### A.7.3 Attainable sizes

The largest sizes $L$ attainable in the numerical work are essentially limited by the exponentially growing dimensions of the corresponding transfer matrices. For the first method (see section 4.3.1) there is however an additional complication, since we do not obtain the amplitudes $A_i$ directly but acquire them by solving the linear system (4.31).

Suppose first that there are $K$ eigenvalues coupling to $P_P$, and call $\Lambda_{\text{max}}$ and $\Lambda_{\text{min}}$ the largest and smallest of those. If all computations are done to $d$ decimal digits, and we wish to obtain the final values of the amplitudes to $d_0$ digits, then we must chose $d$ large enough that $(\Lambda_{\text{min}}/\Lambda_{\text{max}})^K > 10^{d_0-d}$. Indeed, we will need the separation between operators to take the values $l = 1, 2, \dots, K$ in order to have enough equations to solve the linear system for $A_i$. And assuming all $A_i$ to be of the order unity, the smallest term on the right-hand side of (4.31) should not be below the level of numerical precision.

Similar considerations can be made for the case where $K$ is so large that we are unable (or do not want) to determine all the amplitudes. In that case, we truncate the sum in (4.31) at some $i_{\text{max}}$, and provide $i_{\text{max}}$ different values of $l$. Since the formula is no longer exact, we must make sure to use it in the proper asymptotic regime. We therefore take $l = i_{\text{min}}$, $i_{\text{min}} + 1, \dots, i_{\text{max}}$ for some suitable large $i_{\text{min}}$. In practice we have chosen $i_{\text{min}} = 100$ and repeated the computation for a slightly higher value (e.g., $i_{\text{min}} = 120$), in order to check that the $A_i$ were unchanged to the desired numerical precision.

In our largest computations, for $L = 7$, we determined more than 500 amplitudes in this way. This computation needed as much as $d = 4000$ digits of numerical precision and required the work of $\sim 100$ computers simultaneously for a period of $\sim 3$ months. See section 5.2.2 for further details on this computation.

For the second scalar product method (see section 4.3.2) it was enough to work with standard double precision ($d = 16$ digits). This method is altogether more efficient and could be done up to $L = 11$ in a much more reasonable time (at most a few days on a single computer).

### A.7.4 Jordan blocks

With the first method it is possible to determine the Jordan block structure of correlation functions, by using a fit of the form (4.32). To do this in practice requires some reasonable prior knowledge about where to look for the Jordan blocks, and about their expected order $r_i$. The analytical understanding of indecomposability in the Temperley-Lieb algebra has grown steadily over recent years, including in the affine TL case [28], so the search for Jordan blocks is certainly not devoid of any guidance. However, one can also take a more naive numerical approach to the problem.

Imagine that we are interested in some specific value $Q = Q_0$, where indecomposability is known to occur. Consider a correlation function $P_P$ for which we have previously established, for *generic* $Q$, which transfer matrix sectors $V_{\ell,k,m}$ occur with a non-zero amplitude. At the non-generic $Q_0$, some of the eigenvalues from different sectors will collide, as can be seen by explicit diagonalisation using the methods of section A.4 (or predicted analytically). The number of eigenvalues that collide at a given value $\Lambda_i$ is a natural guess
for the largest possible rank $r_i$ of the Jordan block corresponding to the generalised amplitude (4.32). Trying the corresponding Ansatz for different values of $i_{\min}$ (see above), we can examine whether the amplitudes are consistently determined (and non-zero) and arrive at a quantitative understanding of the Jordan block structure.

### A.7.5 Extrapolation of amplitudes

Whichever method is used for obtaining the eigenvalue amplitudes $A_i$ on the cylinder, the results need to be extrapolated to infinite size ($L \to \infty$) in order to be compared with predictions of conformal field theory. We shall see in appendix B that in many cases the finite-size results are quite far away from their CFT limits, more often than not by a factor in the range 2–5. Obtaining reliable extrapolations is important, in particular because we only possess rather small sizes ($L \le 11$). Moreover, most amplitudes exhibit mod 2 parity effects, so that even and odd $L$ must be treated separately.

Some guidance can be taken from our results for the Ising model where larger sizes can be obtained ($L \le 16$). We find the following method to provide quite accurate results. First we produce finite-size results with $2a/L$ equal to or close to a constant, typically $1/2$. Since it turns out that some amplitudes vanish exactly, by symmetry reasons, for $2a/L = 1/2$ precisely, we often take $2a = (L-1)/2$ for odd $L$, and $2a = (L-2)/2$ for even $L$. These results are then corrected by a conformal factor, namely the powers of $\sin(2\pi a/L)$ appearing in (4.13). Finally, we extrapolate the corrected results, separately for odd and even $L$, by fitting all available data against a polynomial in $1/L$ (or in some cases leaving out the first or first few data points). The reliability of the extrapolation can then be estimated by comparing the two parities, which are expected to give identical results.

## A.8 The split-attach algebra

The scalar product method described in section 4.3.2 and appendix A.2 requires us to find the left eigenvectors of $T$. One way to attain this is to take the transpose of the right eigenvectors of the transposed matrix $T^t$.

The scalar product method described in section 4.3.2 and appendix A.2 requires us to find the left eigenvectors of $T$. One way to attain this is to take the transpose of the right eigenvectors of the transposed matrix $T^t$.

In the join-detach algebra for the FK cluster model, $T$ acts on basis states which are set partitions of $L$ points. The basic building block of the sparse matrix factorisation scheme (3.1) is the action of the elementary join and detach operators, $J_i$ and $D_i$, on these states. If this was presented in explicit matrix form, it would of course be trivial to take the transpose of the corresponding matrices. The point is, however, that when we use an iterative diagonalisation scheme (such as the Arnoldi method; see section A.7.1), it is inefficient to represent $J_i$ and $D_i$ as explicit matrices. Rather, we just need to provide their action on any state: for any in-state $v_{in}$, the definition of $J_i$ (or $D_i$) in terms of set partitions provides the corresponding out-state $v_{out} = J_i v_{in}$ (or $D_i v_{in}$). To treat the transpose in a similar setting means that we should instead answer the question: given $v_{out}$, what are the possible $v_{in}$ that could lead to it (and with which transition weights)?
We therefore formally define the split operator $S_i = J_i^t$ and the attach operator $A_i = D_i^t$ as the transpose of the join and detach operator, respectively, and provide now their transition elements in terms of the basis states.

### A.8.1 Without marked clusters

We first consider the analogue of the usual join-detach algebra, that is, without any marked clusters.

The operator $S_i$ performs a split between sites $i$ and $i + 1$. In particular, $S_i$ is zero on states in which sites $i$ and $i + 1$ are not connected (i.e., in the same block of the set partition). If the two sites are connected, $S_i$ acts by breaking up the block containing $i$ and $i + 1$ in all possible ways. To be precise, assume the block consist of $k \ge 2$ points, ordered cyclically: $i_1 < i_2 < \dots < i_k < i_1$ (the inequalities being considered modulo $L$), with $i_1 = i$ and $i_2 = i + 1$. Place one “separator” between $i_1$ and $i_2$. Place a second separator at the position of any of the above inequalities, i.e., between $i_\ell$ and $i_{\ell+1}$ for any $\ell = 1, 2, \dots, k$, with the subscripts considered mod $k$. Note that this includes the case where the two separators are at identical positions. Then break up the block by cutting it at the positions of the two separators. This produces the $k$ possible out-states of $S_i$, which each occur with transition weight 1. Note that the block is being broken into precisely two pieces in the $k-1$ cases where the two separators are different,<fnref n="11" /> whereas nothing happens (i.e., $S_i$ acts as the identity) in the remaining case where the separators coincide.

The operator $A_i$ attaches a singleton at site $i$ to certain other blocks of the set partition, as we now describe. In particular, $A_i$ is zero on states in which site $i$ is not a singleton. If $i$ is an singleton, $A_i$ leaves the state unchanged with weight $Q$, and connects $i$ to any other “visible” block in the set partition with weight 1. To determine whether a block is visible by $i$, draw the set partition as a hypergraph with vertex set $1, 2, \dots, L$ embedded in the half-infinite cylinder (concretely, in figure 1 this would be the part of the cylinder situated to the left of the current time slice, with the vertices on the slice). If a block is adjacent to the face where $i$ resides, the block is said to be visible. Another, less formal way to state the same, is that we connect $i$ to any block that is not nested within another block, seen from the position of $i$.

It is straightforward (albeit somewhat laborious) to verify that the split-detach algebra thus defined satisfies the algebraic relations which are the transpose of those defining the join-detach algebra:

$$S_i^2 = S_i, \qquad (A.21a)$$

$$A_i^2 = AD_i, \qquad (A.21b)$$

$$S_i A_j S_i = S_i \text{ for } j = i, i + 1, \qquad (A.21c)$$

$$A_i S_j A_i = A_i \text{ for } j = i - 1, i. \qquad (A.21d)$$

<footnote n="11">¹¹To be precise, if one separator is placed between $i_{s_1-1}$ and $i_{s_1}$ and the other between $i_{s_2-1}$ and $i_{s_2}$ (with $s_2 \neq s_1$), then the resulting blocks are $\{i_{s_1}, i_{s_1+1}, \dots, i_{s_2-1}\}$ and $\{i_{s_2}, i_{s_2+1}, \dots, i_{s_1-1}\}$, with all subscripts considered mod $k$.</footnote>
### A.8.2 With marked clusters

We now provide the definition of the split-attach algebra in the representation with marked clusters having a fixed spin label. This is analogous to section A.2.

The split operator $S_i$ acts as zero, unless sites $i$ and $i+1$ belong to the same block (which can be unmarked or marked). Otherwise:

* If $i$ and $i+1$ belong to the same unmarked block, $S_i$ leaves the block unchanged or splits it into two unmarked blocks, as before, with a total of $k$ possibilities for a block of size $k$.

* If $i$ and $i+1$ belong to the same marked block, $S_i$ leaves the block unchanged or splits it into two blocks, of which one is unmarked and the other keeps the same mark as the original block. Both choices for which of the two blocks should carry the mark are realised.

The attach operator $A_i$ acts as zero, unless site $i$ is an unmarked singleton. Otherwise:

* With weight $Q$, the operator $A_i$ leaves the singleton block at $i$ unchanged and unmarked.

* With weight 1, it marks site $i$ by each possible colour of the mark which is used in none of the other blocks in the partition.

* With weight 1, it attaches site $i$ to each of its visible blocks (which can be marked or unmarked).

The two first rules cover the cases where the corresponding detach operation is trivial, in the sense that $D_i$ would turn a (marked or unmarked) singleton into an unmarked singleton. The third rule covers the non-trivial case, where the corresponding detach operation acts on a (marked or unmarked) block of size $\ge 2$.

With these modifications one can again check that the defining relations (A.21) are satisfied, but within the larger representation that allows clusters to be marked.

