  The Coulomb repulsion between ions in a linear Paul trap give rise to
anharmonic terms in the potential energy when expanded about the equilibrium
positions. We examine the effect of these anharmonic terms on the accuracy of a
quantum simulator made from trapped ions. To be concrete, we consider a linear
chain of $\text{Yb}^{171+}$ ions stabilized close to the zigzag transition. We
find that for typical experimental temperatures, frequencies change by no more
than a factor of $0.01\%$ due to the anharmonic couplings. Furthermore, shifts
in the effective spin-spin interactions (driven by a spin-dependent optical
dipole force) also tend to be small for detunings to the blue of the transverse
center-of-mass frequency. However, detuning the spin interactions near other
frequencies can lead to nonnegligible anharmonic contributions to the effective
spin-spin interactions. We also examine an odd behavior exhibited by the
harmonic spin-spin interactions for a range of intermediate detunings, where
nearest neighbor spins with a larger spatial separation on the ion chain
interact more strongly than nearest neighbors with a smaller spatial
separation.
