Classical Heisenberg model

Definition

Properties

• 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 gives the following equation of motion

  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 .
• 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 .
• 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