The conformal field theory was developed a long time ago [1], and the modern numerical approach was initiated in Ref. [2]. Recent studies using this conformal bootstrap method have obtained some remarkable results for the 3D Ising model [3,4], Yang–Lee edge singularities [5,6], models [7–10], and self-avoiding walks [11].

A brief summary of the determinant method for the conformal bootstrap theory is as follows. The conformal bootstrap theory is based on the conformal group , and the conformal block is the eigenfunction of the Casimir differential operator . The eigenvalue of this Casimir operator is : (1)

The solutions of the Casimir equation have been studied in Refs. [18–20]. The conformal block has two variables and , which denote the cross ratios, , , where ( is a 2D coordinate). They are expressed as and . For the particular point , the conformal block for the spin-zero () case has a simple expression: (2)

The conformal bootstrap determines by the condition of the crossing symmetry . For practical calculations, the point is chosen and, by the recursion relation derived from the Casimir equation of Eq. (1), at can be obtained [3].

The conformal bootstrap analysis with a small matrix size has been investigated for the Yang–Lee edge singularity with accurate results for the scale dimensions by Gliozzi and Rago [5,6]. For other models, this bootstrap method for the determinant has been considered [12,15,21,22]. In this paper, we emphasize the importance of the structure of minors along the Plüker relations, as shown in the appendix, and we apply it to the Yang–Lee model for a scale dimension of . The intersection point of the zero loci of minors was evaluated and it gave a remarkable value of the critical exponent in Ref. [5]. In these minors, if the value of the space dimension is given, the value of is determined from the intersection point. However, if one goes to larger minors, for instance or minors, one need the values of additional scale dimensions of the operator product expansion (OPE). In the study of such large minors, however, the values of deviate from the expansion [6]. This discrepancy should be improved by the determinant method or by analysis of the expansion, since the conformal bootstrap method should be consistent with the expansion [24,25]. Recently, consistent results were obtained in the vector model through the Mellin amplitude [13].

We consider this discrepancy by repeating Gliozzi’s evaluation of the minors [5] and find that minors give accurate values that agree with the improved expansion (see Fig. 8). The nature of the determinant method is still not known, and the convergence to the true value seems to be slow. As we mentioned before, we have to assume the values of the scale dimensions of the OPE, such as spin 4, spin 6, etc., for the large minors. Unfortunately we do not know precisely these higher spin values at present. Therefore, we concentrate on minors without any assumption of the other scale dimensions of the OPE.

The four-point correlation function is given by (3) = \frac{g(u,v)}{|x_{12}|^{2\Delta_\phi}|x_{34}|^{2\Delta_\phi}} \end{equation} ]]> and the amplitude is expanded as the sum of conformal blocks: (4)

The crossing symmetry of implies (5)

The minor method consists of the derivatives at the symmetric point of Eq. (5). By a change of variables , , derivatives are taken about and . Since the number of equations becomes larger than the number of truncated variables , we need to consider the minors for the determination of the values of . The matrix elements of minors are expressed by (6) and the minors of , , e.g., , , are determinants such as (7) where are numbers chosen differently from (), following the concise correspondence to as , , , , , and .

In the Yang–Lee case, a fusion rule is simply written as [5] (8)

The scalar () term in the bootstrap equation of Eq. (6) is denoted as (9)

We use the notation ( is a number, related to the degree of the differential, is a vector, and is a scalar) for the derivatives of the rule in Eq. (6). For we use (), and for we use . is for .

The scalar () case is expressed by the hypergeometric function ( for the Yang–Lee edge singularity) (10) where (11) and is derivative of . We consider the derivative at a point .

For instance, we have minors: (12)

In this Yang–Lee edge singularity, we find that the minor, , becomes zero at , and it provides exact values of .

The intersection points of zero loci of minors are decomposed to minors at the critical dimensions for the free theory. In the appendix (relation 1), this decomposition is shown. Among several minors, the most fundamental minors may be and , which are made of lower derivatives of and . Note that, in the Yang–Lee model, the constraint is taken, and the minor analysis corresponds to the minor analysis of another model such as the Ising model, in which . From the point of the truncation error in the OPE, as discussed in Ref. [15], it is interesting to start from minors. The Yang–Lee singularity is connected to supersymmetry through the dimensional reduction of a branched polymer, where we have an exact relation of [14,16,17]. We will discuss the dimensional reduction behavior in the Yang–Lee model in minors.

