Models exhibiting order by projection

We present calculations on a simple spin model which shows the phenomenon o f ol cie, by projection. We discuss a two-level atomic system coupled to th e radiation field to explore the possible relevance of order by projection in quantum optics. We point out the similarity between the effective Hamilt onian for the two-level atoms and the reduced BCS Hamiltonian used in the s tudy 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 sys tems on a lattice, exhibiting order by projection. This relates a nonextens ive change in the "kinetic energy" due to a projection term to the extensiv e expectation value of the interaction in the ground state without projecti on.