Multiloop functional renormalization group for the two-dimensional Hubbard model: Loop convergence of the response functions

TL;DR

This paper advances the functional renormalization group method for the 2D Hubbard model by including full momentum and frequency dependence, self-energy feedback, and summing all parquet diagrams through a multiloop extension, leading to more reliable response function calculations.

Contribution

It introduces a multiloop fRG algorithm that sums all parquet diagrams with exact weights, improving the accuracy and convergence of response function computations in the 2D Hubbard model.

Findings

Loop convergence is achieved with affordable computational effort.

Results are independent of cutoff scheme and susceptibility computation method.

Refined fRG scheme enables more accurate analysis of correlated electron systems.

Abstract

We present a functional renormalization group (fRG) study of the two dimensional Hubbard model, performed with an algorithmic implementation which lifts some of the common approximations made in fRG calculations. In particular, in our fRG flow; (i) we take explicitly into account the momentum and the frequency dependence of the vertex functions; (ii) we include the feedback effect of the self-energy; (iii) we implement the recently introduced multiloop extension which allows us to sum up {\emph{all}} the diagrams of the parquet approximation with their exact weight. Due to its iterative structure based on successive one-loop computations, the loop convergence of the fRG results can be obtained with an affordable numerical effort. In particular, focusing on the analysis of the physical response functions, we show that the results become {\emph{independent}} from the chosen cutoff scheme and from the way the fRG susceptibilities are computed, i.e., either through flowing couplings to external fields, or through a "post-processing" contraction of the interaction vertex at the end of the flow. The presented substantial refinement of fRG-based computation schemes paves a promising route towards future quantitative fRG analyses of more challenging systems and/or parameter regimes.