  We calculate the zero temperature electrostatic properties of charged one and
two dimensional arrays of rings, in the classical and quantum limits.
  Each ring is assumed to be an ideal ring of negligible width, with exactly
one electron on the ring that interacts only with nearest neighbor rings. In
the classical limit we find that if the electron is treated as a point
particle, the 1D array of rings can be mapped to an Ising antiferromagnet,
while the 2D array groundstate is a four-fold degenerate "stripe" phase. In
contrast, if we treat the electrical charge as a continuous fluid, the
distribution will not spontaneously break symmetry, but will develop a charge
distribution reflecting the symmetry of the array. In the quantum limit, the
competition between the kinetic energy and Coulomb energy allows for a
transition between unpolarized and polarized states as a function of the ring
parameters. This allows for a new class of polarizable materials whose
transitions are based on geometry, rather than a structural transition in a
unit cell.
