  We propose loading trapped ions into microtraps formed by an optical lattice.
For harmonic microtraps, the Coulomb coupling of the spatial motions of
neighboring ions can be used to construct a broad class of effective
short-range Hamiltonians acting on an internal degree of freedom of the ions.
For large anharmonicities, on the other hand, the spatial motion of the ions
itself represents a spin-1/2 model with frustrated dipolar XY interactions. We
illustrate the latter setup with three systems: the linear chain, the zig-zag
ladder, and the triangular lattice. In the frustrated zig-zag ladder with
dipolar interactions we find chiral ordering beyond what was predicted
previously for a next-nearest-neighbor model. In the frustrated anisotropic
triangular lattice with nearest-neighbor interactions we find that the
transition from the one-dimensional gapless spin-liquid phase to the
two-dimensional spiraling ordered phase passes through a gapped spin-liquid
phase, similar to what has been predicted for the same model with Heisenberg
interactions. Further, a second gapped spin-liquid phase marks the transition
to the two-dimensional Neel-ordered phase.
