Prog. Theor. Exp. Phys. 2018, 053I01 (15 pages) DOI: 10.1093/ptep/pty054 Conformal bootstrap analysis for the Yang–Lee edge singularity S. Hikami Mathematical and Theoretical Physics Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Onna 904-0495, Japan E-mail: email@example.com Received January 18, 2018; Revised April 4, 2018; Accepted April 16, 2018; Published May 30, 2018 ................................................................................................................... The Yang–Lee edge singularity is investigated by the determinant method of the conformal ﬁeld theory. 3 × 3 minors are used for the evaluation of the scale dimension. The agreement with the Padé of expansion in the region 3 < D < 6 is improved. The critical dimension D , where the scale dimension of scalar is vanishing, is used for the improvement of the Padé. For the understanding of the intersection point of zero loci of 3 × 3 minors, 2 × 2 minors are investigated in detail; these are connected through Plüker relations. ................................................................................................................... Subject Index I10 1. Introduction The conformal ﬁeld theory was developed a long time ago , and the modern numerical approach was initiated in Ref. . 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], O(N ) models [7–10], and self-avoiding walks . 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 O(D,2), and the conformal block G ,L is the eigenfunction of the Casimir differential operator D . The eigenvalue of this Casimir operator is C : D G = C G , 2 ,L 2 ,L C = [( − D) + L(L + D − 2)]. (1) The solutions of the Casimir equation have been studied in Refs. [18–20]. The conformal block G (u, v) has two variables u and v, which denote the cross ratios, u = (x x /x x ) , ,L 12 34 13 24 v = (x x /x x ) , where x = x − x (x is a 2D coordinate). They are expressed as u = zz ¯ and 14 23 13 24 ij i j i v = (1 − z)(1 −¯z). For the particular point z =¯z, the conformal block G (u, v) for the spin-zero ,L (L = 0) case has a simple expression: /2 2 2 z D D + 1 D z G (u, v)| = F , , − + 1; , − + 1; . (2) ,0 z=¯z 3 2 1 − z 2 2 2 2 2 2 4(z − 1) The conformal bootstrap determines by the condition of the crossing symmetry x ↔ x .For 1 3 practical calculations, the point z =¯z = 1/2 is chosen and, by the recursion relation derived from the Casimir equation of Eq. (1), G (u, v) at z =¯z = 1/2 can be obtained . ,L © The Author 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 Funded by SCOAP by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami 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 3 × 3 minors was evaluated and it gave a remarkable value of the critical exponent in Ref. . In these 3 × 3 minors, if the value of the space dimension D is given, the value of = is determined from the intersection point. However, if one goes to larger minors, for instance 4 × 4or5 × 5 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 . 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 O(N ) vector model through the Mellin amplitude . We consider this discrepancy by repeating Gliozzi’s evaluation of the 3 × 3 minors  and ﬁnd that 3 × 3 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 3 × 3 minors without any assumption of the other scale dimensions of the OPE. The four-point correlation function is given by g(u, v) φ(x )φ (x )φ (x )φ (x ) = (3) 1 2 3 4 2 2 φ φ |x | |x | 12 34 and the amplitude g(u, v) is expanded as the sum of conformal blocks: g(u, v) = 1 + p G (u, v). (4) ,L ,L ,L The crossing symmetry of x ↔ x implies 1 3 φ φ v G (u, v) − u G (v, u) ,L ,L p = 1. (5) ,L φ φ u − v ,L The minor method consists of the derivatives at the symmetric point z =¯z = 1/2 of Eq. (5). √ √ By a change of variables z = (a + b)/2, z ¯ = (a − b)/2, derivatives are taken about a and b. 