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 excludes spontaneous continuous-symmetry order at positive temperature in important one- and two-dimensional systems with sufficiently short-range interactions. The original isotropic Heisenberg model result rules out both ferromagnetic and antiferromagnetic order under its assumptions.[ref]
Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models by N. D. Mermin and H. Wagner. 1966 abstract: nonzero temperature, finite-range exchange and one/two dimensions
The physical obstruction is strong long-wavelength orientational fluctuation. In a two-dimensional Gaussian angle model, the fluctuation integral includes , which grows logarithmically with size. A locally stiff orientation need not remain aligned across an infinite sample. This leaves room for quasi-long-range order and a Berezinskii–Kosterlitz–Thouless transition, rather than forbidding every finite-temperature phenomenon in two dimensions. Discrete anisotropy, long-range interactions, interlayer coupling and finite size change the applicability of the ideal theorem.