  We study theoretically the collective E$\otimes$e Jahn-Teller-Dicke
distortion in a system of trapped ions. We focus in the limit of infinite range
interactions in which an ensemble of effective spins interacts with two
collective vibrational modes with U(1) symmetric couplings. Our model is
exactly solvable in the thermodynamical limit and it is amenable to be solved
by exact numerical diagonalization for a moderate number of ions. We show that
trapped ions are ideally suited to study the emergence of spontaneous symmetry
breaking of a continuous symmetry and magnetic structural phase transition in a
mesoscopic system.