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 φ φ v G (u, v) − u G (v, u) (m,n) ,L ,L m n f = ∂ ∂ | (6) a=1,b=0 ,L a b φ φ u − v and the minors of 2 × 2, 3 × 3, e.g., d , d , are determinants such as ij ijk (m,n) (m,n) d = det f , d = det f , (7) ij ijk ,L ,L 2/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami where i, j, k are numbers chosen differently from (1, ... , 6), following the concise correspondence to (m, n) as 1 → (2, 0),2 → (4, 0),3 → (0, 1),4 → (0, 2),5 → (2, 1), and 6 → (6, 0). In the Yang–Lee case, a fusion rule is simply written as  [ ]×[ ]= 1 +[ ]+[D,2]+[ ,4] + ··· . (8) φ φ φ 4 The scalar (L = 0) term in the bootstrap equation of Eq. (6) is denoted as φ φ v G (u, v) − u G (v, u) ,0 ,0 vs0 = . (9) φ φ u − v We use the notation (vsk) (k is a number, related to the degree of the differential, v is a vector, and s is a scalar) for the derivatives of the (vs0) rule in Eq. (6). For L = 2 we use (vtk), and for L = 4we use (vqk). (vxk) is for L = 6. The scalar (L = 0) case is expressed by the hypergeometric function ( = for the Yang–Lee edge singularity) −1−/2 −/2 3 × 2 2 −/2 vs0 =−2 ys + ys + ys , (10) 2 2 where D 1 D a ys = F , , − + 1; + , − + 1: (11) 3 2 2 2 2 2 2 2 2 8(−2 + a) and ys is derivative of ys. We consider the derivative at a point a = 1. For instance, we have 2 × 2 minors: vs1 vs3 vs2 vs3 d = det , d = det . (12) 13 23 vt1 vt3 vt2 vt3 In this Yang–Lee edge singularity, we ﬁnd that the 2 × 2 minor, d , becomes zero at D = 6, and it provides exact values of = 2. 2. 2 × 2 minors The intersection points of zero loci of 3 × 3 minors are decomposed to 2 × 2 minors at the critical dimensions D = 6 for the free theory. In the appendix (relation 1), this decomposition is shown. Among several 2 × 2 minors, d , d , d , ... , the most fundamental minors may be d and d , 12 13 23 13 23 which are made of lower derivatives of a and b. Note that, in the Yang–Lee model, the constraint = is taken, and the 2 × 2 minor analysis corresponds to the 3 × 3 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. , it is interesting to start from 2 × 2 minors. The Yang–Lee singularity is connected to supersymmetry through the dimensional reduction of a branched polymer, where we have an exact relation of = + 1[14,16,17]. We will discuss the dimensional reduction D + 2 → D behavior in the Yang–Lee model in 2 × 2 minors. The 2 × 2 minors consist of two parameters, dimension D and the scaling dimension , since we have 2 = ( = 2 ) for the Yang–Lee case, due to the φ theory. φ φ 3/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 1. The intersection of zero loci of 3 × 3 minors (d , d , d , d , d ) for dimension D = 6. The 123 135 134 234 235 axes are (x, y) = ( , Q). The intersection point of the ﬁve lines is D = 6, = 2 with Q = 8, which is φ φ decomposed to the 2 × 2 minor d . The upper ﬁxed point is related to the minor d . 13 23 (i) Free ﬁeld theory at D = 6 The zero loci of d gives exactly the scale dimension = 2.0 at D = 6. The minors for vq, 13 φ vx, and vs (with a scalar scale dimension ) can be zero: vs1 vq1 vs1 vx1 vs1 vs1 det = 0, det = 0, det = 0. (13) vs3 vq3 vs3 vx3 vs3 vs3 For the value = 2, they yield Q(= ) = 8, = 10, and = 4, respectively. These φ 4 6 , , , are determined exactly by 2 × 2 minors of d . This is due to the free theory, i.e., φ 4 6 13 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 D . Other minors yield different values; for instance, d gives = 2.158 at D = 6. These two 23 φ different values of are understood when we evaluate 3 × 3 minors, as shown in Figs. 1 and 2. There are two ﬁxed points of zero loci of 3 × 3 minors in the six dimensions. The 3 × 3 minors are denoted by d in Eq. (7); for instance, d means ijk 123 ⎛ ⎞ vs1 vt1 vq1 ⎜ ⎟ d = det vs2 vt2 vq2 . (14) 123 ⎝ ⎠ vs3 vt3 vq3 (ii) D = 3 The zero loci of 2 × 2 minor do not provide good results except for D = 6 dimensions. This is due to the situation where the free theory breaks down in general dimensions. We have to consider the loci of 2 × 2 minors of a ﬁnite small value due to the interactions, instead of a zero value, for the non-trivial ﬁxed point. The small non-vanishing value of the minor might have an important meaning for the bound of . The loci