Non-linear sigma model
In quantum field theory, a nonlinear σ model describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The non-linear σ-model was introduced by Gell-Mann & Lévy (1960, section 6), who named it after a field corresponding to a spinless meson called σ in their model.
The target manifold T is equipped with a Riemannian metric g. Σ is a differentiable map from Minkowski space M (or some other space) to T.
The Lagrangian density in contemporary chiral form is given by
where we have used a + − − − metric signature and the partial derivative ∂Σ is given by a section of the jet bundle of T×M and V is the potential.
In the coordinate notation, with the coordinates Σa, a = 1, ..., n where n is the dimension of T,
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 and in the double expansion originally proposed by Kenneth G. Wilson.
In both approaches, the non-trivial renormalization-group fixed point found for the 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 ferromagnets 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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