ADG: Automated generation and evaluation of many-body diagrams II. Particle-number projected Bogoliubov many-body perturbation theory

TL;DR

The paper introduces ADG v2.0.0, a Python code that automates the generation and evaluation of many-body diagrams in particle-number projected Bogoliubov perturbation theory, enabling efficient high-order calculations.

Contribution

It presents an improved version of ADG that systematically generates and evaluates off-diagonal BMBPT diagrams at any order, including complex three-body interactions, with a new diagrammatic rule for algebraic evaluation.

Findings

Automated generation of all valid off-diagonal BMBPT diagrams.

Implementation of a new diagrammatic rule for algebraic evaluation.

Flexible code structure for future many-body formalism extensions.

Abstract

We describe the second version (v2.0.0) of the code ADG that automatically (1) generates all valid off-diagonal Bogoliubov many-body perturbation theory diagrams at play in particle-number projected Bogoliubov many-body perturbation theory (PNP-BMBPT) and (2) evaluates their algebraic expression to be implemented for numerical applications. This is achieved at any perturbative order $p$ for a Hamiltonian containing both two-body (four-legs) and three-body (six-legs) interactions (vertices). All valid off-diagonal BMBPT diagrams of order $p$ are systematically generated from the set of diagonal, i.e., unprojected, BMBPT diagrams. The production of the latter were described at length in https://doi.org/10.1016/j.cpc.2018.11.023 dealing with the first version of ADG. The automated evaluation of off-diagonal BMBPT diagrams relies both on the application of algebraic Feynman's rules and on the identification of a powerful diagrammatic rule providing the result of the remaining $p$-tuple time integral. The new diagrammatic rule generalizes the one already identified in https://doi.org/10.1016/j.cpc.2018.11.023 to evaluate diagonal BMBPT diagrams independently of their perturbative order and topology. The code ADG is written in Python3 and uses the graph manipulation package NetworkX. The code is kept flexible enough to be further expanded throughout the years to tackle the diagrammatics at play in various many-body formalisms that already exist or are yet to be formulated.