Classical Heisenberg model
In statistical physics, the classical Heisenberg model, developed by Werner Heisenberg, is the case of the n-vector model, one of the models used to model ferromagnetism and other phenomena.
Definition
The classical Heisenberg model can be formulated as follows: take a d-dimensional lattice, and place a set of spins of unit length,on each lattice node.
The model is defined through the following Hamiltonian:
where
is a coupling between spins.
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 the case of a long-range interaction,, the thermodynamic limit is well defined if ; the magnetization remains zero if ; but the magnetization is positive, at a 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 a long-range interaction,, the thermodynamic limit is well defined if ; the magnetization remains zero if ; but the magnetization is positive at a low enough temperature if .
- Polyakov has conjectured that, as opposed to the classical XY model, there is no dipole phase for any ; namely, at non-zero temperatures the correlations cluster exponentially fast.
Three and higher dimensions
Independently of the range of the interaction, at a low enough temperature the magnetization is positive.Conjecturally, in each of the low temperature extremal states the truncated correlations decay algebraically.