  The properties of a system of charged particles on a 2D lattice, subject to
an anisotropic Jahn-Teller-type interaction and 3D Coulomb repulsion are
investigated. In the mean-field approximation without Coulomb interaction, the
system displays a phase transition of first order. When the long range Coulomb
interaction is included, Monte Carlo simulations show that the system displays
very diverse mesoscopic textures, ranging from spatially disordered pairs to
ordered arrays of stripes, or charged clusters, depending only on the ratio of
the two interactions (and the particle density). Remarkably, charged objects
with even number of particles are more stable than with odd number of
particles. We suggest that the diverse functional behaviour - including
superconductivity - observed in oxides can be thought to arise from the
self-organization of this type.
