  The physics of interacting integer-spin chains has been a topic of intense
theoretical interest, particularly in the context of symmetry-protected
topological phases. However, there has not been a controllable model system to
study this physics experimentally. We demonstrate how spin-dependent forces on
trapped ions can be used to engineer an effective system of interacting spin-1
particles. Our system evolves coherently under an applied spin-1 XY Hamiltonian
with tunable, long-range couplings, and all three quantum levels at each site
participate in the dynamics. We observe the time evolution of the system and
verify its coherence by entangling a pair of effective three-level particles
(`qutrits') with 86% fidelity. By adiabatically ramping a global field, we
produce ground states of the XY model, and we demonstrate an instance where the
ground state cannot be created without breaking the same symmetries that
protect the topological Haldane phase. This experimental platform enables
future studies of symmetry-protected order in spin-1 systems and their use in
quantum applications.
