Random Bond perturbations of the $O(2)$ vector model

We investigate the impact of quenched disorder in the critical $O(2)$ vector model. We first review, in the modern language of Conformal Perturbation Theory, the random temperature perturbation in $4-\epsilon$. Then, we present a direct computation in $3d$. The pure CFT is now strongly interacting, and its CFT data are determined in recent numerical bootstrap studies. We then explore the anisotropic disorder associated with the lowest-dimension charge-$2$ scalar, which is relevant in $3d$. This perturbation breaks locally $O(2)$. However, the symmetry is restored in the IR in a weakly coupled new fixed point.