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

• Polyakov has conjectured that, in dimension 2, as opposed to the classical XY model, there is no dipole phase for any $T>0$; i.e. at non-zero temperature the correlations cluster exponentially fast.

• 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

${\vec {S}}_{t}={\vec {S}}\wedge {\vec {S}}_{xx}.$

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.

See also

• Heisenberg model (quantum)
• Ising model
• Classical XY model
• Magnetism
• Ferromagnetism
• Landau-Lifshitz equation
• Ishimori equation

External links

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