Ising model: Difference between revisions

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(2-dimensional Ising model, link to Peierls)
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* [[1-dimensional Ising model]] (exact solution)
* [[1-dimensional Ising model]] (exact solution)
==2-dimensional Ising model==
==2-dimensional Ising model==
Solved by [[Lars Onsager]] in 1944. [[Rudof Peierls]] had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.
Solved by [[Lars Onsager]] in 1944. [[Rudolf Peierls]] had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.
*[http://dx.doi.org/10.1103/PhysRev.65.117 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review '''65''' pp. 117 - 149 (1944)]
*[http://dx.doi.org/10.1103/PhysRev.65.117 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review '''65''' pp. 117 - 149 (1944)]



Revision as of 08:34, 21 April 2008

The Ising model is also known as the Lenz-Ising model. For a history of the Lenz-Ising model see Refs. 1 and 2. The Ising model is commonly defined over an ordered lattice. Each site of the lattice can adopt two states: either UP (S=+1) or DOWN (S=-1).

The energy of the system is the sum of pair interactions between nearest neighbors.

where is the Boltzmann constant, is the temperature, indicates that the sum is performed over nearest neighbors, and indicates the state of the i-th site, and is the coupling constant.

1-dimensional Ising model

2-dimensional Ising model

Solved by Lars Onsager in 1944. Rudolf Peierls had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.

3-dimensional Ising model

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice:

ANNNI model

The axial next-nearest neighbour Ising (ANNNI) model is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.

References

  1. S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics 39 pp. 883-893 (1967)
  2. Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences 59 pp. 267-318 (2005)