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

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.

References in corpus (4)

• Reduced Density Matrix Functional for Many-Electron Systems
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• Breakdown of traditional many-body theories for correlated electrons
• The self-consistent Dyson equation and self-energy functionals: failure or new opportunities?