# Critical points of coupled vector-Ising systems. Exact results

## TL;DR

This paper uses scale invariant scattering theory to exactly identify critical points in two-dimensional coupled vector-Ising systems, revealing complex phase diagrams and critical exponents for various models.

## Contribution

It provides the first exact determination of critical points in coupled $O(N)$ and Ising systems, including loop gas criticality and BKT transitions.

## Key findings

- Identifies three critical lines intersecting at BKT transition for N=1.
- Classifies critical points in XY-Ising model for N=2.
- Provides exact critical exponents for fully frustrated XY model.

## Abstract

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.