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Journal of Physics C Solid State Physics

On the completeness of Wiegmann's solution to the Kondo problem

C C Lai, WU YONG-SHI

Published Oct 30, 1981• 5 citations
Publisher

Abstract

The recent solution by Wiegmann of the Kondo problem is re-examined to some detail. Special attention is paid to the problem of completeness which is essential to thermodynamic calculations. After obtaining the most general solution to the Kondo Hamiltonian, it is proved rigorously that solutions obtained with the Bethe ansatz do form a complete set. It is also shown that the Wiegmann solutions, being a subset of the Bethe solutions, are not complete although they are obtainable from the latter via some physical arguments.

DOI
10.1088/0022-3719/14/30/004

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The recent solution by Wiegmann of the Kondo problem is re-examined to some detail. Special attention is paid to the problem of completeness which is essential to thermodynamic calculations. After obtaining the most general solution to the Kondo Hamiltonian, it is proved rigorously that solutions obtained with the Bethe ansatz do form a complete set. It is also shown that the Wiegmann solutions, being a subset of the Bethe solutions, are not complete although they are obtainable from the latter via some physical arguments.
The Kondo problem concerns a simple physical model of magnetic impurities in metals that has required the most sophisticated of mathematical techniques. To theorists, the model not only represents a first step towards understanding the far more complex problem of ferromagnetism but also appears tantalizingly close to exact solution by the techniques employed for renormalizable field theories. In 1975 Kenneth Wilson of Cornell University successfully used renormalization-group techniques to calculate numerically a key parameter in the Kondo problem. This past year Natan Andrei of New York University and independently P. B. Wiegmann of the Landau Institute in Moscow executed a second coup by exactly diagonalizing the Hamiltonian to allow a solution in closed form.
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.
We consider the generalized exclusion statistics in the Kondo problem. The thermodynamic Bethe ansatz equations have been used for a multicomponent system of particles obeying the generalized exclusion principle. We have found a relation between the derivative of the phase shift of the scattering matrix for Fermi particles and for particles characterized by generalized exclusion statistics. We show that the statistical matrix in the Kondo problem has a universal form in the high and low temperature limits.
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.