Discrete Lehmann representation of three-point functions

TL;DR

This paper extends the discrete Lehmann representation to three-point functions in imaginary time and Matsubara frequency, enabling efficient and accurate evaluation of complex diagrammatic expressions with reduced computational resources.

Contribution

The authors introduce a systematic algorithm for representing three-point functions using a universal basis, improving computational efficiency and accuracy in many-body physics calculations.

Findings

Efficient representation of three-point functions using a universal basis.

Ability to evaluate infinite Matsubara sums with high accuracy.

Reduced computational cost and memory footprint for diagrammatic calculations.

Abstract

We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.