Abstract
We consider various sufficiently nonlinear vector models of ferromagnets, of nematic liquid crystals and of nonlinear lattice gauge theories with continuous sym- metries. We show, employing the method of Reflection Positivity and Chessboard Esti- mates, that they all exhibit first-order transitions in the temperature, when the nonlinearity parameter is large enough. The results hold in dimension 2 or more for the ferromagnetic models and the RP N−1 liquid crystal models and in dimension 3 or more for the lattice gauge models. In the two-dimensional case our results clarify and solve a recent con- troversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.
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