  We demonstrate that a two-dimensional finite and periodic array of Ising
spins coupled via RKKY-like exchange can exhibit tunable magnetic states
ranging from three distinct magnetic regimes: (1) a conventional ferromagnetic
regime, (2) a glass-like regime, and (3) a new multi-well regime. These
magnetic regimes can be tuned by one gate-like parameter, namely the ratio
between the lattice constant and the oscillating interaction wavelength. We
characterize the various magnetic regimes, quantifying the distribution of low
energy states, aging relaxation dynamics, and scaling behavior. The glassy and
multi-well behavior results from the competing character of the oscillating
long-range exchange interactions. The multi-well structure features multiple
attractors, each with a sizable basin of attraction. This may open the possible
application of such atomic arrays as associative memories.
