The Linear Dispersion s-d Model

In 1980 exact solutions for the s-d model were found independently by Andrei (1980) and Wiegmann (1980), and later also exact solutions for the Anderson model (Wiegmann, 1981; Kawakami & Okiji, 1981). This was a rather surprising development, more particularly, because the methods of solution were based on the ansatz used by Bethe as early as 1931 in his solution of the one dimensional Heisenberg model. Though a quantitative understanding of the Kondo problem was obtained in the 70s using the renormalization group and Fermi liquid approaches, the exact solutions have produced new results, analytic formulae for the behaviour in weak and strong magnetic fields, the form of the electronic specific heat over the full temperature range. The results give complete confirmation of the physical picture that emerged with the earlier work. The approach has proved to be generalizable to some of the more physically relevant models, the s-d model with spin greater than ½ the degenerate models for rare earth impurities (we will discuss these in the next chapter), and models including crystal fields, providing a greater range of predictions for comparison with experiment. The solutions have proved to be immensely valuable in testing some of the approximate methods, ones which can also be used to calculate dynamic as well as thermodynamic properties (it has not proved possible to calculate dynamics directly from the Bethe ansatz), methods which may prove useful in tackling multiple impurity and lattice problems.