## B Basic checks

In this appendix we check our general method for computing the amplitudes $A_i$ against a series of exact results. First we discuss the easy case of two-point functions and quantify the conformal content of the lattice spin operator. Next we compare our results for the $s$-channel spectrum of four-point functions against exact solutions for $Q = 2, Q = 4$, and the limit $Q \to 0$.

### B.1 Lattice observables and scaling fields

As discussed in the main text, one of the basic difficulties in our problem is that lattice observables correspond, in the scaling limit, to combinations of scaling fields weighed with appropriate powers of the cut-off. The question we want to investigate briefly here is how this might affect the measurement of amplitudes.
According to (2.8), the two-point function of the lattice spin operator becomes, in the geometrical formulation, proportional to the probability that two points belong to the same cluster. The leading dimension controlling the large distance behavior of this probability is $h_{1/2,0}$ and thus we expect, in the scaling limit, to have on the cylinder

$$ P_{aa} \xrightarrow{\text{scaling}} \left( \frac{\xi}{(1-\xi)^2} \frac{\bar{\xi}}{(1-\bar{\xi})^2} \right)^{h_{1/2,0}} . \qquad (B.1) $$

We have here used the same notation (4.11) as for the four-point function, so $w_1 = ia$, $w_3 = i(a+x) + l$, and $w_{13} = -ix - l$. As before, we set $\xi = e^{-2\pi(l+ix)/L}$, and the above dependence on $\xi, \bar{\xi}$ follows from a reasoning similar to the one leading to (4.12). We find numerically that only the sector $F_{0,-1}$ contributes, with

$$ F_{0,-1} = \frac{q^{-c/24} \bar{q}^{-c/24}}{P(q)P(\bar{q})} \sum_{e \in \mathbb{Z}} q^{h_{e+1/2,0}} \bar{q}^{h_{e+1/2,0}}. \qquad (B.2) $$

As is usually the case in lattice models, the order operator on the lattice is not purely represented by a conformal field of weight ($h_{1/2,0}, h_{1/2,0}$). In general, one expects instead that this field is only the first in a sum of the type

$$ \mathcal{O}_a(\sigma_i) \sim \sum_{e \in \mathbb{Z}} A_e \epsilon^{2h_{e+1/2,0}} \Phi_{e+1/2,0}(z) \Phi_{e+1/2,0}(\bar{z}) + \text{descendants}, \qquad (B.3) $$

where $z, \bar{z}$ are the complex coordinates corresponding to $\sigma_i$, we have introduced the notation $\Phi_{r,s}$ for conformal chiral fields with weight $h_{rs}$, and $\epsilon$ stands for the lattice cutoff ($\epsilon$ was set equal to unity so far). In the limit where $L, l \to \infty$ (that is, $L/\epsilon, l/\epsilon \to \infty$ if $L, l$ are measured in units of length) the contributions of the excited (higher) spin fields will scale away. For finite values of the parameters, they are unavoidably there. Note that

$$ h_{3/2,0} - h_{1/2,0} = \frac{m+1}{2m}. \qquad (B.4) $$

This is larger for small values of $m$, i.e., smaller values of $Q$: therefore, the closer we will get to $Q=0$, the faster these contributions will decrease with $\epsilon$ ($\epsilon/L$ on the cylinder).

We do not know how the $Q$-dependent amplitudes $A_e$ in (B.3) behave a priori, but we have studied numerically the two-point function in order to understand the amount of “mixing” of the order operator with the leading correction at weight $h_{3/2,0}$. We have checked that this mixing is small and decreases significantly with increasing $L$.

As an example, we give here results for $Q = 3/2$ — a case for which $m$ is irrational, and all operators in the spectrum can be uniquely identified. The dimension of the order parameter is $h_{1/2,0} = 0.0583892$, and by (B.1) the two-point function in the conformal limit expands as

$$ P_{aa}^{\text{scaling}} \propto \xi^{h_{1/2,0}} \bar{\xi}^{h_{1/2,0}} (1 + 0.116778(\xi + \bar{\xi}) + 0.0136372\xi\bar{\xi} + 0.0652078(\xi^2 + \bar{\xi}^2) + \dots), \qquad (B.5) $$

where all numerical constants have been given to six-digit precision. The leading terms at momentum $h - \bar{h} = 1$ and $h - \bar{h} = 2$ are easily identified as the ground states of the
corresponding momentum sectors. With no knowledge of the possible mixing with the term $\xi^{h_{3/2,0}}\bar{\xi}^{h_{3/2,0}}$ one would look for the $\xi\bar{\xi}$ term in (B.5) in the lattice data as the contribution of the first excited state in the sector at vanishing momentum, but a careful analysis of the amplitudes as well as the scaling dimensions shows this is not what happens. On the lattice, the two point function contains, in addition to the terms in (B.5), a term in the bracket proportional to $(\xi\bar{\xi})^{h_{3/2,0}-h_{1/2,0}}$. This follows from (B.3), which leads to

$$ P_{aa}^{\text{lattice}} \propto \frac{1}{|z|^{4h_{1/2,0}}} \left[ 1 + \left( \frac{A_1}{A_0} \right)^2 \left( \frac{\epsilon}{|z|} \right)^{4(h_{3/2,0}-h_{1/2,0})} + \dots \right], \quad (B.6) $$

and, after mapping on the cylinder, to

$$ P_{aa}^{\text{lattice}} \propto \left(\frac{2\pi}{L}\right)^{4h_{1/2,0}} \left[ \left(\frac{\xi}{(1-\xi)^2}\right)^{h_{1/2,0}} \left(\frac{\bar{\xi}}{(1-\bar{\xi})^2}\right)^{h_{1/2,0}} \right. \\ \left. + \left(\frac{A_1}{A_0}\right)^2 \left(\frac{2\pi\epsilon}{L}\right)^{4(h_{3/2,0}-h_{1/2,0})} \left(\frac{\xi}{(1-\xi)^2}\right)^{h_{3/2,0}} \left(\frac{\bar{\xi}}{(1-\bar{\xi})^2}\right)^{h_{3/2,0}} + \dots \right]. \quad (B.7) $$

The amplitude ratio $A_1/A_0$ is non-universal, but, for a given lattice, it is a fixed quantity. The term $L^{-4(h_{3/2,0}-h_{1/2,0})}$ goes to zero as $L$ goes to infinity, guaranteeing the disappearance of the unwanted terms in this limit. The question is how much this might affect the results for $L$ finite.

Measurement of the amplitudes — or rather the ratio with respect to the leading term — gives the following results:

<table><thead><tr><th>L</th><th>ξ + ξ̄</th><th>ξξ̄</th><th>ξ² + ξ̄²</th><th>(h<sub>3/2,0</sub>, h<sub>3/2,0</sub>)</th></tr></thead><tbody><tr><td>5</td><td>0.1009697881</td><td>0.004693387836</td><td>0.05633421252</td><td>-0.0007594762012</td></tr><tr><td>6</td><td>0.1044947727</td><td>0.006252504965</td><td>0.05534757764</td><td>-0.0006074838333</td></tr><tr><td>7</td><td>0.1070768500</td><td>0.007528362715</td><td>0.05578623767</td><td>-0.0004817486169</td></tr><tr><td>8</td><td>0.1089775943</td><td>0.008549110109</td><td>0.05665273685</td><td>-0.0003834115301</td></tr><tr><td>9</td><td>0.1103983091</td><td>0.009361639062</td><td>0.05759817991</td><td>-0.0003078006915</td></tr><tr><td>10</td><td>0.1114793562</td><td>0.010010585922</td><td>0.05849501339</td><td>-0.0002497183860</td></tr><tr><td>11</td><td>0.1123166454</td><td>0.010532656068</td><td>0.05930089355</td><td>-0.0002048250690</td></tr><tr><td>12</td><td>0.1129759889</td><td>0.010956414072</td><td>0.06000781281</td><td>-0.0001698028289</td></tr><tr><td>∞</td><td>0.11679</td><td>0.13644</td><td>0.06519</td><td>&lt; 10<sup>-6</sup></td></tr><tr><td>CFT</td><td>0.116778</td><td>0.0136372</td><td>0.0652078</td><td>0</td></tr></tbody></table>

The extrapolation to $L \to \infty$ was made via a polynomial of order 7 in $1/L$, using all data points, since there are no discernable parity effects in this case. We observe that on one hand the expected ratios converge to their conformal values to a very good precision (4 or 5 digits). On the other, the one that is not expected — corresponding to the coupling to $(h_{3/2,0}, h_{3/2,0})$ — decreases fast, and converges to a value close to zero.<fnref n="12" /> This example

<footnote n="12">¹²To make this last extrapolation, we have *not* used the knowledge of (B.7) concerning the exact power-law dependence on $L$. Rather we performed just the usual polynomial fit, in order to establish the same methodology for other less trivial cases.</footnote>
shows that one can obtain fine extrapolations, even though for the sizes considered in this paper (and the value $Q = 3/2$ taken here) the mixing of the field ($h_{3/2,0}$, $h_{3/2,0}$) with the order parameter has a rather large relative coefficient of the order of $10^{-2}$.

## B.2 The case $Q = 2$

We now consider the case $Q = 2$ (and later $Q = 4$) to check the consistency of our approach, and assess the quality of convergence of the amplitudes in the four-point functions. While in general, the geometrical probabilities cannot be expressed in terms of simple four-point functions in rational CFTs, the situation is better for these two values of $Q$. This has to do with the relationship [18] between correlation functions of spins in the Potts model and the geometrical objects, see eq. (2.9). We stress that for $Q$ arbitrary, the left-hand sides of these equations are only formally defined: it is in fact the right-hand sides that give them a meaning. In the $Q = 2$ case the first relation (2.9a) reads simply

$$G_{aaaa} = P_{aaaa} + P_{aabb} + P_{abba} + P_{abab}, \quad \text{for } Q = 2. \qquad (B.8)$$

On the other hand, recall from (2.7) that

$$G_{aaaa} = \langle \prod_{i=1}^{4} (Q \delta_{\sigma_i, a} - 1) \rangle \qquad (B.9)$$

As discussed in A.6, for $Q = 2$, the order parameter $Q\delta_{\sigma_i, a} - 1$ coincides with the Ising spins $S_i = \pm 1$, so this four-point function is nothing but the four-point function of the spin operator in the Ising model [65]

$$\begin{aligned} \langle \sigma(z_1)\sigma(z_2)\sigma(z_3)\sigma(z_4) \rangle &= \frac{1}{2} \left| \frac{z_{13}z_{24}}{z_{12}z_{23}z_{34}z_{41}} \right|^{1/4} \left( |1 + \sqrt{1-z}| + |1 - \sqrt{1-z}| \right) \quad (B.10) \\ &= \frac{1}{2} |z_{13}|^{-1/4} |z_{24}|^{-1/4} \frac{1}{|z(1-z)|^{1/4}} \left( |1 + \sqrt{1-z}| + |1 - \sqrt{1-z}| \right). \end{aligned}$$

