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Systems with long-range interactions show a variety of intriguing properties:they typically accommodate many meta-stable states, they can give rise tospontaneous formation of supersolids, and they can lead to counterintuitivethermodynamic behavior. However, the increased complexity that comes withlong-range interactions strongly hinders theoretical studies. This makes aquantum simulator for long-range models highly desirable. Here, we show that achain of trapped ions can be used to quantum simulate a one-dimensional modelof hard-core bosons with dipolar off-site interaction and tunneling, equivalentto a dipolar XXZ spin-1/2 chain. We explore the rich phase diagram of thismodel in detail, employing perturbative mean-field theory, exactdiagonalization, and quasiexact numerical techniques (density-matrixrenormalization group and infinite time evolving block decimation). We findthat the complete devil's staircase -- an infinite sequence of crystal statesexisting at vanishing tunneling -- spreads to a succession of lobes similar tothe Mott-lobes found in Bose--Hubbard models. Investigating the melting ofthese crystal states at increased tunneling, we do not find (contrary tosimilar two-dimensional models) clear indications of supersolid behavior in theregion around the melting transition. However, we find that inside theinsulating lobes there are quasi-long range (algebraic) correlations, opposedto models with nearest-neighbor tunneling which show exponential decay ofcorrelations.

Complete devil's staircase and crystal--superfluid transitions in a dipolar XXZ spin chain: A trapped ion quantum simulation