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

Picco, Marco; Santachiara, Raoul

doi:10.1088/1742-5468/2010/07/p07027

Cited by 8 publications

(17 citation statements)

References 49 publications

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“…One of these value, corresponding to a certain interface, was found in agreement with the one proposed on the basis of CFT computation in [9]. This scenario, which establishes for the first time a connection between geometrical objects and extended CFT, has been made particularly clear in [8] where the spin cluster interfaces of the Z 4 model model were studied.…”

Section: Introductionsupporting

confidence: 84%

“…But, so far, the most important insights into this problem come from numerical measurements of the fractal dimensions associated to the spin interfaces for Z 4 and Z 5 spin models [7]. By measuring systematically all the different bulk and boundary spin interfaces, it was found that there is a limited number of possible values for the fractal dimension which can be understood on the basis of the classification of Z N conformal boundary conditions [8]. One of these value, corresponding to a certain interface, was found in agreement with the one proposed on the basis of CFT computation in [9].…”

Section: Introductionmentioning

confidence: 99%

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Critical interfaces and duality in the Ashkin-Teller model

Picco

Santachiara

2011

Phys. Rev. E

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We report on the numerical measures on different spin interfaces and Fortuin-Kasteleyn (FK) cluster boundaries in the Askhin-Teller (AT) model. For a general point on the AT critical line, we find that the fractal dimension of a generic spin cluster interface can take one of four different possible values. In particular we found spin interfaces whose fractal dimension is d(f)=3/2 all along the critical line. Furthermore, the fractal dimension of the boundaries of FK clusters was found to satisfy all along the AT critical line a duality relation with the fractal dimension of their outer boundaries. This result provides clear numerical evidence that such duality, which is well known in the case of the O(n) model, exists in an extended conformal field theory.

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Section: Introductionsupporting

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Critical interfaces and duality in the Ashkin-Teller model

Picco

Santachiara

2011

Phys. Rev. E

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“…• The case (1|234) is not understood for general couplings, but for g = 3/4 (where the AT model coincides with the integrable Fateev-Zamolodchikov Z 4 spin model), the associated operator satisfies a null-state equation in the Z 4 -parafermionic CFT, and is argued [13] The fractal dimension cannot be simply obtained from the CG analysis. However, based on the above exact values and Monte-Carlo simulations (in the region between Ising and Potts points), the following expression was proposed [9] for d f :…”

Section: Spin Interfaces In the Ashkin-teller Modelmentioning

confidence: 99%

Spin interfaces in the Ashkin–Teller model and SLE

Ikhlef¹,

Rajabpour²

2012

J. Stat. Mech.

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We investigate the scaling properties of the spin interfaces in the Ashkin-Teller model. These interfaces are a very simple instance of lattice curves coexisting with a fluctuating degree of freedom, which renders the analytical determination of their exponents very difficult. One of our main findings is the construction of boundary conditions which ensure that the interface still satisfies the Markov property in this case. Then, using a novel technique based on the transfer matrix, we compute numerically the left-passage probability, and our results confirm that the spin interface is described by an SLE in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of Ashkin-Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.Another current research direction is the study of "extended" SLEs, which are curve models determined by two random processes: the driving process of the Loewner chain, and an additional process corresponding, in the associated statistical model, to local variables which are not fixed locally by the presence of the curve. It was argued [6,7] that the martingale conditions for this pair of processes correspond to null-state equations in an extended CFT. In particular, the presence of the additional process affects the relation between the central charge c and the SLE parameter κ. An important point that needs to be considered in this context, is that in the presence of additional variables, the lattice interface no longer satisfies the Markov property, which is an essential ingredient of the SLE model.The Ashkin-Teller (AT) model is a simple local spin model, with a critical line of constant central charge c = 1 and varying exponents. We believe it presents the two features described above: first, although it has been much studied with both lattice and CFT techniques, the scaling theory for its spin interfaces is not identified, and second, these interfaces leave some fluctuating variables, which makes them good candidates for extended SLEs.This paper presents two advances in the study of spin interfaces, on the particular example of the AT model. First, we introduce some specific boundary conditions (BCs) which ensure that the spin interface satisfies the Markov property, even in the presence of an additional variable. We then verify numerically that our interface relates to SLE, and for this purpose, we introduce a new algorithm allowing the measurement of the left-passage probability [8] in the transfer-matrix formalism. Our results also include an accurate numerical determination of the fractal dimension d f of spin interfaces, also by means of the transfer matrix, which supports a conjecture for d f stated in [9]. Second, we identify a specific value on the critical line of the AT model, where the temperature tends to zero, and at this point, we describe an exact mapping of the AT model onto an integrable 19-vertex model. Although this second result does not give direct access to spin interface ...

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“…There are different possibilities to define critical curves in the Ashkin-Teller model [17,18,19,20,21]. In the spin representation one can think about the domain walls between one spin and the other three, or the domain walls between two definite spins and the other two.…”

Section: Introductionmentioning

confidence: 99%

Critical domain walls in the Ashkin–Teller model

Caselle¹,

Lottini²,

Rajabpour³

2011

J. Stat. Mech.

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Abstract. We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE κ , except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE κ in this case.

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

Cited by 8 publications

References 49 publications

Critical interfaces and duality in the Ashkin-Teller model

Critical interfaces and duality in the Ashkin-Teller model

Spin interfaces in the Ashkin–Teller model and SLE

Critical domain walls in the Ashkin–Teller model

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