$\mathbb{Z}_n$ modified XY and Goldstone models and vortex confinement transition

TL;DR

This paper investigates vortex solutions in modified XY and Goldstone models with ${f Z}_n$ symmetry, revealing a vortex confinement transition and classifying stable vortex molecules for different n values.

Contribution

It systematically classifies all stable and metastable vortex solutions in ${f Z}_n$ modified models and elucidates the vortex confinement transition mechanism.

Findings

Identified stable vortex molecules for n=2,3,4.

Discovered vortex confinement transition from integer to fractional vortices.

Characterized the most stable vortex configurations for different n.

Abstract

The modified XY model is a modification of the XY model by addition of a half-periodic term. The modified Goldstone model is a regular and continuum version of the modified XY model. The former admits a vortex molecule, that is, two half-quantized vortices connected by a domain wall, as a regular topological soliton solution to the equation of motion while the latter admits it as a singular configuration. Here we define the ${\mathbb Z}_n$ modified XY and Goldstone models as the $n=2$ case to be the modified XY and Goldstone models, respectively. We exhaust all stable and metastalble vortex solutions for $n=2,3$ and find a vortex confinement transition from an integer vortex to a vortex molecule of $n$ $1/n$-quantized vortices, depending on the ratio between the term of the XY model and the modified term. We find for the case of $n=3$, a rod-shaped molecule is the most stable while a Y-shaped molecule is metastable. We also construct some solutions for the case of $n=4$.The vortex confinement transition can be understood in terms of the ${\mathbb C}/{\mathbb Z}_n$ orbifold geometry.