High-order terms in the renormalized perturbation theory for the Anderson impurity model

TL;DR

This paper develops a systematic method to compute high-order terms in the renormalized perturbation theory for the Anderson impurity model, enabling accurate calculations of self-energy and spectral density up to fifth order.

Contribution

It introduces an automated diagrammatic generation and integration approach, including a generalized propagator to simplify calculations, advancing the precision of perturbative analysis.

Findings

Results agree with Numerical Renormalization Group calculations

Efficient computation of self-energy up to fifth order

Applicable to various model asymmetries

Abstract

We study the renormalized perturbation theory of the single-impurity Anderson model, particularly the high-order terms in the expansion of the self-energy in powers of the renormalized coupling $\tilde{U}$. Though the presence of counter-terms in the renormalized theory may appear to complicate the diagrammatics, we show how these can be seamlessly accommodated by carrying out the calculation order-by-order in terms of skeleton diagrams. We describe how the diagrams pertinent to the renormalized self-energy and four-vertex can be automatically generated, translated into integrals and numerically integrated. To maximize the efficiency of our approach we introduce a generalized $k$-particle/hole propagator, which is used to analytically simplify the resultant integrals and reduce the dimensionality of the integration. We present results for the self-energy and spectral density to fifth order in $\tilde{U}$, for various values of the model asymmetry, and compare them to a Numerical Renormalization Group calculation.