  We derive a Lieb-Robinson bound for the propagation of spin correlations in a
model of spins interacting through a bosonic lattice field, which satisfies
itself a Lieb-Robinson bound in the absence of spin-boson couplings. We apply
these bounds to a system of trapped ions, and find that the propagation of spin
correlations, as mediated by the phonons of the ion crystal, can be faster than
the regimes currently explored in experiments. We propose a scheme to test the
bounds by measuring retarded correlation functions via the crystal
fluorescence.
