Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model

TL;DR

This study applies the multiloop functional renormalization group to the Anderson impurity model, demonstrating convergence to parquet results at weak to intermediate coupling and analyzing the fulfillment of sum rules and Ward identities.

Contribution

It provides a detailed analysis of the convergence and accuracy of the multiloop FRG method in the AIM, including improvements in sum rule and Ward identity compliance with higher loop orders.

Findings

Multiloop FRG converges to parquet approximation in weak/intermediate coupling.

Higher loop orders improve sum rule and Ward identity fulfillment.

Convergence deteriorates at strong coupling, with oscillations increasing.

Abstract

We investigate several fundamental characteristics of the multiloop functional renormalization group (mfRG) flow by hands of its application to a prototypical many-electron system: the Anderson impurity model (AIM). We first analyze the convergence of the algorithm in the different parameter regions of the AIM. As no additional approximation is made, the multiloop series for the local self-energy and response functions converge perfectly to the corresponding results of the parquet approximation (PA) in the weak- to intermediate-coupling regime. Small oscillations of the mfRG solution as a function of the loop order gradually increase with the interaction, hindering a full convergence to the PA in the strong-coupling regime, where perturbative resummation schemes are no longer reliable. By exploiting the converged results, we inspect the fulfillment of (i) sum rules associated to the Pauli principle and (ii) Ward identities related to conservation laws. For the Pauli principle, we observe a systematic improvement by increasing the loop order and including the multiloop corrections to the self-energy. This is consistent with the preservation of crossing symmetries and two-particle self-consistency in the PA. For the Ward identities, we numerically confirm a visible improvement by means of the Katanin substitution. At weak coupling, violations of the Ward identity are further reduced by increasing the loop order in mfRG. In this regime, we also determine the precise scaling of the deviations of the Ward identity as a function of the electronic interaction. For larger interaction values, the overall behavior becomes more complex, and the benefits of the higher-loop terms are mostly present in the contributions at large frequencies.