Simulation of the Jahn-Teller-Dicke magnetic structural phase transition with trapped ions

We study theoretically the collective E circle times e Jahn-Teller-Dicke distortion in a system of trapped ions. In a previous work (Porras et al 2012 Phys. Rev. Lett. 108 235701) we have focused on cooperative Jahn-Teller models consisting of an ensemble of effective spins coupled to a set of many vibrational modes. Here, we show that under suitable conditions the spin ensemble may interact only with a single vibrational mode in each radial direction 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 spontaneous breaking of a continuous symmetry as well as magnetic structural phase transitions in a mesoscopic spin-boson system.