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

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 Arthuis et al. (2019) 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 Arthuis et al. (2019) 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.

本文介绍了代码ADG的第二版（v2.0.0），该代码可自动实现两项功能：(1) 生成粒子数投影玻戈留波夫多体微扰理论（PNP-BMBPT）中涉及的所有有效非对角玻戈留波夫多体微扰理论费曼图；(2) 推导其代数表达式以适配数值应用的实现。
该代码可针对包含两体（四腿）与三体（六腿）相互作用（顶点）的哈密顿量，在任意微扰阶数p下完成上述任务。所有p阶有效非对角BMBPT费曼图，均可通过对角（即未投影）BMBPT费曼图的集合系统性生成。关于对角BMBPT费曼图的生成方法，已于Arthuis等人（2019）的研究中详细阐述，该工作针对ADG的首个版本展开。
非对角BMBPT费曼图的自动化求值，同时依赖代数费曼规则的应用，以及对一种高效图解规则的识别——该规则可直接给出剩余p重时间积分的结果。这一新的图解规则，将Arthuis等人（2019）中提出的对角BMBPT费曼图求值规则进行了推广，使其可不受微扰阶数与拓扑结构的限制，独立完成对角BMBPT费曼图的求值。
ADG代码采用Python3编写，并使用了图操作工具包NetworkX。该代码预留了充足的灵活性，可在后续开发中持续扩展，以处理各类已有或待提出的多体形式体系中涉及的图解问题。