Theory of the Anderson impurity model: The Schrieffer--Wolff transformation re--examined

Abstract

We apply the method of infinitesimal unitary transformations recently introduced by Wegner to the Anderson single impurity model. It is demonstrated that this method provides a good approximation scheme for all values of the on-site interaction $U$, it becomes exact for $U=0$. We are able to treat an arbitrary density of states, the only restriction being that the hybridization should not be the largest parameter in the system. Our approach constitutes a consistent framework to derive various results usually obtained by either perturbative renormalization in an expansion in the hybridization~$\Gamma$, Anderson's ``poor man's" scaling approach or the Schrieffer--Wolff unitary transformation. In contrast to the Schrieffer--Wolff result we find the correct high--energy cutoff and avoid singularities in the induced couplings. An important characteristic of our method as compared to the ``poor man's" scaling approach is that we continuously decouple modes from the impurity that have a large energy difference from the impurity orbital energies. In the usual scaling approach this criterion is provided by the energy difference from the Fermi surface.