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

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) × (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. PROGRAM SUMMARYProgram Title: ADG Licensing provisions: GPLv3 Programming language: Python2.7 Nature of problem: As formal and numerical developments in many-bodyperturbation-theory-based ab initio methods make higher orders reachable, producing and evaluating all the diagrams become rapidly undoable on a handmade basis as both their number and complexity grows quickly, making it prone to mistakes and oversights. Solution method: BMBPT diagrams are encoded as square matrices known as oriented adjacency matrices in graph theory, and then turned into graph objects using the NetworkX package. Checks on the diagrams and evaluation of their time-integrated expression is then done on a purely diagrammatic basis. HF-MBPT diagrams are produced and evaluated as well using the same principle.

A Guided Tour of ab initio Nuclear Many-Body Theory

“…Efforts in that direction have been in progress for some time, and while some methods are at a more advanced stage than others, the improved truncations should be available for regular use within the next couple of years [10,12,86,100,116,244,265]. In part, this is owing to the development of computer tools that automate tasks like diagrammatic evaluation or angular momentum coupling [266,267]. The computational scaling of these approaches will be of order O(N 8 ) or O(N 9 ), which makes applications a task for leadership-class computing resources for the foreseeable future.…”

Many-Body Perturbation Theories for Finite Nuclei

“…This translates into the difficulty to both generate all allowed diagrams at a given order without missing any and to evaluate their expression in a quick and errorsafe way. Consequently, the last tool introduced to tackle this difficulty consists of an automatized generation and evaluation of diagrams [88][89][90][91][92][93][94][95]. All these technical, yet crucial, aspects of MBPT are not addressed in the present article and the interested reader is referred to the references.…”

Normal-ordered k-body approximation in particle-number-breaking theories

“…(46a). 5 In energy density functional (EDF) calculations, particlenumber projection typically lowers the energy, which is not the case with the presently used chiral 2N interaction. Because PHFB calculations with realistic nuclear interactions are rather novel, this feature is briefly analyzed in App.…”