Instabilities in self-consistent diagrammatic approaches and how to cure them

The first text provides a high-level overview of a novel stabilization technique for self-consistent diagrammatic approaches based on the boson-exchange (BE) formalism, while the second text offers specific numerical observations and insights related to the instabilities encountered in these methods.

Here is a detailed, synthesized summary combining both analyses:

Comprehensive Research Summary: Stabilizing Self-Consistent Diagrammatic Approaches via Jacobian Analysis

This research addresses a critical and persistent issue in applying self-consistent diagrammatic approaches—specifically those derived within the Boson-Exchange (BE) formalism (including TRILEX, Single-Boson Exchange (SBE), and Multi-Boson Exchange (MBE))—to compute physical properties of correlated quantum materials. The core finding is that these methods frequently suffer from misleading convergence, where iterative schemes fail to converge to the physically correct fixed point in regions of parameter space where the most physically interesting physics, such as intermediate-to-strong coupling regimes, is expected to manifest.

The Nature of the Problem: Misleading Convergence

The authors rigorously demonstrate that this misleading convergence is not an intrinsic breakdown of the underlying many-electron functional theory or diagrammatic description itself. Instead, it is systematically traced back exclusively to the stability condition of the iterative algorithms employed in their calculations. In these problematic regions, conventional damped iteration schemes fail because the eigenvalues associated with the fixed point have an absolute value greater than unity, leading to divergence or convergence toward unphysical solutions.

Crucially, this instability is observed independently of whether one starts from a nonperturbative/exact input or a weak-coupling/perturbative one. This robustness across different starting points confirms that the issue lies within the iterative procedure's stability rather than the method's fundamental description. Furthermore, this problem is distinct from intrinsic convergence issues plaguing other related frameworks, such as those derived in the Luttinger–Ward functional (LWF).

The Proposed Remedy: Jacobian-Based Stabilization

The central contribution of this work is the proposal and demonstration of a simple and general remedy: modifying the iterative procedure by inverting the unstable eigendirections of the Jacobian associated with the fixed point. This stabilization matrix, derived from the Jacobian's eigenvectors, successfully transforms conventional damped iterations into a scheme that reliably converges to physical fixed points across all parameter ranges, encompassing both perturbative and nonperturbative/strong-coupling regimes.

The authors confirm this stabilization is effective even when approaching regions where vertex divergences occur. They show that the proposed scheme allows for the stabilization of the physical fixed point even after the crossing of one or more vertex divergences.

Analysis of Physical Manifestations and Vertex Divergences

The investigation delves into two key types of irreducibilities relevant to these schemes: two-particle irreducibility (2PI) and U irreducibility. While 2PI vertices can indeed diverge in nonperturbative regimes, the study establishes a vital distinction: vertex divergences are a sufficient but not necessary condition for fixed-point instability; instabilities can occur independently of vertex divergences.

Further numerical analysis highlights specific behaviors observed near these instabilities:

1. Contradictions with Exact Results: The conventional damped iteration yields converged results for quantities like Re that become finite behind the instability boundaries (red dashed lines), directly contradicting exact solutions (e.g., Eq. B2 at half-filling).

2. Oscillations and Imaginary Parts: The converged solutions for Hedin vertices oscillate around the exact solution but possess a finite imaginary part, contrasting with the exact solution (Eq. B5), which is purely real in this context.

3. Bosonic Propagators: The imaginary part of bosonic propagators (W r) at omega=0 remains finite behind the first instability, whereas it vanishes in the exact result (cf. Eq. B4).

Practical Implications and Significance

The successful implementation of this Jacobian-based stabilization scheme is presented as a crucial missing ingredient for cutting-edge, performant BE-based diagrammatic algorithms, such as diagrammatic extensions of DMFT and cluster-DMFT. This stabilization provides a useful compass that enables these algorithms to:

• Access the hardest regions of the parameter space, where strong electronic correlations drive exotic phenomena.

• Ensure reliable convergence regardless of the input coupling regime (perturbative vs. nonperturbative).

In essence, this work shifts the understanding of misleading convergence from a fundamental flaw in diagrammatic theory to a solvable numerical/algorithmic problem, providing a robust tool to unlock the physically rich, strongly correlated regimes that are currently inaccessible due to computational instability.

1. Improved convergence for self-consistent diagrammatic approaches by implementing a Jacobian-based stabilization procedure, specifically stabilized iteration (Eq. 18). This allows for a simple and general remedy to cure misleading convergence, ensuring that the physical fixed point can be stabilized across the entire parameter range, including nonperturbative/strong-coupling regimes.

2. Enhanced reliability in strong-coupling regimes by demonstrating that the stabilization scheme effectively cures convergence issues even when vertex divergences are present. This is achieved by showing that the physical fixed point of all the BE-diagrammatic schemes considered can be stabilized, de facto, across the entire parameter range, including the most challenging nonperturbative/strong-coupling regimes.

3. Robust handling of numerical bottlenecks through optimized Jacobian inversion and Schur decomposition algorithms for determining the stabilization matrix P. This reduces the condition numbers of U and P compared to direct computation by UDU−1, thereby making the determination of stabilization parameters more robust, especially for large state sizes (up to 240000 points).

4. Distinction between intrinsic theory breakdown and algorithmic instability in convergence. The system can reliably distinguish between misleading convergence arising from the stability condition of iterative algorithms and intrinsic breakdown of their self-consistent diagrammatic description, providing a fundamental insight into the problem's origin.

5. Predictive capability for physical solutions across parameter spaces by using stabilized iteration with exact/nonperturbative inputs. This enables the system to converge to the physical fixed point even when conventional methods fail, which is crucial for applications like diagrammatic extensions of DMFT where the exact input is replaced by a numerically obtained (cluster-)DMFT result.