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We investigate several fundamental characteristics of the multiloopfunctional renormalization group (mfRG) flow by hands of its application to aprototypical many-electron system: the Anderson impurity model (AIM). We firstanalyze the convergence of the algorithm in the different parameter regions ofthe AIM. As no additional approximation is made, the multiloop series for thelocal self-energy and response functions converge perfectly to thecorresponding results of the parquet approximation (PA) in the weak- tointermediate-coupling regime. Small oscillations of the mfRG solution as afunction of the loop order gradually increase with the interaction, hindering afull convergence to the PA in the strong-coupling regime, where perturbativeresummation schemes are no longer reliable. By exploiting the convergedresults, we inspect the fulfillment of (i) sum rules associated to the Pauliprinciple and (ii) Ward identities related to conservation laws. For the Pauliprinciple, we observe a systematic improvement by increasing the loop order andincluding the multiloop corrections to the self-energy. This is consistent withthe preservation of crossing symmetries and two-particle self-consistency inthe PA. For the Ward identities, we numerically confirm a visible improvementby means of the Katanin substitution. At weak coupling, violations of the Wardidentity are further reduced by increasing the loop order in mfRG. In thisregime, we also determine the precise scaling of the deviations of the Wardidentity as a function of the electronic interaction. For larger interactionvalues, the overall behavior becomes more complex, and the benefits of thehigher-loop terms are mostly present in the contributions at large frequencies.

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