Statistical mechanics, structure of matter Lattice spin systems Theorem

Mermin–Wagner theorem

Builds on Lattice model Gibbs measure Peierls argument

Statement

Let $d \in \set{1, 2}$ and $N \ge 2$ , and let $S^{N-1} \subset \R^N$ be the unit sphere. Consider the classical lattice model on $\Z^d$ with spins $\sigma_x \in S^{N-1}$ and nearest-neighbour coupling $J > 0$ . Add a field $h \ge 0$ along the first unit vector $e_1$ , so that the energy on a finite box $\Lambda$ is $H_\Lambda(\sigma) = -J \sum_{\set{x, y}} \sigma_x \cdot \sigma_y - h \sum_{x \in \Lambda} \sigma_x \cdot e_1.$ The first sum runs over the neighbouring pairs in $\Lambda$ . The reference measure at each site is the uniform measure on $S^{N-1}$ . Let $m(\beta, h)$ be the limit of $\langle \sigma_0 \cdot e_1 \rangle$ under the Gibbs distribution on boxes growing to $\Z^d$ , at inverse temperature $\beta$ . Then for every $\beta < \infty$ , the spontaneous magnetization vanishes: $\lim_{h \to 0^+} m(\beta, h) = 0.$

Remarks

N. David Mermin and Herbert Wagner proved the theorem in 1966 for the quantum Heisenberg model. They allowed couplings $J(x)$ of longer range, under the condition $\sum_x \abs{x}^2 \abs{J(x)} < \infty$ , and the classical case follows in the same way. Their proof uses the Bogoliubov inequality to bound the squared magnetization by a constant divided by an integral over wave vectors $k$ . As $h \to 0$ this integral behaves like $\int d^d k / \abs{k}^2$ near $k = 0$ , which diverges in $d \le 2$ . In words, long-wavelength spin waves cost little energy in low dimension, and at any positive temperature they destroy long-range order. The continuous symmetry is necessary: the two-dimensional Ising model, with discrete symmetry, is ordered at low temperature by the Peierls argument. The dimension is necessary: in $d \ge 3$ these models have spontaneous magnetization at low temperature, by the infrared bounds of Fröhlich, Simon and Spencer. Later work showed that in $d = 2$ every infinite-volume Gibbs measure of such models is invariant under rotations of the spins.

Going deeper

For a student

The theorem applies to the n-vector models, including the XY and classical Heisenberg models, and its versions cover superfluids and crystals. It rules out spontaneous magnetization but not every phase transition: the two-dimensional XY model has a Berezinskii–Kosterlitz–Thouless transition, where correlations change from exponential to power-law decay. Its proof is a good model of how fluctuations at long wavelengths decide whether order survives. Comparing with the Peierls argument shows the difference between discrete and continuous symmetries.

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Gibbs measure concept
Ising model example
Lattice model concept
Peierls argument statement
Mermin–Wagner theorem statement

Tag	`5C0QQFXSD7`
URI	`m2://5C0QQFXSD7`
Kind	statement, theorem
Writing	W0 stub
MSC	82B20, 82B26, 82D40
Status	This copy of the site has no build data. Run `m2 status 5C0QQFXSD7`.
About	n-vector model, Spontaneous magnetization
Uses	Lattice model, Gibbs measure, Peierls argument
Same as	Wikidata Q1547975 Wikipedia Mermin–Wagner theorem
Changed	3 November 2026

Mermin–Wagner theorem

Statement

Remarks

Going deeper

For a student

Reading path

Further reading

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