  Periodic nanostructures can display the dynamics of arrays of atoms while
enabling the tuning of interactions in ways not normally possible in Nature. We
examine one dimensional arrays of a ``synthetic atom,'' a one dimensional ring
with a nearest neighbor Coulomb interaction. We consider the classical limit
first, finding that the singly charged rings possess antiferroelectric order at
low temperatures when the charge is discrete, but that they do not order when
the charge is treated as a continuous classical fluid. In the quantum limit
Monte Carlo simulation suggests that the system undergoes a quantum phase
transition as the interaction strength is increased. This is supported by
mapping the system to the 1D transverse field Ising model. Finally we examine
the effect of magnetic fields. We find that a magnetic field can alter the
electrostatic phase transition producing a ferroelectric groundstate, solely
through its effect of shifting the eigenenergies of the quantum problem.
