Communications in Mathematical Physics

, Volume 50, Issue 1, pp 79–95 | Cite as

Infrared bounds, phase transitions and continuous symmetry breaking

  • J. Fröhlich
  • B. Simon
  • T. Spencer


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%).


Neural Network Phase Transition Lower Bound Quantum Field Theory Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • J. Fröhlich
    • 1
  • B. Simon
    • 1
  • T. Spencer
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsThe Rockefeller UniversityNew YorkUSA

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