Mermin–Wagner theorem

In quantum field theory and statistical mechanics, the Hohenberg–Mermin–Wagner theorem or Mermin–Wagner theorem (also known as 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≤ 2 dimensional infinite systems was rigorously proved by David Mermin and Herbert Wagner (1966), citing a more general unpublished proof by Pierre Hohenberg (published later in 1967) in statistical mechanics. It was also reformulated later by Sidney Coleman (1973) for quantum field theory. The theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model.