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Improving matrix-product state techniques based on the purification of thedensity matrix, we are able to accurately calculate the finite-temperaturedynamic response of the infinite spin-1 XXZ chain with single-ion anisotropy inthe Haldane, large-$D$ and antiferromagnetic phases. Distinct thermallyactivated scattering processes make a significant contribution to the spectralweight in all cases. In the Haldane phase intraband magnon scattering isprominent, and the onsite anisotropy causes the magnon to split into singletand doublet branches. In the large-$D$ phase response, the intraband signal isseparated from an exciton-antiexciton continuum. In the antiferromagneticphase, holons are the lowest-lying excitations, with a gap that closes at thetransition to the Haldane state. At finite temperatures, scattering betweendomain-wall excitations becomes especially important and strongly enhances thespectral weight for momentum transfer $\pi$.

Finite-temperature dynamic structure factor of the spin-1 XXZ chain with single-ion anisotropy