May

8

Heisenberg model (classical)

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The Heisenberg model is the <math>n = 3</math> 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

<math>\vec{s}_i \in \mathbb{R}^3, |\vec{s}_i|=1</math>,

each one placed on a lattice node.

The model is defined through the following Hamiltonian:

<math>\mathcal{H} = -\sum_{i,j} \mathcal{J}_{ij} \vec{s}_i \cdot \vec{s}_j</math>

with

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

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.


See also

  • Heisenberg model (quantum)
  • Ising model
  • XY model


External links

  • Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models
  • The Heisenberg Model - a Bibliography

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