  It is known that arrays of trapped ions can be used to efficiently simulate a
variety of many-body quantum systems. Here, we show how it is possible to build
a model representing a spin chain interacting with bosons which is exactly
solvable. The exact spectrum of the model at zero temperature and the ground
state properties are studied. We show that a quantum phase transition occurs
when the coupling between spins and bosons reaches a critical value, which
corresponds to a level crossing in the energy spectrum. Once the critical point
is reached, the number of bosonic excitations in the ground state, which can be
assumed as an order parameter, starts to be different from zero. The population
of the bosonic mode is accompanied by a macroscopic magnetization of the spins.
This double effect could represent an useful resource for the phase transition
detection since a measure on the phonon can give information about the phase of
the spin system. A finite temperature phase diagram is also given in the
adiabatic regime.
