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The modified XY model is a variation of the XY model extended by a halfperiodic term, exhibiting a rich phase structure. As the Goldstone model, alsoknown as the linear O(2) model, can be obtained as a continuum and regularmodel for the XY model, we define the modified Goldstone model as that of themodified XY model. We construct a vortex, a soliton (domain wall), and amolecule of two half-quantized vortices connected by a soliton as regularsolutions of this model. Then we investigate its phase structure in twoEuclidean dimensions via the functional renormalization group formalism andfull numerical simulations. We argue that the field dependence of the wavefunction renormalization factor plays a crucial role in the existence of theline of fixed points describing the Berezinskii-Kosterlitz-Thouless (BKT)transition, which can ultimately terminate not only at one but at two endpoints in the modified model. This structure confirms that a two-step phasetransition of the BKT and Ising types can occur in the system. We compare ourrenormalization group results with full numerical simulations, which alsoreveal that the phase transitions show a richer scenario than expected.

Vortex confinement transitions in the modified Goldstone model