Models exhibiting order by projection

We present calculations on a simple spin model which shows the phenomenon of ol cie, by projection. We discuss a two-level atomic system coupled to the radiation field to explore the possible relevance of order by projection in quantum optics. We point out the similarity between the effective Hamiltonian for the two-level atoms and the reduced BCS Hamiltonian used in the study of the ultrasmall metallic grains, and we present some calculations to show the transition from a nonsuperconducting to a superconducting ground state in the grains. We present a sum rule for the interacting electron systems on a lattice, exhibiting order by projection. This relates a nonextensive change in the "kinetic energy" due to a projection term to the extensive expectation value of the interaction in the ground state without projection.