Hidden frustrated interactions and quantum annealing in trapped-ion spin-phonon chains

We show that a trapped-ion chain interacting with an optical spin-dependent force shows strong frustration effects due to the interplay between long range interactions and the dressing by optical phases. We consider a strong spin-phonon coupling regime and predict a quantum phase diagram with different competing magnetic and structural orders. In a highly frustrated region, the system shows enhanced quantum fluctuations and entanglement, which are characteristic of spin-liquid phases. We propose and describe within a mean-field approach a quantum annealing process to induce a quasi-adiabatic evolution towards the ground state.