Mermin–Wagner theorem
In quantum field theory and statistical mechanics, the Mermin–Wagner theorem (also known as Mermin–Wagner–Hohenberg theorem, Mermin–Wagner–Berezinskii theorem, or Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.
This is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function.
The absence of spontaneous symmetry breaking in d ≤ 2 dimensional systems was rigorously proved by Sidney Coleman (1973) in quantum field theory and by David Mermin, Herbert Wagner and Pierre Hohenberg in statistical physics. That the theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model.
Consider the free scalar field φ of mass m in two Euclidean dimensions. Its propagator is:
For small m, G is a solution to Laplace's equation with a point source:
This is because the propagator is the reciprocal of ∇2 in k space. To use Gauss's law, define the electric field analog to be E = ∇G. The divergence of the electric field is zero. In two dimensions, using a large Gaussian ring:
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