Abstract
These lectures are a survey of some mathematically rigorous results concerning phase transitions and symmetry breaking in statistical mechanics and quantum field theory. This subject has a rather long history: The first results on phase transitions were obtained by R. Peierls in 1936, [Pe]. He showed that the Ising model in two or more dimensions has spontaneous magnetization at low temperatures. His argument was later reformulated in various ways and applied to many model systems. See the references in [Gr]. We shall apply a variant of the Peierls argument [GJS3] to prove that there is a phase transition in the two dimensional, anisotropic (→φ·→φ)2 quantum field model in two dimensions and we explain how, in this model, a phase transition gives rise to soliton sectors. We show that the mass gap on the soliton sector is bounded below by the “surface tension” of this model.
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