We also recall the structure constant $C_{\sigma\sigma\epsilon} = \frac{1}{2}$, where $\epsilon$ denotes the energy operator; the latter appears as the non-trivial part of the scaling limit of $S_iS_{i+1}$. The four-point function (B.10) involves two conformal blocks, corresponding to the fusion channels $\sigma\sigma \sim 1$ and $\sigma\sigma \sim \epsilon$. Expanding the function G as in (4.6) gives then

$$\begin{aligned} |z|^{1/4} G(z, \bar{z}) = 1 &+ \frac{1}{4} z^{1/2} \bar{z}^{1/2} + \frac{1}{16} (z^{1/2} \bar{z}^{3/2} + z^{3/2} \bar{z}^{1/2}) \\ &+ \frac{1}{64} z^{3/2} \bar{z}^{3/2} + \frac{1}{64} (z^2 + \bar{z}^2) + \dots \end{aligned} \qquad (B.11)$$

The normalised correlation function on the lattice should then be

$$\begin{aligned} &1 + s^2(\xi\bar{\xi})^{1/2} + (s^2 - s^4)(\xi\bar{\xi})^{1/2}(\xi + \bar{\xi}) + \frac{s^4}{4}(\xi^2 + \bar{\xi}^2) \\ &\qquad + (s^2 - 2s^4 + s^6)(\xi\bar{\xi})^{3/2} + \frac{s^8}{16}(\xi\bar{\xi})^2 + \dots, \end{aligned} \qquad (B.12)$$

where $s = \sin \frac{2\pi a}{L}$, and the extra terms arise from the conformal mapping as in (4.12).
<figure><img src="image_10.png" /><figcaption>Figure 9. The simple modules $\mathcal{X}_{0,i}$ and $\mathcal{X}_{0,-1}$ are the tops of these two diagrams representing the structure of the standard modules for $q = e^{i\pi/4}$.</figcaption></figure>

## B.2.1 Reduction from the generic case

We first checked — by computing the four-spin correlation function numerically for the ordinary (spin representation) Ising model on the cylinder — that the identity (B.8) holds exactly in finite size; details are given in section A.6. This computation is certainly a rather stringent test of the program that determines the $P_{a_1,a_2,a_3,a_4}$ numerically. Note that the (large) set of eigenvalues that contribute to the geometrical correlations for generic values of $Q$ reduces drastically — as expected — when we consider the combination (B.8). In algebraic terms, generically irreducible representations of the Jones algebra become reducible, and a complex structure of submodules of the relevant $\overline{W}_{j,z^2}$ develops. It turns out that only two simple quotients contribute to the Ising model: $\mathcal{X}_{0,i}$ and $\mathcal{X}_{0,-1}$, which are obtained as the ‘tops’ of chains of modules according to the diagrams in figure 9 (for a detailed discussion of the emergence of simple modules of the Jones algebra describing minimal models when $q$ is a root of unity, see [29]).

The continuum limit of the simple modules of the Jones algebra is fully expressed in terms of irreducible modules $X_{r,s}$ of the Virasoro algebra

$$ \mathcal{X}_{0,i} \mapsto X_{1,1} \otimes \bar{X}_{1,1} \bigoplus X_{21} \otimes \bar{X}_{2,1} = X_{h=0} \otimes \bar{X}_{h=0} \bigoplus X_{h=1/2} \otimes \bar{X}_{h=1/2}, \quad (B.13a) $$

$$ \mathcal{X}_{0,-1} \mapsto 2X_{12} \otimes \bar{X}_{12} = 2X_{h=1/16} \otimes \bar{X}_{h=1/16}, \quad (B.13b) $$

where the factor 2 indicates that, in fact, the representation splits into two isomorphic direct summands for the subalgebra generated by $e_i$ and $u^2$, and $X_{r,s}$ denotes the irreducible Virasoro module with conformal weight $h_{rs}$ (and, e.g., character given by the well-known Rocha-Caridi formula [66]).

We checked that the combination (B.8) receives contributions only from the tops in the diagrams in figure 9, which corresponds, in the continuum limit, to the required channels of the minimal Ising model. Note however that individual probabilities in (B.8), unlike their sum, do involve contributions outside the simple representations.
## B.2.2 Numerical checks of the amplitudes

We now turn to the question of the convergence of the amplitudes measured on the lattice. Surprisingly, it seems this question has not been investigated much in the past (see [67] for early work). Since the Ising model is much easier to handle numerically than the general FK cluster model, we can study in this case much larger sizes (up to $L_{max} = 16$), and explore in particular the nature of the convergence of the coefficients in the expansion (B.12).

We present in the following figures 10–11 two different ways of handling the data. First, we consider the case where $a$ is changed as $L$ increases. For definiteness we consider $2a = \frac{L}{2}$ when $L$ is even and $2a = \frac{L-1}{2}$ when $L$ is odd. In both cases, $\sin \frac{2\pi a}{L} \to 1$ as $L \to \infty$, but we should of course expect even/odd parity effects in $L$ because of the different choices of $2a$.

A priori the aim would be to compute the amplitude ratios, with respect to the ground state amplitude, corresponding to the five last terms in (B.12), namely the contributions to $G_{aaaa}$ of $\epsilon$, $(L_{-1} + \bar{L}_{-1})\epsilon$, $T + \bar{T}$, $\bar{L}_{-1}L_{-1}\epsilon$, and $T\bar{T}$, respectively. However it turns out that $(L_{-1} + \bar{L}_{-1})\epsilon$ and $T + \bar{T}$ correspond to eigenvalues that are exactly degenerate in finite size, so the corresponding contributions cannot be disentangled. We thus list the amplitude ratios in (B.12) that we can access numerically along with their corresponding analytical predictions:

$$ \epsilon : s^2 \qquad (B.14a) $$

$$ (L_{-1} + \bar{L}_{-1})\epsilon + (T + \bar{T}) : 2 \left( s^2 - \frac{3}{4}s^4 \right) \qquad (B.14b) $$

$$ \bar{L}_{-1}L_{-1}\epsilon : s^2 - 2s^4 + s^6 \qquad (B.14c) $$

$$ T\bar{T} : \frac{s^8}{16} \qquad (B.14d) $$

The four panels of figure 10 show the corresponding amplitudes ratios, where in each case we have divided the ratio by the expected analytical result just given. It is seen that in each case the result tends to 1 after extrapolation, thus confirming the whole analysis.

A number of remarks can be made about figure 10. First, the extrapolations have to be carried out using some amount of common sense. In some cases the obvious non-monotonicity of the finite-size results (panel d) makes it clear that one should leave out the first few points from the fits. In other cases, (panel c) including all points in a high-order polynomial fit leads to the best results. Second, the comparison between fits through even and odd sizes appears to be a good measure for the “error bar” on the extrapolated value. Third, there is a tendency for the finite-size effects to grow as one goes to higher descendents. Thus, if we were limited to smaller sizes — such as $L_{max} = 11$, as would be the case for the FK cluster model at generic values of $Q$ — it should be expected that the last data point might very well be off the true extrapolated value by a factor of 2 or more.

It is also interesting to study what happens when $a$ is fixed while $L$ is increased. In this case, one does not expect the amplitude ratios to converge to the four-point CFT values reported in (B.14), since the latter require $a, L \gg 1$ (in units of the lattice spacing) with fixed value of the ratio $\frac{a}{L}$ (and hence $s$ fixed). To illustrate this, we consider for instance
<figure><img src="image_11.png" /><figcaption>**Figure 10.** Measures of the first four amplitude ratios in $G_{aaaa}$ for the Ising model, divided by the expected analytical results. The respective panels show the contributions from (a) the energy operator $\epsilon$, (b) its first descendent $(L_{-1} + \bar{L}_{-1})\epsilon$ along with the stress tensor $T + \bar{T}$, (c) its second descendent $\bar{L}_{-1}L_{-1}\epsilon$, and (d) the quantity $T\bar{T}$. All data is shown against $1/L$. The extrapolations for even and odd $L$ are shown respectively as blue and orange curves.</figcaption></figure>

the case of $2a = 2$ fixed, with $L$ increasing — note that we do not expect parity effects using this protocol. To make the analysis comparable to the one above, we again divide the ratio by the CFT predictions (B.14). The results are shown in figure 11, to be compared with the previous figure 10. Clearly the convergence is no longer towards one. It is however remarkable that the numerical data still appear to converge to a finite value. For the energy operator and its first two descendents (panels a, b, c) we get fine extrapolations to the values 0.9292, 0.9289 and 0.9292 respectively, which are all compatible among themselves. For $T\bar{T}$ the extrapolations are more problematic, but still appear to converge to a finite value ≈ 0.53, which definitely appears to be different from the previous one.

We are not sure whether these values can be interpreted within CFT — neither whether the observed convergence is real or only apparent. Repeating the computations for $2a = 3$, a similar extrapolation gives the consistent values 0.9686 and 0.9680 for the energy field and its first descendent, whereas $T\bar{T}$ gives another value ≈ 0.71.

### B.3 The case $Q = 4$

As pointed out in [1], the Potts model at $Q = 4$ is particularly interesting, since in this case all geometrical correlation functions can be expressed in terms of spin correlation functions. This provides analytical results for all the $P_{abcd}$, and thus provides a simple benchmark
<figure><img src="image_12.png" /><figcaption>**Figure 11.** Same as figure 10, but for a fixed distance $2a = 2$ between each pair of operators. The panels are as in figure 10. The extrapolations, shown as blue curves, now uses sizes $L$ of both parities.</figcaption></figure>

against which to test our approach. There is however a drawback to this idea: numerics for the Potts model at $Q = 4$ are known to be affected by stronger than usual corrections to scaling due to the presence of a marginal operator. As a result, convergence to the expected amplitudes appears to be slower than for other values of $Q$, especially small ones.

