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 ZN 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 3 states 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, 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 ZN 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 3 states Potts model case, where the parafermionic blocks coincide with the Virasoro ones, we rederive the crossing formula found by S.M.Flores et al., 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.

arXiv comment25 pages, 5 figures

References in corpus (17)

• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• A Cluster Method for the Ashkin--Teller Model
• Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap
• On four-point connectivities in the critical 2d Potts model
• Stochastic Loewner evolution for conformal field theories with Lie-group symmetries
• Frozen into stripes: fate of the critical Ising model after a quench
• Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
• A formula for crossing probabilities of critical systems inside polygons
• SLE in self-dual critical Z(N) spin systems: CFT predictions
• Numerical study on Schramm-Loewner Evolution in nonminimal conformal field theories
• Two-point connectivity of two-dimensional critical $Q-$ Potts random clusters on the torus
• Geometrical properties of parafermionic spin models
• Critical interfaces and duality in the Ashkin Teller model
• Critical interfaces of the Ashkin-Teller model at the parafermionic point
• Rigid Fuchsian systems in 2-dimensional conformal field theories
• Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus
• Multiple Schramm-Loewner evolutions for coset Wess-Zumino-Witten models