Spin interfaces in the Ashkin-Teller model and SLE

TL;DR

This paper studies the spin interfaces in the Ashkin-Teller model, demonstrating they follow Schramm-Loewner Evolution (SLE) in the scaling limit through novel boundary conditions and transfer matrix techniques.

Contribution

It introduces boundary conditions that preserve the Markov property of spin interfaces and confirms their SLE description via numerical transfer matrix methods.

Findings

Spin interfaces satisfy the Markov property with new boundary conditions.

Numerical transfer matrix results confirm SLE description of interfaces.

Mapping of the Ashkin-Teller model to an integrable 19-vertex model and dilute Brauer model.

Abstract

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 an SLE in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of Ashkin-Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.