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Provable first-order transitions for nonlinear vector and gauge models with continuous symmetries. (English) Zbl 1074.82031
Summary: We consider various sufficiently nonlinear vector models of ferromagnets, of nematic liquid crystals and of nonlinear lattice gauge theories with continuous symmetries. We show, employing the method of Reflection Positivity and Chessboard Estimates, 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 controversy 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.

82D40 Statistical mechanical studies of magnetic materials
82D25 Statistical mechanical studies of crystals
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
81T25 Quantum field theory on lattices
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