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Quantum magnets admit more than one classical limit and $N$-level systemswith strong single-ion anisotropy are expected to be described by a classicalapproximation based on SU($N$) coherent states. Here we test this hypothesis bymodeling finite temperature inelastic neutron scattering (INS) data of theeffective spin-one antiferromagnet \bfso{}. The measured dynamic structurefactor is calculated with a generalized Landau-Lifshitz dynamics for SU($3$)spins. Unlike the traditional classical limit based on SU($2$) coherent states,the results obtained with classical SU($3$) spins are in good agreement withthe measured temperature-dependent spectrum. The SU($3$) approach developedhere provides a general framework to understand the broad class of materialscomprising weakly coupled antiferromagnetic dimers, trimers, or tetramers, andmagnets with strong single-ion anisotropy.

Understanding temperature-dependent SU($3$) spin dynamics in the $S=1$ antiferromagnet Ba$_2$FeSi$_2$O$_7$