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We develop the theory to describe the equilibrium ion positions and phononmodes for a trapped ion quantum simulator in an oblate Paul trap that createstwo-dimensional Coulomb crystals in a triangular lattice. By coupling theinternal states of the ions to laser beams propagating along the symmetry axis,we study the effective Ising spin-spin interactions that are mediated via theaxial phonons and are less sensitive to ion micromotion. We find that the axialmode frequencies permit the programming of Ising interactions with inversepower law spin-spin couplings that can be tuned from uniform to $r^{-3}$ withDC voltages. Such a trap could allow for interesting new geometricalconfigurations for quantum simulations on moderately sized systems includingfrustrated magnetism on triangular lattices or Aharonov-Bohm effects on iontunneling. The trap also incorporates periodic boundary conditions around loopswhich could be employed to examine time crystals.

Creation of two-dimensional coulomb crystals of ions in oblate Paul traps for quantum simulations