of two important 2 × 2 minors d and d are shown φ 13 23 in Fig. 2, in which the value of d is changed by small values from −0.007 to 0.004, and the zero loci of d are also shown in Fig. 3 as a guide. The correct value of is realized in some 23 φ 4/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 2. The loci of 2 × 2 minors d with values from −0.007 to 0.004 are shown; d = 0 is also shown by a 13 23 green line. The axes are (x, y) = (D, ). Fig. 3. A close-up of the contour map in Fig. 2 near the = 0 loci of minor d . The green line shows the φ 13 zero loci of d as a guide. The axes are (x, y) = (D, ). 23 φ ﬁnite-value contours of d for D < 6. Near D = 6, deviation from the zero loci of d is expected 13 13 since deviates from 2 as the = 6 − D expansion shows: 2 − 0.5555. φ φ In Fig. 2, the ﬁnite-value curves of d converge to a curve starting from = 0.25 (at D = 1) 13 φ to 2.0 (at D = 4). This curve resembles the correct contour if the space dimension is shifted from D to D + 2. (The curve is translated to the right in Fig. 2 by two space dimensions by keeping the same value of .) This dimensional reduction will be discussed in Sect. 4. The ﬁnite-value contour map of d shows a valley shape, and one line is located at D = 2 near the exact value =−0.4, as shown in Fig. 3. In Fig. 3, the value of minor d is 0.0034, which φ 13 gives =−0.4. The cusp point moves when the value of minor d is changed and its trace φ 13 may give the bound of . 5/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 4. The zero loci of d[ , ] for different dimensions: D = 6 (dark blue), D = 5.5 (red), D = 5 φ 13 (green), D = 4.5 (yellow), and D = 4 (blue). The axes are (x, y) = ( , ). In Figs. 2 and 3, contour maps of d with small non-vanishing values are shown. 3. Critical dimension for = 0 In the Yang–Lee edge singularity, the scale dimension of φ becomes zero at some dimension between 2 and 3, since it is −0.4 in two dimensions and approximately 0.2 in three dimensions. We call this dimension the critical dimension D , on which the scale dimension vanishes. In the c φ determinant method, it is known that even small minors give accurate values of the scale dimensions [5,6]. We discuss here small determinants of 2 × 2, 3 × 3, and 4 × 4 matrices for this critical dimension D . Figure 2 shows Z-shaped curves for zero or very small values of the loci of d . This ﬁgure can be understood when we plot versus for various values of ﬁxed dimensions. The minor of d φ 13 has so far been considered as a function of D and with the condition of = due to the φ φ Yang–Lee model. However, in general we are able to consider and as two free parameters for d , and, for the Yang–Lee case, we put = . 13 φ In Fig. 4, we plot the zero loci of d[ , ] for different values of D. This minor then becomes φ 13 the usual one including the Ising case. A complicated contour for the zero loci of d is obtained especially around D = 4.4, where it has three solutions for = . This solution corresponds to the Z shape around (D, ) = (4, 1.5) in Fig. 2. In addition, we ﬁnd that the line of = of the Yang–Lee condition goes through very near the zero loci of d , but it does not intersect. This corresponds to Fig. 2 with the ﬁnite-value loci of d around D = 3. With some value of D, the contours in Fig. 4 (bottom left) and Fig. 5 intersect at the point ( , ) = (0, 0). We ﬁnd that this critical dimension from Fig. 5 is D = 2.6199. (15) As seen in Fig. 3, for a small ﬁnite value of d , the contour of d approaches the line = 0 13 13 φ from both sides, above > 0 and < 0. The peak of the contours approaches the critical φ φ 6/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 5. A contour plot of the zero loci of d[ , ] for different dimensions: D = 2.6 (blue), D = 2.61 φ 13 (yellow), D = 2.6199 (green). The axes are (x, y) = ( , ). dimension D . We consider the derivative of d by the dimension D, and we estimate D as the c 13 c point at which this derivative becomes zero. For = 0.000 001, we obtain D = 2.605 74. φ c The derivative of d by the space