Skeleton series and multivaluedness of the self-energy functional in zero space-time dimensions

TL;DR

This paper explores the multivalued nature of the self-energy functional in a simplified zero-dimensional model, revealing fundamental algebraic phenomena related to diagrammatic series convergence and multiple solution branches.

Contribution

It demonstrates that the multivaluedness and convergence issues observed numerically in higher dimensions also occur in elementary algebraic models, providing fundamental insights.

Findings

Self-energy functional is multivalued in zero-dimensional models.

Skeleton series converges to incorrect branches above critical interaction.

Similar phenomena occur with pair propagators in the formalism.

Abstract

Recently, Kozik, Ferrero and Georges have discovered numerically that for a family of fundamental models of interacting fermions, the self-energy $\Sigma[G]$ is a multi-valued functional of the fully dressed single-particle propagator G, and that the skeleton diagrammatic series $\Sigma_{\rm bold}[G]$ converges to the wrong branch above a critical interaction strength. We consider the zero space-time dimensional case, where the same mathematical phenomena appear from elementary algebra. We also find a similar phenomenology for the fully bold formalism built on fully dressed single-particle propagator and pair propagator.