High-frequency asymptotics of the vertex function: diagrammatic parametrization and algorithmic implementation

TL;DR

This paper develops a diagrammatic parametrization scheme for the high-frequency behavior of vertex functions in many-body physics, enabling more efficient and accurate numerical calculations in quantum many-body methods.

Contribution

It introduces a novel parametrization capturing high-frequency vertex behavior for arbitrary interactions and densities, with algorithmic implementation details for many-body solvers.

Findings

Validated against exact diagonalization for a single impurity Anderson model.

Enhances the efficiency of vertex-based diagrammatic methods.

Applicable to local and non-local interactions in many-body calculations.

Abstract

Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet-equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and non-local static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.