# 2d Ising model in CFT and statistical mechanics

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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 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.

This post imported from StackExchange Physics at 2014-08-11 14:59 (UCT), posted by SE-user MKO

edited Sep 19, 2014
@Hunter: I am familiar with this book a little bit. It does not discuss the statistical mechanics approach at all, and rather takes the central charge and OPE for granted.

This post imported from StackExchange Physics at 2014-08-11 14:59 (UCT), posted by SE-user MKO
Oh ok, I thought it did, but I'm might be wrong. Sorry

This post imported from StackExchange Physics at 2014-08-11 14:59 (UCT), posted by SE-user Hunter

This is one of the major difficulties in conformal field theory, deriving the relation between specific models and the conformal field description, which comes from another direction entirely. It is largely open, but it is extremely important, because right now as far as I know, it is still true that the identifications of conformal field theories with particular lattice models is done heuristically, by comparing central charges and numerical identity of correlation functions. It is of central importance to do it more rigorously, but I haven't kept up with the math literature here, and it is very active in the last decade, and they might have some new insight regarding this (although from the scattering of papers I read, I don't think so).

@RonMaimon, it reminds me that,  I heard rumors saying that there is a strong indication that 3D Ising model is going to be solved using CFT techniques, as suggested by this paper. I'm not equipped with the key techniques used in the paper, so I'd appreciate it if you can take a look and do a review of its viability. I'll propose a import request.[Imported]

I highlighted in bold the part of the question that is not yet clearly answered in the two existing answers.

Having looked at section 12.1 of Senechal it looks like it is the same Ising model in conformal field theory as in statistical mechanics, only they study the continuum limit and not the exact solution, i.e. instead of a lattice it is a continuous space...

This post imported from StackExchange Physics at 2014-08-11 14:59 (UCT), posted by SE-user bolbteppa
@bolbteppa: You are right. But the relation is non-trivial, and this is what I want to understand in greater detail.

This post imported from StackExchange Physics at 2014-08-11 14:59 (UCT), posted by SE-user MKO

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Most of the recent mathematical progress are based on the Schramm-Loewner evolution (SLE), a random process describing a random curve in the plane. A physicist friendly introduction by Cardy is http://arxiv.org/abs/cond-mat/0503313 This paper contains a review of the relation with 2d CFT. In particular, the central charge and critical exponents can be derived from SLE (work of Lawler, Schramm, Werner). So to rigorously derive some results on some lattice models, it is enough to relate the scaling limit of the lattice model to some SLE. This was done by Smirnov for the 2d Ising model in http://arxiv.org/abs/0708.0039 (Werner, resp. Smirnov, received a Fields medal in 2006, resp. 2010, for this and related works)

As I don't think that the full data of a 2d CFT has been completely mathematically axiomatized, there is actually no rigorous derivation that the critical point of 2d lattice models is described by a 2d CFT. Nevertheless, the work described in the preceding paragraph enables us to identiy the critical behavior of many quantities, to prove their conformal invariance and to extract from them some rigorous numerical results.

EDIT: In fact, it seems that the existence of the scaling limit of n-points correlation functions of 2d Ising and their conformal invariance have been recently proved: see  http://www.math.columbia.edu/~hongler/icmp-spin-spin.pdf for a review.

answered Aug 11, 2014 by (4,890 points)
edited Sep 19, 2014 by 40227

SLE is extremely indirect, it's a classification of conformally invariant statistically growing paths in 2d. The identifications based on SLE are rigorous, but they are not the same thing as constructing the coarse-grained fields directly from the lattice quantities. A "proper answer" to this question should directly identify the Virasoro generators and field operators from an Ising model description, and this is still not done anywhere.

Ron Maimon : I agree.

Scale invariance plus unitarity implies conformal invariance in $d=2$ is the key idea. See the discussion here where Riva and Cardy present an example where scale invariance does not imply conformal invariance (their example is not reflection positive). For the lattice counterpart  $c<1$ unitary minimal models such as the 2d Ising model, at criticality one has scale invariance and unitarity. All one needs to do is to extract the central charge as well as the critical exponents (aka as scaling dimensions of the chiral primaries) from the lattice description. Hasn't this been done, at the very least for the 2d Ising model?

