# Fermi-edge singularity and the functional renormalization group

## TL;DR

This paper investigates the Fermi-edge singularity using the functional renormalization group, comparing various fRG methods to parquet equations, and analyzing the limitations of leading log approximations in this context.

## Contribution

It demonstrates how fRG can reproduce the leading log formula for the X-ray-edge singularity and highlights the limitations of this approach for more general cases.

## Key findings

- fRG reproduces the leading log formula for X-ray-edge singularity
- Leading log derivation relies on special cancellations not generalizable
- Comparison of fRG implementations with parquet equations

## Abstract

We study the Fermi-edge singularity, describing the response of a degenerate electron system to optical excitation, in the framework of the functional renormalization group (fRG). Results for the (interband) particle-hole susceptibility from various implementations of fRG (one- and two- particle-irreducible, multi-channel Hubbard-Stratonovich, flowing susceptibility) are compared to the summation of all leading logarithmic (log) diagrams, achieved by a (first-order) solution of the parquet equations. For the (zero-dimensional) special case of the X-ray-edge singularity, we show that the leading log formula can be analytically reproduced in a consistent way from a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic structure, we show that this derivation relies on fortuitous partial cancellations special to the form of and accuracy applied to the X-ray-edge singularity and does not generalize.