Analytical solution for time-integrals in diagrammatic expansions: application to real-frequency diagrammatic Monte Carlo

TL;DR

This paper introduces an analytical solution for time-integrals in Feynman diagrams, enhancing diagrammatic Monte Carlo methods by improving convergence and broadening applicability to various correlation functions.

Contribution

It provides a closed-form analytical solution for imaginary-time integrals in Feynman diagrams, enabling more efficient and general Monte Carlo algorithms for interacting fermions.

Findings

Improved convergence of self-energy calculations with fewer perturbation orders.

Enhanced Monte Carlo algorithm versatility for different correlation functions.

Broader applicability of the analytical solution to various Monte Carlo methods.

Abstract

The past years have seen a revived interest in the diagrammatic Monte Carlo (DiagMC) methods for interacting fermions on a lattice. A promising recent development allows one to now circumvent the analytical continuation of dynamic observables in DiagMC calculations within the Matsubara formalism. This is made possible by symbolic algebra algorithms, which can be used to analytically solve the internal Matsubara frequency summations of Feynman diagrams. In this paper, we take a different approach and show that it yields improved results. We present a closed-form analytical solution of imaginary-time integrals that appear in the time-domain formulation of Feynman diagrams. We implement and test a DiagMC algorithm based on this analytical solution and show that it has numerous significant advantages. Most importantly, the algorithm is general enough for any kind of single-time correlation function series, involving any single-particle vertex insertions. Therefore, it readily allows for the use of action-shifted schemes, aimed at improving the convergence properties of the series. By performing a frequency-resolved action-shift tuning, we are able to further improve the method and converge the self-energy in a non-trivial regime, with only 3-4 perturbation orders. Finally, we identify time integrals of the same general form in many commonly used Monte Carlo algorithms and therefore expect a broader usage of our analytical solution.