Classical Heisenberg model
It can be formulated as follows: take a d-dimensional lattice, and a set of spins of the unit length
each one placed on a lattice node.
The model is defined through the following Hamiltonian:
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.
- In the continuum limit the Heisenberg model (2) gives the following equation of motion
- 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 
- In case of long range interaction, , the thermodynamic limit is well defined if ; the magnetization remains zero if ; but the magnetization is positive, at low enough temperature, if (infrared bounds).
- As in any 'nearest-neighbor' n-vector model with free boundary conditions, if the external field is zero, there exists a simple exact solution.
Two dimensions 
- In the case of long range interaction, , the thermodynamic limit is well defined if ; the magnetization remains zero if ; but the magnetization is positive at low enough temperature if (infrared bounds).
- Polyakov has conjectured that, as opposed to the classical XY model, there is no dipole phase for any ; i.e. at non-zero temperature the correlations cluster exponentially fast.
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 
- Heisenberg model (quantum)
- Ising model
- Classical XY model
- Landau-Lifshitz equation
- Ishimori equation
- Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models
- The Heisenberg Model - a Bibliography
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