Two-dimensional critical Potts and its tricritical shadow

Janke, Wolfhard; Schakel, Adriaan M. J.

doi:10.1590/s0103-97332006000500023

Cited by 9 publications

(14 citation statements)

References 41 publications

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“…To make the connection with the minimal model M 5 [6] with central charge 4/5 one has to take into account relations of the kind χ I = χ (1,1) + χ (4,1) or χ ε = χ (1,2) + χ (1,3) where χ (r,s) is the Virasoro character of the operator φ (r,s) in the minimal Kac table. The boundary operator ψ ε transforming in the representation ε generates then the boundary conditions (1|2 + 3) and thus the interface SLE 24/5 discussed in [31].…”

Section: A Known Example: Three-states Potts Modelmentioning

confidence: 99%

“…Among these models, we mention for instance the critical percolation, the self-avoiding walks, the loop erased random walks, or again spin lattice models such as the Potts models [1]. These studies have benefited from a great amount of numerical work [4] supporting the proposed theoretical scenario. In general, the critical models whose geometrical properties are well understood, even if there are often no rigorous proofs, can be associated to the critical phases of a one-parameter family of statistical models, the O(n) loop models, the parameter n representing the loop fugacity.…”

Section: Introductionmentioning

confidence: 99%

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Critical interfaces of the Ashkin–Teller model at the parafermionic point

Picco¹,

Santachiara²

2010

J. Stat. Mech.

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We present an extensive study of interfaces defined in the Z 4 spin lattice representation of the Ashkin-Teller (AT) model. In particular, we numerically compute the fractal dimensions of boundary and bulk interfaces at the Fateev-Zamolodchikov point. This point is a special point on the self-dual critical line of the AT model and it is described in the continuum limit by the Z 4 parafermionic theory. Extending on previous analytical and numerical studies [10,12], we point out the existence of three different values of fractal dimensions which characterize different kind of interfaces. We argue that this result may be related to the classification of primary operators of the parafermionic algebra. The scenario emerging from the studies presented here is expected to unveil general aspects of geometrical objects of critical AT model, and thus of c = 1 critical theories in general.

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Section: A Known Example: Three-states Potts Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Critical interfaces of the Ashkin–Teller model at the parafermionic point

Picco¹,

Santachiara²

2010

J. Stat. Mech.

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“…This mathematical approach, which is built on random conformally invariant fractal curves, entails the identification of the models (1.5) with SLE κ (Schramm-Loewner Evolution) with κ = 4p ′ p . For percolation, with (p, p ′ ) = (2, 3) and κ = 6, the fractal dimensions d = 2(1 − ∆) of various fractal geometric curves are known [29,33,[49][50][51][52] including those of chordal SLE paths, hulls (H), cluster mass (C), external perimeter (EP) and red bonds (RB):…”

Section: Introductionmentioning

confidence: 99%

Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Morin-Duchesne¹,

Klümper²,

Pearce³

2017

J. Stat. Mech.

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Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as LM(2, 3) in the Yang-Baxter integrable family of logarithmic minimal models LM(p, p ′ ). We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y -system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D 3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights ∆ 1,s for s ∈ Z 1 with ∆ r,s = (3r − 2s) 2 − 1 /24. For the periodic case, the finite-size corrections agree with the conformal weights ∆ 0,s , ∆ 1,s with s ∈ 1 2 Z 0 . These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying q-binomial and q-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine u(1) characters given by the u(1) partition function Z 2,3 (q) = Z

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“…The belief that spin clusters are indeed conformally invariant for other values of Q = 2 in the critical regime 0 ≤ Q ≤ 4 has even been challenged at times, but seems however well established by now [6,7]. Some progress has been accomplished in the Q = 3 case [5,8] by speculating that the spin clusters in the critical Potts model would be equivalent to FK clusters in the tricritical Potts model [8,9]. This equivalence has however not been proven, and is moreover restricted so far to the simplest geometrical questions [10].…”

mentioning

confidence: 99%

Critical exponents of domain walls in the two-dimensional Potts model

Dubail¹,

Jacobsen²,

Saleur³

2010

J. Phys. A: Math. Theor.

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We address the geometrical critical behavior of the two-dimensional Q-state Potts model in terms of the spin clusters (i.e., connected domains where the spin takes a constant value). These clusters are different from the usual Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross and branch. We develop a transfer matrix technique enabling the formulation and numerical study of spin clusters even when Q is not an integer. We further identify geometrically the crossing events which give rise to conformal correlation functions. This leads to an infinite series of fundamental critical exponents h 1 − 2 ,2 1 , valid for 0 ≤ Q ≤ 4, that describe the insertion of 1 thin and 2 thick domain walls.

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Two-dimensional critical Potts and its tricritical shadow

Cited by 9 publications

References 41 publications

Critical interfaces of the Ashkin–Teller model at the parafermionic point

Critical interfaces of the Ashkin–Teller model at the parafermionic point

Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Critical exponents of domain walls in the two-dimensional Potts model

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Two-dimensional critical Potts and its tricritical shadow

Cited by 9 publications

References 41 publications

Critical interfaces of the Ashkin–Teller model at the parafermionic point

Critical interfaces of the Ashkin–Teller model at the parafermionic point

Conformal partition functions of critical percolation fromD3thermodynamic Bethe Ansatz equations

Critical exponents of domain walls in the two-dimensional Potts model

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Product

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Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations