Multiloop flow equations for single-boson exchange fRG

TL;DR

This paper develops multiloop functional renormalization group flow equations for the single-boson exchange decomposition of the four-point vertex in fermionic systems, enabling more flexible and accurate many-body calculations.

Contribution

It re-derives the SBE decomposition within a generalized framework and derives multiloop RG flow equations for both the parquet and SBE approximations.

Findings

Derived mfRG flow equations for SBE ingredients in different approximations

Established structural similarity between parquet and SBE mfRG flow equations

Extended the SBE framework for potential applications to inhomogeneous and real-frequency systems

Abstract

The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction $U$, instead of a classification based on two-particle reducibility within the widely-used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully $U$-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure.