  Trapped-ion quantum simulators have demonstrated a long history of studying
the physics of interacting spin-lattice systems using globally addressed
entangling operations. Here, we seek to broaden and delimit the classes of
effective spin-spin interactions achievable using exclusively global driving
fields. We find that new categories of interaction graphs become achievable
with perfect or near-perfect theoretical fidelity by tailoring the coupling to
each vibrational mode of the ion crystal, or by shaping the trapping potential
to include specific anharmonic terms. We also derive a rigorous test to
determine whether a desired interaction graph is accessible using only globally
driven fields. These tools broaden the reach of trapped-ion quantum simulators
so that they may more easily address open questions in materials science and
quantum chemistry.
