The Classical Heisenberg model, developed by

Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...

, is the

n = 3

case of the

n-vector model In statistical mechanics, the ''n''-vector model or O(''n'') model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model. ...

, one of the models used in

statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxi ...

to model

ferromagnetism Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...

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

, each one placed on a lattice node. The model is defined through the following Hamiltonian: :

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

with :

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

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 phen ...

. * In the

continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...

the Heisenberg model (2) gives the following equation of motion ::

\vec_=\vec\wedge \vec_.

:This equation is called the

continuous classical Heisenberg ferromagnet equation Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuou ...

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 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 case of 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 low enough temperature, if

1< \alpha < 2

(infrared bounds). *As in any 'nearest-neighbor'

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,

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 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 for any

T>0

; i.e. at non-zero temperature the correlations cluster exponentially fast.

Three and higher dimensions

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.

References

External links

Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models
{{Webarchive, url=https://web.archive.org/web/20200608023338/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.17.1133 , date=2020-06-08

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