group theory of the quantum sine-Gordon model, we straightforwardly reproduce previous results and derive the higher-order terms and a new universal correction in the flow equations.
We explicitly evaluate the infinite series of integrals that appear in the "Anderson-Yuval" reformulation of the anisotropic Kondo model in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo model of arbitrary spin with the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid and tunneling in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.
We analyze the single-channel Kondo model using the recently developed unitary renormalization-group (URG) method and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in good agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both kkk-space number diagonal (Fermi liquid) and off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The URG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local-moment and strong-coupling ground states and is related to a change in the Luttinger volume of the conduction bath under the crossover to strong coupling. Our results offer fresh insight into the nature of the emergent many-particle entanglement within the Kondo cloud and pave the way for further investigations in more exotic contexts such as the fixed point of the overscreened multichannel Kondo problem.
We analyze the single-channel Kondo model using the recently developed unitary renormalization group (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in excellent agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both k-space number-diagonal (Fermi liquid) as well off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number-diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The unitary RG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local moment and strong-coupling ground states, and is related to a change in the Luttinger volume of the conduction bath. Our results offer fresh insight on the nature of the emergent many-particle entanglement within the Kondo cloud, and pave the way for further investigations in more exotic contexts such as the fixed point of the over-screened multi-channel Kondo problem.