dimension D gives the approximate solution of D in the limit 13 c → 0. The minor d is proportional to in the limit → 0, since vs1, vs3 ∼ , and φ 12 φ φ −1 vt1, vt3 ∼ . We consider how the minors become zero in the limit → 0. For instance, at D = 2.61, we have, for small , 2 2 2 vs1 = 0.188 , vs2 = 0.139 , vs3 = 0.225 φ φ φ 1.055 0.667 1.078 vt1 = , vt2 = , vt3 = . (16) φ φ φ The coefﬁcients p , p are obtained from φ t vs0 vt0 p 1 = . (17) vs1 vt1 p 0 Since vs0 =− , vt0 = 0.7185/ in the limit → 0, we obtain a solution of p =−4, p = φ φ φ t a log (a is a constant), which gives the value of the central charge C = 0. With this choice of p = p , the central charge C becomes C = /p = 0 for this critical dimension D . This means t (D,2) t c that the energy–momentum tensor operator T = can be neglected since the OPE coefﬁcient (D,2) of this operator p becomes zero. This is the well known c catastrophe, which leads to logarithmic conformal ﬁeld theory . When neglecting the term, and only considering vs0 in Eq. (10), (D,2) we obtain the critical dimension D = 2.748 632 ··· . 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 D by another sophisticated method by taking higher operators. 4 × 4 minors The numbers of zero loci of minors are C = 6!/4!2!= 15. We investigated 6 4 the critical dimensions D by changing D between D = 2.58 and D = 2.59. Remarkably, at 7/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 6. The intersection of zero loci of minors at D = 2.589 53, = 0. The axes are (x, y) = (Q, ). D = 2.589 53, all 15 lines intersect at a single point in the contour map of (Q, ) under the condition = 0. If we take this as the value of the critical dimension D , we ﬁnd the following φ c set of results: D = 2.589 53, = 0, Q = 4.403 97, = 3.871 27. (18) c φ This critical dimension D obtained with 4 × 4 minors is close to D = 2.6199 from the 2 × 2 c c minor d with the intersection point of = = 0. In Fig. 6, all loci coincide but, if we include 13 φ more relevant operators, this value may change. For the = 0 case, in any dimension D, d can become zero. When = 0, by the deﬁnition φ ij φ of the scale dimension , the two-point correlation function G(r) = 1/r becomes a constant in the long-range limit r →∞. In the Yang–Lee edge singularity, the exponent of the density σ is related to as σ = . (19) D − From this relation, σ is vanishing for = 0. 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. 4. Dimensional reduction The zero loci of the 2 × 2 minor d shows an interesting characteristic linear behavior for D < 4, which is approximated as = (3D − 2)/5. In Fig. 7, the shift of the zero loci of d φ 13 for D → D + 2 is shown (translation of two dimensions to the right in Fig. 7). The blue line of Fig. 7 for the zero loci of d almost coincides with the red line of the Yang–Lee edge singularity analyzed by the Padé approximation. The red line represents the result of 3 × 3 minors and it can be 8/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 7. The D → D + 2 shifted line of the zero loci of d (yellow line) is shown by the blue line and is consistent with the estimated Yang–Lee line (red) by Padé analysis. The blue and red lines are almost the same for 1 < D < 5.5. approximated as 3D − 8 = . (20) This expression satisﬁes the exact values of D = 1( =−1), D = 2( =−0.4), and D = 6 φ φ ( = 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 D + 2 dimensions is equivalent to a D-dimensional Yang– Lee edge singularity for the critical phenomena. This dimensional reduction has been explained by supersymmetry . A rigorous mathematical proof of this dimensional reduction was shown in Ref. . We will study this dimensional reduction in a separate paper using the conformal bootstrap in a determinant method . The Yang–Lee edge singularity requires (= 2 ) = . For a branched polymer due to dimensional reduction there is a relation = + 1. This relation is a manifestation of the supersymmetry of the system . Since the blue line is very close to the red line in Fig. 7, one can make an estimation of the critical dimension D , in which = 0. The intersection of d with the = 0 line can be evaluated c φ 13 φ very precisely as D = 0.599 547 1444. Adding 2 to this value for the dimensional reduction, it gives the estimation of the critical dimension D as D = 2.599 547 1444, which is very close to other c c estimations of this paper. We discussed this value in the previous section as D = 2.6199 as in Eq. (18). 