3 - Spontaneous symmetry breaking, mean field

Summary

In many statistical models (but not all) one can introduce a local order parameter associated with a finite, discrete or continuous group of symmetries. The higher the temperature, the more important the fluctuations. One expects therefore in general to find pure phases at low temperature, with a reduced symmetry group, as a consequence of a nonvanishing expectation value of some order parameter. This situation is referred to as spontaneous symmetry breaking. A typical example is the classical Heisenberg model describing short-range interactions of an n-vector field ϕ, with an orthogonal O(n) symmetry group. For n = 1, 2, 3, this can account for a Curie transition from a ferromagnetic to a paramagnetic phase. In particle physics the σ-model of Gell-Mann and Levy involves a spontaneous symmetry breaking of chiral invariance, typical of massless spinor fields, accompanied by soft excitation modes, the so-called Goldstone modes, associated to a π-meson triplet and leading to a nonvanishing dynamical fermion mass. In a first and rather crude approximation, one can analyse the action itself or an effective one incorporating some fluctuation effects, and look for extrema as a function of field configurations, generally translationally invariant. The remaining fluctuations are then treated perturbatively. This mean field method is common to a great variety of domains, ranging from the Clausius–Mossotti formula for a polarizable medium, the Weiss molecular field in the theory of magnetism, Landau's effective action in various statistical contexts, the effective medium approximation in disordered systems, the Hartree–Fock method in atomic or many-body physics, to the semiclassical approximation in the study of quantum systems.