Why is it interesting?

A theorem due to Mermin and Wagner states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. […] The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. page 525 in Quantum Field Theory by Claude Itzykson, ‎Jean-Bernard Zuber

Layman

The Mermin-Wagner theorem explains, why the existence of phase transitions like Magnetization or Freezing depends on the dimension of the material undergoing the transition. Imagine water molecules close to the freezin point in the following configurations:

• a one-dimensional chain
• a two dimensional monolayer of watermolecules
• a cube

The molecules are moving around because of their thermal energy but also interact with each other due to hydrogen bonding.

For water to freeze, the temperature has to be low enough such that the bonding between the molecules becomes strong enough to keep the molecules fixed at their place. This means that the number of water molecules that are neighbours determines how hard it is for a water molecule to move away from its position. We can imagine that there are more neighbours, the higher the dimension of the configuration is. Mermin-Wagner predicts that for certain problems there is a lower critical dimension below which there are not enough neighbors to build up a strong enough bond such that all the water molecules "freeze" at their position. This behaviour is also called cooperativity.

Water freezing is actually a very complex problem and the above explanation is only meant to simplify the visualization and should not be taken to be physically correct. But for simple problems like the continuous Ising model it is possible to calculate this lower critical dimension to be .

Student

Researcher

Take note that the Mermin-Wagner theorem is not so universal as it is presented most of the time:

In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin–Wagner theorem. Hohenberg, Mermin, and Wagner, in a series of papers, proved in the 1960s that two-dimensional systems with a continuous symmetry cannot have a broken symmetry at finite temperature. At least, that is the English phrase everyone quotes when they discuss the theorem; they actually prove it for several particular systems, including superfluids, superconductors, magnets, and translational order in crystals. Indeed, crystals in two dimensions do not break the translational symmetry; at finite temperatures, the atoms wiggle enough so that the atoms do not sit in lock-step over infinite distances (their translational correlations decay slowly with distance). But the crystals do have a broken orientational symmetry: the crystal axes point in the same directions throughout space. (Mermin discusses this point in his paper on crystals.) The residual translational correlations (the local alignment into rows and columns of atoms) introduce long-range forces which force the crystalline axes to align, breaking the continuous rotational symmetry.http://pages.physics.cornell.edu/~sethna/StatMech/EntropyOrderParametersComplexity.pdf

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Examples

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History