# HEISENBERG MODEL (CLASSICAL)

Werner Heisenberg Model Schrodinger and Heisenberg Model Heisenberg Quotes Heisenberg Effect Heisenberg Walter White Heisenberg's Uncertainty Principle Heisenberg Jewelry Heisenberg's Atomic Theory

# 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.

## Heisenberg model (classical) 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.

## Heisenberg model (classical) 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 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 Ishimori equation and so on.

### Heisenberg model (classical) One dimension

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

### Heisenberg model (classical) Two dimensions

• In the case of long range interaction, $J_{{x,y}}\sim |x-y|^{{-\alpha }}$, the thermodynamic limit is well defined if $\alpha >2$; the magnetization remains zero if $\alpha \geq 4$; but the magnetization is positive at low enough temperature if $2<\alpha <4$ (infrared bounds).
• Polyakov has conjectured that, as opposed to the [1]

### Heisenberg model (classical) 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.

## Heisenberg model (classical) References

1. ^ Polyakov, A.M. (1975). Phys. Lett. B 59. Bibcode:1975PhLB...59...79P. doi:10.1016/0370-2693(75)90161-6 http://www.sciencedirect.com/science/article/pii/0370269375901616 |url= missing title (help).

# Werner Heisenberg Model Schrodinger and Heisenberg Model Heisenberg Quotes Heisenberg Effect Heisenberg Walter White Heisenberg's Uncertainty Principle Heisenberg Jewelry Heisenberg's Atomic Theory

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