Talk:Ising model

e & maximal evenness

Is that e^... term the natural base e or is it the e from the rest of the equation? BenFrantzDale 06:48, Jan 31, 2005 (UTC)

It's the natural base. I think this could be made clearer. On an unrelated note, I think the reference to "Maximally Even Sets: A Discovery in Mathematical Music Theory" is bizarre. Was this just put here by the author of that work? The Ising model has an absolutely huge literature. I propose removing that, or at least heavily diluting it with other references to restore some balance. --128.40.213.241 20:06, 20 Feb 2005 (UTC)

Note: You can check who added the referance by clicking "Page history" at the bottom of the article.

What sort of balance is required? People usually talk of balance inregards to NPOV (neutral point of view), but I cannot tell what the POV is. Hyacinth 21:57, 20 Feb 2005 (UTC)

Not that it's a "point of view" in any controversial sense. I think hardly anyone who knows about the Ising model will know about that, so it's funny having it as a further reference without any more general links. It's not something most people charged with writing a sensible summary of the model would include; just someone who's very interested in music. I'll add another link to the page and shut up. --128.40.213.241 18:53, 21 Feb 2005 (UTC)

Ising exactly solved on other lattices and boundary conditions

The Ising model has been exactly solved in 2D on other lattices beyond the square lattice. It has also been solved on the honeycomb lattice, the triangular lattice, and the Kagome lattice. In the isotropic case where all the interaction energies are equal, they reduce in the thermodynamic limit to the square lattice on the positive temperature axis. However, the Ising model has been solved in these other lattices in their anisotropic cases, and so they do not simply reduce to the square lattice result. As an example of a benefit of considering other lattices like the triangular lattice is that you can get expressions for the diagonal correlations which are simpler than those derived for the square lattice by first solving for the expression in the triangular lattice and then taking the limit where the diagonal interaction energy goes to zero. See McCoy and Wu's 1973/2014 book for details. Therefore I think they deserve a discussion of their own.

Furthermore, even for the square lattice, there is the question of boundary conditions. The fully periodic case, on the torus, is the case that was initially solved by Onsager and others. However, the square lattice Ising model has also been solved for cylindrical boundary conditions and even with a non-zero magnetic field interacting with an end row of the cylinder by McCoy and Wu. See their 1973/2014 book for instance. For bulk properties of the square lattice model, the effects of the cylindrical boundary conditions and the boundary magnetic field decay exponentially. However, boundary correlation functions can be considered and explored, something that obviously cannot be done for fully periodic (toroidal) boundary conditions.

Triangular Lattice Ising and Antiferromagnetism

The square lattice Ising model is invariant under the change from ferromagnetism to antiferromagnetism. The same is true for Ising on the honeycomb lattice. This is because both the square and honeycomb lattices are bi-partite. The triangular lattice, however, is not bi-partite, and its exact solution differs whether you're considering a ferromagnetic or antiferromagnetic interaction energy. For example, its critical point is at a finite positive temperature for ferromagnetism but at T=0 for antiferromagnetism. Furthermore, at T=0 there is a finite entropy since the model is geometrically frustrated. This is not seen in the square lattice solution.

Naming

Why is it called the Ising model if it was invented by Lenz? Shouldn't it at least be called the Lenz-Ising model, or even the Lenz-Ising-Onsager model? — Preceding unsigned comment added by 69.222.178.0 (talk) 02:50, 30 July 2018 (UTC)

Because the first appearance of the model is in a paper published by Ising? If people start publishing lots of papers calling the model with names you suggest, perhaps in another 100 years the name "Lenz-Ising model" will become standard. PhysicsAboveAll (talk) 07:14, 9 September 2020 (UTC)

This question is related to the name of the model, so I've written it here under this naming section. The article starts by introducing two different ways to pronounce Ising, the "English" one and the German one. Isn't using an English pronounciation of a German name just wrong? Doesn't the English pronounciation just come from the misunderstanding that the name refers to an English word instead of a German name? Wouldn't it be correct to remove the English pronounciation from the article, or at least show the German pronounciation first? — Preceding unsigned comment added by 88.114.119.5 (talk) 11:55, 22 March 2021 (UTC)

removed the ipa for the english pronounciation. Spacebusdriver (talk) 12:26, 28 June 2022 (UTC)

In dimensions greater than four, the phase transition of the Ising model is described by mean field theory

I think a reference is needed here. Pibieta (talk) 16:20, 8 September 2020 (UTC)

This has been proven rigorously, I believe. By Michael Aizenman perhaps? There should be references in Glimm and Jaffe's book. PhysicsAboveAll (talk) 07:14, 9 September 2020 (UTC)

In dimensions 5 and above, mean field behaviour was proven in 1982 by Aizenman (CMP, vol 86) and Fröhlich (Nuclear Phys. B, vol 200). In dimension 4, this is a recent result by Duminil-Copin and Aizenman. Hairer (talk) 12:58, 9 September 2020 (UTC)