Ising model, same interaction with all spins

[maajdl]

Hello,

I just read a question about the Ising model, and this reminds me of an old interrogation I had long ago. It is simply that:

The Ising model deals with spins interacting only with close neighbors.
I would be interested in a model where all spins interact with each other in exactly the same way.
Would you know if that has been studied, and if there are some references?​

Thanks!

 

[ZapperZ]

... a model where all spins interact with each other in exactly the same way.

What does that mean? The coupling strength with the nearest neighbor is identical to nearest neighbor, to the next nearest neighbor, to the next-next nearest neighbor, etc... etc? This doesn't sound physically reasonable, does it?

Zz.

 

[maajdl]

You are right: this seems not physically reasonable.
This idea came to me 30 years ago, by pure curiosity.
However, it could be that such models have been considered in quantum information theory.

 

[ZapperZ]

You are right: this seems not physically reasonable.
This idea came to me 30 years ago, by pure curiosity.
However, it could be that such models have been considered in quantum information theory.

One can consider ANY toy model that one wants, but the question is, is it physically reasonable? As I've often asked my grad students whenever they think that they want to pursue a line of inquiry, it may be interesting, but is it important?

Zz.

 

[Hypersphere]

I believe this tends to be called the "Ising model on a complete graph". See chapter 3 in these http://ipg.epfl.ch/lib/exe/fetch.php?media=en:courses:doctoral_courses_2010-2011:lecture3.pdf. Since the atomic orbitals tend to be quite localized (few papers even consider next next nearest neighbor interactions) this case tends to be more popular in mathematics (judging by those lecture notes, communication and computer science may have found some applications too).

 

[maajdl]

Thanks Hypersphere!

 

[aro]

There are systems that behave approximately in this way:

Consider a localized (e.g. bound to a defect) electron in a solid whose wavefunction stretches out over many lattice sites. Assume each nucleus (lattice site) has nonzero spin. Now each nuclear spin couples to the spin of the electron (by Fermi contact interaction), with strength proportional to the wavefunction overlap. Hence each pair of nuclear spins experiences a coupling (mediated by the electron spin).
The nuclear-nuclear spin coupling is then proportional to the product of the wavefunction overlap (with the electron) of both nuclei. This yields a long range exchange interaction between nuclear spins.

This situation arises e.g. in semiconductor quantum dots. The model in this particular case is called 'central spin model', you can search for this and find plenty of references.

Also the RKKY interaction describes a similar type of long range coupling, in this case it is coupling between nuclear spins mediated by conduction electrons.

 

[tshuxun]

I have read a book -- statistical mechanics（writen by Tsung-Dao Lee ，1957 Nobel prize）in Chinese(but as I know this book have English edition),he say consider the next neighbor in 1D Ising model can be claculated by matrix method,even a short discussion ,we can know that there should someone else have study that

 

[t!m]

I strongly disagree that the model is irrelevant because it is unphysical (though some have given examples of when it may be physically relevant). The model described is an incredibly important, well-known model system, sometimes known as the "infinite range" Ising model. It is important from a fundamental point of view in statistical mechanics, because it can be solved exactly and the solution is equivalent to that obtained by mean-field theory. Analysis of the model can provide crucial insights into the applicability and success of mean-field solutions of more "physical" model Hamiltonians, in particular related to the idea that mean-field theory becomes exact in infinite dimensions.