SLE in self-dual critical [formula omitted] spin systems: CFT predictions

Abstract

The Schramm–Loewner evolution (SLE) describes the continuum limit of domain walls at phase transitions in two-dimensional statistical systems. We consider here the SLE in Z ( N ) spin models at their self-dual critical point. For N = 2 and N = 3 these models correspond to the Ising and three-state Potts model. For N ⩾ 4 the critical self-dual Z ( N ) spin models are described in the continuum limit by non-minimal conformal field theories with central charge c ⩾ 1 . By studying the representations of the corresponding chiral algebra, we show that two particular operators satisfy a two level null vector condition which, for N ⩾ 4 , presents an additional term coming from the extra symmetry currents action. For N = 2 , 3 these operators correspond to the boundary conditions changing operators associated to the SLE 16 / 3 (Ising model) and to the SLE 24 / 5 and SLE 10 / 3 (three-state Potts model). We suggest a definition of the interfaces within the Z ( N ) lattice models. The scaling limit of these interfaces is expected to be described at the self-dual critical point and for N ⩾ 4 by the SLE 4 ( N + 1 ) / ( N + 2 ) and SLE 4 ( N + 2 ) / ( N + 1 ) processes.