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Physical review. B, Condensed matter

Role of open boundaries in the Bethe ansatz solution of the Kondo problem

Andrei A. Zvyagin

Published Nov 18, 2002• 2 citations
Publisher

Abstract

The Bethe ansatz solution for the Kondo problem with open boundary conditions is presented. We show that one of the features of the generic Kondo effect---the finite magnetic susceptibility at low energies (present in the Bethe ansatz solution for periodic boundary conditions for the same model)---is absent due to the effect of open edges, if one considers the total effect of the impurity and the open edge together. However, we show that the behavior of the dynamical magnetic impurity itself is similar for periodic and open boundaries. It is shown that a real static boundary local potential (magnetic field) cannot model all features of the behavior of the dynamical magnetic impurity in the generic Kondo situation.

DOI
10.1103/physrevb.66.174430

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The Bethe ansatz solution for the Kondo problem with open boundary conditions is presented. We show that one of the features of the generic Kondo effect---the finite magnetic susceptibility at low energies (present in the Bethe ansatz solution for periodic boundary conditions for the same model)---is absent due to the effect of open edges, if one considers the total effect of the impurity and the open edge together. However, we show that the behavior of the dynamical magnetic impurity itself is similar for periodic and open boundaries. It is shown that a real static boundary local potential (magnetic field) cannot model all features of the behavior of the dynamical magnetic impurity in the generic Kondo situation.
We review some exact results on Kondo impurity systems derived from Bethe-ansatz solutions and boundary conformal field theory with particular emphasis on universal aspects of the phenomenon. The finite-size spectra characterizing the low-energy fixed point are computed from the Bethe-ansatz solutions of various models related to the Kondo problem. Using the finite-size scaling argument, we investigate their exact critical properties. We also discuss that a universal relation between the Kondo effect and the impurity effect in one-dimensional quantum systems usefully expedites our understanding of these different phenomena.
In this paper, the Kondo magnetic effect is studied for the XXZ\mathrm{XXZ}XXZ spin chain where the impurities are coupled to the edges of the system. The Hamiltonian of our model can be constructed from the transfer matrix. It is exactly solvable and the exchange constants between the bulk and the boundary impurities are arbitrary. The finite-size corrections to the ground-state energy are obtained for the magnetic impurities and the boundary condition. The specific heat and the susceptibility contributed by the impurities are derived and the Kondo temperature is given explicitly by the use of the Luttinger-Fermi liquid picture and the Bethe ansatz method.
We propose an integrable Kondo problem in a one-dimensional $t\ensuremath{-}J$ model. With the open boundary condition of the wave functions at the impurity sites, the model can be exactly solved via Bethe ansatz for a set of JL,R{J}_{L,R}JL,R​ (Kondo coupling constants) and VL,R{V}_{L,R}VL,R​ (impurity potentials) parametrized by a single parameter ccc. The integrable value of JL,R{J}_{L,R}JL,R​ runs from negative infinity to positive infinity, which allows us to study both the ferromagnetic Kondo problem and the antiferromagnetic Kondo problem in a strongly correlated electron system. Generally, there is a residual entropy for the ground state, which indicates a typical non-Fermi liquid behavior.
To study the impurity potential effect to the Kondo problem in a Luttinger liquid, we introduce a model of ferromagnetic single-impurity Kondo problem in a multi-band one-dimensional δ-potential Fermi gas, which is exactly solvable for a special Kondo coupling constant via Bethe ansatz. It is found that the boundary potential modifies the local moment seriously due to the charge-spin coupling and the strong backward scattering off the impurity, and thus strongly affects the fixed point of the system. Generally, their is a residual entropy for the ground state which depends on the strength of the boundary potential. In addition, the tunneling effect through the impurity and the breakdown of the open boundary induced by the impurity are also discussed.