The Mermin-Wagner Theorem states that a continuous symmetry cannot be spontaneously broken at nonzero temperature in a system with sufficiently short-range interactions confined to one or two spatial dimensions, so the long-range magnetic order familiar from three-dimensional magnets cannot occur there. Named for N. David Mermin and Herbert Wagner, it is a foundational no-go result of statistical mechanics constraining what kinds of order low-dimensional systems can sustain.
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StatementContinuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions of two or fewer. 1 Classification
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Lead section
In quantum field theory and statistical mechanics, the Hohenberg-Mermin-Wagner theorem or Mermin-Wagner theorem (also known as Mermin-Wagner-Berezinskii theorem or Mermin-Wagner-Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2.
Lead section
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≤ 2 dimensional infinite systems was rigorously proved by David Mermin and Herbert Wagner (1966), citing a more general unpublished proof by Pierre Hohenberg (published later in 1967) in statistical mechanics.
In Branch: Mathematical Physics, Lead paragraph
In quantum field theory and statistical mechanics, the Hohenberg-Mermin-Wagner theorem or Mermin-Wagner theorem (also known as Mermin-Wagner-Berezinskii theorem or Mermin-Wagner-Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2.
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