In any case, we start by observing that the Potts model partition function, can be expressed in the continuum limit as the sum [11, 68],

$$Z_{\text{Potts } Q=4} = \sum_{n \in \mathbb{Z}} F_{2n,1} - \frac{1}{2} F_{2n+1,1} + \frac{3}{2} F_{2n,-1}, \qquad (\text{B.15})$$

where the generating functions are obtained by taking the limit $m \to \infty$ of our general result (4.24)

$$F_{j,1} = \frac{1}{\eta \bar{\eta}} \sum_{k \in \mathbb{Z}} q^{(j+k)^2/4} \bar{q}^{(j-k)^2/4}, \qquad (\text{B.16a})$$

$$F_{j,-1} = \frac{1}{\eta \bar{\eta}} \sum_{k \in \mathbb{Z}} q^{(j+k+1/2)^2/4} \bar{q}^{(j-k-1/2)^2/4}. \qquad (\text{B.16b})$$

In contrast with the case of generic $Q$ [11], we see that only a very small subset of the $F_{j,e2i\pi p/M}$ generating functions contribute. This corresponds to the fact — which we checked explicitly in finite size, and which has a simple representation theoretic interpretation; see below — that of all the usual Jones algebra modules, only those corresponding to $z^2 = \pm 1$ play a role at $Q = 4$.
### B.3.1 Exact results via the Ashkin-Teller model

After these preliminaries, we now consider four-point functions per se. In [1], some of the probabilities at $Q = 4$ are given based on results in [18], combined with the careful reading of a paper by Al.B. Zamolodchikov [69]. Calling $\tau_1, \tau_2$ the two spins of the Ashkin-Teller model, we introduce, following [69], the four quantities

$$R_1 = \langle \tau_1 \tau_2 \tau_2 \tau_1 \rangle, \qquad (B.17a)$$

$$R_2 = \langle \tau_1 \tau_1 \tau_2 \tau_2 \rangle, \qquad (B.17b)$$

$$R_3 = \langle \tau_1 \tau_2 \tau_1 \tau_2 \rangle, \qquad (B.17c)$$

$$G = \langle \tau_1 \tau_1 \tau_1 \tau_1 \rangle; \qquad (B.17d)$$

note that $G$ here must not be confused with $G$ in (4.1). It is then expected that<fnref n="13" />

$$P_{aabb} \propto G + R_2 - R_1 - R_3, \qquad (B.18a)$$

$$P_{aaaa} \propto -G + R_1 + R_2 + R_3, \qquad (B.18b)$$

$$P_{abab} - P_{abba} \propto R_1 - R_3. \qquad (B.18c)$$

The quantities on the right-hand side are then calculated in [69]. We introduce first

$$F_0(z) \equiv [z(1-z)]^{-1/8} \theta_3^{-1}(q), \qquad (B.19)$$

where $z = \frac{z_{12}z_{34}}{z_{13}z_{24}}$ as usual. We shall also need the Jacobi theta functions

$$\theta_3(q) = \sum_{n=-\infty}^{\infty} q^{n^2}, \qquad (B.20a)$$

$$\theta_2(q) = \sum_{n=-\infty}^{\infty} q^{(n+1/2)^2}. \qquad (B.20b)$$

These two functions are used to define $q(z)$ implicitly via

$$z = \frac{\theta_2^4(q)}{\theta_3^4(q)} \qquad (B.21)$$

Note that the function $q(z)$ is analytic in $z$:<fnref n="14" />

$$q(z) = \frac{z}{16} + \frac{z^2}{32} + \frac{21z^3}{1024} + \frac{31z^4}{2048} + \frac{6257z^5}{524288} + \frac{10293z^6}{1048576} + \frac{279025z^7}{33554432} + \frac{483127z^8}{67108864} + \dots \qquad (B.22)$$

<footnote n="13">¹³Note there is a slight change of notations between [1] and [69]: $R_1$ and $R_2$ are interchanged.</footnote>

<footnote n="14">¹⁴Thus $q$ must not be confused with either the nome $q$ in generating functions of conformal weights, not with the quantum group deformation parameter $q$.</footnote>
We then set

$$G = F_0 \bar{F}_0 \sum_{m,n=-\infty}^{\infty} q^{(\beta_+ m + \beta_- n)^2} \bar{q}^{(\beta_+ m - \beta_- n)^2}, \qquad (B.23a)$$

$$R_1 = F_0 \bar{F}_0 \sum_{m,n=-\infty}^{\infty} q^{(\beta_+ m + \beta_- (n+1/2))^2} \bar{q}^{(\beta_+ m - \beta_- (n+1/2))^2}, \qquad (B.23b)$$

$$R_2 = F_0 \bar{F}_0 \sum_{m,n=-\infty}^{\infty} (-1)^m q^{(\beta_+ m + \beta_- n)^2} \bar{q}^{(\beta_+ m - \beta_- n)^2}, \qquad (B.23c)$$

$$R_3 = F_0 \bar{F}_0 \sum_{m,n=-\infty}^{\infty} (-1)^m q^{(\beta_+ m + \beta_- (n+1/2))^2} \bar{q}^{(\beta_+ m - \beta_- (n+1/2))^2}. \qquad (B.23d)$$

Finally we choose

$$\beta_+ = 1, \qquad (B.24a)$$

$$\beta_- = \frac{1}{2}. \qquad (B.24b)$$

The claim is then that the $z, \bar{z}$ dependent part of the four-point function (the function $G$ of section 4.1) is given by corresponding combinations of the functions $G$ and $R_i$. So for instance

$$P_{aabb} \propto |z_{12}z_{34}|^{-1/4} G_{aabb}(z, \bar{z}), \qquad (B.25)$$

with

$$G_{aabb}(z, \bar{z}) = G + R_2 - R_1 - R_3 \qquad (B.26) \\ = 2F_0 \bar{F}_0 \left( \sum_{m,n=-\infty}^{\infty} q^{(2m+n/2)^2} \bar{q}^{(2m-n/2)^2} - q^{(2m+n/2+1/4)^2} \bar{q}^{(2m-n/2-1/4)^2} \right).$$

What we must do then is to expand the combinations (B.23) in powers of $z$. This gives expressions of the form

$$(z\bar{z})^{-1/8} \sum_{\Delta\bar{\Delta}} z^{\Delta}\bar{z}^{\bar{\Delta}}. \qquad (B.27)$$

The point is that the set of weights $(\Delta, \bar{\Delta})$ is the set of conformal weights appearing in the s-channel expansion of the four point function. (It is not necessarily the set of primary fields, as the exact result does not distinguish between primaries and descendants. Hence our use of the notation $\Delta$ is slightly abusive when compared with the general case.) This must be the same set as the set of conformal weights contributing to the lattice observable. Hence, we must compare the sets occurring in (B.23) with the sets determined directly. These sets can be determined numerically, as for $Q$ generic, and are found to be:

$$\mathrm{Spec} \, P_{aabb} \subset \mathrm{Spec} \, W_{0,1}/W_{1,1} + \mathrm{Spec} \, W_{0,-1} + \mathrm{Spec} \, W_{2,1}/W_{3,1} + \dots, \qquad (B.28a)$$

$$\mathrm{Spec} \, P_{aaaa} \subset \mathrm{Spec} \, W_{0,-1} + \mathrm{Spec} \, W_{2,1}/W_{3,1} + \dots, \qquad (B.28b)$$

$$\mathrm{Spec} \, (P_{abab} - P_{abba}) \subset \mathrm{Spec} \, W_{2,-1} + \dots, \qquad (B.28c)$$

$$\mathrm{Spec} \, (P_{abab} + P_{abba}) \subset \mathrm{Spec} \, W_{2,1}/W_{3,1} + \dots. \qquad (B.28d)$$
Here the quotients of modules have a direct meaning in terms of subtracting eigenvalues common to various sectors in finite size. Like for other non-generic values of $q$, the modules contributing to the probabilities are no longer irreducible, and the $Q = 4$ results correspond to taking the simple irreducible tops. Indeed we find in general that $\text{Spec } W_{2k+1,1} \subset \text{Spec } W_{2k,1}$ for $k = 1, 2, \dots$.

We have moreover established numerically that $\text{Spec } (P_{abab} - P_{abba})$ does not contain $\text{Spec } W_{4,-1}$, which would otherwise have been a viable candidate in view of symmetries and the results for generic $Q$. For these reasons we conjecture that the complete result generalising (B.28) should in fact read:

$$ \text{Spec } P_{aabb} = \text{Spec } W_{0,1}/W_{1,1} + \text{Spec } W_{0,-1} + \sum_{k \ge 1} \text{Spec } W_{2k,1}/W_{2k+1,1}, \quad (\text{B.29a}) $$

$$ \text{Spec } P_{aaaa} = \text{Spec } W_{0,-1} + \sum_{k \ge 1} \text{Spec } W_{2k,1}/W_{2k+1,1}, \quad (\text{B.29b}) $$

$$ \text{Spec } (P_{abab} - P_{abba}) = \text{Spec } W_{2,-1}, \quad (\text{B.29c}) $$

$$ \text{Spec } (P_{abab} + P_{abba}) = \sum_{k \ge 1} \text{Spec } W_{2k,1}/W_{2k+1,1} + \dots \quad (\text{B.29d}) $$

### B.3.2 Probability $P_{aaaa}$

First we consider $P_{aaaa}$. The spectrum predicted from the decomposition in (B.28) is

$$ (q\bar{q})^{1/24} F_{0,-1} = 2q^{1/16} \bar{q}^{1/16} (1 + q + \bar{q} + (q\bar{q})^{1/2} + q\bar{q} + \dots), \quad (\text{B.30a}) $$

$$ (q\bar{q})^{1/24} (F_{2,1} - F_{3,1} + \dots) = q\bar{q} + q^{1/4}\bar{q}^{9/4} + q^{9/4}\bar{q}^{1/4} + \dots \quad (\text{B.30b}) $$

Corresponding to this we find

$$ -G + R_1 + R_2 + R_3 = (z\bar{z})^{-1/8} \left( \begin{aligned} & \sqrt{2}z^{1/16}\bar{z}^{1/16} + \frac{1}{16\sqrt{2}}(z^{1/16}\bar{z}^{17/16} + z^{17/16}\bar{z}^{1/16}) \\ & + \frac{1}{8\sqrt{2}}(z\bar{z})^{9/16} + \dots - \frac{1}{128}z\bar{z} - \frac{1}{512}(z^{1/4}\bar{z}^{9/4} + z^{9/4}\bar{z}^{1/4}) + \dots \end{aligned} \right), \quad (\text{B.31}) $$

and thus, after mapping to the cylinder

$$ P_{aaaa} \propto (\xi\bar{\xi})^{1/16} + \frac{\cos^2 \frac{2\pi a}{L}}{4\sqrt{2}} (\xi\bar{\xi})^{1/16} (\xi + \bar{\xi}) + \frac{\sin^2 \frac{2\pi a}{L}}{2\sqrt{2}} (\xi\bar{\xi})^{9/16} + \dots \\ - \frac{(4 \sin^2 \frac{2\pi a}{L})^{15/8}}{128\sqrt{2}} \xi\bar{\xi} - \frac{(4 \sin^2 \frac{2\pi a}{L})^{19/8}}{512\sqrt{2}} (\xi\bar{\xi})^{1/4} (\xi^2 + \bar{\xi}^2) + \dots, \quad (\text{B.32}) $$

where we recall that

$$ \xi = e^{-2\pi(l+ix)/L}, \quad \bar{\xi} = e^{-2\pi(l-ix)/L}. \quad (\text{B.33}) $$

