the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Feb 2026
Scalable Modeling of Multi-spin Ensembles in SABRE Hyperpolarization: a Symmetry-based Framework for Zero and Ultralow Fields
Abstract. This work presents a theoretical framework for quantitative, scalable modeling of SABRE (Signal Amplification by Reversible Exchange) experiments under zero- and ultralow-field (ZULF) conditions. SABRE exploits the singlet spin order of parahydrogen to hyperpolarize nuclear spins of substrates without chemical modification, enhancing NMR signals. In ZULF SABRE method polarization transfer occurs in ultralow magnetic fields where Zeeman interactions are comparable to or weaker than scalar couplings, enabling coherent mixing of spin states and revealing interactions often suppressed at high fields. Our approach captures the full quantum dynamics of SABRE, including coherent evolution, chemical exchange, and relaxation, within a Liouville-space formalism. We demonstrate that the Hamiltonian, relaxation, and exchange superoperators possess symmetry with respect to the total spin, allowing the dynamics to be rigorously restricted to the zero-quantum coherence subspace. This symmetry-based reduction yields a scalable framework for efficient simulation of multi-spin SABRE systems. The approach is validated against full Liouville-space calculations for small systems and is further applied to a 14-spin SABRE complex, demonstrating its ability to treat spin systems of a complexity well beyond the reach of conventional full Liouville-space simulations. The framework thus provides a predictive tool for optimal polarization fields, ZULF NMR spectra and the design of novel hyperpolarization experiments.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Magnetic Resonance.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
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RC1: 'Comment on mr-2026-4', Anonymous Referee #1, 21 Mar 2026
- AC1: 'Reply on RC1', Danil Markelov, 27 Mar 2026
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RC2: 'Comment on mr-2026-4', Anonymous Referee #2, 23 Mar 2026
The manuscript, Scalable Modeling (2 l...) of Multi-Spin Ensembles in SABRE Hyperpolarization: A Symmetry-Based Framework for Zero and Ultralow Fields, presents a technically sound and well-structured approach to reducing the dimensionality of Liouville space for the simulation of reversible exchange processes in SABRE under ultralow-field conditions. The symmetry-based formalism is clearly articulated and provides a rigorous mathematical framework to identify invariant subspaces of the spin dynamics, enabling a substantial reduction of computational complexity while preserving the essential features of the system.
The main strength of the work lies in the formalisation of symmetry constraints acting on the Hamiltonian, relaxation, and exchange superoperators, which leads to a compact and computationally efficient description of the dynamics. This represents a valuable contribution, particularly in the context of ZULF and SABRE simulations, where the dimensionality of the Liouville space rapidly becomes prohibitive.
At the same time, the presentation would benefit from improved accessibility. While the mathematical structure is solid, the physical interpretation of the reduced space is not always immediately transparent, particularly to non-expert readers. In particular, additional schematic figures illustrating the structure of the basis states, the relevant coherence pathways, and how the dynamics evolve within the reduced Liouville space would significantly broaden the readability of the manuscript. Figure 2 goes in this direction, but in its current form it remains too minimal to effectively guide the reader through the underlying physics.
Another aspect, admittedly not straightforward to quantify in general, that would strengthen the manuscript is a more explicit discussion of computational scaling. Providing, where possible, estimates of computational cost as a function of the number of spins and of the relaxation model would enhance the practical relevance of the work and allow a clearer assessment of the advantages over full Liouville-space simulations.
Finally, it would be important to better position the present work within the existing literature. A closely related numerical framework has recently been presented at Euromar 2025 in Oulu (FI) and published (DOI: 10.1039/D5CP01773D), where Liouville-space reduction is achieved not only through symmetry considerations of the Hamiltonian and superoperators, but also by explicitly exploiting molecular symmetry and incorporating a dipolar (non–random field) relaxation model. In that case, the reduction emerges naturally from both the spin system topology and the underlying physical relaxation mechanisms, rather than from symmetry arguments alone.
Overall, this is a solid and timely contribution that provides a clear mathematical framework for symmetry-based reduction of Liouville space in SABRE simulations. With minor improvements in terms of visualization and contextualization with respect to existing approaches, the manuscript would be well suited for publication.
Citation: https://doi.org/10.5194/mr-2026-4-RC2 - AC2: 'Reply on RC2', Danil Markelov, 27 Mar 2026
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The authors of “Scalable Modelling of Multi-Spin Ensembles in SABRE Hyperpolarization: A Symmetry-Based Framework for Zero and Ultralow Field” present an informative guide for highly efficient numerical simulations of spin dynamics in SABRE, employing the pseudo-spin approach for equivalent spins.
The manuscript is carefully prepared and has no obvious errors. However, in its current detailed form, it is not always clear under which conditions the reduction of the Liouville space to a subspace is valid, and more importantly under which circumstances this approximation may break down. Clarifying these limits of applicability would strengthen the work.
Furthermore, the manuscript appears to focus primarily on ZULF NMR, although the authors suggest that the proposed methodology could be applied more broadly. It would therefore be beneficial to elaborate on how this approach could be extended to high-field NMR experiments, where, for example, the equivalence of methyl and methylene groups might also be exploited.
In this context, the authors should also reference prior efforts aimed at simplifying spin dynamics calculations through permutation symmetry. Relevant examples include, but are not limited to, the work of I. Kuprov (https://doi.org/10.1016/j.jmr.2007.09.014) and S. I. Doronin (https://doi.org/10.1134/S1063776111130036).
Finally, a minor point - line 410 appears to be missing an important reference and should be corrected.