Critical domain walls in the Ashkin-Teller model

TL;DR

This paper investigates the fractal properties of interfaces in the 2D Ashkin-Teller model, revealing that symmetric interfaces are not described by simple SLE except at the Ising point, while non-symmetric interfaces at the four-states Potts model are compatible with SLE.

Contribution

It provides a detailed analysis of interface fractal dimensions in the Ashkin-Teller model and clarifies their relation to Schramm-Loewner Evolution (SLE) across different critical points.

Findings

Symmetric interfaces' fractal dimension varies along the critical line.

Symmetric interfaces are not described by simple SLE except at the Ising point.

Non-symmetric interfaces at the four-states Potts model are compatible with simple SLE.

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$_\kappa$, 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$_\kappa$ in this case.