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I am not completely sure that this question is appropriate for this mathematical site. But since in the past I did get on MO couple of times nice answers to rather physical questions, I will try. Sorry if it is irrelevant here.

Some time ago I started to read basic things about conformal field theory. One of the basic examples there is the Ising model. It is characterized by certain specific collection of fields on the plane acted by the Virasoro algebra with certain central charge, and by a specific operator product expansion. In the conformal fields literature I read it is claimed that this model comes from the statistical mechanics.

In the literature on statistical mechanics what is called the Ising model is something completely different: one fixes a discrete lattice on the plane, and there is just one field which attaches numbers $\pm 1$ to each vertex of the lattice. (At the beginning I was not even sure that it is the same model of the same Ising :-))

As far as I heard there is a notion of scaling limit when the lattice spacing tends to zero. At this limit (at the critical temperature?) some important quantities (correlation functions?) converge to a limit. My guess is that this scaling limit should be somehow relevant to connect the two Ising models I mentioned above.

Question. Is there a good place to read about explicit relation between the two Ising models? In particular I would be interested to understand how to obtain the operator product expansion and the central charge starting from the statistical mechanics description.

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2 Answers 2

The relationship between the Ising model (spins on a lattice) and conformal field theory holds only in the immediate vicinity of the critical point, when correlation lengths go to infinity and all details on the scale of the lattice constant become irrelevant.

The relationship is explained, for example, in these lecture notes. Let me walk you through them.

The correspondence starts from the Ising model of classical spins on a two-dimensional lattice, which is equivalent to a one-dimensional model of quantum mechanical spin-$1/2$ degrees of freedom (Ising chain). In the lecture notes this mapping is derived from quantum to classical in chapter 2, with the inverse mapping shown in Appendix B. The space and time dimension of the quantum spin chain become the two spatial dimensions of the classical Ising model.

Chapter 3 of the lecture notes then explains how the quantum spin chain can be transformed into a quantum field theory of free spinless fermions in the continuum limit, which is the appropriate limit near the critical point. The path-integral formulation of the field theory then produces the correlation functions by conformal invariance and operator product expansion (chapter 4).

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Thanks a lot, that is very useful guide. But some step is still probably missing for me: I heard that the Ising model in CFT is described by two fields $\sigma$ and $\varepsilon$, where $\sigma$ is the scaling limit of the field $\sigma_i$ on the lattice (in the classical stat. mechanics Ising model), and $\varepsilon$ is limit of something like $\sigma_i\sigma_{i+1}$ (I am not sure). If there is an explanation of this in the literature, I would be very curious. –  semyon alesker May 27 at 8:55
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The energy field $\varepsilon$ (which is the primary field $\psi$) is constructed in section 4.4 and the spin field $\sigma$ (also known as the twist field) in section 4.5. –  Carlo Beenakker May 27 at 10:16

There has been quite a few developments as far as making this relation mathematically rigorous. See this paper and this one by Camia, Garban and Newman as well as this paper by Chelkak, Hongler and Izyurov for the scaling limit of the $\sigma$. As for the scaling limit of the $\varepsilon$ see the thesis by Hongler.

It would be very nice if one could combine these results in order to prove the beginning of the OPE, namely obtaining the correlations for the $\varepsilon$ continuum field by collapsing the continuum $\sigma$'s in pairs. As far as I know this has not been proved yet. In a simplified hierarchical $\phi^4$ model, me and my collaborators are close to deriving a similar OPE thanks to the results in this paper.

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Thank you. In the papers you mentioned the treatment is mathematically rigorous? –  semyon alesker Jul 3 at 2:56
    
Yes. That's what I said. –  Abdelmalek Abdesselam Jul 4 at 13:30

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