  The antiferromagnetic spin-one chain is considerably one of the most
fundamental quantum many-body systems, with symmetry protected topological
order in the ground state. Here, we present results for its dynamical spin
structure factor at finite temperatures, based on a combination of exact
numerical diagonalization, matrix-product-state calculations and quantum Monte
Carlo simulations. Open finite chains exhibit a sub-gap band in the thermal
spectral functions, indicative of localized edge-states. Moreover, we observe
the thermal activation of a distinct low-energy continuum contribution to the
spin spectral function with an enhanced spectral weight at low momenta and its
upper threshold. This emerging thermal spectral feature of the Haldane spin-one
chain is shown to result from intra-band magnon scattering due to the thermal
population of the single-magnon branch, which features a large bandwidth-to-gap
ratio. These findings are discussed with respect to possible future studies on
spin-one chain compounds based on inelastic neutron scattering.
