Many-Body Vacuum Diagrams and Their Recursive Graphical Construction

TL;DR

This paper introduces a recursive graphical method to generate all connected vacuum diagrams in many-body physics, providing a systematic approach to compute the grand-canonical potential as a functional of correlation functions.

Contribution

It develops a recursive graphical construction for vacuum diagrams, enabling systematic generation and weighting of diagrams in many-body perturbation theory.

Findings

Derived a nonlinear functional differential equation for the grand-canonical potential.

Implemented a recursive graphical solution to generate all connected vacuum diagrams.

Applied the method to produce Hugenholtz diagrams for a weakly interacting Bose gas.

Abstract

The grand-canonical potential of many-body physics can be considered as a functional of the interaction-free correlation function. As such it obeys a nonlinear functional differential equation which can be turned into a recursion relation. This is solved graphically order by order in the two-particle interaction to find all connected vacuum diagrams with their proper weights. As a special case, the procedure is applied to generate the Hugenholtz diagrams for a weakly interacting Bose gas.