Columnar phase in quantum dimer models

The quantum dimer model, relevant for short-range resonant valence bond physics, is rigorously shown to have long range order in a crystalline phase in the attractive case at low temperature and not too large flipping term. This term flips horizontal dimer pairs to vertical pairs (and vice versa) and is responsible for the word 'quantum' in the title. In addition to the dimers, monomers are also allowed. The mathematical method used is 'reflection positivity'. The model and proof can easily be generalized to dimers or plaquettes in three-dimensions.