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

Picco, Marco; Ribault, Sylvain; Santachiara, Raoul

doi:10.21468/scipostphys.7.4.044

Cited by 28 publications

(115 citation statements)

References 52 publications

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“…Its dependence on β 2 ∈ >0 is very singular, and differs from the smooth dependence that we expect in critical statistical systems. This difference was even used for distinguishing the limit CFT from the critical Potts model, although they looked identical from the point of view of the numerical behaviour of certain correlation functions [8,16].…”

Section: Discussionmentioning

confidence: 99%

The non-rational limit of D-series minimal models

Ribault

2020

SciPost Phys. Core

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We study the limit of D-series minimal models when the central charge tends to a generic irrational value c\in (-\infty, 1)c∈(−∞,1). We find that the limit theory’s diagonal three-point structure constant differs from that of Liouville theory by a distribution factor, which is given by a divergent Verlinde formula. Nevertheless, correlation functions that involve both non-diagonal and diagonal fields are smooth functions of the diagonal fields’ conformal dimensions. The limit theory is a non-trivial example of a non-diagonal, non-rational, solved two-dimensional conformal field theory.

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Section: Discussionmentioning

confidence: 99%

The non-rational limit of D-series minimal models

Ribault

2020

SciPost Phys. Core

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“…The first factor takes into account the non-universal, small distance effects due to the lattice. We refer the reader to [39,41] for a more detailed discussion of these ultraviolet corrections. The values of d 0 are reported in Table 1.…”

Section: Plane Limitmentioning

confidence: 99%

“…Contrary to statistical models with local and positive Boltzmann weights, whose critical points are described by the unitary minimal models [36], the critical point of pure percolation is described by a non-unitary and logarithmic CFT [37,38]. This CFT is not fully known, but very recent results have paved the way to its complete solution [37][38][39][40][41][42][43]. The line of new critical points shown in Figure 2 remains by far less understood.…”

Section: Introductionmentioning

confidence: 99%

Topological effects and conformal invariance in long-range correlated random surfaces

et al. 2020

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We consider discrete random fractal surfaces with negative Hurst exponent H<0H<0. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level hh. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter (HH) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value h=h_ch=hc and for H\leq-\frac{3}{4}H≤−34 the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For -\frac{3}{4}<H< 0−34<H<0 instead, there is a line of critical points with continously varying exponents. The universality class of these points, in particular concerning the conformal invariance of the level clusters, is poorly understood. By combining the Conformal Field Theory and the numerical approach, we provide new insights on these phases. We focus on the connectivity function, defined as the probability that two sites belong to the same level cluster. In our simulations, the surfaces are defined on a lattice torus of size M\times NM×N. We show that the topological effects on the connectivity function make manifest the conformal invariance for all the critical line H<0H<0. In particular, exploiting the anisotropy of the rectangular torus (M\neq NM≠N), we directly test the presence of the two components of the traceless stress-energy tensor. Moreover, we probe the spectrum and the structure constants of the underlying Conformal Field Theory. Finally, we observed that the corrections to the scaling clearly point out a breaking of integrability moving from the pure percolation point to the long-range correlated one.

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“…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%

“…Ratio A 1 /A 2 between the amplitudes of the two singlet fields (see table 2), corresponding to the lines with i 13 = 24 and i 13 = 35 (see table14), plotted against 1/L. The curve is a secondorder polynomial fit to the last three data points.…”

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

Cited by 28 publications

References 52 publications

The non-rational limit of D-series minimal models

The non-rational limit of D-series minimal models

Topological effects and conformal invariance in long-range correlated random surfaces

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

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