  We study the linear response to time-dependent probes of a symmetry-protected
topological phase of bosons in one-dimension, the Haldane insulator (HI). This
phase is separated from the ordinary Mott insulator (MI) and density-wave (DW)
phases by continuous transitions, whose field theoretical description is here
reviewed. Using this technique, we compute the absorption spectrum to two types
of periodic perturbations and relate the findings to the nature of the critical
excitations at the transition between the different phases. The HI-MI phase
transition is topological and the critical excitations possess trivial quantum
numbers: they correspond to particles and holes at zero momentum. Our findings
are corroborated by a non-local mean-field approach, which allows us to
directly relate the predicted spectrum to the known microscopic theory.
