✎ Mermin-Wagner theorem [The Physics Travel Guide]

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====== Mermin-Wagner theorem ======

<tabbox Why is it interesting?> 
<blockquote>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. <cite>page 525 in Quantum Field Theory by Claude Itzykson, ‎Jean-Bernard Zuber </cite> </blockquote>

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Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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<tabbox Student>

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
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Take note that the Mermin-Wagner theorem is not so universal as it is presented most of the time:

<blockquote>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.<cite>http://pages.physics.cornell.edu/~sethna/StatMech/EntropyOrderParametersComplexity.pdf</cite></blockquote>

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