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

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

each one placed on a lattice node.

The model is defined through the following Hamiltonian:

$\mathcal{H} = -\sum_{i,j} \mathcal{J}_{ij} \vec{s}_i \cdot \vec{s}_j\quad (2)$

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. Note that in the continuum limit the Heisenberg model (2) gives the following equation of motion

$\vec{S}_{t}=\vec{S}\wedge \vec{S}_{xx}.\quad (3)$

This equation is called the continuous classical Heisenberg ferromagnet equation or shortly Heisenberg model and is integrable in the soliton sense. It admits several integrable and nonintegrable generalizations like Landau-Lifshitz equation, Ishimori equation and so on.