# Classical Heisenberg model

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Classical Heisenberg model

The Classical 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.

## Definition

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.

## Properties

• Polyakov has conjectured that, in dimension 2, as opposed to the classical XY model, there is no dipole phase for any T > 0; i.e. at non-zero temperature the correlations cluster exponentially fast.
• The general mathematical formalism used to describe and solve the Heisenberg model and certain generalizations is developed in the article on the Potts model.
• In the continuum limit the Heisenberg model (2) gives the following equation of motion $\vec{S}_{t}=\vec{S}\wedge \vec{S}_{xx}.$
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.

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