Resolve multivaluedness-related convergence problems of self-consistent perturbation theory

Resolve the convergence problems directly linked to the multivaluedness of the Luttinger–Ward functional in self-consistent perturbative expansions.

Background

The paper distinguishes algorithmic instabilities in boson-exchange diagrammatic approaches from convergence failures caused by the intrinsic multivaluedness of the Luttinger–Ward functional. Although the proposed Jacobian-based stabilization method addresses the former, the authors explicitly state that the latter remains unresolved. This problem concerns the fundamental breakdown of self-consistent perturbation theory rather than the stability of a particular iteration scheme.

References

In fact, while the convergence issues directly linked to the multivaluedness of the LWF , still represent an unsolved problem, a clear strategy can be exploited to stabilize the fixed points of all the iterative schemes considered.

— Instabilities in self-consistent diagrammatic approaches and how to cure them  (2609.11405 - Gievers et al., 10 Sep 2026) in Section 1, Introduction

It is left for future studies if those are connected to the unstable region in TRILEX that we observe.

— Instabilities in self-consistent diagrammatic approaches and how to cure them  (2609.11405 - Gievers et al., 10 Sep 2026) in Section 3.2, Identification of unstable regions

This raises the question whether also GW+EDMFT , which can be seen as a further approximation of the fermion-boson vertex in TRILEX , might suffer from the same problem in specific parameter regimes.

— Instabilities in self-consistent diagrammatic approaches and how to cure them  (2609.11405 - Gievers et al., 10 Sep 2026) in Section 5.1, Consequences for practical calculations

Second order introduces two separate demands on top of the classification itself: a combinatorial increase in the number of cutting-rule terms to evaluate, and the need for degenerate-state (quasi-degenerate) perturbation theory, which is a standard requirement of second-order many-body treatments independent of this classification. We defer the treatment of those issues to future work.

— A QED Functional Formulation of Relativistic Many-Body Perturbation Theory in Atoms  (2610.22216 - Jonge et al., 15 Oct 2026) in Section ‘Perturbative Treatment of the Second-Order Skeletons’