Rigorous Estimates for Correlation Functions and Existence of Phase Transitions in Some Models of Interacting Bosons

Using the so-called method of infrared bounds and a Roepstorff's inequality we obtain a lower bound for the amount of condensate and derive an upper bound for the anomalous average [Formula: see text] of Huang–Davies (HD) model. A lower bound for the static structure factor is also obtained. We generalize the HD model and prove the existence of Bose condensation for this kind of model systems. Finally we derived upper and lower bounds for correlation functions associated to a system of Fermi-particles interacting with a field of phonons.

LONG RANGE ORDER IN THE BLUME-EMERY-GRIFFITHS MODEL

Using the Infrared Bounds method it ws demonstrated that a first order phase transition takes place in the m-dimensional (m≥3) Blume-Emery-Griffiths model.

BEHAVIOR IN LARGE DIMENSIONS OF THE POTTS AND HEISENBERG MODELS

We extend to the Potts and Heisenberg models some of the results proven in [6] for the Ising model. For both these models we prove that if the interaction is properly normalized, then as the space dimensionality goes to ∞, the spontaneous magnetization converges to the value given by the corresponding Curie-Weiss model (except possibly at the Curie-Weiss transition point, in the case of the Potts model.) For the Potts model we prove also that the ordered phases approach product measures. The proofs are based on the convergence of the free energy to the Curie-Weiss value and on infrared bounds. A consequence of our result for the q-state Potts model is an asymptotic upper bound for the transition temperature, which for q > 2 is better than the one obtained by the conventional use of infrared bounds, or comparison inequalities between different Potts models.