The first and third terms in this expression are associated with the largest eigenvalue in $F_{0,-1}$ at zero momentum (this is denoted $V_{100}$ in section A.4). The second term also belongs to $F_{0,-1}$, but at non-zero momentum (namely $V_{101}$). Finally, the fourth and fifth terms are
the leading ones in $F_{2,1}$ (in respective momentum sectors $V_{200}$ and $V_{202}$). To associate these terms to definite eigenvalues, for each size $L$, requires going through the detailed analysis of section A.5. We moreover note that one has to be careful in the comparison, since on the lattice we do not individually observe the amplitudes of conjugate terms, such as the two terms in $(\xi\bar{\xi})^{1/16}(\xi + \bar{\xi})$, since they correspond to the same eigenvalue. Therefore, to make the comparison we must divide the numerically observed amplitude by two in such cases (cf. remark 6)—this has been done tacitly below and in subsequent similar cases. This leads to the following results for the second to fifth terms in (B.32):

<table><thead><tr><th>L</th><th></th><th>(ξξ)<sup>1/16</sup>(ξ + ξ̄)</th><th>(ξξ̄)<sup>9/16</sup></th><th>ξξ̄</th><th>(ξξ̄)<sup>1/4</sup>(ξ<sup>2</sup> + ξ̄<sup>2</sup>)</th></tr></thead><tbody><tr><td rowspan="3">5</td><td>lattice</td><td>0.01025716807</td><td>0.2499038005</td><td>-0.1702401995</td><td>-0.05899812501</td></tr><tr><td>CFT</td><td>0.01688067229</td><td>0.3197920460</td><td>-0.0615759664</td><td>-0.02928111207</td></tr><tr><td>ratio</td><td>0.6076</td><td>0.7815</td><td>2.7647</td><td>2.0149</td></tr><tr><td rowspan="3">7</td><td>lattice</td><td>0.005692994702</td><td>0.2821337922</td><td>-0.1820689962</td><td>-0.06496561673</td></tr><tr><td>CFT</td><td>0.008753198131</td><td>0.3360469943</td><td>-0.0675747438</td><td>-0.03294025194</td></tr><tr><td>ratio</td><td>0.6504</td><td>0.8396</td><td>2.6943</td><td>1.9722</td></tr><tr><td rowspan="3">9</td><td>lattice</td><td>0.003595596367</td><td>0.2949643343</td><td>-0.1820598325</td><td>-0.06747242144</td></tr><tr><td>CFT</td><td>0.005330469599</td><td>0.3428924513</td><td>-0.0701787272</td><td>-0.03455627732</td></tr><tr><td>ratio</td><td>0.6745</td><td>0.8602</td><td>2.5942</td><td>1.9525</td></tr></tbody></table>

We see that the first two columns come out satisfactorily, and the ratios can be rather convincingly extrapolated to a number close to one, despite of the small number of sizes. For the higher-order terms (the last two columns of the table) the convergence is definitely slower, but still goes in the right direction. We have already seen (witness figure 10 in the Ising case) that ratios of the order of 2 at small sizes ($L \le 9$ here) are not uncommon, and it is quite plausible that also these columns could be extrapolated to one, provided one could obtain a few more sizes. In conclusion, the test of $P_{aaaa}$ is fully compatible with the exact results.

### B.3.3 Probability $P_{aabb}$

We next consider the probability $P_{aabb}$ of a pair of “short” clusters. The spectrum predicted from the decomposition in (B.28) is

$$ (q\bar{q})^{1/24} F_{0,-1} = 2q^{1/16} \bar{q}^{1/16} (1 + q + \bar{q} + (q\bar{q})^{1/2} + q\bar{q} + \dots), \\ (q\bar{q})^{1/24} (F_{0,1} - F_{1,1} + F_{2,1} - F_{3,1} + \dots) = 1 + (q\bar{q})^{1/4} + (q\bar{q})^{1/4}(q + \bar{q}) \\ + q\bar{q} + (q\bar{q})^{1/4}(q^2 + \bar{q}^2) + \dots \quad (B.34) $$

Correspondingly, we find

$$ G - R_1 + R_2 - R_3 = (z\bar{z})^{-1/8} \left( -\sqrt{2}(z\bar{z})^{1/16} - \frac{1}{16\sqrt{2}}(z\bar{z})^{1/16}(z + \bar{z}) + \dots \right. \\ \left. + 1 + \frac{1}{2}(z\bar{z})^{1/4} + \frac{1}{16}(z\bar{z})^{1/4}(z + \bar{z}) + \dots \right), \quad (B.35) $$

and thus

$$ P_{aabb} \propto -\sqrt{2}(\xi\bar{\xi})^{1/16} + \dots \\ + 1 + \frac{(4 \sin^2 \frac{2\pi a}{L})^{3/8}}{2} (\xi\bar{\xi})^{1/4} + \dots \quad (B.36) $$
The first term is the leading eigenvalue in $F_{0,-1}$, corresponding in finite size to the largest eigenvalue in $V_{100}$. The second and third terms are the two leading eigenvalues in $\bar{F}_{0,1}$, corresponding to the two largest eigenvalues in $V_{000}$. The comparison with numerics — presented in the by now familiar format — here comes out as:

<table><thead><tr><th>L</th><th></th><th>(ξξ̄)<sup>1/16</sup></th><th>(ξξ̄)<sup>1/4</sup></th></tr></thead><tbody><tr><td rowspan="3">5</td><td>lattice</td><td>-1.342154491</td><td>0.6006793776</td></tr><tr><td>CFT</td><td>-1.414213562</td><td>0.8098363105</td></tr><tr><td>ratio</td><td>0.9490</td><td>0.7417</td></tr><tr><td rowspan="3">7</td><td>lattice</td><td>-1.400211084</td><td>0.7065845314</td></tr><tr><td>CFT</td><td>-1.414213562</td><td>0.8250340627</td></tr><tr><td>ratio</td><td>0.9901</td><td>0.8564</td></tr><tr><td rowspan="3">9</td><td>lattice</td><td>-1.431374714</td><td>0.7598690826</td></tr><tr><td>CFT</td><td>-1.414213562</td><td>0.8312967743</td></tr><tr><td>ratio</td><td>1.0121</td><td>0.9141</td></tr></tbody></table>

It again appears convincing that the ratios will converge to one, i.e., that the numerical results confirm the analytical ones.

### B.3.4 The combination $P_{abab} - P_{abba}$

As a last example we discuss antisymmetric combination $P_{abab} - P_{abba}$ of the probabilities of having two “long” propagating clusters, as shown in figure 2.

It is easy to expand the spectrum of exponents arising from (B.28) on the one hand: the first few terms are

$$(q\bar{q})^{1/24} F_{2,-1} = (q\bar{q})^{9/16} (q + \bar{q} + q^2 + \bar{q}^2 + 2q\bar{q} + \dots) + (q\bar{q})^{1/16} (q^3 + \bar{q}^3 + \dots).$$

The powers of $q, \bar{q}$ on the right-hand side are expected to be exactly the exponents appearing in the sum (B.27). Using the foregoing discussion and the expressions of $R_1, R_3$, we can on the other hand identify these exponents by performing a direct expansion whose first few terms are:

$$R_1 - R_3 = \frac{(z\bar{z})^{-1/8}}{128\sqrt{2}} \left\{ (z\bar{z})^{9/16} \left( z + \bar{z} + \frac{25}{32}(z^2 + \bar{z}^2) + \frac{9}{16}z\bar{z} + \dots \right) \right. \\ \left. \qquad + (z\bar{z})^{1/16} \left( \frac{1}{16}(z^3 + \bar{z}^3) + \dots \right) \right\}.$$

This plays the role of the function $G$ in our general discussion (4.12). It is easy to check that the two sets agree. Moreover, we have also checked that there is no gap in the spectrum, that is, all exponents predicted from the generating function (B.28) are indeed present with non-zero coupling constant.

To check the numerical values of the amplitudes, we need as usual to map the four-point function on the cylinder. We find the first few terms

$$P_{abab} - P_{abba} \propto (\xi\bar{\xi})^{9/16}(\xi + \bar{\xi}) + \left[ \frac{9}{4} - \frac{9 \sin^2 \frac{2\pi a}{L}}{4} \right] (\xi\bar{\xi})^{25/16} \quad (B.37) \\ + \left[ \frac{25}{8} - \frac{25}{8} \sin^2 \frac{2\pi a}{L} \right] (\xi\bar{\xi})^{9/16}(\xi^2 + \bar{\xi}^2) + \frac{\sin^2 \frac{2\pi a}{L}}{4} (\xi\bar{\xi})^{1/16}(\xi^3 + \bar{\xi}^3) + \dots.$$
The first term corresponds to the lowest eigenvalue in the $F_{2,-1}$ sector with conformal spin $h - \bar{h} = 1$, originating from $V_{211}$ in finite size. The second and third term correspond to the leading eigenvalue with $h - \bar{h} = 0$ (resp. $h - \bar{h} = 2$), originating from $V_{210}$ (resp. $V_{212}$)—note that since the scaling dimension $h + \bar{h} = \frac{25}{8}$ is the same for these two terms their ordering in the finite-size data cannot be guessed straightaway, and one has to make use of the lattice momentum to reveal the correct conformal spin. Finally, the fourth term corresponds to the lowest eigenvalue in the sector with $h - \bar{h} = 3$ — we notice that this state is not present for $L = 5$, since this size is yet too small to accommodate the high value of the momentum. The numerical results for the last three terms being discussed (taking ratios with respect to the first term) now run as follows:

<table><thead><tr><th>L</th><th></th><th>(ξξ)<sup>25/16</sup></th><th>(ξξ)<sup>9/16</sup>(ξ<sup>2</sup> + ξ<sup>2</sup>)</th><th>(ξξ)<sup>1/16</sup>(ξ<sup>3</sup> + ξ<sup>3</sup>)</th></tr></thead><tbody><tr><td rowspan="3">5</td><td>lattice</td><td>0.05719383440</td><td>0.2474873290</td><td>0</td></tr><tr><td>CFT</td><td>0.10742794066</td><td>0.2984109462</td><td>0</td></tr><tr><td>ratio</td><td>0.5324</td><td>0.8294</td><td></td></tr><tr><td rowspan="3">7</td><td>lattice</td><td>0.03267744359</td><td>0.1238111030</td><td>0.1246090804</td></tr><tr><td>CFT</td><td>0.05570501180</td><td>0.1547361439</td><td>0.2376211084</td></tr><tr><td>ratio</td><td>0.5866</td><td>0.8001</td><td>0.5244</td></tr><tr><td rowspan="3">9</td><td>lattice</td><td>0.02135602073</td><td>0.07680979880</td><td>0.1355927174</td></tr><tr><td>CFT</td><td>0.03392290080</td><td>0.09423028002</td><td>0.2424615776</td></tr><tr><td>ratio</td><td>0.6295</td><td>0.8151</td><td>0.5592</td></tr></tbody></table>

Finally, observe that the Kac parametrisation

$$h_{rs} = \frac{[(m+1)r - ms]^2 - 1}{4m(m+1)} \qquad (B.38)$$

becomes, in the limit $m \to \infty$, $h_{rs} = \frac{(r-s)^2}{4}$. The exponents appearing in the various Ashkin-Teller correlators are thus in agreement with the spectra conjectured in [1]

$$ \begin{aligned} G &: S_{\mathbb{Z},2\mathbb{Z}}, \\ R_1 &: S_{\mathbb{Z}+\frac{1}{2},2\mathbb{Z}}, \\ R_2 &: S_{\mathbb{Z},2\mathbb{Z}} \\ R_3 &: S_{\mathbb{Z}+\frac{1}{2},2\mathbb{Z}} \end{aligned} \qquad (B.39) $$

after the switch of $r, s$ labels mentioned earlier in remark 5.

