It can be formulated as follows: take a d-dimensional lattice, and a set of spins of the unit length

${\displaystyle {\vec {s}}_{i}\in \mathbb {R} ^{3},|{\vec {s}}_{i}|=1\quad (1)}$ ,

each one placed on a lattice node.

The model is defined through the following Hamiltonian:

${\displaystyle {\mathcal {H}}=-\sum _{i,j}{\mathcal {J}}_{ij}{\vec {s}}_{i}\cdot {\vec {s}}_{j}\quad (2)}$ 

with

${\displaystyle {\mathcal {J}}_{ij}={\begin{cases}J&{\mbox{if }}i,j{\mbox{ are neighbors}}\\0&{\mbox{else.}}\end{cases}}}$ 

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

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

This equation is called the continuous classical Heisenberg ferromagnet equation or shortly Heisenberg model and is integrable in the sense of soliton theory. It admits several integrable and nonintegrable generalizations like Landau-Lifshitz equation, Ishimori equation and so on.

One dimension edit

• In case of long range interaction, ${\displaystyle J_{x,y}\sim |x-y|^{-\alpha }}$ , the thermodynamic limit is well defined if ${\displaystyle \alpha >1}$ ; the magnetization remains zero if ${\displaystyle \alpha \geq 2}$ ; but the magnetization is positive, at low enough temperature, if ${\displaystyle 1<\alpha <2}$  (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 edit

• In the case of long-range interaction, ${\displaystyle J_{x,y}\sim |x-y|^{-\alpha }}$ , the thermodynamic limit is well defined if ${\displaystyle \alpha >2}$ ; the magnetization remains zero if ${\displaystyle \alpha \geq 4}$ ; but the magnetization is positive at low enough temperature if ${\displaystyle 2<\alpha <4}$  (infrared bounds).
• Polyakov has conjectured that, as opposed to the classical XY model, there is no dipole phase for any ${\displaystyle T>0}$ ; i.e. at non-zero temperature the correlations cluster exponentially fast.^[1]

Three and higher dimensions edit

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.

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

• Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models Archived 2020-06-08 at the Wayback Machine
• The Heisenberg Model - a Bibliography
• Monte-Carlo simulation of the Heisenberg, XY and Ising models with 3D graphics (requires WebGL compatible browser)