Analytic results for the Anderson Impurity Model

In the Renormalised Perturbation Theory (RPT) the Anderson impurity model is interpreted in terms of renormalised parameters $\boldsymbol{\tilde{\mu}}= (\tilde{\epsilon}_d, \tilde{\Delta}, \tilde{U})$ which are in a one-to-one correspondence with the bare impurity level $\epsilon_d$, hybridisation $\Delta$ and on-site interaction $U$. Though the renormalised parameters suffice to describe the low-energy fixed point of the model, the parameters themselves usually have to be determined using the Numerical Renormalisation Group. In this paper we address this issue by applying the flow equation method to the particle-hole symmetric model to derive a system of differential equations for $\textrm{d}\boldsymbol{\tilde{\mu}}/ \textrm{d} U$. To leading order in $\tilde{U}$ this assumes a particularly simple form, permitting its analytic solution in this approximation. We find that our results for the renormalised parameters are in good agreement with the NRG and use them to calculate the Wilson ratio, spin susceptibility and conductance of the impurity.