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

\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.

One dimension

Two dimensions

Three and higher dimensions

Independently of the range of the interaction, at low enough temperature the magnetization is positive.

Conjecturally, in each of the low temperature extremal states the truncated correlations decay algebraically.

See also

References

External links