  The vibrations of a chain of trapped ions can be considered, under suitable
experimental conditions, as an ensemble of interacting phonons, whose quantum
dynamics is governed by a Bose--Hubbard Hamiltonian. In this work we study the
quantum phases which appear in this system, and show that thermodynamical
properties, such as critical parameters and critical exponents, can be measured
in experiments with a limited number of ions. Besides that, interacting phonons
in trapped ions offer us the possibility to access regimes which are difficult
to study with ultracold bosons in optical lattices, like models with attractive
or site--dependent phonon-phonon interactions.
