ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory

TL;DR

The paper introduces ADG, a Python code that automates the generation and evaluation of many-body diagrams in Bogoliubov perturbation theory, streamlining complex calculations for Hamiltonians with two- and three-body interactions.

Contribution

It presents a novel diagrammatic rule that simplifies the summation of time-ordered diagrams, enhancing the automation and efficiency of many-body perturbation theory calculations.

Findings

Automated diagram generation using graph theory and adjacency matrices.

Introduction of a new diagrammatic rule for summing time-ordered diagrams.

Code capable of handling Hamiltonians with two- and three-body interactions.

Abstract

We describe the first version (v1.0.0) of the code ADG that automatically (1) generates all valid Bogoliubov many-body perturbation theory (BMBPT) diagrams 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). The automated generation of BMBPT diagrams of order $p$ relies on elements of graph theory, i.e., it is achieved by producing all oriented adjacency matrices of size $(p+1) \times (p+1)$ satisfying topological Feynman's rules. The automated evaluation of BMBPT diagrams of order $p$ 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 diagrammatic rule in question constitutes a novel finding allowing for the straight summation of large classes of time-ordered diagrams at play in the time-independent formulation of BMBPT. Correspondingly, the traditional resolvent rule employed to compute time-ordered diagrams happens to be a particular case of the general rule presently identified. The code ADG is written in Python2.7 and uses the graph manipulation package NetworkX. The code is also able to generate and evaluate Hartree-Fock-MBPT (HF-MBPT) diagrams and is made flexible enough to be expanded throughout the years to tackle the diagrammatics at play in various many-body formalisms that already exist or are yet to be formulated.