, Volume 50, Issue 1, pp 79-95

Infrared bounds, phase transitions and continuous symmetry breaking

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We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k −2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).

Research supported by USNSF under grants GP-38048 and MPS-74-13252
A. Sloan Fellow; also in the Department of Physics
Communicated by J. L. Lebowitz