Competing many-body interactions in systems of trapped ions

We propose and theoretically analyze an experimental configuration in which lasers induce three-spin interactions between trapped ions. By properly choosing the intensities and frequencies of the lasers, three-spin couplings may be dominant or comparable to two-spin terms and magnetic fields. In this way, trapped ions can be used to study exotic quantum phases which do not have a counterpart in nature. We study the conditions for the validity of the effective three-spin Hamiltonian and predict qualitatively the quantum phase diagram of the system.