The minors consist of two parameters, dimension and the scaling dimension , since we have () for the Yang–Lee case, due to the theory.

(i) Free field theory at

The zero loci of gives exactly the scale dimension at . The minors for , , and (with a scalar scale dimension ) can be zero: (13)

For the value , they yield , , and , respectively. These are determined exactly by minors of . This is due to the free theory, i.e., no interactions between the scale dimensions. It is known that the six dimensions have algebraic identities, which are valid for a free theory in any dimension [23].

Other minors yield different values; for instance, gives at . These two different values of are understood when we evaluate minors, as shown in Figs. 1 and 2. There are two fixed points of zero loci of minors in the six dimensions. The minors are denoted by in Eq. (7); for instance, means (14)

In the Yang–Lee edge singularity, the scale dimension of becomes zero at some dimension between 2 and 3, since it is in two dimensions and approximately 0.2 in three dimensions. We call this dimension the critical dimension , on which the scale dimension vanishes. In the determinant method, it is known that even small minors give accurate values of the scale dimensions [5,6]. We discuss here small determinants of , , and matrices for this critical dimension .

Figure 2 shows -shaped curves for zero or very small values of the loci of . This figure can be understood when we plot versus for various values of fixed dimensions. The minor of has so far been considered as a function of and with the condition of due to the Yang–Lee model. However, in general we are able to consider and as two free parameters for , and, for the Yang–Lee case, we put .

In Fig. 4, we plot the zero loci of for different values of . This minor then becomes the usual one including the Ising case. A complicated contour for the zero loci of is obtained especially around , where it has three solutions for . This solution corresponds to the shape around in Fig. 2.

In addition, we find that the line of of the Yang–Lee condition goes through very near the zero loci of , but it does not intersect. This corresponds to Fig. 2 with the finite-value loci of around .

With some value of , the contours in Fig. 4 (bottom left) and Fig. 5 intersect at the point . We find that this critical dimension from Fig. 5 is (15)

As seen in Fig. 3, for a small finite value of , the contour of approaches the line from both sides, above 0$]]> and . The peak of the contours approaches the critical dimension . We consider the derivative of by the dimension , and we estimate as the point at which this derivative becomes zero. For , we obtain . The derivative of by the space dimension gives the approximate solution of in the limit . The minor is proportional to in the limit , since , and .

We consider how the minors become zero in the limit . For instance, at , we have, for small , (16)

The coefficients are obtained from (17)

Since in the limit , we obtain a solution of ( is a constant), which gives the value of the central charge . With this choice of , the central charge becomes for this critical dimension . This means that the energy–momentum tensor operator can be neglected since the OPE coefficient of this operator becomes zero. This is the well known catastrophe, which leads to logarithmic conformal field theory [26]. When neglecting the term, and only considering in Eq. (10), we obtain the critical dimension . However, this value is too large compared to the expected value and might not be correct. We need further operators to get the correct value. We hope to get the correct value of the critical dimension by another sophisticated method by taking higher operators.

minors The numbers of zero loci of minors are . We investigated the critical dimensions by changing between and . Remarkably, at , all 15 lines intersect at a single point in the contour map of under the condition . If we take this as the value of the critical dimension , we find the following set of results: (18)

This critical dimension obtained with minors is close to from the minor with the intersection point of . In Fig. 6, all loci coincide but, if we include more relevant operators, this value may change.

For the case, in any dimension , can become zero. When , by the definition of the scale dimension , the two-point correlation function becomes a constant in the long-range limit .

In the Yang–Lee edge singularity, the exponent of the density is related to as (19)

From this relation, is vanishing for . This means that the density is constant at the transition point. Several interesting systems are known in which the density is constant but a phase transition occurs. One example is a localization problem under the random potential.

The zero loci of the minor shows an interesting characteristic linear behavior for , which is approximated as . In Fig. 7, the shift of the zero loci of for is shown (translation of two dimensions to the right in Fig. 7). The blue line of Fig. 7 for the zero loci of almost coincides with the red line of the Yang–Lee edge singularity analyzed by the Padé approximation. The red line represents the result of minors and it can be approximated as (20)

This expression satisfies the exact values of (), (), and (= 2).