## B.4 The case $Q = 0$

The case $Q = 0$ (or rather the limit $Q \to 0$) is connected to the combinatorics of spanning trees and forests (see, e.g., [70–72]). It is interesting in the present context for two reasons. On the one hand, it gives rise to Jordan cells in the transfer matrix, and thus provides an opportunity to study their effect on the determination of amplitudes. On the other hand, it also turns out that the combination $P_{abab} - P_{abba}$ can be explicitly calculated in this
case, providing another non-trivial check of our method, this time in the region of small values of $Q$.

While for $Q$ generic, the two-point function of the Potts spin operator is proportional to the probability that the two points belong to the same cluster [see (2.8)], the limit $Q \to 0$ needs to be handled with care, since the partition function itself vanishes. At leading order, the only terms left in the partition function are spanning trees: a naive definition — requiring that the two points still belong to the same cluster — would then lead to a trivially constant two-point function. To obtain non-trivial results it is better to change the normalisation by one factor of $Q$, that is, to redefine the partition function as the number of spanning trees. A natural redefinition of the two-point function will be given below; it follows by using the equivalence of the $Q \to 0$ limit with symplectic fermions [73, 74] and the theory of spanning trees.

The simplest, in fact, is to start by discussing the combination $P_{abab} - P_{abba}$. Indeed, consider four points labelled 1, 2, 3, 4 in the plane. As usual, we consider the square lattice in concrete calculations. We denote by $\mathcal{N}_{13,24}$ (resp. $\mathcal{N}_{14,23}$) the numbers of configurations of spanning trees where one tree connects points 1, 3 and a different tree connects points 2, 4 (resp. one tree connects points 1, 4 and a different ones points 2, 3). It is possible to show, by generalising Kirchoff’s original discussion [75] along the lines of [76], that

$$ \mathcal{N}_{13,24} - \mathcal{N}_{14,23} = \det \Delta_{(12)(34)}, \qquad (B.40) $$

where the right-hand side is the determinant of the lattice Laplacian after having removed the lines corresponding to points 1, 2 and columns corresponding to points 3, 4. Here, the Laplacian is defined as follows. It is a matrix denoted $\Delta$ whose linear size is the number of vertices of the graph, here a planar lattice, that we suppose loopless (i.e., no vertex is connected to itself) for simplicity. The diagonal elements $\Delta_{ii}$ are the number of edges incident on vertex $i$. The off-diagonal elements $\Delta_{ij}$ are equal to minus the number of edges connecting the vertices $i$ and $j$. We also denote by $\Delta^{(kl)}$ the minor of $\Delta$ obtained by erasing row $k$ and column $l$. We recall that $\det \Delta = 0$, since by definition the sum of all rows (or the sum of all columns) is zero. Moreover, by the Kirchhoff matrix-tree theorem [75], $\det \Delta^{(kl)}$ is equal to the number of spanning trees on the graph. If we now go back to the $Q \to 0$ limit of the Potts model and the definition (2.6) of the probabilities $P_{a_1,a_2,a_3,a_4}$ in terms of the cluster expansion, we see that (B.40) is the leading contribution to $P_{abab} - P_{abba}$ as $Q \to 0$: in this limit indeed, any non-connected additional cluster gets cancelled by a power of $Q$, and so do cycles within the clusters (so each cluster is a tree indeed).

Meanwhile, (B.40) can be calculated in the continuum limit, which is simply described by a pair of symplectic fermions $(\theta^+, \theta^-)$ with Euclidean action

$$ S = \int d^2x \, d_{\alpha\beta} \partial_{\mu} \theta^{\alpha} \partial_{\mu} \theta^{\beta}, \qquad (B.41) $$

subject to summation over repeated indices, and with $d_{+-} = 1, d_{-+} = -1$. The quantity in (B.40) is then nothing but $\langle \theta^+(1)\theta^+(2)\theta^-(3)\theta^-(4) \rangle$ and gives$^{15}$ [77, 78]

$$ \langle \theta^+(1)\theta^+(2)\theta^-(3)\theta^-(4) \rangle = \ln \left| \frac{z_{14}z_{23}}{z_{13}z_{24}} \right|^2. \qquad (B.42) $$

<footnote>$^{15}$Note that Wick’s theorem has to be handled with care, since the partition function vanishes.</footnote>
### B.4.1 Two-point function

Before studying $P_{abab} - P_{abba}$, we can extract from this a useful definition of the two-point function as well. Indeed imagine sending point 1 to 3, and point 2 to 4. In this case, the configurations in $N_{14,23}$ become negligible, while those in $N_{13,24}$ describe a situation where 1 = 3 belongs to a tree and 2 = 4 to a different tree. We use this to define a two-point function:

$$g_2(1, 2) = \frac{N_{1,2}}{N_1}, \qquad (B.43)$$

where $N_{1,2}$ is the number of configurations of spanning forests with two trees only, with points 1 and 2 belonging to different trees, while $N_1$ denotes the number of spanning trees. In the continuum limit we therefore have from (B.42)

$$g_2(z, \bar{z}) = 2 \ln |z/\epsilon|^2, \qquad (B.44)$$

where $\epsilon$ is a short-distance cutoff. Note that $\epsilon$ cannot be eliminated by multiplicative renormalisation, as is usually done in CFT. We could, alternatively, set $\epsilon = 1$, and say that the two-point function takes the form (B.44) up to a non-universal additive constant.

On the cylinder, we expect the following behavior in the conformal limit [79]:

$$g_2(w, \bar{w}) = 2 \ln \left( \frac{L}{\pi \epsilon} \sinh \frac{\pi w}{L} \right) + 2 \ln \left( \frac{L}{\pi \epsilon} \sinh \frac{\pi \bar{w}}{L} \right). \qquad (B.45)$$

We shall measure $w, L$ in units of the lattice spacing. This means that $\epsilon$ will be a (non-universal) numerical constant of order unity. We now expand $g_2$ to see how it connects with the results from the transfer matrix:

$$g_2(\xi, \bar{\xi}) = 4 \ln \frac{L}{2\pi\epsilon} + \frac{4\pi l}{L} - 2 \sum_{n=1}^{\infty} \left( \frac{\xi^n}{n} + \frac{\bar{\xi}^n}{n} \right). \qquad (B.46)$$

It is seen that $g_2$ consists of a linear term and a sum of exponentials. All the amplitudes can be compared with lattice calculations: the sum of exponentials arises from a sum over eigenvalues of the transfer matrix as usual, while the linear term arises because of the presence of a Jordan cell of rank two. Write (B.46) as

$$g_2 \propto Al + \sum_{i \ge 0} B_i (\Lambda_i / \lambda_0)^l, \qquad (B.47)$$

where $\Lambda_0$ denotes the ground-state eigenvalue. We get from (B.46) that $A = \frac{4\pi}{L}$ while $B_0 = 4 \ln \frac{L}{2\pi\epsilon}$, $B_1 = -4$, and $B_2 = -2$ (where the last term comes from $n = 2$ in the sum). However, if we change the normalisation of (B.47) so that $A = \frac{1}{L}$, we get instead:

$$B_0 = \frac{1}{\pi} \ln \frac{L}{2\pi\epsilon}, \quad B_1 = -\frac{1}{\pi}, \quad B_2 = -\frac{1}{2\pi}, \quad B_3 = -\frac{1}{3\pi}, \dots \qquad (B.48)$$

As usual we can confront this with the numerical computations. We find that $A = \frac{1}{L}$ exactly in finite size, which motivates the above choice of normalisation. Moreover, $g(1, 3)$ is found to couple only to a very small set of eigenvalues: apart from the rank-two Jordan
<figure><img src="image_13.png" /><figcaption>Figure 12. Structure of standard modules for $\mathfrak{q} = e^{i\pi/2}$.</figcaption></figure>

cell parameterised by $(B_0, A)$, there are only two (resp. three) simple eigenvalues for $L = 5$ and $2a = 2$ (resp. $L = 7$ and $2a = 3$). Presented in the usual table form we find:

<table><thead><tr><th>L</th><th></th><th>B<sub>0</sub></th><th>B<sub>1</sub></th><th>B<sub>2</sub></th><th>B<sub>3</sub></th></tr></thead><tbody><tr><td rowspan="3">5</td><td>lattice</td><td>0.4457211803</td><td>-0.2933396913</td><td>-0.1523814889</td><td>-</td></tr><tr><td>CFT</td><td>0.6560432383</td><td>-0.3183098862</td><td>-0.1591549431</td><td></td></tr><tr><td>ratio</td><td>0.6794</td><td>0.9215</td><td>0.9574</td><td></td></tr><tr><td rowspan="3">7</td><td>lattice</td><td>0.5509145352</td><td>-0.3020464547</td><td>-0.1439480976</td><td>-0.1049199829</td></tr><tr><td>CFT</td><td>0.7631456776</td><td>-0.3183098862</td><td>-0.1591549431</td><td>-0.1061032954</td></tr><tr><td>ratio</td><td>0.7219</td><td>0.9489</td><td>0.9045</td><td>0.9888</td></tr></tbody></table>

Note that we have here set $\epsilon = 1$ in the CFT result for $B_0$ in order to get an order of magnitude estimate. Since this term is not expected to be universal, the agreement for $B_0$ is here seen to be reasonable. The agreement for the other terms ($B_1, B_2, B_3$) is seen to be very good, even at these small sizes.

We now comment more generally on the structure of the two-point function in terms of the transfer matrix. The linear term in (B.46) indicates the presence of a Jordan cell of rank two (since we get a term linear in $l$), and that the two corresponding (pseudo) eigenvalues must be in the ground-state sector since there is no exponential decay. The sum of exponentials involves only integer conformal weights, and corresponds simply to coupling to descendants of the identity. Remarkably, note that we couple only to purely chiral or purely antichiral fields. This explains the scarcity of non-zero amplitudes observed numerically.

It is also useful here to recall the structure of some of the modules for $Q = 0$. Since $\mathfrak{q}$ is a root of unity, the $W_{j,z^2=e^{2i\kappa}}$ are reducible, and have a structure of submodules that depends on the values of $j, z$. For instance we get the structure shown in figure 12.

