Scaling analysis of the extended single impurity Anderson model: Renormalization due to valence fluctuations

In this paper we have explored the role of valence fluctuations in an extended Anderson impurity model (e-SIAM) in which there is an additional Hubbard repulsion between conduction and impurity electrons, employing perturbative renormalization methods. We have calculated the scaling equations for the model parameters and solved them both analytically and numerically to find how valence fluctuations renormalize these parameters. Analytical solutions of the scaling equations yielded scaling invariants of the model which we find to be different than the Anderson impurity model signifying different kind of scaling trajectories of e-SIAM. The strong coupling physics of SIAM is known to be governed by spin fluctuations and we hence analysed how the strong coupling regime of e-SIAM is renormalized by valence fluctuations. Doing a third order perturbative renormalization of the model gave us access to the Kondo scale. We not only confirmed that valence fluctuations enhance the Kondo scale but we also calculated the functional form of this dependence. Benchmarking some of our results with numerical renormalization group calculations, we found excellent agreement. Applying Schrieffer-Wolff transformation, we found that the strong coupling regime of the model is governed by spin-charge Kondo model unlike the Anderson impurity model. Our results suggest that spin Kondo effect can co-exist with valence fluctuation mediated charge Kondo effect.