  Laser-cooled and trapped atomic ions form an ideal standard for the
simulation of interacting quantum spin models. Effective spins are represented
by appropriate internal energy levels within each ion, and the spins can be
measured with near-perfect efficiency using state-dependent fluorescence
techniques. By applying optical fields that exert optical dipole forces on the
ions, their Coulomb interaction can be modulated to produce long-range and
tunable spin-spin interactions that can be reconfigured by shaping the spectrum
and pattern of the laser fields, in a prototypical example of a quantum
simulator. Here we review the theoretical mapping of atomic ions to interacting
spin systems, the preparation of complex equilibrium states, the study of
dynamical processes in these many-body interacting quantum systems, and the use
of this platform for optimization and other tasks. The use of such quantum
simulators for studying spin models may inform our understanding of exotic
quantum materials and shed light on the behavior of interacting quantum systems
that cannot be modeled with conventional computers.