Meanwhile, we can read the dimension content from the general formulae:

$$F_{j,1} = \frac{1}{(q\bar{q})^{-2/24} P(q)P(\bar{q})} \sum_{e \in \mathbb{Z}} q^{\frac{(j-2e)^2-1}{8}} \bar{q}^{\frac{(j+2e)^2-1}{8}}. \qquad (B.49)$$
Let us consider in more detail $F_{1,1}$. This contains the exponents $(h, \bar{h})$, here ordered by the total value of the scaling dimension $h + \bar{h}$:

$$ (0,0); (0,1) \times 2; (1,0) \times 2; (1,1) \times 3; (2,0) \times 3; (0,2) \times 3; \dots \qquad (B.50) $$

We see that the first eigenvalue with $h - \bar{h} = 1$ is $(h, \bar{h}) = (1,0)$, the first eigenvalue with $h - \bar{h} = 2$ is $(h, \bar{h}) = (2,0)$, etc. This agrees with the expansion (B.46). Since only purely chiral contributions appear in this expansion, this means that, for a given value of $n = h - \bar{h}$, we have only the smallest realisation, $(h, \bar{h}) = (n,0)$. The same applies to all the other sectors.

## B.4.2 The four-point function $P_{abab} - P_{abba}$

We now go back to the combination $P_{abab} - P_{abba}$, which we write in terms of the variable $z \equiv \frac{z_{12}z_{34}}{z_{13}z_{24}}$:

$$ P_{abab} - P_{abba} \propto -[\ln(1-z) + \ln(1-\bar{z})], \qquad (B.51) $$

and expanding this we get

$$ P_{abab} - P_{abba} \propto \sum_{n=1}^{\infty} \frac{z^n + \bar{z}^n}{n}. \qquad (B.52) $$

Note that, if we exchange points 1 and 2, we have $z = \frac{z_{12}z_{34}}{z_{13}z_{24}} \rightarrow \frac{z_{21}z_{34}}{z_{23}z_{14}} = \frac{z}{z-1}$. This corresponds to sending $1-z \rightarrow \frac{1}{1-z}$, and thus to changing the sign of $P_{abab} - P_{abba}$, as required from the geometrical interpretation. This does *not mean* that only odd spins appear in the expression (4.6), because in this case there are large degeneracies (in fact, only a few of the terms correspond to primary fields).

Going to the cylinder as usual we find now

$$ P_{abab} - P_{abba} \propto \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \left( 4 \sin^2 \frac{2\pi a}{L} \right)^n \frac{\xi^n}{(1-\xi)^{2n}} + \text{h.c.} \qquad (B.53) $$

To compare with the numerical work, we rewrite (B.53) as

$$ P_{abab} - P_{abba} \propto \sum_{n=1}^{\infty} C_n \xi^n + \text{h.c.} \qquad (B.54) $$

and choose the normalisation $C_1 = 1$, so that the amplitude ratios can be read off directly from the following coefficients. The first few then read explicitly

$$ C_2 = 2 \left( \cos \frac{2\pi a}{L} \right)^2, \qquad (B.55a) $$

$$ C_3 = \frac{1}{3} \left( 1 + 2 \cos \frac{4\pi a}{L} \right)^2, \qquad (B.55b) $$

$$ C_4 = \left( \cos \frac{2\pi a}{L} + \cos \frac{6\pi a}{L} \right)^2. \qquad (B.55c) $$
As above, the numerical work was done for $L = 5$ and $2a = 2$ (resp. $L = 7$ and $2a = 3$). The contributing eigenvalues are then found to be exactly the same as for the two-point function $g_2(1, 2)$, corresponding to the terms $n = 1, 2$ (resp. $n = 1, 2, 3$) in (B.54). There is no longer any Jordan cell (the term $n = 0$). Note also that in both cases, we expect at all orders to have only chiral or antichiral contributions. The numerical results compared with the conformal predictions run as follows:

<table><thead><tr><th>L</th><th></th><th>C<sub>2</sub>/C<sub>1</sub></th><th>C<sub>3</sub>/C<sub>1</sub></th></tr></thead><tbody><tr><td rowspan="3">5</td><td>lattice</td><td>0.1984202998</td><td>—</td></tr><tr><td>CFT</td><td>0.1909830056</td><td></td></tr><tr><td>ratio</td><td>1.0389</td><td></td></tr><tr><td rowspan="3">7</td><td>lattice</td><td>0.09439172580</td><td>0.2233910496</td></tr><tr><td>CFT</td><td>0.09903113209</td><td>0.2143680440</td></tr><tr><td>ratio</td><td>0.9532</td><td>1.0421</td></tr></tbody></table>

Once again the agreement is very reasonable, given the rather small sizes.

A peculiarity of this four-point function — as well as of the two-point function $g_2$ considered above — is that only fields with $h = 0$ or $\bar{h} = 0$ appear. Since the spin $h - \bar{h}$ is limited to values $0, 1, \dots, L - 1$ for a finite size $L$, this means only a very small number of eigenvalues contribute to the correlation function at finite $L$. This is quite different from the usual case, where having $h - \bar{h}$ fixed does not preclude eigenvalues with larger $h + \bar{h}$ from contributing. Of course, it could be that some eigenvalues would contribute in finite $L$, although their amplitude would tend to zero when $L$ increases, because they would not be present in the continuum limit. But this does not seem to be the case here — a fact that can probably be proven exactly in finite size using the Laplacian on the cylinder.

### B.4.3 The combination $P_{aabb} - P_{abba}$

It is also interesting to observe that, under crossing $z_2 \leftrightarrow z_3$, ie $z \to \frac{1}{z}$, the combination $P_{abab} - P_{abba}$ that we have just discussed becomes $P_{aabb} - P_{abba}$. We have studied the latter combination independently. From our analytical result (B.51) we now find

$$P_{aabb} - P_{abba} \propto \ln z - \ln(1-z) + \ln \bar{z} - \ln(1-\bar{z}). \quad (B.56)$$

Hence on the cylinder we expect

$$P_{aabb} - P_{abba} \propto \ln(4 \sin^2(2\pi a/L)) + \ln \xi + \sum_{n=1}^{\infty} \left( 2 \frac{\xi^n}{n} + \frac{(-4 \sin^2(2\pi a/L))^n}{n} \frac{\xi^n}{(1-\xi)^{2n}} \right) + \text{h.c.} \quad (B.57)$$

It is easy to ascertain that the correct sectors are observed in the numerical study. Indeed we observe now again a Jordan cell on the ground state, plus the same 2 (resp. 3) simple eigenvalues as before for the case $L = 5$ and $2a = 2$ (resp. $L = 7$ and $2a = 3$). The indecomposability parameter, i.e., the coefficient multiplying $l$ in the correlation function, is found to be $A = \frac{1}{L}$ exactly in finite size, like it was for the two-point function. Let us
write the remaining terms, concerning the simple eigenvalues, as $\sum_{n=1}^{\infty} D_n \xi^n + \text{h.c.}$ The comparison with numerics can then be summarised as follows:

<table><thead><tr><th>L</th><th></th><th>D<sub>2</sub>/D<sub>1</sub></th><th>D<sub>3</sub>/D<sub>1</sub></th></tr></thead><tbody><tr><td rowspan="3">5</td><td>lattice</td><td>-0.1984202998</td><td>—</td></tr><tr><td>CFT</td><td>-0.1909830056</td><td></td></tr><tr><td>ratio</td><td>1.0389</td><td></td></tr><tr><td rowspan="3">7</td><td>lattice</td><td>-0.3298008395</td><td>0.08579175620</td></tr><tr><td>CFT</td><td>-0.3460107358</td><td>0.08232653457</td></tr><tr><td>ratio</td><td>0.9532</td><td>1.0421</td></tr></tbody></table>

It will not have escaped the reader's attention that the ratios are exactly the same as for the previously considered case $P_{abab} - P_{abba}$. This presumably extends to arbitrary $L$ and implies that the symmetry $z \to \frac{1}{z}$ is respected exactly on the lattice — something that we again suspect can be proven by a careful study of the Laplacian on the cylinder. Per se, this close relationship between the foregoing two four-point functions is pretty remarkable, since after all, in the lattice study of $P_{abab}$ we have two long clusters, while in $P_{aabb}$ we have two short ones.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

[1] M. Picco, S. Ribault and R. Santachiara, *A conformal bootstrap approach to critical percolation in two dimensions*, SciPost Phys. 1 (2016) 009 [arXiv:1607.07224] [INSPIRE].

[2] C.M. Fortuin and P.W. Kasteleyn, *On the Random cluster model. 1. Introduction and relation to other models*, Physica 57 (1972) 536 [INSPIRE].

[3] R.B. Potts, *Some generalized order-disorder transformations*, Math. Proc. Cambr. Phil. Soc. 48 (1952) 106.

[4] G. Delfino and J. Viti, *On three-point connectivity in two-dimensional percolation*, J. Phys. A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE].

[5] M. Picco, R. Santachiara, J. Viti and G. Delfino, *Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator*, Nucl. Phys. B 875 (2013) 719 [arXiv:1304.6511] [INSPIRE].

[6] Y. Ikhlef, J.L. Jacobsen and H. Saleur, *Three-point functions in $c \le 1$ Liouville theory and conformal loop ensembles*, Phys. Rev. Lett. 116 (2016) 130601 [arXiv:1509.03538] [INSPIRE].

[7] P. Di Francesco, D. Sénéchal and P. Mathieu, *Conformal field theory*, Springer, Germany (1997).

[8] G. Gori and J. Viti, *Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance*, JHEP 12 (2018) 131 [arXiv:1806.02330] [INSPIRE].
[9] A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].

[10] L. Luo, B.-Q. Xia and Y.-F. Cao, Peakon solutions to supersymmetric Camassa-Holm equation and Degasperis-Procesi equation, Commun. Theor. Phys. 59 (2013) 73 [INSPIRE].

[11] P. Di Francesco, H. Saleur and J.B. Zuber, Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models, J. Stat. Phys. 49 (1987) 57.

[12] R.J. Baxter, Potts model at the critical temperature, J. Phys. C 6 (1973) L445.

[13] R.J. Baxter, S.B. Kelland and F.Y. Wu, Equivalence of the Potts model or Whitney polynomial with an ice-type model, J. Phys. A 9 (1976) 397.

[14] H.N.V. Temperley and E.T. Lieb, Relation between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with planar lattices: some exact results for the ‘percolation’ problem, Proc. Roy. Soc. London A 322 (1971) 251.

[15] R. Vasseur and J.L. Jacobsen, Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model, J. Phys. A 45 (2012) 165001 [arXiv:1111.4033].

[16] J. Dubail, J.L. Jacobsen and H. Saleur, Critical exponents of domain walls in the two-dimensional Potts model, J. Phys. A 43 (2010) 482002 [arXiv:1008.1216].

[17] J. Dubail, J.L. Jacobsen and H. Saleur, Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model, J. Stat. Mech. 12 (2010) 12026 [arXiv:1010.1700].

[18] G. Delfino and J. Viti, Potts q-color field theory and scaling random cluster model, Nucl. Phys. B 852 (2011) 149 [arXiv:1104.4323] [INSPIRE].