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The 2-dimensional Ising model without external magnetic field is exactly solvable, hence one can calculate essentially everything about it. (But see the discussion below.) In particular, the critical exponents are known exactly. This has been used by Belavin et al. (1984, Appendix E) to identify the critical scaling limit of the Ising model with the minimal conformal field theory that carries a unitarizable representation of the Virasoro algebra with central charge 1/2. They also refer to the book [15] by McKoy and Wu on the two-dimensional Ising model, which gives the state in 1973.

The same can be done with other exactly solvable 2-dimensional lattice theories; the above paper mentions in a footnote on p.365 the case of the 3-state Potts model.

Something similar seems to happen with the (not exactly solvable) Ising model in 3 dimensions; see http://physicsoverflow.org/22085/ and http://www.physicsoverflow.org/23095/

answered Sep 17, 2014 by (12,890 points)
edited Sep 19, 2014

These references are well known but do not really answer the question. The question is about the relation between the 2d lattice Ising model and the 2d "Ising" CFT. Even if the 2d lattice Ising model is often called "exactly solvable", we do not know how to calculate everything about it. It is possible to compute the partition function, to show that there exists a phase transition, to compute the critical exponents and in particular the two spins correlation functions. But, as far as I know, it is not known  how to compute the three spins correlations and to check that at the critical point, they have the typical form required by conformal invariance. Rather, the insight of Polyakov and al was to assume that the 2d lattice model at the critical point has a scaling limit which is a 2d CFT. Once one is in the world of 2d CFT, the things are kind of easy because ot the conformal symmetry and a 2d minimal CFT is indeed kind of "exactly solvable". The correct  2d CFT scaling limit of a given lattice model is guessed by comparing the type of observables and the critical exponents. But the rigorous derivation of the existence of the scaling limit and the proof of the conformal invariance of this limit are not known in complete generality (see my answer and for example the introduction of http://arxiv.org/abs/0910.2045 for recent progresses).

Things are similar for the 3d Ising model. People assume that the critical point is described by a 3d CFT, they try to identify this 3d CFT and once it is done, they study this 3d CFT to predict results about the 3d Ising model. But they do not even try, until now, to really show that the scaling limit of the critical lattice 3d Ising model is the identified 3d CFT.

What is the typical form required for the 3 spins correlation function to prove conformal invariance? Also, I do not understand what the (principle or analytical?) difficulties in calculating it are, as the partition function is available?

@Dilaton: it is possible to recover correlation functions from the partition function only if one knows the partition function of the system with external sources. In this case, it is possible to recover the correlation functions by derivating the partition function with respect to sources. For the Ising model, to have external sources is equivalent to have an external magnetic field.  The partition function of the Ising model with an external magnetic field is not known.

It is a remark due to Polyakov that in a CFT, the form of three points correlations function is determined up to a constant, $\frac{1}{(x_1 - x_2)^{\Delta_1 + \Delta_2 - \Delta_3}(x_1 - x_3)^{\Delta_1 + \Delta_3 - \Delta_2}(x_2 - x_3)^{\Delta_2 + \Delta_3 - \Delta_1}}$.

In fact, it seems that the existence of the scaling limit of n-points correlation functions of 2d Ising and their conformal invariance have been recently rigorously proved: see  http://www.math.columbia.edu/~hongler/icmp-spin-spin.pdf for a review, but it is a consequence of the recent work mentionned in my answer and not of old exact results.

I agree with @40227 that there are issues that remain to match the two descriptions of the 2D Ising model. Solving the 2D Ising model in a magnetic field has the same difficulty level as that of the 3D Ising model. (see the last section of Boris Kastening' s paper) The awesome computation of the spontaneous magnetisation by Yang is the best one can do at this point (I think). Another interesting result is that of Zamolodchikov who showed that the critical Ising model perturbed by the magnetic field can be solved using methods of integrable QFTs. I wonder if some version of this result can be realised on the lattice.

While the 2D Ising model in a magnetic field is not known to be solvable, the dilute A3 model is a lattice model with a critical point in the Ising universality class, and is solvable in a magnetic field. See Physics Letters B322 (1994) 198-206. The continuum limit is an $E_8$-based solvable field theory. The quantum version of the dilute A3 model gives a reasonably realistic description of a quasi-one-dimensional Ising ferromagnet realizable in cobalt niobate. See R. Coldea et al., Science 327 (2010), 177-180.

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