Journal of Statistical Physics, Vol. 106, Nos. 3/4, February 2002. p. 723-783.
We study a large class F of models of the quantum statistical mechanics dealing with two types of particles. First the spinless electrons are quantum particles obeying to the Fermi statistics, they can hop. Secondly the ions which cannot move, are classical particles. The Falicov-Kimball (FK) model is a well known model belonging to F, for which the existence of an antiferomagnetic phase transition was proven in the seminal paper of Kennedy and Lieb. This result was extended by Lebowitz and Macris. A new approach to this problem based on quantum selection of the ground states was proposed in ref. 4. In this paper we extend this approach to show that, under the "strong insulating condition," any hamiltonian of the class F admits, at every temperature, an effective hamiltonian, which governs the behaviour of the ions interacting through forces mediated by the electrons. The effective hamiltonians are long range many body Ising hamiltonians, which can be computed by a cluster expansion expressed in term of the quantum fluctuations. Our main result is that we can apply the powerfull results of the classical statistical mechanics to our quantum models. In particular we can use the classical Pirogov-Sinai theory to establish a hierarchy of phase diagrams, we can also study of the behaviour of the quantum interfaces, and so on...