Quantum phases of interacting phonons in ion traps

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.