Theoretical basis for quantum simulation with a planar ionic crystal in a Penning trap using a triangular rotating wall

One of the challenges with quantum simulation in ion traps is that the effective spin-spin exchange couplings are not uniform across the lattice. This can be particularly important in Penning-trap realizations where the presence of an ellipsoidal boundary at the edge of the trap leads to dislocations in the crystal. By adding an additional anharmonic potential to better control interion spacing, and a triangular-shaped rotating wall potential to reduce the appearance of dislocations, one can achieve better uniformity of the ionic positions. In this work, we calculate the axial phonon frequencies and the spin-spin interactions driven by a spin-dependent optical dipole force, and discuss what effects the more uniform ion spacing have on the spin simulation properties of Penning-trap quantum simulators. Indeed, we find that the spin-spin interactions behave more like a power law for a wide range of parameters.