5. 3 × 3 minors and Padé analysis Up to now, we have mainly discussed 2×2 minors. If we take a spin-4 operator and its scale dimension Q = , we need to do an analysis of the intersection of the zero loci of the 3 × 3 minors d . 4 ijk 9/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami 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. Q (Padé) φ φ D = 3.0 0.174 343 (0.187 825) 4.341 06 (3.771 24) 0.229 95 D = 3.5 0.499 401 (0.500 969) 5.041 95 (4.375 56) 0.531 53 D = 4.0 0.823 283 (0.815 623) 5.711 52 (4.922 83) 0.831 75 D = 4.5 1.137 55 (1.123 71) 6.333 95 (5.446 45) 1.1300 D = 5.0 1.438 07 (1.419 87) 6.917 16 (5.959 72) 1.4255 D = 5.5 1.724 69 (1.746 82) 7.469 85 (6.466 72) 1.7165 D = 6.0 2.0 8.0 2.0 From the formula of the minors in a 3 × 6 matrix in the appendix, we have a Plüker formula such as (Eq. (A.12)) −+−= 0. (21) For instance, at D = 3, we ﬁnd that the zero loci of d =, d =, d =, and 123 126 124 d = intersect at a point, and the above Plüker formula is satisﬁed. In Table 1, we present the intersection of three zero loci of minors d , d , and d . The 123 134 124 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 D. In Table 1, we present in parentheses such nearby different intersection points of the three zero loci of the minors. In the = 6 − D expansion, is known up to the four-loop level : 2 3 4 = 2 − 0.555 55x − 0.029 4925x + 0.021 845x − 0.039 4773x , (22) where x = = 6 − D. Including the critical dimension D , for which is vanishing, the above c φ expansion becomes 2 2 0.55555 = (6 − x − D ) + − x + O(x ) . (23) φ c 6 − D (6 − D ) 6 − D c c c Inserting the value of D = 2.6199 from Sect. 4, it becomes = (3.3801 − x)[0.591 70 + 0.010 695x − 0.005 561 25x 3 4 + 0.004 817 55x − 0.010 2541x + ···]. (24) This expansion is approximated by the [2,2] Padé method, a + a x + a x 0 1 2 = (3.3801 − x) , (25) 1 + b x + b x 1 2 with a = 0.591 70, a = 2.3916, a = 1.0090, b = 4.023 85, b = 1.6419. The curve of this 0 1 2 1 2 Padé is incorporated in Fig. 8. For the values of of the Yang–Lee singularity in D dimensions, the previous results obtained by Gliozzi  are very close to the values of Table 1; for instance, at D = 4, = 0.823. In another evaluation with more primary operators , the values = 0.8466 in D = 4 and = 1.455 in φ φ 10/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Fig. 8. is estimated by a [2,2] Padé with D = 2.6199. The dots are the values from Table 1 fora3 × 3 φ c minor analysis of . D = 5 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. . Our new analysis of Padé with a ﬁxed value at the critical dimension = 0 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 D . Unfortunately, what we have done is a numerical estimation of D by the bootstrap determinant method and it may be an approximation. It is important to obtain an exact value of D , and the comparison with expansion will be improved with an exact value of D . 6. Summary We have considered theYang–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 3×3 determinant result of Gliozzi , we have improved the consistency with the result of the expansion. We ﬁnd that the basic 2 × 2 minor d 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 3 × 3 minors. We have shown that the value obtained by the intersection of the zero loci of 3 × 3 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 D . We obtained the critical dimension D from the minor d for the = 0 and → 0 limits. The c 13 φ intersection of the zero loci of d[D, , ] at the point = = 0 gives the critical dimension φ 13 φ D = 2.6199. We have estimated the critical dimension approximately by other methods, by the