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

TL;DR

This paper uses nonperturbative renormalization group techniques to analyze $bZ_4$-symmetric perturbations in the XY model across dimensions 2 to 4, revealing insights into critical behavior and symmetry effects.

Contribution

It provides the first detailed estimates of the eigenvalues associated with $bZ_4$ perturbations in the XY model using second-order derivative expansion.

Findings

Accurate estimates of the leading irrelevant eigenvalue in 3D.

Approximate recovery of Kosterlitz-Thouless physics near 2D.

Analysis of symmetry interplay and derivative expansion implementations.

Abstract

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.