A conformal bootstrap approach to critical percolation in two dimensions

Picco, Marco; Ribault, Sylvain; Santachiara, Raoul

doi:10.21468/scipostphys.1.1.009

Cited by 56 publications

(210 citation statements)

References 28 publications

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“…In our previous work [9], we have performed Monte-Carlo computations of four-point connectivities, with sufficient precision for being tested against CFT predictions or guesses. Moreover, we have introduced a numerical implementation of the conformal bootstrap approach that tests whether an exact ansatz for the spectrum gives rise to crossingsymmetric four-point functions.…”

Section: Monte-carlo Computations and The Bootstrap Approachmentioning

confidence: 99%

“…In [9], we have proposed a simple ansatz S 2Z,Z+ 1 2 for the spectrum. This ansatz leads to crossing-symmetric four-point functions, so it must be the spectrum of a consistent CFT, which was later called the odd CFT in [11].…”

Section: The Partition Function and The Spectrummentioning

confidence: 99%

“…This is barely enough for seeing a difference between S 2Z,Z+ 1 2 and S Potts , and insufficient for probing S Potts in detail. A much stronger test could come from probing crossing symmetry of four-point functions built from S Potts , using the techniques of [9]. But this would first require determining the full structure of the representations, not just their characters.…”

Section: The Partition Function and The Spectrummentioning

confidence: 99%

“…eigenvalues of the Virasoro generator L 0 , and do not capture the full action of the Virasoro algebra. This information is potentially within reach of their lattice algebraic methods, or of CFT-motivated guesses as in [9] (Appendix A.3). This issue will however not affect our comparison, because for generic Q our spectrum S 2Z,Z+ 1 2 is made of irreducible Verma modules.…”

Section: The Spectrum Of the Potts Modelmentioning

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: Monte-carlo Computations and The Bootstrap Approachmentioning

confidence: 99%

Section: The Partition Function and The Spectrummentioning

confidence: 99%

Section: The Partition Function and The Spectrummentioning

confidence: 99%

Section: The Spectrum Of the Potts Modelmentioning

confidence: 99%

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

Ribault²,

2019

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“…The critical exponents and three point-functions are well known, and closely related with Liouville CFT at c ≤ 1 (sometimes called time-like Liouville) [23,24,25]. The question of higher correlation functions in the model remains still partly open, but has recently witnessed important progress [26,27]. The antiferromagnetic (AF) critical line is not self-dual, 1 and given by…”

Section: Critical Linesmentioning

confidence: 99%

Conformally invariant boundary conditions in the antiferromagnetic Potts model and the SL(2, ℝ)/U(1) sigma model

Robertson

Jacobsen

Saleur

2019

J. High Energ. Phys.

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We initiate a study of the boundary version of the square-lattice Q-state Potts antiferromagnet, with Q ∈ [0, 4] real, motivated by the fact that the continuum limit of the corresponding bulk model is a noncompact CFT, closely related with the SL(2, R) k /U (1) Euclidian black-hole coset model.While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the SL(2, R) k /U (1) coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum.The CBC we find are of two types. The first is free boundary Potts spins, for which we confirm an old conjecture for the generating functions of conformal levels, and show them to be related to characters in a non-linear deformation of the W∞ algebra.The second type of CBC -which corresponds to restricting the values of the Potts spins to a subset of size Q1, or its complement of size Q − Q1, at alternating sites along the boundary -is new, and turns out to be conformal in the antiferromagnetic case only. Using algebraic and numerical techniques, we show that the corresponding spectrum generating functions produce all the characters of discrete representations for the coset CFT. The normalizability bounds of the associated discrete states in the coset CFT are found to have a simple interpretation in terms of boundary phase transitions in the lattice model. In the two-boundary case, with two distinct alt conditions, we obtain similar results, at least in the case when the corresponding boundary condition changing operator also inserts a number of defect lines.For √ Q = 2 cos π k , with k ≥ 3 integer, we show also how our boundary conditions can be reformulated in terms of a RSOS height model. The spectrum generating functions are then identified with string functions of the compact SU (2) k−2 /U (1) parafermion theory (with symmetry Z k−2 ). The new alt conditions are needed to cover all the string functions. We provide an algebraic proof that the two-boundary alt conditions correctly produce the fusion rules of string functions. We expose in detail the special case of Q = 3 and its link with three-colourings of the square lattice and a corresponding boundary six-vertex model.Finally, we discuss the case of an odd number of sites (in the loop model) and the relation with wired boundary conditions (in the spin model). In this case the RSOS restriction produces the disorder operators of the parafermion theory. arXiv:1906.07565v2 [cond-mat.stat-mech]

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Conformal covariance of connection probabilities and fields in 2D critical percolation

Camia

2023

Comm Pure Appl Math

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Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well‐defined scaling limit for every . Moreover, the limiting functions transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and for any . This implies that and , for some constants C2 and C3.We also define a site‐diluted spin model whose n‐point correlation functions Cn can be expressed in terms of percolation connection probabilities and, as a consequence, have a well‐defined scaling limit with the same properties as the functions . In particular, . We prove that the magnetization field associated with this spin model has a well‐defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) 5/48. A heuristic analysis of the four‐point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension 5/4, which counts the number of percolation four‐arm events and can be identified with the so‐called “four‐leg operator” of conformal field theory.

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A conformal bootstrap approach to critical percolation in two dimensions

Cited by 56 publications

References 28 publications

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

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

Conformally invariant boundary conditions in the antiferromagnetic Potts model and the SL(2, ℝ)/U(1) sigma model

Conformal covariance of connection probabilities and fields in 2D critical percolation

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