Ion chains in periodic potentials: Devil's staircases and quantum effects

We theoretically analyse the ground state of the Frenkel-Kontorova model where the particles interact via repulsive power-law forces. We show that the classical ground state can be mapped to the one of a long-range, antiferromagnetic Ising model and is a complete devil staircase as a function of the discommensuration. We then show that in the full quantum regime the commensurate-incommensurate transition is described by the Thirring model with power-law interactions. We analyse the mean-field phase diagram for Coulomb chains in relevant experimental regimes and determine the relevant spectroscopic features signalling the transition.