Uniqueness of the $2d$ critical Ising model

This post is motivated by a request from JHEP to review a recent article by Anton de la Fuente. I am grateful to the author for stimulating correspondence.

 

The conformal bootstrap: analytic vs numerical

 

The critical Ising model is described by a unitary conformal field theory. In two dimensions, that theory is part of a family called minimal models, which can be exactly solved in the analytic bootstrap framework of Belavin, Polyakov and Zamolodchikov. Minimal models are parametrized by two coprime integers 2 ≤ p < q, they are unitary when q = p + 1, and the Ising model is the case (p, q)=(3, 4).

These 2d bootstrap results date back to the 1980s. More recently, the bootstrap method has been successfully used in higher dimensional CFTs, such as the 3d Ising model. While the basic ideas are the same, there are important technical differences between 2d and higher d.
In 2d, the infinite-dimensional Virasoro symmetry algebra yields enough equations for exactly solving minimal models. In higher d, the conformal algebra is finite-dimensional, and yields fewer equations. A popular bootstrapping technique is to also use the inequalities that follow from the assumption of unitarity. In the space of parameters (parameters such as the energy levels), these inequalities carve a region of consistent theories. The game is to make enough assumption for this region to be very small, in which case we can determine the values of the parameters for a given model. In this sense, solving a CFT means showing that it is unique under certain assumptions. In practice, a CFT has infinitely many parameters, and we focus on a finite number of parameters, getting rid of the others using inequalities: the method is approximate, and yields numerical results.

These higher d methods can be applied to 2d models, and their results compared to known exact 2d results. This is a priori interesting not only for testing the methods, but also for learning more about 2d CFT. Not all 2d CFTs are expected to be exactly solvable: minimal models are defined by the assumption that the spectrum is made of finitely many irreducible representations of the Virasoro algebra, but it is less well-known what happens if we allow infinitely many representations.

Case of the 2d Ising model

 

In his recent article, de la Fuente applies the higher d bootstrap techniques to the 2d Ising model. He finds that the model is unique under a number of assumptions: unitarity, conformal and ℤ₂ symmetry, and bounds on the dimensions of a few operators. This allows him to compute the dimensions of these operators with about three significant digits, which is convincing evidence that we can recover the known results with this method.

Aside from validating numerical bootstrap techniques, have we learned something new? To answer this question, let me give a more detailed reminder of known analytic results. The Virasoro algebra comes with a parameter called the central charge, whose value is $c=\frac12$ for the Ising model. We know all unitary representations of the Virasoro algebra with c < 1: so, if we had a Virasoro symmetry algebra with c < 1, we could easily get uniqueness results for the Ising model, optimize them by relaxing the assumptions as much as possible, and understand what happens when uniqueness breaks down. This raises two questions:

• Do we have a Virasoro symmetry algebra? In order to use the numerical bootstrap techniques, de la Fuente assumes that we have global conformal symmetry. But Virasoro symmetry is known to follow from 2d global conformal symmetry, unitarity, and discreteness of the spectrum. Discreteness is not assumed, as it is not needed for the numerical bootstrap. Discreteness is probably not quite necessary for Virasoro symmetry either: it might be possible to do with weaker assumptions, such as de la Fuente’s gap in the spin 2 sector.
• If we have a Virasoro algebra, what is its central charge? We do not know, but maybe we could find out using inequalities from unitarity, as I will argue below.

This suggests that we might seek to prove Virasoro symmetry and find bounds on the central charge, rather than directly constrain parameters such as energy levels. By combining analytic and numerical bootstrap techniques, one might obtain more powerful results. It is not clear how easy this would be, and I will now sketch how one might get started.

 

Combining analytic and numerical bootstrap techniques?

 

We want bounds on the central charge, but the known unitary bootstrap techniques give bounds on conformal dimensions (= energy levels) and on OPE coefficients. So let us relate the central charge to OPE coefficients. Let σ be a scalar primary field, and T the energy-momentum tensor. If we had Virasoro symmetry we would have the OPEs
$$T(y)\sigma(z) = \frac{\frac12 \Delta_\sigma\sigma(z)}{(y-z)^2} + \cdots$$
$$T(y)T(z) = \frac{\frac12 c }{(y-z)^4} + \cdots$$
where Δ_σ is the total (left + right) conformal dimension. The interpretation is that we have the OPE coefficient $\lambda_{\sigma\sigma T}= \frac12 \Delta_\sigma$, with T normalized such that $\left<TT\right> = \frac12 c$. However, if we forget about Virasoro symmetry and use global conformal symmetry instead, the natural normalization becomes $\left<TT\right> =1$, and the OPE coefficient becomes
$$\lambda_{\sigma\sigma T} = \frac{\Delta_\sigma}{\sqrt{2c}} \qquad \text{or equivalently} \qquad c = \frac12 \left(\frac{\Delta_\sigma}{\lambda_{\sigma\sigma T}}\right)^2$$
In order to get an upper bound on c, we therefore need an upper bound on Δ_σ, and a lower bound on λ_σσT. In the article we have marginality bounds such as Δ_σ < 2 for some scalar primary fields: the remaining task is to find lower bounds on λ_σσT.

The bounds can afford to be far from optimal, as we only need to show c < 1 while the Ising model has $c=\frac12$. The numerical implementation of the unitary bootstrap is designed for finding bounds that are very close to exact values: maybe the coarse bounds that we need can be obtained analytically. The hope is that such techniques can lead to uniqueness statements for all unitary minimal models. (Non-unitary CFTs would require different techniques.) Ultimately, the interesting and difficult problem is to discover new unitary CFTs with c ≥ 1.