I find it surprisingly difficult to find researches/papers on systematic "many-body interaction" extensions of the Ising model. Can somebody tell me a good review/article etc on this matter that goes through possible extensions of the Ising model in a somewhat systematic way?
The usual $s_is_j$ interaction can be extended in a few different ways:
- Maintaining the product form. In this case, we can imagine interactions like $s_is_js_k$ which would now break the $Z_2$ symmetry, but would recover it when we have even numbers of spins in the interaction. I know that the three-body Ising model has a first-order transition in the fully-connected mean-field theory, and also know that there's a glassy phase for the sparsely-connected mean-field version of it. When we get to four-body interactions in this form, e.g. $s_is_js_ks_l$ with $\langle i,j,k,l \rangle$ in plaquettes, this really resembles the $Z_2$ gauge theory but is slightly different since the model we get by simply extending the normal Ising model would have spins on the sites of the lattice. Still, I was able to find papers talking about the "plaquette Ising model" which I think is exactly the same as what I have in mind here.
Anyway, it would be good to know if there's a systematically sorted reference of these extension models, instead of some bunch of facts like I wrote here.
Also, another way of extending the model could be...
- Maintaining its trend to let spins point in the same direction. We can have energetic terms that favor $k$ spins pointing in the same direction, so the normal Ising model is $k=2$. I think this would also be a natural extension, but as far as I know, I don't even know if this model has a name. My intuition says that when k is large enough the $Z_2$ symmetry breaking should become a first-order transition, but couldn't find any actual study about this. Does anybody know anything about such kind of models?