Feynman diagrammatics based on discrete pole representations: A path to renormalized perturbation theories

TL;DR

This paper introduces a novel analytic method combining Matsubara integration and discrete pole representations to accurately evaluate impurity diagrams, simplifying calculations and avoiding sign problems, with applications to real-frequency and self-consistent schemes.

Contribution

The paper presents a new analytic diagrammatic technique using discrete pole representations combined with Matsubara integration, improving precision and computational efficiency in impurity problem evaluations.

Findings

Analytic sums of pole representations yield highly precise results on the Matsubara axis.

The approach avoids the numerical sign problem commonly encountered in diagram evaluations.

Real-frequency evaluations are feasible but require noise mitigation at low temperatures.

Abstract

By merging algorithmic Matsubara integration with discrete pole representations we present a procedure to generate fully analytic closed form results for impurity problems at fixed perturbation order. To demonstrate the utility of this approach we study the Bethe lattice and evaluate the second order self-energy for which reliable benchmarks exist. We show that, when evaluating diagrams on the Matsubara axis, the analytic sums of pole representations are extremely precise. We point out the absence of a numerical sign problem in the evaluation, and explore the application of the same procedure for real-frequency evaluation of diagrams. We find that real-frequency results are subject to noise that is controlled at low temperatures and can be mitigated at additional computational expense. We further demonstrate the utility of this approach by evaluating dynamical mean-field and bold diagrammatic self-consistency schemes at both second and fourth order and compare to benchmarks where available.