# Reference for the 3d Ising model

+ 3 like - 0 dislike
466 views

Before I read the papers on the numerical solutions of the 3d Ising model (such as the work of El Showk et. al) I would like to know a bit more about the correspondende between the 3d Ising model and conformal field theory, in specific the field content, the scaling dimensions, correlation functions etc (I would like to see why they are what they are rather than a list as shown in the work I include above).

Therefore, are there any good introductory references for the 3d Ising model and CFT? I am quite familiar with the 2d case and its solution (also there are many references available for this) so I am only interested in 3d.

+ 4 like - 0 dislike

Section 3.8 of Cardy's book Scaling and Renormalization in Statistical Physics (only the first few pages are publicly online) describes scaling fields and scaling dimensions for the 3D Ising model, giving in particular the relations $\Delta_\sigma=1/2+\eta/2$ and $\Delta_\epsilon=3-1/\nu$ figuring on p.25 of the paper Bootstrapping mixed correlators in the 3D Ising model by Filip Kos, David Poland, David Simmons-Duffin. The analogous relations to the fields in More constraining conformal bootstrap by Ferdinando Gliozzi are $\Delta_\phi=1/2+\eta/2$ and $\Delta_{\phi^2}=3-1/\nu$. Numerical values for comparison are given in [12,13] of the latter paper.

Together with the general outline of the relations between the renormalization flow and CFTs in the last chapter of Cardy's book, this seems enough to make the identifications on a heuristic level. I haven't seen any more definite work on the relations, and would be interested in anything better that you can come up with in your own search.

answered Mar 25, 2016 by (15,608 points)

@ArnoldNeumaier I do not do research on this field, I am just purely interested in learning about it. I have asked for references also some senior faculty in my department and if they provide me with something good I will let you know.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.