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Classical Heisenberg model - Wikipedia

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In statistical physics, the classical Heisenberg model, developed by Werner Heisenberg, is the $n=3$ case of the n-vector model, one of the models used to model ferromagnetism and other phenomena.

The classical Heisenberg model can be formulated as follows: take a d-dimensional lattice, and place a set of spins of unit length,

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

on each lattice node.

The model is defined through the following Hamiltonian:

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

where

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

is 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

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

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, and so on.

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.

^ Polyakov, A.M. (1975). "Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields". Phys. Lett. B 59 (1): 79–81. Bibcode:1975PhLB...59...79P. doi:10.1016/0370-2693(75)90161-6.