The Schramm-Loewner evolution (SLE) is a powerful tool to describe fractal interfaces in 2D critical statistical systems. Yet the application of SLE is well established for statistical systems described by quantum field theories satisfying only conformal invariance, the so called minimal conformal field theories (CFTs). In this talk I will consider some interface which can be defined in N-states spin models with cyclic $Z_N$ symmetry. The phase diagram of these lattice models presents self-dual critical points described by non-minimal CFTs where the role of the $Z_N$ symmetry beside the conformal one should be taken into account. Using CFT predictions, we show that these interfaces are suitable candidates for SLE curves in non-minimal CFTs.

LPT, ENS