(\mathbb{Z}_4\)-symmetric perturbations to the XY model from functional renormalization

We employ the second order of the derivative expansion of the nonperturbative renormalization group to study cubic (\(\mathbb{Z}_4\)-symmetric) perturbations to the classical \(XY\) model in dimensionality \(d\in [2,4]\). In \(d=3\) we provide accurate estimates of the eigenvalue \(y_4\) corresponding to the leading irrelevant perturbation and follow the evolution of the physical picture upon reducing spatial dimensionality from \(d=3\) towards \(d=2\), where we approximately recover the onset of the Kosterlitz-Thouless physics. We analyze the interplay between the leading irrelevant eigenvalues related to \(O(2)\)-symmetric and \(\mathbb{Z}_4\)-symmetric perturbations and their approximate collapse for \(d\to 2\). We compare and discuss different implementations of the derivative expansion in cases involving one and two invariants of the corresponding symmetry group.