# The Fierz convergence criterion: a controlled approach to strongly-interacting systems with small embedded clusters

## TL;DR

This paper introduces a new embedded-cluster method based on the TRILEX formalism that uses the Fierz ambiguity as a convergence criterion, achieving near-exact results for the 2D Hubbard model with small clusters.

## Contribution

It develops a controlled approach leveraging the Fierz ambiguity as a convergence criterion within the TRILEX formalism for strongly-interacting systems.

## Key findings

- Small clusters (size four) yield results close to benchmark data.
- The Fierz parameter serves as a quality control measure.
- The method outperforms GW+EDMFT in certain regimes.

## Abstract

We present an embedded-cluster method, based on the TRILEX formalism. It turns the Fierz ambiguity, inherent to approaches based on a bosonic decoupling of local fermionic interactions, into a convergence criterion. It is based on the approximation of the three-leg vertex by a coarse-grained vertex computed by solving a self-consistently determined multi-site effective impurity model. The computed self-energies are, by construction, continuous functions of momentum. We show that, in three interaction and doping regimes of parameters of the two-dimensional Hubbard model, self-energies obtained with clusters of size four only are very close to numerically exact benchmark results. We show that the Fierz parameter, which parametrizes the freedom in the Hubbard-Stratonovich decoupling, can be used as a quality control parameter. By contrast, the $GW$+extended dynamical mean field theory approximation with four cluster sites is shown to yield good results only in the weak-coupling regime and for a particular decoupling. Finally, we show that the vertex has spatially nonlocal components only at low Matsubara frequencies.