Variational structure of Luttinger-Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory

TL;DR

This paper rigorously establishes the well-defined nature of the Luttinger-Ward functional in Euclidean lattice field theory and derives the bold diagrammatic expansion without formal assumptions.

Contribution

It provides a rigorous proof of the Luttinger-Ward functional's existence and a solid derivation of the bold diagrammatic expansion in Euclidean lattice field theory.

Findings

Luttinger-Ward functional is well-defined for all physical Green's functions.

The free energy can be variationally minimized using the Luttinger-Ward functional.

The bold diagrammatic expansion is derived rigorously without formal assumptions.

Abstract

The Luttinger-Ward functional was proposed more than five decades ago to provide a link between static and dynamic quantities in a quantum many-body system. Despite its widespread usage, the derivation of the Luttinger-Ward functional remains valid only in the formal sense, and even the very existence of this functional has been challenged by recent numerical evidence. In a simpler and yet highly relevant regime, namely the Euclidean lattice field theory, we rigorously prove that the Luttinger-Ward functional is a well-defined universal functional over all physical Green's functions. Using the Luttinger-Ward functional, the free energy can be variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.