Critical 2d Potts model

In two dimensions, the critical Potts model is believed to have an infinite-dimensional symmetry algebra, which makes it accessible to analytic methods of two-dimensional conformal field theory. There may be a realistic prospect of solving it, in the sense of analytically (or semi-analytically) computing observables such as cluster connectivities.

Motivations

The Potts model is a statistical model that generalizes the Ising model, and includes percolation as a special case. Therefore, it generalizes and unifies statistical models of fundamental interest.

In two dimensions and in the critical limit, the model may be solvable. The torus partition function is already known analytically. Powerful methods such as the conformal bootstrap can be used. In higher dimensions, the model's lack of unitarity would be an obstacle to using standard conformal bootstrap methods. But in two dimensions, conformal symmetry is more powerful and non-unitary models can be solved.

Type of project

Tools: Given the known information on the spectrum that was obtained by other means, it may be possible to complete the solution of the model using the bootstrap method, analytic and/or semi-analytic. Interpreting the results in terms of statistical observables such as cluster connectivities and/or spin correlation functions may however require going back to the statistical description.

Chances of success: The known results on describing some observables exactly or approximately can surely be generalized to some extent. However, it is not known whether it is possible to build a consistent CFT from observables in the Potts model, and if so, whether that CFT includes all interesting observables. The scope of the project is therefore not precisely defined.

Length and difficulty: The known spectrum contains several qualitatively different infinite series of states. The statistical model contains various interesting observables. And even after solving the model for generic numbers of states, understanding special cases would require a lot of work. Solving the 2d critical Potts model is something in between a project and a field of research.

Known results

Some aspects of the model are reviewed in the literature, in the loop model approach or in the conformal booststrap approach.

Spectrum

The torus partition function is known exactly, see ref. or ref. for reviews. We write $\chi _{\langle r,s\rangle }$ the character of a diagonal degenerate representation with $r,s\in \mathbb {N} ^{*}$. We write $\chi _{(r,s)}^{N}$ the character of the generally non-diagonal representation ${\mathcal {V}}_{\Delta _{(r,s)}}\otimes {\bar {\mathcal {V}}}_{\Delta _{(r,-s)}}$. The torus partition function reads

$Z^{\text{Potts}}=\sum _{s=1}^{\infty }\chi _{\langle 1,s\rangle }+(Q-1)\sum _{s\in \mathbb {N} +{\frac {1}{2}}}\chi _{(0,s)}^{N}+\sum _{r=2}^{\infty }\sum _{s\in {\frac {1}{r}}\mathbb {Z} }D'_{r,s}\chi _{(r,s)}^{N}$

where we define

$D'_{r,s}={\frac {1}{r}}\sum _{r'=0}^{r-1}e^{2\pi ir's}w(r\wedge r')$

and

$w(d)=q^{2d}+q^{-2d}+(-1)^{d}(Q-1)\quad {\text{with}}\quad {\sqrt {Q}}=q+q^{-1}$

However, in contrast to what happens in CFTs such as minimal models, the partition function does not fully characterize the spectrum. The partition function characterizes the eigenvalues of the generator $L_{0}$ of the Virasoro algebra, but this is not enough for characterizing the action of the full algebra when that generator is not diagonalizable. And the spectrum of the Potts model is known to involve some representations where $L_{0}$ is not diagonalizable.

Observables

In addition to the spectrum, the statistical model's observables include spin correlation functions and cluster connectivities. Spin correlation functions are a priori defined only when the number of states is an integer $Q=2,3,4,\dots$. Spin clusters allow the model to be reformulated such that $Q$ is an arbitrary complex number.

It is natural to assume that in the critical limit, spin correlation functions tend to correlation functions of primary fields in a CFT. Somewhat less naturally, this assumption is also extended to cluster connectivities, at least for 2, 3 and 4-point connectivities.

Fusion rules

In order to compute correlation functions, we should know not only the spectrum, but also which states contribute to which correlation functions: in other words, the fusion rules. A piece of information is already known: which states contribute to four-point connectivities in various channels. However, four-point connectivities are only a very special type of correlation functions.

Correlation functions

Four-point connectivities can be compared to results from the lattice approach or from Monte-Carlo calculations. They have been determined using a semi-analytic bootstrap approach. The connectivity $P_{aaaa}$ is the unique solution where the field $(0,{\tfrac {1}{2}})$ propagates in all three channels, in addition to non-diagonal fields. The connectivity $P_{aabb}$ is characterized by adding three fields in the s-channel: $(0,{\tfrac {1}{2}})$, and the degenerate fields $(1,2)$ and $(1,3)$.

Work to be done

Determining the spectrum

The structure of Virasoro representations is already known. What is missing is the action of the symmetric group $S_{Q}$. In the closely related case of the $O(n)$ model, this has been determined using methods that should also work in the Potts model.

Bootstrapping four-point functions

Connectivites are very special types of four-point functions. But in order to solve the model, we should in principle compute four-point functions of arbitrary fields. Once the spectrum is known and bootstrap techniques are available, the remaining missing ingredients are the fusion rules.

It is possible to adopt the null assumption that all states can appear in all channels of all four-point functions, but this may lead to unresolvable ambiguities. It would probably be better to guess the fusion rules, based on the known cases and on algebraic constraints from Virasoro symmetry, and to test the guesses by checking crossing symmetry.

Interpreting correlation functions

Assuming that crossing-symmetric four-point functions are found, we have a consistent CFT. There remains the problem of interpreting its correlation functions in terms of the original Potts model.

The CFT will a priori include four-point functions that correspond to four-point connectivities. Linear combinations of four-point functions may correspond to spin four-point functions. Spin correlation functions are originally defined only if the number of states $Q$ is integer, but they may well make sense of any complex $Q$ by analytic continuation. The interpretation of higher correlation functions is however less clear.

Spin clusters

Spin clusters differ from Fortuin-Kasteleyn clusters. Spin cluster connectivities are described by fields with Kac indices $({\tfrac {1}{2}},0)$, versus $(0,{\tfrac {1}{2}})$ for FK cluster connectivities. There should also exist an energy field, i.e. a diagonal field with Kac indices $(1,2)$, such that

$\left\langle ({\tfrac {1}{2}},0)^{3}\right\rangle \neq 0\quad ,\quad \left\langle ({\tfrac {1}{2}},0)^{2}(1,2)\right\rangle \neq 0$

By standard bootstrap results, this implies that the $(1,2)$ field is not degenerate. This may not contradict the existence of another field with indices $(1,2)$, which would be degenerate. This would then imply that the field $({\tfrac {1}{2}},0)$ is diagonal, otherwise its fusion with the degenerate field would produce non-diagonal fields with half-integer spins. This would also imply that the energy field belongs to a logarithmic representation with Jordan cells of rank three.

To bootstrap these non-vanishing three-point functions, one could take inspiration from FK cluster connectivities, and try to compute the spin cluster connectivity $P_{aabb}^{S}$ by adding three fields in the s-channel: $({\tfrac {1}{2}},0)$, the degenerate field $(1,3)$, and the non-degenerate field $(1,2)$.

Limits and special cases

Cases of particular interest, including percolation, occur for rational values of the central charge, where the algebraic and analytic structures of CFT become more complicated. Understanding the behaviour of the spectrum in such cases is already challenging. Moreover, in correlation functions, we expect that the limit is finite due to cancellations of infinitely many singularities.