Fractal Ground State of Ion Chains in Periodic Potentials

Trapped ions in a periodic potential are a paradigm of a frustrated Wignercrystal. The dynamics is captured by a long-range Frenkel-Kontorova model. Theclassical ground state can be mapped to the one of an antiferromagnetic spinchain with long-range interactions in a magnetic field, whose strength isdetermined by the mismatch between chain's and substrate lattice's periodicity.The mapping is exact when the substrate potential is a piecewise harmonicpotential and holds for any two-body interaction decaying as 1/r^α withthe distance r. The ground state is a devil's staircase of regular, periodicstructures as a function of the mismatch, whose range of stability depends alsoon the coefficient α. While the staircase is well defined in thethermodynamic limit for α>1, for Coulomb interactions, α=1, itdisappears and the sliding-to-pinned transitions becomes crossovers. However,due to the logarithmic convergence to the thermodynamic limit characteristic ofthe Coulomb potential, the staircase is found for any finite number of ions. Wediscuss the experimental parameters as well as the features that allow one toobserve and reveal our predictions in experimental platforms. These dynamicsare a showcase of the versatility of trapped ion platforms for exploring theinterplay between frustration and interactions.