Criticality of the $O(2)$ model with cubic anisotropies from nonperturbative renormalization

We study the $O(2)$ model with ${\mathbb{Z}}_4$-symmetric perturbations within the framework of nonperturbative renormalization group (RG) for spatial dimensionality $d=2$ and $d=3$. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In $d=3$ the system is controlled by the $XY$, Ising, and low-$T$ fixed points in presence of a dangerously irrelevant anisotropy coupling $\lambda$. In $d=2$ the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed points, giving rise to nonuniversal critical behavior; and an isolated Ising fixed point. In addition to inducing crossover behavior in universal properties, the presence of the Ising fixed point yields a generic, abrupt change of critical temperature at a specific value of the anisotropy field.