The Ashkin-Teller (AT) model is a 2D statistical model, consisting of two coupled Ising models.  It has a critical line with constant central charge c=1 and varying critical exponents, containing in particular the Z_4 spin model of Fateev-Zamolodchikov.  We construct a discretely holomorphic parafermion existing for the whole critical line, identify the corresponding interface in terms of the cluster representation of the AT model, and find that this interface has a constant fractal dimension d_f=3/2 along the critical line.  In this talk, I will explain this construction and discuss attempts to relate the Z_N and AT interfaces to SLE. I will also compare our results to recent works by M. Picco and R. Santachiara (LPTHE/LPTMS, Paris).

- Y. Ikhlef, M. A. Rajabpour, J. Phys. A: Math. Theor. 44 (2011) 042001