Statistical mechanics, structure of matter Lattice spin systems Example

Classical Heisenberg model

Example

Let $G = (V, E)$ be a finite graph, let $J \in \R$ be a coupling constant and let $\beta \ge 0$ be the inverse temperature. A configuration assigns to each vertex $x$ a unit vector $s_x$ of $\R^3$ , that is, a point of the sphere $S^2$ . Its energy is $H(s) = -J \sum_{\set{x, y} \in E} s_x \cdot s_y.$ The classical Heisenberg model on $G$ is the probability measure on $(S^2)^V$ with density $e^{-\beta H(s)} / Z$ with respect to the product of the uniform measures. Here $Z$ is the normalizing constant. For $J > 0$ the model is ferromagnetic and for $J < 0$ it is antiferromagnetic.

Remarks

The energy is invariant under every rotation of $\R^3$ applied to all spins at once, so the model has $O(3)$ symmetry. It is the case $n = 3$ of the n-vector model. It is the classical limit of the quantum Heisenberg model, where the spins are operators. On $\Z^2$ the Mermin–Wagner theorem shows that there is no spontaneous magnetization at any positive temperature. On $\Z^3$ with $J > 0$ , the infrared bounds of Fröhlich, Simon and Spencer show long-range order at low temperature.

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Tag	`B9N542W1CN`
URI	`m2://B9N542W1CN`
Kind	example
Writing	W0 stub
MSC	82B20, 82D40
Instance of	Lattice model
Uses	Graph, Boltzmann distribution
Same as	Wikidata Q5128287 Wikipedia Classical Heisenberg model
Changed	3 November 2026

Classical Heisenberg model

Example

Remarks

Reading path

Further reading

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