Critical points of coupled vector-Ising systems. Exact results

We show that scale-invariant scattering theory allows to exactly determine the critical points of two-dimensional systems with coupled O(N) and Ising order parameters. The results are obtained for N continuous and include criticality of the loop gas type. In particular, for N = 1 we exhibit three critical lines intersecting at the Berezinskii Kosterlitz Thouless transition point of the Gaussian model and related to the Z4 symmetry of the isotropic Ashkin Teller model. For N = 2 we classify the critical points that can arise in the XY-Ising model and provide exact answers about the critical exponents of the fully frustrated XY model.