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In more than two dimensions, nonlinear ''σ'' models contain a dimensionful coupling constant and are thus not perturbatively renormalizable.
Nevertheless, they exhibit a non-trivial ultraviolet fixed point of the renormalization group both in the lattice formulation<ref>{{cite book | last = Zinn-Justin | first= Jean | title= Quantum Field Theory and Critical Phenomena | publisher = Oxford University Press | date = 2002 }}</ref><ref>{{ cite book | last= Cardy | first= John L. | title = Scaling and the Renormalization Group in Statistical Physics | publisher = Cambridge University Press | date = 1997 }}</ref> and in the double expansion originally proposed by [[Kenneth G. Wilson]].<ref>{{cite journal
In both approaches, the non-trivial renormalization-group fixed point found for the [[n-vector model|''O(n)''-symmetric model]] is seen to simply describe, in dimensions greater than two, the critical point separating the ordered from the disordered phase. In addition, the improved lattice or quantum field theory predictions can then be compared to laboratory experiments on [[critical phenomena]], since the ''O(n)'' model describes physical [[Heisenberg ferromagnet]]s and related systems. The above results point therefore to a failure of naive perturbation theory in describing correctly the physical behavior of the ''O(n)''-symmetric model above two dimensions, and to the need for more sophisticated non-perturbative methods such as the lattice formulation.
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