The Mermin-Wagner theorem is a result in statistical mechanics and condensed matter physics that addresses the behavior of certain types of physical systems at low temperatures, specifically those defined by continuous symmetry. The theorem, which was formulated by N. D. Mermin and H. Wagner in the 1960s, states that in two-dimensional systems with continuous symmetry, spontaneous symmetry breaking and long-range order cannot occur at finite temperatures.
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The Mermin-Wagner theorem forbids spontaneous breaking of a continuous internal symmetry at positive temperature in one- and two-dimensional systems with sufficiently short-range interactions.