Stabilizing the parquet problem
Abstract
We systematically analyze the stability of the iterative solution of the parquet equations by studying the spectrum of the Jacobian associated with the commonly used damped fixed-point iteration procedure. In this context, we provide an explicit criterion that determines when the physical fixed point of the parquet iteration becomes unstable. Importantly, we demonstrate that misleading convergence issues, observed in parquet calculations at intermediate-to-high interaction values, are not restricted to parameter regions where the two-particle irreducible vertex diverges, but can also arise in absence of vertex divergences. Hence, the misleading convergence issues of parquet-based algorithms are not directly caused by the crossings of two solutions of the (multivalued) Luttinger-Ward functional, that are associated with vertex divergences. Building on these insights, we apply the controlled stabilization strategy of Phys. Rev. Lett. 137, 016502 (2026) to the challenging case of the parquet iteration. This strategy allows the convergence to the physical solution in the instability regimes. We apply this procedure to the zero-point model and the Hubbard model in the atomic limit, where we successfully stabilize the physical solution deep in the non-perturbative regime, even across multiple divergence lines. Further, we compare our stabilization strategy to alternative fixed-point solvers that can be used to solve the parquet problem.
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BibTeX
@misc{eßl2026stabilizing,
title = {{Stabilizing the parquet problem}},
author = {Herbert Eßl and Stefan Rohshap and Marcel Gievers and Markus Wallerberger and Alessandro Toschi and Anna Kauch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.04936},
eprint = {2606.04936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04936},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stabilizing the parquet problem
AU - Herbert Eßl
AU - Stefan Rohshap
AU - Marcel Gievers
AU - Markus Wallerberger
AU - Alessandro Toschi
AU - Anna Kauch
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2606.04936
UR - https://arxiv.org/abs/2606.04936
AB - We systematically analyze the stability of the iterative solution of the parquet equations by studying the spectrum of the Jacobian associated with the commonly used damped fixed-point iteration procedure. In this context, we provide an explicit criterion that determines when the physical fixed point of the parquet iteration becomes unstable. Importantly, we demonstrate that misleading convergence issues, observed in parquet calculations at intermediate-to-high interaction values, are not restricted to parameter regions where the two-particle irreducible vertex diverges, but can also arise in absence of vertex divergences. Hence, the misleading convergence issues of parquet-based algorithms are not directly caused by the crossings of two solutions of the (multivalued) Luttinger-Ward functional, that are associated with vertex divergences. Building on these insights, we apply the controlled stabilization strategy of Phys. Rev. Lett. 137, 016502 (2026) to the challenging case of the parquet iteration. This strategy allows the convergence to the physical solution in the instability regimes. We apply this procedure to the zero-point model and the Hubbard model in the atomic limit, where we successfully stabilize the physical solution deep in the non-perturbative regime, even across multiple divergence lines. Further, we compare our stabilization strategy to alternative fixed-point solvers that can be used to solve the parquet problem.
ER -