Geometrical four-point functions in the two-dimensional critical Q-state Potts model: connections with the RSOS models

He, Yifei; Grans-Samuelsson, Linnea; Jacobsen, Jesper Lykke; Saleur, Hubert

doi:10.1007/jhep05(2020)156

Cited by 19 publications

(79 citation statements)

References 42 publications

(232 reference statements)

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“…σ,σ . In the case of pure percolation, for instance, the spectrum is known but not the structure constants, even if very recent progresses have paved the way to their determination [43]. The plane limit M , N = ∞ is recovered by noting that all the one-point functions V ∆,∆ q vanish but the identity one…”

Section: Three Main Assumptionsmentioning

confidence: 99%

“…In general, the presence of degenerate fields is a crucial feature of a CFT [64], which in some cases allow to solve the theory [65][66][67][68]. For pure percolation, the energy field is degenerate, which leads to relations between the different structure constants of the theory [43,68,69]. Table 4.…”

Section: Spectrum and Structure Constantsmentioning

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%

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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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Section: Three Main Assumptionsmentioning

confidence: 99%

Section: Spectrum and Structure Constantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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%

“…It is also expected that this does not hold for all fields with degenerate weights. In fact, it was suggested in [3,15] that, in the Potts-model case, only fields with weights (h r,1 , h r,1 ) give rise to null descendants. Since the spectrum of the model is expected to contain non-diagonal fields with weights (h r,s , h r,−s ) and (h r,−s , h r,s ) for r, s ∈ N * , this means that the theory should contain fields with degenerate (left or right) weights whose null descendants are nonzero, even though their two-point function vanishes.…”

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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The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

Grans-Samuelsson

Jacobsen

Saleur

2021

J. High Energ. Phys.

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We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of $$ {W}_{0,{\mathfrak{q}}^{\pm 2}} $$ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $$ {\mathfrak{q}}^{\pm 2} $$ q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

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Geometrical four-point functions in the two-dimensional critical Q-state Potts model: connections with the RSOS models

Cited by 19 publications

References 42 publications

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

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

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

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