Zn modified XY and Goldstone models and vortex confinement transition

The modified XY model is a modification of the XY model by the 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 Zn modified XY and Goldstone models as the n=2 case to be the modified XY and Goldstone models, respectively. We exhaust all stable and metastable 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 that, 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 C/Zn orbifold geometry.