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arXiv:1005.0493 · Journal of Statistical Mechanics Theory and Experiment

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

Marco Picco, Raoul Santachiara

Published Jul 30, 2010• 11 citations
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Abstract

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.

DOI
10.1088/1742-5468/2010/07/p07027

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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.
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.
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 a Schramm–Loewner evolution (SLE) in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of the Ashkin–Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.
We present measurements of the fractal dimensions associated to the geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters in these models but we discovered that these clusters do not percolate at the critical point of the model under consideration. These results clearly mark a difference in the behaviour of these non local objects compared to the Ising model or the 3-state Potts model which corresponds to the simplest cases of Z_N spin models with N=2 and N=3 respectively. We compare these fractal dimensions with the ones obtained for SLE interfaces.
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 a Schramm–Loewner evolution (SLE) in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of the Ashkin–Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.
The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength JJJ, and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where d=2d=2d=2 is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.
We find the critical surface of the Ashkin–Teller model on the generic iso-radial graphs by using the results for the anisotropic Ashkin–Teller model on the square lattice. Different geometrical aspects of this critical surface are discussed, especially their connection to the anisotropy angle. The free energy of the model on the generic iso-radial graph is extracted using the inversion identities. In addition, lattice holomorphic variables are discussed at some particular points of the critical line. We check our conjectures numerically for the anisotropic triangular lattice Ashkin–Teller model.