  We study a spin-boson chain that exhibits a local Z2 symmetry. We investigate
the quantum phase diagram of the model by means of perturbation theory, mean
field theory and the Density Matrix Renormalization Group method. Our
calculations show the existence of a first order phase transition in the region
where the boson quantum dynamics is slow compared to the spin-spin
interactions. Our model can be implemented with trapped ion quantum simulators,
leading to a realization of minimal models showing local gauge invariance and
first order phase transitions.
