Summing parquet diagrams using the functional renormalization group: X-ray problem revisited

TL;DR

This paper introduces a straightforward approach using the functional renormalization group and partial bosonization to sum parquet diagrams in fermionic systems, effectively addressing competing instabilities and revisiting the X-ray problem.

Contribution

It presents a novel, simplified method for summing parquet diagrams via partial bosonization and a specific truncation of flow equations, applicable to complex many-body problems.

Findings

Derived the frequency dependence of the X-ray response function.

Calculated the particle-particle susceptibility in the X-ray problem.

Demonstrated the method's general applicability to systems with strong fluctuations.

Abstract

We present a simple method for summing so-called parquet diagrams of fermionic many-body systems with competing instabilities using the functional renormalization group. Our method is based on partial bosonization of the interaction utilizing multi-channel Hubbard-Stratonovich transformations. A simple truncation of the resulting flow equations, retaining only the frequency-independent parts of the two-point and three-point vertices amounts to solving coupled Bethe-Salpeter equations for the effective interaction to leading logarithmic order. We apply our method by revisiting the X-ray problem and deriving the singular frequency dependence of the X-ray response function and the particle-particle susceptibility. Our method is quite general and should be useful in many-body problems involving strong fluctuations in several scattering channels.