SLE in self-dual critical Z(N) spin systems: CFT predictions

TL;DR

This paper investigates the scaling limits of domain wall interfaces in self-dual Z(N) spin models at criticality, linking them to specific SLE processes and conformal field theories, extending understanding beyond minimal models.

Contribution

It identifies SLE descriptions for interfaces in Z(N) models at criticality, including non-minimal CFTs for N>5, and proposes a definition of interfaces in lattice models.

Findings

For N=2,3, the interfaces correspond to known SLE processes.

For N>=4, interfaces are described by SLE_{4(N+1)/(N+2)} and SLE_{4(N+2)/(N+1)}.

Operators satisfy null vector conditions with additional symmetry current terms.

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 SLEs in the self-dual Z(N) spin models at the critical point. For N=2 and N=3 these models correspond to the Ising and three-state Potts model. For N>5 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.