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 pameters. The results are obtained for $N$ continuous and include criticality of 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 $Z_4$ 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.