  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 has on the spin simulation
properties of Penning trap quantum simulators. Indeed, we find the spin-spin
interactions behave more like a power law for a wide range of parameters.
