Four spins correlation function of the $q$ states Potts model, for general values of $q$. Its percolation model limit $q \rightarrow 1$

Under the assumption that the product of two spin operators decomposes uniquely into the degenerate conformal fields $\{\Phi_{n',n}\}$, the general expression for the correlation function of four spins is defined for the $q$ states Potts model with $q$ taking general values in the interval $1 \leq q \leq 4$. The limit of $q \rightarrow 1$ is considered in detail and the four spins function is obtained for the percolation model.


Introduction.
The renewed interest in the critical Potts model is related to the studies of three and four points cluster connectivities, which are connected to the coorresponding spin correlation functions [1] - [5].
In the context of 2D conformal field theory, of the q states Potts model [6,7], the four spins function: is a complicated object. This is because the spin operator is not a degenerate field, for general values of q, unlike for instance the energy operator which is the Φ 1,2 degenerate field, in the conformal field theory classification, for general values of q in the interval For the degenerate operators, the conformal field theory provides well defined methods for calculation, in particular, of four-point functions [8,7,9]. For non-degenerate operators such techniques are absent.
In the next Section we shall suggest the method which allows to turn around this difficulty, in order to define the four spins function (1.1) for general values of q.
To be more precise, this is done under the assumption, yet to be justified, that the product of two spin operators decomposes uniquely into the degenerate conformal fields {Φ n ′ ,n }. This assumption is presented in some more details further below, in the next Section, in the set of remarks which follow the equation (2.36).
In the third Section we shall consider the limit q → 1 for the function (1.1), which is the percolation model four spins function. This limit turns out to be very delicate.
The Section 4 is devoted to the discussion. is covered by the fields [8]: Above is used the Coulomb Gas (CG) based parametrisation. α − , α + are the charges of the screening operators, α 0 is the background charge of the vacuum [7]. The CG vertex operators representing the primary fields (2.2) have the form: V n ′ ,n (z,z) ≡ V α n ′ ,n (z,z) = exp{iα n ′ ,n ϕ(z,z)} (2.7) ϕ(z,z) is a free scalar field and the corresponding CG charges {α n ′ ,n } have the form: If, in turn, α 2 + is parametrized as then, for the parameter p taking odd integer values p = 3, 5, 7, ...∞ , (2.10) the corresponding conformal field theories, with the central charge c p = 1 − 6 p(p + 1) , (2.11) correspond to minimal models representing the unitary set of Potts models, with q taking a discrete (infinite) set of values ranging from q = 2 (p = 3, c = 1/2) to q = 4 (p → ∞, c = 1) [10]. More precisely, q is related to the parameter p by the formula: More details could be found in [6,7,10].
To come to c p in (2.11), from the central charge expression in (2.6), the standard relations are to be used: For a particular unitary minimal model, with p = 2s + 1 (2.14) s being positive integer, a finite set of primary fields (2.2) decouple, from the rest, by the operator algebra . They form a finite table (Kac table): More explicitly, with the parametrization by the parameter p, this formula is of the form: For n ′ = (p + 1)/2, n = (p − 1)/2, of the first field in (2.17), and also for n ′ = (p + 1)/2, n = (p + 1)/2, for the second field in (2.17), one gets the same value: -the conformal dimension of the spin field of the Potts model, for general values of the parameter p.
The fact that the spin operator has two representations, as in (2.17), for general values of p (or c, or q), in the same way as it was the case for the minimal models when p was odd integer, this fact will be important in the following.
In the context of a particular minimal model M p , with p being odd integer, the parameter s = p − 1 2 (2.21) in (2.18) being positive integer, the correlation function of four spins (1.1) could readily be calculated with the Coulomb Gas technique [7,9]. But for general values of p this way of calculating the four spin function is blocked: one would need to use a fractional number of screening operators, a fractional number of integrations, in the technique of [7,9].
