Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part I: Perturbative approach

Abstract

Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In Part I of this two-part series, we illustrate how the combinatorial properties of Feynman diagrams in MBPT can be studied in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature) and provide a self-contained explanation of Feynman diagrams in this setting. We prove the combinatorial validity of the bold diagrammatic expansion, with methods generalizable to several types of field theories and interactions. Our treatment simplifies the presentation and numerical study of known techniques in MBPT such as the self-consistent Hartree-Fock approximation (HF), the second-order Green's function approximation (GF2), and the GW approximation. The bold diagrams are closely related to the Luttinger-Ward (LW) formalism, which was proposed in 1960 but whose analytic properties have not been rigorously established. The analytical study of the LW formalism in the setting of Gibbs measures will be the topic of Part II.

References in corpus (9)

• The numerical renormalization group method for quantum impurity systems
• First principles approach to the electronic structure of strongly correlated systems: combining GW and DMFT
• Non-existence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for Hubbard-like models
• The self-consistent Dyson equation and self-energy functionals: failure or new opportunities?
• DCA${}^{+}$: Dynamical Cluster Approximation with continuous lattice self-energy
• The correlation energy functional within the GW-RPA approximation: exact forms, approximate forms and challenges
• Renormalization group flow of the Luttinger-Ward functional: conserving approximations and application to the Anderson impurity model
• Bold Diagrammatic Monte Carlo in the Lens of Stochastic Iterative Methods
• Comment on `Non-existence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for Hubbard-like models'

Cited by in corpus (1)

• Diagrammatic Monte Carlo Method for Impurity Models with General Interactions and Hybridizations