Analytic results for the Anderson Impurity Model

TL;DR

This paper derives analytic expressions for the renormalised parameters of the Anderson Impurity Model using flow equations, providing insights into low-energy properties and matching numerical results.

Contribution

It introduces an analytic approach to determine renormalised parameters in the Anderson model via flow equations, reducing reliance on numerical methods.

Findings

Renormalised parameters agree with NRG results

Calculated Wilson ratio, spin susceptibility, and conductance

Provided analytic solutions in the particle-hole symmetric case

Abstract

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.