Origin of misleading convergence in self-consistent many-electron theories: Fundamental aspects and practical implications

Alessandro Toschi; Evgeny Kozik; Herbert E{\ss}l; Matthias Reitner

arxiv: 2502.01420 · v2 · submitted 2025-02-03 · ❄️ cond-mat.str-el

Origin of misleading convergence in self-consistent many-electron theories: Fundamental aspects and practical implications

Herbert E{\ss}l , Matthias Reitner , Evgeny Kozik , Alessandro Toschi This is my paper

Pith reviewed 2026-05-23 04:25 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords self-consistent methodsmany-electron theoriesLuttinger-Ward functionalirreducible vertex functionmisleading convergencemultivaluednessstrongly correlated regimesstabilization procedure

0 comments

The pith

A stability condition derived from the equations shows misleading convergence in self-consistent many-electron theories can occur without irreducible vertex divergences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Self-consistent schemes for many-electron problems commonly iterate to unphysical solutions in strongly correlated regimes. The paper derives a mathematical condition that identifies when the physical solution is stable against small perturbations in the iteration. This condition clarifies the link between misleading convergence and multivaluedness of the Luttinger-Ward functional, both tied to divergences of the irreducible vertex function, yet demonstrates that misleading convergence remains possible even when those divergences are absent. The authors also outline a systematic procedure to enforce convergence to the physical solution.

Core claim

The mathematical condition for the stability of the physical solution establishes that misleading convergence and the multivaluedness of the Luttinger-Ward functional, while fundamentally linked through divergences of the irreducible vertex function, are distinct phenomena, with misleading convergence able to arise even in the absence of such divergences. A systematic procedure for stabilizing the physical solution is proposed.

What carries the argument

The stability condition for the physical solution, extracted from the underlying equations of the many-electron theory to separate convergence behavior from functional multivaluedness.

If this is right

Misleading convergence and Luttinger-Ward multivaluedness remain distinct even though both connect to vertex divergences.
Misleading convergence can be diagnosed and avoided in regimes without vertex divergences.
The proposed stabilization procedure provides a general route to enforce the physical solution in self-consistent schemes.
The distinction allows convergence issues to be addressed without resolving the full multivaluedness problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The stability condition might be used to monitor and correct iterations in existing numerical codes without requiring full vertex information.
Similar separation of convergence and multivaluedness questions could be examined in other self-consistent frameworks that rely on Luttinger-Ward-like functionals.

Load-bearing premise

The derivation assumes that a mathematical condition for the stability of the physical solution exists and can be extracted from the equations to separate convergence behavior from functional multivaluedness.

What would settle it

A concrete self-consistent calculation in a model where the irreducible vertex function remains finite throughout the relevant parameter range yet the iteration still converges to an unphysical fixed point would test whether misleading convergence occurs independently of vertex divergences.

Figures

Figures reproduced from arXiv: 2502.01420 by Alessandro Toschi, Evgeny Kozik, Herbert E{\ss}l, Matthias Reitner.

**Figure 2.** Figure 2: FIG. 2. Misleading convergence for the scheme of Eq. (1) in the ZP model (upper panels) and HA (lower panels). Left panels: [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Iteration in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

read the original abstract

Self-consistent approaches in many-electron problems typically converge to an unphysical solution in strongly correlated regimes. By deriving the mathematical condition for the stability of the physical solution, we unveil the precise relation between two distinct issues previously considered equivalent: the misleading convergence in self-consistent schemes and the multivaluedness of the Luttinger-Ward functional. Although these problems are fundamentally linked through the divergences of the irreducible vertex function, we show that misleading convergence can occur even in the absence of such divergences. Eventually, a systematic procedure for stabilizing the physical solution is proposed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper separates misleading convergence from vertex divergences and Luttinger-Ward multivaluedness via a derived stability condition.

read the letter

The key point is that misleading convergence in self-consistent many-electron theories is not the same as Luttinger-Ward functional multivaluedness and can occur without irreducible vertex divergences. The authors derive a mathematical condition for the stability of the physical solution and use it to make this separation. They also outline a systematic stabilization procedure for the physical solution in strongly correlated regimes. This refines earlier assumptions that treated the two problems as equivalent. The work does well by linking the issues through vertex divergences while showing they are not identical, which addresses a recurring practical failure mode in methods like DMFT. The proposed procedure targets exactly the kind of convergence trouble people encounter in numerical calculations. Soft spots are limited. The abstract is high-level, so the full paper must demonstrate that the stability condition is extracted cleanly without circularity or model-specific restrictions. If the derivation relies on unstated assumptions about the underlying equations, that would need checking, but nothing in the description suggests a load-bearing flaw. The practical implications would be stronger with explicit tests on simple models. This paper is for researchers running self-consistent diagrammatic calculations on correlated electrons. Readers who have hit unphysical fixed points in their own work would get direct value from the stabilization steps. It deserves a serious referee because the distinction is specific, the problem is real, and the claims are checkable against existing literature. I would send it to peer review.

Referee Report

0 major / 0 minor

Summary. The paper derives the mathematical condition for the stability of the physical solution in self-consistent many-electron theories. It distinguishes misleading convergence from the multivaluedness of the Luttinger-Ward functional, showing they are linked via divergences of the irreducible vertex function but that misleading convergence can occur even without such divergences. A systematic procedure to stabilize the physical solution is proposed.

Significance. If the derivation holds, the work clarifies a fundamental distinction in many-body theory with direct implications for the reliability of self-consistent methods (e.g., DMFT) in strongly correlated regimes. The explicit separation of convergence behavior from functional multivaluedness, together with the proposed stabilization procedure, would be a useful advance if the supporting equations and proofs are rigorous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for acknowledging the potential significance of distinguishing misleading convergence from Luttinger-Ward functional multivaluedness, along with the proposed stabilization procedure. We appreciate the conditional positive assessment and address the overall report below. No specific major comments were enumerated in the report, so we provide a concise response to the referee's summary and recommendation. We maintain that the derivations are rigorous and the claims are supported by the explicit mathematical conditions derived in the paper.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and reader's summary describe a derivation of a stability condition that separates misleading convergence from Luttinger-Ward multivaluedness and vertex divergences, without any supplied equations, self-citations, or fitted inputs that reduce the claimed result to its own premises by construction. No load-bearing steps are visible in the provided text that match the enumerated circularity patterns. The derivation is presented as extracted from the underlying many-electron equations, and the paper is treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, specific axioms, or invented entities are identifiable. The work builds on standard concepts such as the Luttinger-Ward functional and vertex functions from many-body theory.

pith-pipeline@v0.9.0 · 5627 in / 1173 out tokens · 65248 ms · 2026-05-23T04:25:25.010872+00:00 · methodology

discussion (0)

Reference graph

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