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

Ion-trap simulation of the quantum phase transition in an exactly solvable model of spins coupled to bosons