On the other hand, the four-point function could readily be calculated with the CG technique. It could be represented, in the CG technique, by the four-point function The CG charge of the operators V α n ′ ,n (1), V α n ′ ,n (0) is given by the formula (2.8). The charge of the spin operator V ασ (0) is also defined by the formula (2.8), but with the fractional indices n ′ = (p + 1)/2, n = (p − 1)/2 in the case of the first operator in (2.17): The charge of the operator V α + σ (∞), representing the spin operator at infinity, is CG conjugate: The total CG charge of the four vertex operators in (2.23) is According to the CG technique [7,9], this function requires n ′ − 1 screening operators and n − 1 screening operators This means that the four-point function (2.23), and hence the function (2.22), will be expressed by the multiple integral of [9] and it could be calculated. Now, in more detail, the function (2.22) could be represented, in the s -channel decomposition (decomposition in powers of z,z), as follows: in the standard way, by the Virassoro algebra [8]. This function depends also on the four external operators which dependence have been suppressed.
The operator algebra constants in (2.29) -they are given by the multiple integrals of [9]. They are expressed finally by products of Γ functions.
When n ′ is given odd values, n ′ = 2l + 1, l = 0, 1, 2, 3, ..., in the sum of (2.31), corresponds to fusion: But this process could be turned around as in Fig.2, We remind that for the normalised operators, the operator algebra constants are symmetric in all three indices, i.e. in all three operators. In this way, from one particular intermediate channel of the four point function in (2.31), we get the constants for the product of two spin operators producing the fields In other words, we get the operator algebra constants for the intermediate channels of Stating it again, the constants in the decomposition (2.36) above, by using the sym- Several remarks could be added at this point.
The three-point functions could be defined for any three operators [11,3,12]. In general, they correspond to projecting a couple of operators on the third one. They do not, in general, represent the operator algebra constants of a particular theory, the constants which would appear, naturally, in the decomposition of four-point functions.
The constants D σ σ,(2l+1,n) , eq.(2.32), which could also be calculated as three-point functions, they do represent the operator algebra constants, of the Potts model, as they originate in the four-point function (2.31). They could be called physical in the above sense. In this case the constants D (2l+1,n) σ,σ , in (2.34), are also physical, they correspond to real processes in the model. They define, in particular, the four point function < σσσσ >, as in eq.(2.36).
As has been stated above with respect to the four-point function (2.29), the spin operator appears as one of the intermediate channels of this function in the case when the index n ′ is an odd integer. For n ′ even, the spin operator is not amont the intermediate channel operators. Which means, in turn, that the primary operators {Φ n ′ ,n }, with n ′ even, do not appear in the fusion of two spin operators. The corresponding operator algebra constants are zero. In this case the sum in (2.36) could be considered as being taken over all the primary fields, of the Potts model, in the intermediate channels.
We consider, but this is clearly an assumption, that the set of primary fields {Φ n ′ ,n } is complete in the "neutral", or "even" sector: the sector which is generated by fusing an even number of spins, for the model with q general, a real number.
The spin type operators, or "odd" sector fields, are generated by fusing the spin operator with the fields {Φ n ′ ,n }, with the energy operator Φ 1,2 to begin with. The fusion rule is clear from the Coulomb Gas integral representation of the function < σ(∞)Φ n ′ ,n (1)Φ n ′ ,n (z,z)σ(0) >, eq.(2.29). The sum over (t ′ , t), eq.(2.29), is being taken over fractional values: the fractional indices of the operator σ, eq.(2.17), being shifted by integer values.
In other words, the fusion rule σ × Φ n ′ ,n → t ′ ,t Φ t ′ ,t is the same as that of minimal models. The indices of the operators {Φ t ′ ,t }, which we call spin type operators, have fractional values which are the same as those of the spin operator but shifted by integer is the spin operator, σ, and Φp+1 2 , p+3 2 is a particular spin type operator. The conformal dimensions of non-degenerate spin type operators {Φ t ′ ,t } are still given by the formula (2.19), as follows from the Coulomb Gas. The operator algebra coefficients for fusions σ×Φ n ′ ,n → t ′ ,t Φ t ′ ,t could all be calculated by the corresponding Coulomb Gas integrals, the way this is done for a particular fusion channel, σ × Φ n ′ ,n → σ, in the Section 3.
Under the assumption made above, on the completeness of the set of degenerate fields {Φ n ′ ,n } for the even sector, the decomposition (2.36), for the function < σσσσ >, will be exact. It defines the four-spin function of the Potts model.
