Fractal ground state of mesoscopic ion chains in periodic potentials

Trapped ions in a periodic potential are a paradigm of a frustrated Wigner crystal. The dynamics are captured by a long-range Frenkel-Kontorova model. We show that the classical ground state can be mapped to the one of a long-range Ising spin chain in a magnetic field, whose strength is determined by the mismatch between the chain's and substrate lattice's periodicity. The mapping is exact when the substrate potential is a piecewise harmonic potential and holds for any two-body interaction decaying as $1/{r}^{\ensuremath{\alpha}}$ with the distance $r$. The ground state is a devil's staircase of regular, periodic structures as a function of the mismatch and of the interaction exponent $\ensuremath{\alpha}$. While the staircase is well defined in the thermodynamic limit for $\ensuremath{\alpha}>1$, for Coulomb interactions, $\ensuremath{\alpha}=1$, we argue that it disappears and the sliding-to-pinned transition becomes a crossover, with a convergence to the thermodynamic limit scaling logarithmically with the chain's size. Due to this slow convergence, fractal properties can be observed even in chains of hundreds of ions at laser cooling temperatures. These dynamics are a showcase of the versatility of trapped-ion platforms for exploring the interplay between frustration and interactions.