Heisenberg model (classical)

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The Heisenberg model is the $n = 3$ case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.

It can be formulated as follows: take a d-dimensional lattice, and a set of spins of the unit length

$\vec{s}_i \in \mathbb{R}^3, |\vec{s}_i|=1$ ,

each one placed on a lattice node.

The model is defined through the following Hamiltonian:

$\mathcal{H} = -\sum_{i,j} \mathcal{J}_{ij} \vec{s}_i \cdot \vec{s}_j$

with

$\mathcal{J}_{ij} = \begin{cases} J & \mbox{if }i, j\mbox{ are neighbors} \\ 0 & \mbox{else.}\end{cases}$

a coupling between spins.

The general mathematical formalism used to describe and solve the Heisenberg model and certain generalizations is developed in the article on the Potts model.