Exact Logarithmic Four-Point Functions in the Critical Two-Dimensional Ising Model

Abstract

Based on conformal symmetry we propose an exact formula for the four-point connectivities of Fortuin-Kasteleyn clusters in the critical Ising model when the four points are anchored to the boundary. The explicit solution we found displays logarithmic singularities. We check our prediction using Monte Carlo simulations on a triangular lattice, showing excellent agreement. Our findings could shed further light on the formidable task of the characterization of logarithmic conformal field theories and on their relevance in physics.

Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance

“…(4.9) above E(η) and K(η) are the complete elliptic integrals of first and second kind (with Mathematica convention for the modulus). In conclusion [45] the ratio R at Q = 2 is given by…”

“…The conjectures presented in Sec. 4 As done in [45], we map the four points z 1 = 0, z 2 = η, z 3 = 1 and z 4 = ∞ on the boundary of an equilateral triangle by a Schwartz-Christoffel transformation…”

On four-point connectivities in the critical 2d Potts model

“…Although connectivities are non-local quantitites, we work under the basic assumption that cluster connectivities are linearly related to correlation functions of local fields in a CFT that describes the critical Potts model. This assumption is known to hold when the clusters are anchored to a boundary [2][3][4][5] and in some other cases [6,7]. In particular, the three-point connectivity of the two-dimensional Potts model was found to be related to a three-point function in Liouville theory [6].…”

“…V (2,1)(1,1) (0)V (1,2)(1,1) (z)V (2,1)(1,1) (1)V (1,2)(1,1) (∞) (C. 29) for the W A 3 minimal model with central charge c = 4 5 with (p, p ′ ) = (4,5). This is obtained by setting in (C.18):…”

Observation of nonscalar and logarithmic correlations in two- and three-dimensional percolation

“…[8,9] classified the S Q irreducible operators, related them to cluster observables and unravelled the corresponding LCFT contents. We stress that these works and this Letter apply to bulk LCFT, which is more challenging [10][11][12] than its boundary counterpart [13,14]. Remarkably, in this context, some of the most salient structural properties (like: for each Q 0 , which operator mixings produce logarithmic factors, and in which correlators) turn out to be independent of dimension d. For d > 2, this route seems the only known semi-rigorous way of deriving exact results on the logarithmic structure of LCFTs.…”