Critical interfaces of the Ashkin-Teller model at the parafermionic point

15 pages, 3 figures.International audienceWe present an extensive study of interfaces defined in the Z_4 spin lattice representation of the Ashkin-Teller (AT) model. In particular, we numerically compute the fractal dimensions of boundary and bulk interfaces at the Fateev-Zamolodchikov point. This point is a special point on the self-dual critical line of the AT model and it is described in the continuum limit by the Z_4 parafermionic theory. Extending on previous analytical and numerical studies [10,12], we point out the existence of three different values of fractal dimensions which characterize different kind of interfaces. We argue that this result may be related to the classification of primary operators of the parafermionic algebra. The scenario emerging from the studies presented here is expected to unveil general aspects of geometrical objects of critical AT model, and thus of c=1 critical theories in general