Two-particle irreducible functional renormalization group schemes---a comparative study

TL;DR

This paper develops and compares two-particle irreducible functional renormalization group schemes for Fermi systems, analyzing their theoretical foundations, truncation strategies, and performance through application to the quantum anharmonic oscillator.

Contribution

It introduces and compares new IRG schemes based on the two-particle irreducible approach, including truncation strategies and their relation to self-consistent perturbation theory.

Findings

The schemes are equivalent to self-consistent perturbation theory under certain truncations.

Different truncation strategies have varying computational efficiencies.

Application to the quantum anharmonic oscillator demonstrates the schemes' practical performance.

Abstract

We derive functional renormalization group schemes for Fermi systems which are based on the two-particle irreducible approach to the quantum many-body problem. In a first step, the cutoff is introduced in the non-interacting propagator as it is commonly done in functional renormalization group based on one-particle irreducible vertex functions. The most natural truncation of the resulting infinite hierarchy of flow equations is shown to be fully equivalent to self-consistent perturbation theory. An earlier suggested alternative truncation strategy is considered as well. In a second step, the cutoff is introduced in the two-particle interaction. Again two truncation procedures are investigated, one of which was derived before. In the latter, the mean-field solution of the many-body problem is considered as the starting point of the renormalization group flow. We compare the performance and the required numerical resources for solving the coupled flow equations for all the approximate schemes by applying them to the problem of the quantum anharmonic oscillator. In a functional integral representation, this model has a formal similarity to the quantum many-body problem. The perspectives for applying the derived two-particle irreducible functional renormalization group approaches to zero- and one-dimensional systems of correlated fermions are discussed.