[19] B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Statist. Phys. 34 (1984) 731 [INSPIRE].

[20] B. Duplantier and H. Saleur, Exact critical properties of two-dimensional dense self-avoiding walks, Nucl. Phys. B 290 (1987) 291.

[21] R. Vasseur, J.L. Jacobsen and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech. 1207 (2012) L07001 [arXiv:1206.2312] [INSPIRE].

[22] R. Vasseur and J.L. Jacobsen, Operator content of the critical Potts model in [dimensions and logarithmic correlations, Nucl. Phys. B 880 (2014) 435 [arXiv:1311.6143] [INSPIRE].

[23] R. Couvreur, J. Lykke Jacobsen and R. Vasseur, Non-scalar operators for the Potts model in arbitrary dimension, J. Phys. A 50 (2017) 474001 [arXiv:1704.02186] [INSPIRE].

[24] T. Halverson and A. Ram, Partition algebras, Eur. J. Combin. 26 (2005) 869.

[25] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London U.K. (1982).

[26] P.P. Martin, Potts models and related problems in statistical mechanics, World Scientific, Singapore (1991).

[27] P. Martin and H. Saleur, The Blob algebra and the periodic Temperley-Lieb algebra, Lett. Math. Phys. 30 (1994) 189 [hep-th/9302094] [INSPIRE].

[28] J.J. Graham and G.I. Lehrer, The representation theory of affine Temperley-Lieb algebras, L'Ens. Math. 44 (1998) 173.
[29] A.M. Gainutdinov, N. Read, H. Saleur and R. Vasseur, The periodic $sl(2|1)$ alternating spin chain and its continuum limit as a bulk LCFT at $c = 0$, JHEP 05 (2015) 114 [arXiv:1409.0167].

[30] A.M. Gainutdinov, J.L. Jacobsen and H. Saleur, A fusion for the periodic Temperley-Lieb algebra and its continuum limit, JHEP 11 (2018) 117 [arXiv:1712.07076] [INSPIRE].

[31] V.F.R. Jones, Quotient of the affine Hecke algebra in the Brauer algebra, L'Ens. Math. 40 (1994) 313.

[32] N. Read and H. Saleur, Enlarged symmetry algebras of spin chains, loop models and S-matrices, Nucl. Phys. B 777 (2007) 263 [cond-mat/0701259] [INSPIRE].

[33] J.F. Richard and J.L. Jacobsen, Eigenvalue amplitudes of the Potts model on a torus, Nucl. Phys. B 769 (2007) 256 [math-ph/0608055].

[34] J.L. Jacobsen and J. Salas, Phase diagram of the chromatic polynomial on a torus, Nucl. Phys. B 783 (2007) 238 [cond-mat/0703228] [INSPIRE].

[35] J.L. Jacobsen and P. Zinn-Justin, A transfer matrix for the backbone exponent of two-dimensional percolation, J. Phys. A 35 (2002) 2131 [cond-mat/0111374].

[36] J.L. Jacobsen and P. Zinn-Justin, Monochromatic path crossing exponents and graph connectivity in two-dimensional percolation, Phys. Rev. E 66 (2002) 055102(R) [cond-mat/0207063].

[37] Y. Deng, H.W.J. Blöte and B. Nienhuis, Backbone exponents of the two-dimensional $q$-state Potts model: a Monte Carlo investigation, Phys. Rev. E 69 (2004) 026114.

[38] Z. Zhou, J. Yang, Y. Deng and R.M. Ziff, Shortest-path fractal dimension for percolation in two and three dimensions, Phys. Rev. E 86 (2012) 061101 [arXiv:1112.3428].

[39] A.M. Gainutdinov, N. Read and H. Saleur, Continuum limit and symmetries of the periodic $gl(1|1)$ spin chain, Nucl. Phys. B 871 (2013) 245 [arXiv:1112.3403] [INSPIRE].

[40] A.M. Gainutdinov, N. Read and H. Saleur, Bimodule structure in the periodic $gl(1|1)$ spin chain, Nucl. Phys. B 871 (2013) 289 [arXiv:1112.3407] [INSPIRE].

[41] A.M. Gainutdinov, N. Read and H. Saleur, Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the $gl(1|1)$ periodic spin chain, Howe duality and the interchiral algebra, Commun. Math. Phys. 341 (2016) 35 [arXiv:1207.6334] [INSPIRE].

[42] A.M. Gainutdinov, J.L. Jacobsen, H. Saleur and R. Vasseur, A physical approach to the classification of indecomposable Virasoro representations from the blob algebra, Nucl. Phys. B 873 (2013) 614 [arXiv:1212.0093] [INSPIRE].

[43] A.M. Gainutdinov et al., Logarithmic conformal field theory: a lattice approach, J. Phys. A 46 (2013) 494012 [arXiv:1303.2082] [INSPIRE].

[44] J. Belletête et al., On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories, J. Phys. A 50 (2017) 484002 [arXiv:1705.07769] [INSPIRE].

[45] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].

[46] V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B 330 (1990) 523 [INSPIRE].
[47] F.C. Alcaraz, U. Grimm and V. Rittenberg, *The XXZ Heisenberg chain, conformal invariance and the operator content of $c < 1$ systems*, Nucl. Phys. B 316 (1989) 735 [inSPIRE].

[48] M. den Nijs, *Extended scaling relations for the magnetic critical exponents of the Potts model*, Phys. Rev. B 27 (1983) 1674 [inSPIRE].

[49] J.L. Jacobsen and H. Saleur, *Combinatorial aspects of boundary loop models*, J. Stat. Mech. 1 (2008) 01021 [arXiv:0709.0812].

[50] J. de Gier, A. Ponsaing and J.L. Jacobsen, *Finite-size corrections for universal boundary entropy in bond percolation*, SciPost Phys. 1 (2016) 012 [arXiv:1610.04006] [inSPIRE].

[51] B. Estienne and Y. Ikhlef, *Correlation functions in loop models*, arXiv:1505.00585 [inSPIRE].

[52] S. Migliaccio and S. Ribault, *The analytic bootstrap equations of non-diagonal two-dimensional CFT*, JHEP 05 (2018) 169 [arXiv:1711.08916] [inSPIRE].

[53] I. Runkel and G.M.T. Watts, *A Nonrational CFT with $c = 1$ as a limit of minimal models*, JHEP 09 (2001) 006 [hep-th/0107118] [inSPIRE].

[54] S. Ribault and R. Santachiara, *Liouville theory with central charge less than one*, JHEP 08 (2015) 109 [arXiv:1503.02067].

[55] W.M. Koo and H. Saleur, *Representations of the Virasoro algebra from lattice models*, Nucl. Phys. B 426 (1994) 459 [hep-th/9312156] [inSPIRE].

[56] J.L. Jacobsen, S. Ribault, H. Saleur and L.G. Samuelsson, in preparation.

[57] H.W.J. Blöte and M.P. Nightingale, *Critical behaviour of the two-dimensional Potts model with a continuous number of states: a finite size scaling analysis*, Physica A 112 (1982) 405.

[58] J. Salas and A.D. Sokal, *Transfer matrices and partition function zeros for antiferromagnetic Potts models. 1. General theory and square lattice chromatic polynomial*, J. Statist. Phys. 104 (2001) 609 [cond-mat/0004330] [inSPIRE].

[59] S.C. Chang, J. Salas and R. Shrock, *Exact Potts model partition functions for strips of the square lattice*, J. Stat. Phys. 107 (2002) 1207 [cond-mat/0108144].

[60] F.M. Gomes and D.C. Sorensen, *Arpack++. An object-oriented version of ARPACK eigenvalue package*, https://www.caam.rice.edu//software/ARPACK/arpack++.html.

[61] B. Haible and R.B. Kreckel, CLN — Class Library for Numbers, https://www.ginac.de/CLN.

[62] J.L. Jacobsen, *Critical points of Potts and $O(N)$ models from eigenvalue identities in periodic Temperley-Lieb algebras*, J. Phys. A 48 (2015) 454003 [arXiv:1507.03027].

[63] J.L. Jacobsen, C.R. Scullard and A.J. Guttmann, *On the growth constant for square-lattice self-avoiding walks*, J. Phys. A 49 (2016) 494004 [arXiv:1607.02984].

[64] J.L. Jacobsen, J. Salas and C.R. Scullard, *Phase diagram of the triangular-lattice Potts antiferromagnet*, J. Phys. A 50 (2017) 345002 [arXiv:1702.02006].

[65] B. McCoy and T.T. Wu, *The two-dimensional Ising model*, Harvard University Press, U.S.A. (1973).

[66] A. Rocha-Caridi, *Vacuum vector representations of the Virasoro algebra*, in *Vertex operators in mathematics and physics*, J. Lepowsky et al. eds., Springer, Germany (1984).
[67] P. Reinicke and T. Vescan, Finite-size corrections to matrix elements in a conformal theory. *Applications to the magnetisation of the three-state Potts model*, J. Phys. A 20 (1987) L653.

[68] S.-K. Yang, *Modular invariant partition function of the Ashkin-Teller model on the critical line and N = 2 superconformal invariance*, Nucl. Phys. B 285 (1987) 183 [INSPIRE].

[69] Al.B. Zamolodchikov, *Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model*, Sov. Phys. JETP 63 (1986) 1061.

[70] J.L. Jacobsen, J. Salas and A.D. Sokal, *Spanning forests and the q state Potts model in the limit $q \to 0$*, J. Statist. Phys. 119 (2005) 1153 [cond-mat/0401026] [INSPIRE].

[71] S. Caracciolo et al., *Fermionic field theory for trees and forests*, Phys. Rev. Lett. 93 (2004) 080601 [cond-mat/0403271] [INSPIRE].

[72] J.L. Jacobsen and H. Saleur, *The Arboreal gas and the supersphere $\sigma$-model*, Nucl. Phys. B 716 (2005) 439 [cond-mat/0502052] [INSPIRE].

[73] H. Saleur, *Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry*, Nucl. Phys. B 382 (1992) 486 [hep-th/9111007] [INSPIRE].

[74] E.V. Ivashkevich, *Correlation functions of dense polymers and $c = -2$ conformal field theory*, J. Phys. A 32 (1999) 1691 [cond-mat/9801183].

[75] G. Kirchhoff, *Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird*, Ann. Phys. Chem. 72 (1847) 497.

[76] V.B. Priezzhev, *The dimer problem and the Kirchhoff theorem*, Sov. Phys. Usp. 28 (1985) 1125.

[77] H.G. Kausch, *Curiosities at $c = -2$*, hep-th/9510149 [INSPIRE].

[78] H.G. Kausch, *Symplectic fermions*, Nucl. Phys. B 583 (2000) 513 [hep-th/0003029] [INSPIRE].

[79] J.M. Luck, *Finite size scaling and the two-dimensional XY model*, J. Phys. A 15 (1982) L169 [INSPIRE].