peak approaching the ﬁnite d (D = 2.6050), the dimensional reduction value (D = 2.5995), and 4 × 4 13 c c minor analysis (D = 2.589). We emphasize that the estimation of the critical dimension D 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 . We have discussed the dimensional reduction property in the Yang–Lee model, using the fact that a branched polymer in D + 2 dimensions is equivalent to a D-dimensional Yang–Lee edge singularity. Acknowledgements The author is grateful to Ferdinando Gliozzi for discussions on the determinant method and the useful suggestion of the critical dimension. He also thanks Edouard Brézin for discussion of the dimensional reduction problem. 11/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami This work is supported by a Japan Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid 16K05491. The Mathematica11 system of wolfram.com is acknowledged for this research. Funding Open Access funding: SCOAP . Appendix. Plüker formula Relation 1 (3 × 3 minors) For the determinant d , which is deﬁned as ⎛ ⎞ vs1 vs2 vs3 ⎜ ⎟ d = det vt1 vt2 vt3 ⎝ ⎠ vq1 vq2 vq3 vs1 vs2 vs1 vs3 vs2 vs3 = (vq3)det − (vq2)det + (vq1)det , (A.1) vt1 vt2 vt1 vt3 vt2 vt3 if the ﬁrst determinant (d ) and second minor (d ) on the right-hand side are zero, then the third 12 13 minor (d ) should be vanishing when d = 0. 23 123 Relation 2 (2 × 4 matrix) The 2 × 4 matrix is denoted as x x x x 11 12 13 14 M = . (A.2) x x x x 21 22 23 24 The Plüker relation is −+= 0, (A.3) where the minor is denoted as [ij]= (x )(x ) − (x )(x ). The application of this formula to minors 1i 2j 1j 2i of our case can be taken as vs1 vs2 vs3 vs4 M = . (A.4) vt1 vt2 vt3 vt4 Thus, in our notation for 2 × 2 minors, d =, d =, d =, d =, d =, d = (A.5) 12 13 14 23 24 34 and we have an identity d d − d d + d d = 0, (A.6) 12 34 13 24 14 23 which is represented by the tableaux 1 2 1 3 1 4 − + = 0. (A.7) 3 4 2 4 2 3 12/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami Note that the third tableau is not ordered by increasing row and column (4 is larger than 3). Another application of this formula is to take the following matrix: vs1 vt1 vq1 vx1 M = . (A.8) vs3 vt3 vq3 vx3 In the D = 6 case, we ﬁnd that 2 × 2 minors made of the above matrix are vanishing, == = 0 for = 2, = 8, and = 10. Therefore they satisfy the relation of Eq. (A.3). In this φ 4 6 D = 6 case, we also ﬁnd that, for such values of = 2, = 8, = 10, the other three sets of φ 4 6 minors are vanishing at D = 6: vq1 vx1 vt1 vx1 = det = 0, = det = 0, vq3 vx3 vt3 vx3 vt1 vq1 = det = 0. (A.9) vt3 vq3 Relation 3 (3 × 6 matrix) (i) The 3 × 6 matrix in the bootstrap minor method is ⎛ ⎞ vs1 vs2 vs3 vs4 vs5 vs6 ⎜ ⎟ vt1 vt2 vt3 vt4 vt5 vt6 . (A.10) ⎝ ⎠ vq1 vq2 vq3 vq4 vq5 vq6 The Plüker relation is obtained by a mutual exchange of numbers: +−+−+= 0 (A.11) This identity is obtained from the ﬁrst term to the second term by (2 ↔ 6), and from the second to third by (2 ↔ 3), etc. with the sign. We can also use the following relations (obtained similarly): −+−= 0 ++−= 0; (A.12) from these equations, Eq. (A.11) becomes =−3−+. (A.13) In our minor examples, we have d d =−3d d − d d + d d . (A.14) 146 235 123 456 125 346 135 246 The terms on the right-hand side are ordered by increasing row and column. (ii) Another application can be taken as ⎛ ⎞ vs1 vt1 vq1 vx1 vs1 vs1 ⎜ ⎟ . (A.15) ⎝ vs2 vt2 vq2 vx2 vs2 vs2 ⎠ vs3 vt3 vq3 vx3 vs3 vs3 13/15 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053I01/5025800 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053I01 S. Hikami The Plüker relation is the same as before. In a tableau, Eq. (A.13) is expressed as 1 4 6 1 2 3 1 2 5 1 3 5 =−3 − + . (A.16) 2 3 5 4 5 6 3 4 6 2 4 6 The left-hand side is not ordered by increasing column, but the right-hand side is a combination of ordering by both increasing row and column. 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Progress of Theoretical and Experimental Physics – Oxford University Press
Published: May 30, 2018
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