statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

, the classical Heisenberg model, developed by

Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II. He pub ...

, is the

n = 3

case of the ''n''-vector model, one of the models used to model

ferromagnetism Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...

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, :

\vec_i \in \mathbb^3, , \vec_i, =1\quad (1)

, on each lattice node. The model is defined through the following

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

: :

\mathcal = -\sum_ \mathcal_ \vec_i \cdot \vec_j\quad (2)

where :

\mathcal_ = \begin J & \mboxi, j\mbox \\ 0 & \mbox\end

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 statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenom ...

. * In the continuum limit the Heisenberg model (2) gives the following equation of motion ::

\vec_=\vec\wedge \vec_.

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

Ishimori equation The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable . Equation The Ishimori equation has the for ...

, and so on.

One dimension

*In the case of a long-range interaction,

J_\sim , x-y, ^

, the thermodynamic limit is well defined if

\alpha >1

; the magnetization remains zero if

\alpha \ge 2

; but the magnetization is positive, at a low enough temperature, if

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

* In the case of a long-range interaction,

J_\sim , x-y, ^

, the thermodynamic limit is well defined if

\alpha >2

; the magnetization remains zero if

\alpha \ge 4

; but the magnetization is positive at a low enough temperature if

2< \alpha <  4

(infrared bounds). * Polyakov has conjectured that, as opposed to the

classical XY model The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for . Definition ...

, there is no

dipole phase In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: * An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...

for any

T>0

; 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.

References

{{reflist

External links

Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models

Monte-Carlo simulation of the Heisenberg, XY and Ising models with 3D graphics (requires WebGL compatible browser)
Magnetic ordering Spin models Lattice models Werner Heisenberg