We shall finish this Section by giving formulas for the operator algebra coefficients As has been stated above, they are defined by a particular channel in the s-channel This function is well defined by the Coulomb Gas integrals: The cases of values of the index n being odd and even have to be considered separately.
For n taking odd values, n = 2k + 1, the total Coulomb Gas charge in the function (2.37) is given by: Accordingly, the Coulomb Gas function (2.37) (n = 2k + 1) requires 2l screenings α − and 2k screening α + . One obtains: Identification of operators in the intermediate channels of (2.31), of the spin operator in particular, is obtained by calculating the total Coulomb Gas charge of two vertex operators in the product of (2.39), at 0 and (z,z) for instance, and the cloud of screenings running around them. In particular, when the operators at 0 and (z,z), V ασ (0) and V α 2l+1,2k+1 (z,z), are surrounded by l and k screenings, the total charge of this subset of operators is given by: So that the corresponding intermediate channel operator is in fact the spin operator.
In correspondance to the above described factorisation of the total set of operators (in the limit of z → 0), into a subset of operators located around 0 and (z,z) and an another subset located around 1 and ∞, the coefficients, squared, (D σ,(2l+1,2k+1) ) 2 are being expressed as a product of two 3-point functions: The variables z,z of V 2l+1,2k+1 (z,z) in the second 3-point function are assumed to have been factored out by scaling.
In particular, for the spin operator in the intermediate channel: The C constants are the operator algebra coefficients of the Coulomb Gas vertex operators.
The proportionality in (2.42) becomes the equality, when normalisation of the Coulomb Gas operators is taken into account [12]. One gets: Here Z is the Coulomb Gas partition function [12]: . Is used also the relation between the norms of the operators V + α and V α [12]: The Coulomb Gas structure constants are given by [9,12]: -equations (4.4),(4.8) of [12], with the normalization factor 1/Z added. In the equations above l and k are the numbers of screening operators needed by the Coulomb Gas cor- , c are the Coulomb Gas charges of the operators V a , V b , V c . The parameters α ′ , β ′ , γ ′ , α, β, γ are given by: Our excuses for using α (with indexes) for the Coulomb Gas charges is the previous equations, while also as a parameter (without indexes) in the equations (2.47), (2.48).
In the case of n being an even integer, n = 2k, the expressions are differences. In this case the Coulomb Gas function (2.37), will require 2l screenings α − and 2k − 1 screenings α + . The We remind that the spin operator, the first operator in (2.17), has the CG charge: Then, by (2.51), the total charge in the 0, (z,z) region is equal to: which is the charge of the second operator in (2.17), the second copy of the spin operator.
More precisely α p+1 This equality is easily verified. On one hand, On the other hand When projected onto the operator V ασ (∞), to form the corresponding 3-point function, one obtains the contribution of the 0, (z,z) region, which factors out from the rest of the operators, to be given by: The variables z andz of the operator V 2l+1,2k (z,z), initially, are assumed to have been factored out by scaling.
In a similar way as is describe above, one checks that the rest of the operators in (2.50), the operators V α + σ (∞) and V 2l+1,2k (1) and l and k screenings integrated around, that this subset of operators correspond to the intermediate channel CG operator V ασ .
It could be remarked that in the present case the expressions for the structure constants (D (2l+1,n) σ,σ ) 2 , given in terms of CG constants expressed by products of the usual Γ functions, are by far much simpler to analyse and calculate as compared to more general expressions for the 3-point functions in [11,3,12], expressed in terms of Υ function.
It could also be remarked that the general formula (2.36), for the four spins function, might not be so straightforward to use in certain particular cases, for particular values of the parameter p. In particular, the reductions to a finite number of terms, in (2.36), in cases of minimal models, have to be analysed carefully, as certain decouplings are not explicite, not simply given by the corresponding D constants being zero, see the remarks in [11] and [12].
One particular application of the general formula (2.36) will be considered in the next Section, the percolation model limit of the four spins function (2.36). The limit of p → 2, c → 0, or of this limit turns out to be fairly complicated.
The four spins correlation function in the percolation model has already been considered, in particular, in [13]. The limiting procedure for the spin operators, used in the paper [13], was somewhat forced, with the result that just one intermediate channel was selected, out of many, and not the most singular one in the limit q → 1. In a sense, in this paper we shall complete the study of the four spins function which was started in [13].
