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When I recently started to read about conformal field theory, one of the basic examples there is the so called 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. It 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.

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 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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  • $\begingroup$ Kardar's "Statistical Physics of Fields" is an excellent resource for this (as well as Di Francesco's "big yellow book", though it is less pedagogical in my opinion). $\endgroup$ – TotallyRhombus Jan 27 '17 at 15:51
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In CFT, we are interested in the continuum limit, where we can classify classes of models at their critical points.

By means of the Jordan-Wigner transformation one can construct a fermion operator out of the spin operators of the usual 2D Ising model. Then, the continuum critical Ising model is described by a massless real fermion:

$$ S=\frac 12\int d^2 z\left( \psi\bar \partial \psi + \bar \psi \partial \bar \psi \right) $$ from which the correlation function and the stress-energy tensor can be calculated.

You can read more about this in the following references:

(1) Schultz, "Two dimensional Ising model as a soluble problem of many fermions", 1964

(2) Di Francesco, "Conformal Field Theory", 1997

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The correspondence is explained precisely in my answer to What is the difference between scale invariance and self-similarity?

At the level of Euclidean correlations one has for distinct points $x_1,\ldots,x_n$ in $\mathbb{R}^2$, the relation $$ \langle \phi(x_1\cdots\phi(x_n) \rangle=\lim_{\varepsilon\rightarrow 0^+} \epsilon^{-\frac{n}{8}} \langle \sigma_{\lfloor \epsilon^{-1}x_1\rfloor} \cdots\sigma_{\lfloor \epsilon^{-1}x_n\rfloor}\ \rangle\ . $$ Here $\lfloor\cdots \rfloor$ means the integer parts of the coordinates. On the right, one has the correlation of the critical Ising model on the unit lattice. On the left, one has the correlation of the Euclidean Ising CFT.

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