Vortex confinement transitions in the modified Goldstone model

TL;DR

This paper explores the phase transitions in a modified Goldstone model related to the XY model, revealing complex vortex structures and a richer phase transition scenario than previously understood, using both theoretical and numerical methods.

Contribution

It introduces the modified Goldstone model derived from the modified XY model and investigates its phase structure, vortex solutions, and transition behavior using the functional renormalization group and numerical simulations.

Findings

Identification of vortex, soliton, and molecule solutions in the model

Discovery of a two-step BKT and Ising phase transition scenario

Confirmation of a richer phase transition structure than expected

Abstract

The modified XY model is a variation of the XY model extended by a half periodic term, exhibiting a rich phase structure. As the Goldstone model, also known as the linear O(2) model, can be obtained as a continuum and regular model for the XY model, we define the modified Goldstone model as that of the modified XY model. We construct a vortex, a soliton (domain wall), and a molecule of two half-quantized vortices connected by a soliton as regular solutions of this model. Then we investigate its phase structure in two Euclidean dimensions via the functional renormalization group formalism and full numerical simulations. We argue that the field dependence of the wave function renormalization factor plays a crucial role in the existence of the line of fixed points describing the Berezinskii-Kosterlitz-Thouless (BKT) transition, which can ultimately terminate not only at one but at two end points in the modified model. This structure confirms that a two-step phase transition of the BKT and Ising types can occur in the system. We compare our renormalization group results with full numerical simulations, which also reveal that the phase transitions show a richer scenario than expected.