Hidden symmetries of finite-size clusters with periodic boundary conditions

Finite-size clusters with periodic boundary conditions resemble isolated clusters for a small number of sites, and infinite lattices for a large number of sites. The transition from a self-contained system to an infinite lattice passes through an intermediate region with increased (hidden) symmetry. In this high-symmetry region irreducible representations of the space group may stick together to form higher-dimensional representations of the complete symmetry group. This transition is examined for a class of simple-, body-centered-, and face-centered-cubic lattice clusters and the two-dimensional square lattice cluster. The implications of an enlarged symmetry group are also studied for a model of strongly correlated electrons interacting on eight-site clusters.