Four-point geometrical correlation functions in the two-dimensional 𝑄-state Potts model: connections with the RSOS models

This is a review of the preprint by He, Grans-Samuelsson, Jacobsen and Saleur.

Context
This article is a contribution towards solving the critical 2d Potts model. It focuses on four-point connectivities: observables that are not determined by conformal symmetry, even though critical exponents are known, and which encode finer details of the model such as fusion rules and structure constants. The authors' approach is to obtain the critical model as a continuum limit of a lattice model. The lattice model itself is solved numerically thanks to the underlying affine Temperley-Lieb algebra.

In a previous article using the same approach, Jacobsen and Saleur have determined the spectrums of the four-point connectivities, i.e. the states that propagate from two points to two other points. This implies that the Potts model differs from the solvable CFT that was shown to describe connectivities to a good approximation. The present article investigates the lattice interpretation of that CFT, which includes minimal models as special cases.

Approach and results
The article builds on Pasquier's lattice discretization of minimal models as an RSOS model. Reformulating the RSOS model in terms of loops allows a direct comparison with the Potts model: the two models are found to differ only by the weights of certain configurations. In particular, the weight $$M^{D_{1+\frac{p}{2}}}(k=2l)$$ (Eq. (54)) is much more complicated in the RSOS model. While originally computed for the discrete central charges of minimal models, this weight is a smooth function of the central charge, with finitely many poles. However, in order to compute a four-point connectivity, we should sum over the number $$k$$ of non-contractible loops, which can run up to infinity in the critical limit: we recover the singular behaviour of the solvable CFT as a function of the central charge on the half-line $$c\in (-\infty, 1)$$, with poles that are dense on the half-line.

The article also discovers exact expressions for some (combinations of) $$k$$-amplitudes, i.e. contributions to connectivities of configurations with $$k$$ non-contractible loops. These amplitudes are found to be independent of the lattice size, so their critical limits can be determined from any lattice that is large enough to allow for the desired number of loops. ("Facts of type 2".) Similarly, there are exact expressions for ratios of contributions to different connectivities. ("Facts of type 1".) And finally, these exact results are brought together to explain the nontrivial cancellations that make the spectrums of minimal models (and of the solvable CFT) much simpler than the spectrum of the Potts model.

Implications for solving the 2d critical Potts model
In the critical limit, the Potts model's amplitudes are related to structure constants of the resulting CFT. Some of these structure constants are in principle accessible to analytic bootstrap techniques, based on exploiting the existence of degenerate fields. In order to compare this with the preprint's "facts of type 1", let us formally write the relation between the modules $$\mathcal{W}_{j,z^2}$$ of the affine Temperley-Lieb algebra, and highest-weight representations $$\mathcal{V}_{(r,s)}$$ of the Virasoro algebra, parametrized by Kac table indices that are not necessarily integer, with left and right conformal weights $$(h,\bar h)=(h_{(r,s)},h_{(r,-s)})$$:

\mathcal{W}_{r,e^{2\pi is}} \simeq \bigoplus_{s'\in s+\mathbb{Z}} \mathcal{V}_{(r,s')} $$ Due to the existence of a degenerate field with indices $$(1,2)$$, the structure constants for the Virasoro representations that appear in a given $$ \mathcal{W}_{r,e^{2\pi is}}$$ are expected to be related to one another by analytic bootstrap equations. Therefore, the statement in Section 5.2 that ratios of amplitudes depend only on the module (i.e. on $$ \mathcal{W}_{r,e^{2\pi is}}$$) follows from the critical model's Virasoro symmetry, plus the existence of a degenerate field. Moreover, the ratios $$\alpha_{j,z^2}$$ of amplitudes of different modules are inaccessible to the known analytic bootstrap techniques. The analytic determination of some of these ratios as rational functions of $$Q$$ suggests that the model might nevertheless be analytically solvable.

Validity
The article relies on numerical transfer matrix computations, as explained in Appendix D. The code and the raw numerical results are not publicly available. The article's main claims are about infinite series of quantities: only the first few cases can be computed numerically. To compute the next case in one example, "some 30 years of CPU time was wasted on an unsuccessful attempt".

The main technical results, and the general picture, are pretty convincing, and backed by solid numerical evidence. However, there is still some room for surprises in cases that are numerically inaccessible. After all, when it comes to supergravity amplitudes or to $$\mathcal{N}=4$$ supersymmetric Yang-Mills theory, qualitatively new features have been observed to arise at pretty high orders in perturbative computations.

This article pushes numerical transfer matrix computations to their practical limit. For a compelling confirmation of the results, and an extension to higher orders, other methods may be needed.

Interest
By elucidating the relation between the Potts model and the solvable CFT on the lattice, the article provides powerful tools for studying the Potts model itself. Both theories are made of the same building blocks, but the solvable CFT has a much simpler spectrum, and its very existence implies many nontrivial relations between these building blocks. Moreover, the surprisingly weak dependence of these quantities on the lattice size greatly enhances the power of lattice techniques for computing their critical limits, which are in principle accessible only when the lattice becomes large. This also raises the prospect that more theories of the same type might exist, which would be built by changing the weights of lattice configurations.