Interaction of domain walls and vortices in the two-dimensional O(2) and O(3) principal chiral models

Using extensive Monte Carlo simulations, we investigate the critical properties of domain walls, vortices and $\mathbb{Z}_2$ vortices in the Ising-$O(2)$ and Ising-$O(3)\otimes O(2)$ models. We have consider the nontrivial case when disorder in the Ising order parameter induces disorder in the continuous parameter. Such a situation arises when a domain wall becomes opaque for continuous parameter correlations. We find that in this case the vortex density at the BKT transition (or crossover) point turns out to be non-universal, while the wall density at the Ising transition remains universal, i.e. in agreement with the Ising model. An important part of this study is the numerical measurement of defect-defect correlators. We find that the wall-vortex correlator tends to zero in the thermodynamic limit at the Ising point, which explains the universality of the wall density. A possible multicritical behavior of the models is also discussed.