Mermin–Wagner theorem

The Mermin–Wagner theorem rules out spontaneous breaking of continuous symmetries at nonzero temperature in one- and two-dimensional systems with sufficiently short-range interactions.

Goldstone theorem: A theorem connecting spontaneously broken continuous symmetries to gapless excitations. Its would-be gapless modes fluctuate strongly enough to destroy long-range order in low dimensions.
David Mermin: An American physicist whose work included foundational contributions to condensed-matter theory. He co-authored the theorem's influential 1966 proof.
Two-dimensional Heisenberg model: A lattice model of spins with rotationally invariant exchange interactions in two spatial dimensions. It is a central setting where continuous magnetic order is forbidden at positive temperature.
Discrete symmetry: A symmetry with a finite set of transformations, such as exchanging two states. The theorem does not forbid spontaneous breaking of discrete symmetries in two dimensions.
Spin wave: A collective, low-energy fluctuation of ordered spins, described as a propagating wave of spin orientation. Thermally populated long-wavelength spin waves produce the divergent fluctuations behind the theorem.
Herbert Wagner: A German theoretical physicist known for work on statistical mechanics and condensed matter. He co-authored the 1966 result for isotropic Heisenberg models.
One-dimensional Heisenberg model: A chain of spins coupled by rotationally invariant exchange interactions. It illustrates the theorem's restriction in one spatial dimension.
Long-range interaction: An interaction whose strength decays slowly enough with distance to couple distant parts of a system significantly. Sufficiently slowly decaying interactions can evade the theorem's short-range assumption.
Infrared divergence: A divergence caused by contributions from modes with arbitrarily small momentum or energy. The integral over long-wavelength fluctuations diverges in one and two dimensions.
N. David Mermin and Herbert Wagner 1966 paper: A paper proving the absence of spontaneous magnetization in isotropic Heisenberg models in one and two dimensions at positive temperature. This paper established the result that later became known by their names.

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