2.4 Summary and outlook

Summing up, we have shown that the resonant level model with a nontrivial hybridization function $\Delta_{\rm eff}(\epsilon)$ can be used as an effective model for the single impurity Kondo model. The key observation was the fact that the flow equation solutions of both models are approximately identical if a suitable $\Delta_{\rm eff}(\epsilon)$ is chosen for the RLM (Fig. 1). In this mapping the impurity orbital occupation number $n_d-1/2$ plays the role of the Kondo impurity spin $S_z$. The impurity orbital energy corresponds to the local magnetic field acting on $S_z$.

In contrast with the conventional approach where effective models describe the vicinity of the low-energy renormalization group fixed points(3,5), our effective model very accurately describes both certain low- and high-energy properties of the original Kondo model: compare for example our discussion of the dynamical spin-spin correlation function in Fig. 2.3. It also yields thermodynamic quantities that are in excellent agreement with much more expensive numerical methods (see Fig. 3.3). The nontrivial behavior of the effective hybridization function encodes the quasiparticle interaction, which leads to e.g. the correct Wilson ratio for small magnetic fields (with 5% accuracy). Notice, however, that our effective model does not allow the correct evaluation of higher-order correlation functions beyond the low-energy limit,

In conclusion, our approach describes many aspects of the complicated many-body Kondo physics for not too large magnetic fields within a simple noninteracting model. Therefore one can very easily and intuitively understand certain properties of the Kondo model, e.g. the dependence of correlation functions on a local magnetic field (Fig. 2.3). One main prospect of our approach is to look at other correlation functions using this effective model, in particular the $T$-matrix for applications in the framework of DMFT calculations. Future prospects also include cluster problems and the single impurity Anderson model. Work along these lines is in progress.

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