Neutron scattering sum rules, symmetric exchanges, and helicoidal magnetism in MnSb₂O₆

MnSb2O6 is based on the noncentrosymmetric P321 space group with magnetic Mn2+ (S = 5/2, L approximate to 0) spins ordering below TN = 12 K in a cycloidal structure. The spin rotation plane was found to be tilted away from the c axis [Kinoshita et al., Phys. Rev. Lett. 117, 047201 (2016)] resulting as a helicoidal ground state, which we refer as the tilted structure. In our previous diffraction study [Chan et al., Phys. Rev. B 106, 064403 (2022)] we found no evidence that this tilted structure is favored over the pure cycloidal order (referred as the untilted structure). The ground-state magnetic structure, expected to be built and originate from seven nearest-neighbor Heisenberg exchange constants, has been shown to be coupled to the underlying crystallographic chirality with polar domain switching being reported. We apply neutron spectroscopy to extract these symmetric exchange constants. Given the high complexity of the magnetic exchange network, crystallographic structure and complications fitting many parameter linear spin-wave models, we take advantage of multiplexed neutron instrumentation to use the first moment sum rule of neutron scattering to estimate these symmetric exchange constants. The first moment of neutron scattering provides a way of deriving the Heisenberg exchange constant between two neighboring spins if the relative angle and distance of the two ordered spins is known. We show that the first moment sum rule combined with the known magnetic ordering wavevector fixes six of the seven exchange constants. The remaining exchange constant is not determined by this analysis because of the equal spatial bond distances present for different chiral exchange interactions. However, we find this parameter is fixed by the magnon dispersion near the magnetic zone boundary, which is not sensitive to the tilting of the global magnetic structure. We then use these parameters to calculate the low-energy spin-waves in the Neel state to reproduce the neutron response without strong antisymmetric coupling. Using Green's response functions, the stability of long-wavelength excitations in the context of our proposed untilted magnetic structures is then discussed. The results show the presence of strong symmetric exchange constants for the chiral exchange pathways and illustrate an underlying coupling between crystallographic and magnetic "chirality" through predominantly symmetric exchange. We further argue that the excitations can be consistently modelled in terms of an untilted magnetic structure in the presence of symmetric-only exchange constants.