Spin interfaces and crossing probabilities of spin clusters in parafermionic models

Abstract

We consider fractal curves in two-dimensional Z N spin lattice models. These are N states spin models that undergo a continuous ferromagnetic-paramagnetic phase transition described by the Z N parafermionic field theory. The main motivation here is to investigate the correspondence between Schramm-Loewner evolutions (SLE) and conformal field theories with extended conformal algebras (ECFT). By using Monte-Carlo simulation, we compute the fractal dimension of different spin interfaces for the N = 3 and N = 4 spin models that correspond respectively to the Q = 3 Potts model and to the Ashkin-Teller model at the Fateev-Zamolodchikov point. These numerical measures, that improve and complete the ones presented in the previous works [1,2], are shown to be consistent with SLE/ECFT predictions. We consider then the crossing probability of spin clusters in a rectangular domain. Using a multiple SLE approach, we provide crossing probability formulas for Z N parafarmionic theories. The parafermionic conformal blocks that enter the crossing probability formula are computed by solving a Knhiznik-Zamolodchikov system of rank 3. In the Q = 3 Potts model case (N = 3), where the parafermionic blocks coincide with the Virasoro ones, we rederive the crossing formula found in [3] that is in good agreement with our measures. For N ≥ 4 where the crossing probability satisfies a third order differential equation instead of a second order one, our formulas are new. The theoretical predictions are compared to Monte-Carlo measures taken at N = 4 and a fair agreement is found.