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Critical interfaces and duality in the Ashkin-Teller model.

Marco Picco¹, Raoul Santachiara

¹Laboratoire de Physique Théorique et Hautes Energies, CNRS, Université Pierre et Marie Curie, UMR 7589, Paris, France. marco.picco@lpthe.jussieu.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 30, 2011

Summary

Numerical measures reveal four fractal dimensions for spin cluster interfaces in the Ashkin-Teller (AT) model. A duality relation was found between Fortuin-Kasteleyn (FK) cluster boundaries and their outer boundaries along the AT critical line.

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Area of Science:

Statistical mechanics
Condensed matter physics
Conformal field theory

Background:

The Ashkin-Teller (AT) model is a fundamental model in statistical mechanics.
Understanding critical phenomena and fractal properties is key in various physical systems.

Purpose of the Study:

To numerically investigate fractal dimensions of spin interfaces and Fortuin-Kasteleyn (FK) cluster boundaries in the AT model.
To explore duality relations for these fractal dimensions along the AT critical line.

Main Methods:

Numerical simulations were employed to calculate fractal dimensions.
Analysis focused on spin cluster interfaces and FK cluster boundaries.
The study examined properties along the critical line of the AT model.

Main Results:

Four distinct values for the fractal dimension of spin cluster interfaces were identified.
Specific spin interfaces with a fractal dimension of 3/2 were found along the entire critical line.
A duality relation was numerically confirmed between the fractal dimensions of FK cluster boundaries and their outer boundaries.

Conclusions:

The findings provide strong numerical evidence for the existence of a duality relation in an extended conformal field theory.
This duality mirrors known relations in the O(n) model, extending its applicability.
The study clarifies the complex fractal geometry within the Ashkin-Teller model.