Our problem will be to take, properly, the limit q → 0 for the expression of the four spins function in (2.36). To take the limit we shall use the parameter α 2 + ≡ ρ taken in the form The value α 2 + = 3/2 corresponds to the q = 1, c = 0 point, so that the limit q → 1, c → 1 corresponds to taking the limit ǫ → 0 in (3.1). The operator algebra structure constants (D (2l+1,n) σ,σ ) 2 in (2.36) have different limits, when ǫ → 0, dependng on values of l and n.
For some of them the limit is regular, they take finite or zero values when ǫ → 0. For others the limit is singular, they diverge as 1/ǫ or 1/ǫ 2 .
To summerize the possibilities, when ǫ → 0, depending on values of l and n. The expressions for the coefficients (D (2l+1,n) σ,σ ) 2 are given by (2.44), for n odd, n = 2k + 1, and by (2.61) for n even, n = 2k. The tables in Fig.3 and Fig.4 summerize different limits for these coefficients, for different values of l and n, odd and even. Most of the calculations have been done with the use of Mathematica.
Since the principal divergence is that of 1/ǫ 2 , we shall define as the four spins function of the percolation model the limit: In this limit only the 1/ǫ 2 intermediate channels will contribute, the rest will be suppressed.
Although, for ǫ small but different from zero, the other channels will perfectly be present. In particular, the channel of the energy operator Φ 1,2 , which is the cell of l = 0, k = 1 in the Fig.4. This channel has been selected in [13], it diverges as 1/ǫ.
Still in this paper we shall study the four spins function as defined by the limit (3.3).
In this limit an infinity of intermediate channels will contribute, those which correspond to the diagonals 1/ǫ 2 in Fig.3.
The first diagonal is the one which starts with the cell (l = 0, k = 1), the field Φ 1,3 , having the dimension in the limit ǫ → 0.
It is easily checked that all the fields on this diagonal (the cells (0,1), (3,3), (6,5) and so on) have the same dimension, in the limit ǫ → 0, the same as the field Φ 1,3 which is the base of this diagonal.    The second 1/ǫ 2 diagonal is the one which starts with the field Φ 1,7 , the cell l = 0, k = 3, having the dimension ∆ 1,7 = 15 , (3.5) in the limit. All the fields on this diagonal have the same dimension, that of the field Φ 1,7 at the base.
And so on. The 1/ǫ 2 fields which are placed on the left of the first diagonal are the exact copies, by reflections with respect to the solide line diagonal in Fig.3, of the 1/ǫ 2 fields on the right: (0, 1) → (2, 0) and so on. They will not be considered as being different. This is similar to the doubling of primary fields in finite Kac tables, in the case of minimal models.
Then, in view of the above information, resuming the contributions of the 1/ǫ 2 fields along the first diagonal, to the expansion in (2.36), amounts to resuming the coefficients (3.7) over s: σ,σ ) 2 ef f is the effective constant for the first diagonal.
One could provide the whole series, for the expansion (2.36), where the summation which remains is that over the diagonals, with coefficients (D) 2 effective and the conformal block functions |F (z)| 2 for the fields at the base, for each diagonal.
But here the numerical observation comes. One finds that the ratio of the first two (D) 2 ef f coefficients: -this ratio is extremely small. With the use of Mathematica, one finds: and the corresponding conformal block function of the base field F 1,3 (z).

Discussion.
As has already been argued in [13], to realize on the lattice the four spins correlation function, of the conformal theory, might be difficult. The reason is that the lattice spins are, in general, linear combinations of the conformal theory spin-like operators. So that the lattice four spins correlation function will, in general, be a linear combination of different conformal theory four points correlation functions, of spin-like operators.
When doing different linear combinations of lattice defined four points connectivities, the number of possibilities is limited, on the lattice.
One may think of doing the analysis in the opposite direction: taking, in the conformal theory context, a linear combination of different spin-like operators, with undefined coefficients, calculate the corresponding four points functions and then try to fine tune the undefined coefficients so as to match a particular lattice defined four points connectivity.
To realise this approach might be easier in the context of a particular minimal model, where the number of spin-like operators, all known, is finite. Still it requires some work to be done, in the conformal theory context, to calculate a number of four points functions which arise.