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

TL;DR

This paper revisits the Anderson impurity model using Wegner's infinitesimal unitary transformations, providing a more accurate and consistent framework that improves upon traditional scaling and Schrieffer--Wolff methods, especially at high energies.

Contribution

It demonstrates that Wegner's method offers a comprehensive approximation scheme for all interaction strengths, correcting high-energy cutoffs and avoiding singularities in the model.

Findings

Provides a unified framework for deriving impurity model results

Corrects high-energy cutoff issues in traditional methods

Avoids singularities in induced couplings

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.