Jorge Yago Malo

Trapped ions in a periodic potential are a paradigm of a frustrated Wigner crystal. The dynamics is captured by a long-range Frenkel-Kontorova model. The classical ground state can be mapped to the one of an antiferromagnetic spin chain with long-range interactions in a magnetic field, whose strength is determined by the mismatch between 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 $1/r^α$ with the distance $r$. The ground state is a devil's staircase of regular, periodic structures as a function of the mismatch, whose range of stability depends also on the coefficient $α$. While the staircase is well defined in the thermodynamic limit for $α>1$, for Coulomb interactions, $α=1$, it disappears and the sliding-to-pinned transitions becomes crossovers. However, due to the logarithmic convergence to the thermodynamic limit characteristic of the Coulomb potential, the staircase is found for any finite number of ions. We discuss the experimental parameters as well as the features that allow one to observe and reveal our predictions in experimental platforms. These dynamics are a showcase of the versatility of trapped ion platforms for exploring the interplay between frustration and interactions.

A Wigner chain in a periodic potential is a paradigmatic example of geometric frustration with long-range interactions. The dynamics emulates the Frenkel-Kontorova model with Coulomb interactions. In the continuum approximation, dislocations are sine-Gordon solitons with power-law decaying tails. We show that their action is mapped into a massive, long-range (1+1) Thirring model, where the solitons are charged fermionic excitations over an effective Dirac sea. We identify the corresponding mean field theory and show that the Coulomb interactions destabilize structures commensurate with the periodic substrate, suppressing their onset and giving rise to {\it interaction-induced} lubrication. Our study identifies the role of long-range interactions on determining nanofriction. Our predictions can be probed in state-of-the-art trapped ion experiments.