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

Gori, Giacomo; Viti, Jacopo

doi:10.1007/jhep12(2018)131

Cited by 15 publications

(18 citation statements)

References 70 publications

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“…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].…”

Section: Why Four-point Connectivitiesmentioning

confidence: 99%

“…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):…”

Section: B4 An Excursion In the Complex Planementioning

confidence: 99%

“…5) where det ′ [A] is the product of the non-vanishing eigenvalues of A, and det [A(z i ; z i )] is the determinant of the A after removing z i 's row and z i 's column. Given the pairs of points (z 1 , z 2 ) and (z 3 , z 4 ), let Z 12,34 be the sum over graphs made of two disconnected , z 2 ; z 3 , z 4 )] is the determinant of A after removing z 1 and z 2 's rows, and z 3 and z 4 's columns.…”

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confidence: 99%

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

Ribault²,

2019

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We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states Q ∈ (0, 4) that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the 2 to 4 significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases Q = 0, 3, 4. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane {ℜc < 13}, where c is the central charge of the Virasoro symmetry algebra.

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Section: Why Four-point Connectivitiesmentioning

confidence: 99%

Section: B4 An Excursion In the Complex Planementioning

confidence: 99%

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confidence: 99%

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

Ribault²,

2019

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“…p + n), at finite size. For a state of eigenvalue of the Hamiltonian at a given system size N , we consider lattice precursors to its conformal weights, 12 which we also denote (h,h), defined as the solutions to…”

Section: Jhep10(2020)109mentioning

confidence: 99%

“…First, many three-point functions were determined using connections with Liouville theory at c < 1 [9][10][11]. Second, a series of attempts using conformal bootstrap ideas [3,[12][13][14][15][16] led to the determination of some of the most fundamental four-point functions in the problem (namely, those defined geometrically, and hence for generic Q), also shedding light on the operator product expansion (OPE) algebra and the relevance of the partition functions determined in [1]. In particular, the set of operators -the so-called spectrum -required to describe the partition function [1] and correlation functions [15] in the Potts-model CFT was settled.…”

Section: Introductionmentioning

confidence: 99%

The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q

Grans-Samuelsson

Liu

et al. 2020

J. High Energ. Phys.

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The spectrum of conformal weights for the CFT describing the two-dimensional critical Q-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years [1]. However, the exact nature of the corresponding Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ representations has remained unknown up to now. Here, we solve the problem for generic values of Q. This is achieved by a mixture of different techniques: a careful study of “Koo-Saleur generators” [2], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the four-point amplitudes recently determined using the “interchiral conformal bootstrap” in [3] on the analytical side. We find that null-descendants of diagonal fields having weights (hr,1, hr,1) (with r ∈ ℕ*) are truly zero, so these fields come with simple Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ (“Kac”) modules. Meanwhile, fields with weights (hr,s, hr,−s) and (hr,−s, hr,s) (with r, s ∈ ℕ*) come in indecomposable but not fully reducible representations mixing four simple Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ modules with a familiar “diamond” shape. The “top” and “bottom” fields in these diamonds have weights (hr,−s, hr,−s), and form a two-dimensional Jordan cell for L0 and $$ {\overline{L}}_0 $$ L ¯ 0 . This establishes, among other things, that the Potts-model CFT is logarithmic for Q generic. Unlike the case of non-generic (root of unity) values of Q, these indecomposable structures are not present in finite size, but we can nevertheless show from the numerical study of the lattice model how the rank-two Jordan cells build up in the infinite-size limit.

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Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

Jacobsen

Saleur

2019

J. High Energ. Phys.

160

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We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional Q-state Potts model conformal field theory. In a recent work [1], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on infinite cylinders. We find that the spectra in the scaling limit are much richer than those proposed in [1]: they involve in particular fields with conformal weight hr,s where r is dense on the real axis.

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

Cited by 15 publications

References 70 publications

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

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

The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q

Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

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