On the Luttinger-Ward functional and the convergence of skeleton diagrammatic series expansion of the self-energy for Hubbard-like models

TL;DR

This paper investigates the convergence properties of the skeleton diagrammatic series for the self-energy in Hubbard-like models, reaffirming convergence for most cases and analyzing discrepancies with previous studies.

Contribution

It establishes the uniform convergence and uniqueness of the skeleton series for Hubbard-like models and explores reasons for conflicting observations in prior research.

Findings

Series converges uniformly for almost all wave vectors and energies.

Contrary to prior beliefs, energy domain correlation functions are unreliable.

Provides a symbolic formalism for diagram equivalence.

Abstract

We consider a number of questions regarding the Luttinger-Ward functional and the many-body perturbation series expansion of the proper self-energy $\Sigma(\mathbf{k};z)$ specific to uniform ground states (ensemble of states) of interacting fermion systems in terms of skeleton self-energy diagrams and the interacting Green function $G(\mathbf{k};z)$. Utilising a link between the latter series expansion and the classical moment problem (of the Hamburger type), along with the associated continued-fraction expansion, we reaffirm our earlier observation (2007) that for lattice models of fermions interacting through short-range two-body potentials (i.e. for Hubbard-like models) this series is uniformly convergent for almost all wave vectors $\mathbf{k}$ and complex energies $z$. The limit of this series is unique. We inquire into the reasons underlying the contrary observation by Kozik et al. (2015) regarding skeleton-diagrammatic perturbation series expansion of $\Sigma$. In doing so, we make a number of observations of general interest. Chief amongst these, we observe that contrary to general belief thermal correlation functions calculated in the energy domain (i.e. over the set of the relevant Matsubara frequencies) are generally unreliable, in contrast to those calculated in the imaginary-time domain. In an appendix we present a symbolic computational formalism and short programs for singling out topologically distinct proper self-energy diagrams that are algebraically equivalent up to determinate multiplicative constants. This complements the work presented in our previous publication (2019). [Abridged abstract]