This dimensional shift seems accidental and only appears in a lower truncation, but there is a dimensional reduction that has been proven rigorously.

It is known that a branched polymer in dimensions is equivalent to a -dimensional Yang–Lee edge singularity for the critical phenomena. This dimensional reduction has been explained by supersymmetry [28]. A rigorous mathematical proof of this dimensional reduction was shown in Ref. [29]. We will study this dimensional reduction in a separate paper using the conformal bootstrap in a determinant method [14]. The Yang–Lee edge singularity requires . For a branched polymer due to dimensional reduction there is a relation . This relation is a manifestation of the supersymmetry of the system [16].

Since the blue line is very close to the red line in Fig. 7, one can make an estimation of the critical dimension , in which . The intersection of with the line can be evaluated very precisely as . Adding 2 to this value for the dimensional reduction, it gives the estimation of the critical dimension as , which is very close to other estimations of this paper. We discussed this value in the previous section as as in Eq. (18).

Up to now, we have mainly discussed minors. If we take a spin-4 operator and its scale dimension , we need to do an analysis of the intersection of the zero loci of the minors .

From the formula of the minors in a matrix in the appendix, we have a Plüker formula such as (Eq. (A.12)) (21)

For instance, at , we find that the zero loci of , , , and intersect at a point, and the above Plüker formula is satisfied.

In Table 1, we present the intersection of three zero loci of minors , , and . The value of is obtained from the intersection point of the zero loci of the minors. However, the intersection point is not the only one, and we find at least three different intersection points for each fixed dimension . In Table 1, we present in parentheses such nearby different intersection points of the three zero loci of the minors.

Table 1.

Estimated scale dimensions by zero loci of three dimensional minors. Comparison with Pade analysis is shown. The value of parenthesis is estimation from other zero loci nearby.

In the expansion, is known up to the four-loop level [30]: (22) where . Including the critical dimension , for which is vanishing, the above expansion becomes (23)

Inserting the value of from Sect. 4, it becomes (24)

This expansion is approximated by the [2,2] Padé method, (25) with . The curve of this Padé is incorporated in Fig. 8.

For the values of of the Yang–Lee singularity in dimensions, the previous results obtained by Gliozzi [5] are very close to the values of Table 1; for instance, at , . In another evaluation with more primary operators [6], the values in and in are obtained, which are larger than the values in Table 1, and a comparison with expansion shows the apparent deviations, as shown in Fig. 5 of Ref. [6]. Our new analysis of Padé with a fixed value at the critical dimension seems more precise, and the expansion and bootstrap determinant method agree well with each other, as shown in Fig. 8. One of the aims of this paper was to clarify this point by the evaluation of the critical dimension . Unfortunately, what we have done is a numerical estimation of by the bootstrap determinant method and it may be an approximation. It is important to obtain an exact value of , and the comparison with expansion will be improved with an exact value of .

We have considered the Yang–Lee edge singularity in this article by the bootstrap determinant method, initiated by Gliozzi. Although our results for the scale dimension are same as the determinant result of Gliozzi [5], we have improved the consistency with the result of the expansion. We find that the basic minor is important for the determination of the above scale dimensions through the analysis of the intersection points of dimensions one to six, although we used the practical value of obtained by minors. We have shown that the value obtained by the intersection of the zero loci of minors agrees with the result of the expansion. The discrepancy between the expansion and the determinant method is reduced with the new Padé analysis with the introduction of the critical dimension .

We obtained the critical dimension from the minor for the and limits. The intersection of the zero loci of at the point gives the critical dimension . We have estimated the critical dimension approximately by other methods, by the peak approaching the finite , the dimensional reduction value , and minor analysis .

We emphasize that the estimation of the critical dimension is useful in practical terms for the precise analysis of the Yang–Lee edge singularity between two and six dimensions by the Padé analysis of the expansion [30].

We have discussed the dimensional reduction property in the Yang–Lee model, using the fact that a branched polymer in dimensions is equivalent to a -dimensional Yang–Lee edge singularity.