# Ising models with many-body interactions

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:

1. 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...

1. 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?
• +1. Thanks Jun !!! Welcome to the site!!! My book gives what I think are all known methods (about 70 of them) for transforming Hamiltonians with many-spin interactions into Hamiltonians with only 2-spin interactions, without changing the ground energy (or state), but I wish I knew a review paper that covers all the work involving Hamiltonians with multi-spin terms (then I'd know all the work on which my book's material can be applied!). So I love this question! – Nike Dattani Jul 26 '20 at 2:59
• +1 Welcome to the site! @NikeDattani, I think your comment deserves a full answer, it looks like an amazing reference! – ProfM Jul 26 '20 at 8:55
• Is there a specific reason (e.g. algorithmic convenience) that you want to use multi-spin interactions? – taciteloquence Jul 26 '20 at 9:51
• @NikeDattani Thanks for the reference! It's really helpful to have an organized list of all the computational reduction techniques to simulate different systems with 2-body interactions like this. I think there were also some works by Tony Cubitt on proving the "universality" of the TFIM, which is closely related. My focus for this question was on more "physical" models where we have spacial locality and symmetries. While computationally, 2-body interaction is "enough", the models I mentioned can have very different phase diagrams, so I was curious about them. – Jun_Gitef17 Jul 26 '20 at 15:05
• @taciteloquence Thanks for asking. In arxiv.org/abs/2001.10045 I was dealing with an artificial model, and was curious about the effects of the many-body interaction it had (especially how it may change the phase transitions). I thought the Ising model would be a good starting point to systematically think about such effects, and searched for literature but couldn't find any. So while I think even the many-body extension of the Ising model could be interesting in its own right, my final interest is actually about a different model (with many-body interactions). – Jun_Gitef17 Jul 26 '20 at 16:11

As a (very) incomplete answer to this question, here is one paper discussing the Ising FM with a plaquette term. Here specifically chosen because there was not a good cluster algorithm for it (so they could try out a new algorithm: self-learning MC).

Junwei Liu, Yang Qi, Zi Yang Meng & Liang Fu, Phys. Rev. B 95, 041101 (2017)
Also arXiv:1610.03137

• Thanks for your answer! I actually already knew the self-learning MC paper. In this study, the Hamiltonian also has a two-body term, destroying the gauge degrees of freedom that exist when only the plaquette terms are considered. So I think it's safe to say that the model they study is essentially a 2-body Ising model with 4-body perturbation, but it's good to know about these individual examples of course. Thanks! BTW, the arXiv paper says "no simple and efficient global update method is known" for the 4-body term. The Wolff algorithm CAN be applied but it's just not THAT efficient, right? – Jun_Gitef17 Jul 26 '20 at 16:35
• @Jun_Gitef17 "The Wolff algorithm CAN be applied but it's just not THAT efficient, right?" sounds like this could be asked as a separate question! – Nike Dattani Jul 27 '20 at 0:18
• taciteloquence: Could you elaborate on why they included the plaquette term? It's because there's no good cluster algorithm for the plaquette term? Why did they include something if there's no good algorithm to treat it? I'm asking from the perspective of someone that hasn't opened the paper (ideally answers on SE would be sufficiently self-contained so that I wouldn't have to read the paper to understand the SE answer!). – Nike Dattani Jul 27 '20 at 0:21
• @NikeDattani I added a parenthetical remark to explain why the added the plaquette term. Jun: I agree, your comment deserves to be its own question. – taciteloquence Jul 27 '20 at 3:28
• Maybe we could add "self-learning MC" to my question "What are the types of QMC?" – Nike Dattani Jul 27 '20 at 3:30