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 signal amplification by reversible exchange (SABRE) 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 the 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, allowing the treatment of arbitrary spin ensembles, including those containing 15N, 13C, 1H, and other nuclei. 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.