Non-Closing Double-Commutator Flows and the Small-Coupling Limit of Spin-Boson Models
Abstract
Spin-boson models are paradigmatic examples of open quantum systems and serve as the theoretical paradigm for current quantum computers based on single-ion trap technology. Despite their ubiquity, a complete spectral diagonalization of these models remains an open problem, except in highly singular regimes. This paper establishes a rigorous framework for the approximate diagonalization of generalized spin-boson systems. Our approach is inspired by the Brockett-Wegner double-commutator flow, which is a non-linear differential equation governing the evolution of (here unbounded) operators. Unlike relatively recent applications of this flow to quadratic Hamiltonians in quantum field theory, it does not close in the spin-boson context. We overcome this fundamental obstruction by performing a detailed analysis of the resulting non-closed algebraic structure, allowing us to explicitly bound the higher-order error term with respect to the spin-boson coupling strength. Consequently, this work provides the first mathematically rigorous justification for several heuristic diagonalization techniques widely employed in the theoretical physics literature for small-coupling regimes, in the simplest non-trivial cases. More broadly, our framework renders a flow-based algorithm feasible for systematic higher-order diagonalization and self-energy renormalization. This strategy is conceptually akin to multi-scale analysis, or, much more recently, to the iterative, local Lie-Schwinger block-diagonalization method by Fröhlich and Pizzo.
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BibTeX
@misc{bru2026nonclosing,
title = {{Non-Closing Double-Commutator Flows and the Small-Coupling Limit of Spin-Boson Models}},
author = {Jean-Bernard Bru and Nathan Metraud and Walter de Siqueira Pedra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28208},
eprint = {2608.28208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28208},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Non-Closing Double-Commutator Flows and the Small-Coupling Limit of Spin-Boson Models
AU - Jean-Bernard Bru
AU - Nathan Metraud
AU - Walter de Siqueira Pedra
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.28208
UR - https://arxiv.org/abs/2608.28208
AB - Spin-boson models are paradigmatic examples of open quantum systems and serve as the theoretical paradigm for current quantum computers based on single-ion trap technology. Despite their ubiquity, a complete spectral diagonalization of these models remains an open problem, except in highly singular regimes. This paper establishes a rigorous framework for the approximate diagonalization of generalized spin-boson systems. Our approach is inspired by the Brockett-Wegner double-commutator flow, which is a non-linear differential equation governing the evolution of (here unbounded) operators. Unlike relatively recent applications of this flow to quadratic Hamiltonians in quantum field theory, it does not close in the spin-boson context. We overcome this fundamental obstruction by performing a detailed analysis of the resulting non-closed algebraic structure, allowing us to explicitly bound the higher-order error term with respect to the spin-boson coupling strength. Consequently, this work provides the first mathematically rigorous justification for several heuristic diagonalization techniques widely employed in the theoretical physics literature for small-coupling regimes, in the simplest non-trivial cases. More broadly, our framework renders a flow-based algorithm feasible for systematic higher-order diagonalization and self-energy renormalization. This strategy is conceptually akin to multi-scale analysis, or, much more recently, to the iterative, local Lie-Schwinger block-diagonalization method by Fröhlich and Pizzo.
ER -