# Classical Heisenberg model

﻿
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.[1]
• 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.

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Heisenberg model (classical) — The 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.It can be formulated as follows: take a d dimensional lattice, and a set of spins of the unit …   Wikipedia

• Heisenberg model (quantum) — The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically. In the prototypical Ising model,… …   Wikipedia

• Heisenberg model — The Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model (classical), a classical nearest neighbour spin model *Heisenberg model (quantum), a model where the spins are treated quantum mechanically using Pauli… …   Wikipedia

• Heisenberg's microscope — exists only as a thought experiment, one that was proposed by Werner Heisenberg, criticized by his mentor Niels Bohr, and subsequently served as the nucleus of some commonly held ideas, and misunderstandings, about Quantum Mechanics. Basic ideas… …   Wikipedia

• Heisenberg, Werner — ▪ German physicist and philosopher Introduction in full  Werner Karl Heisenberg  born Dec. 5, 1901, Würzburg, Ger. died Feb. 1, 1976, Munich, W.Ger.  German physicist and philosopher who discovered (1925) a way to formulate quantum mechanics in… …   Universalium

• Ising model — The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It has since been used to model diverse phenomena in which bits of information, interacting in pairs, produce collectiveeffects.Definition… …   Wikipedia

• Landau–Lifshitz model — In solid state physics, the Landau Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equations describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space… …   Wikipedia

• Majumdar–Ghosh Model — The Majumdar Ghosh model is an extension of the one dimensional quantum Heisenberg spin model in which an extra interaction is added coupling spins two lattice spacings apart such that second neighbor coupling half as strong the first. It is… …   Wikipedia

• n-vector model — The n vector model or O(n) model is one of the many highly simplified models in the branch of physics known as statistical mechanics. In the n vector model, n component, unit length, classical spins are placed on the vertices of a lattice. The… …   Wikipedia

• N-vector model — The n vector model or O( n ) model is one of the many highly simplified models in the branch of physics known as statistical mechanics. In the n vector model, n component, unit length, classical spins mathbf{s} i are placed on the vertices of a… …   Wikipedia