Parquet approximation and one-loop renormalization group: Equivalence on the leading-logarithmic level

TL;DR

This paper demonstrates that a carefully constructed one-loop functional renormalization group approximation is fully equivalent to the leading-logarithmic parquet approximation for models with logarithmic divergences, such as the Kondo model.

Contribution

It establishes the equivalence between a purely-fermionic one-loop FRG and the leading-logarithmic parquet approximation for certain models, clarifying their relationship.

Findings

One-loop FRG matches parquet approximation at leading-log level.

The equivalence confirms the validity of one-loop RG for these models.

The study clarifies the relationship between different RG approximation schemes.

Abstract

We investigate the functional renormalization group (FRG) flow of the two-particle vertex function of a model for X-ray absorption in metals. Concerning the appearance of logarithmic divergences, the model is prototypical for an important class of mostly zero- and one-dimensional systems which includes the Kondo model and the interacting one-dimensional Fermi gas. For our analysis we formulate the FRG in the framework of the real-time zero-temperature formalism, in which the model was studied before with a parquet-based approach. We establish that a reasonably crafted, purely-fermionic one-loop FRG approximation is fully equivalent on a detailed level to the leading-logarithmic parquet approximation. These two approximation schemes are thus found to just represent different perspectives on the same technical steps. This finding also reconfirms the traditional understanding of the capabilities of one-loop RG approximations for such models, which was recently put into question by an investigation of the X-ray-absorption model with multiloop FRG.