Vortex supersolid in the XY model with tunable vortex fugacity

In this paper, we investigate the XY model in the presence of an additional potential term that independently tunes the vortex fugacity. By increasing the strength of this term and thereby the vortex chemical potential $\mu$, we observe significant changes in the phase diagram with the emergence of a normal vortex-antivortex lattice as well as a superconducting vortex-antivortex crystal (supersolid) phase. We examine the transition lines between these two phases and the conventional non-crystalline one as a function of both the temperature and the chemical potential. Our findings suggest the possibility of a peculiar tricritical point where second-order, first-order, and infinite-order transition lines meet. We discuss the differences between the present phase diagram and previous results for two-dimensional Coulomb gas models. Our study provides important insights into the behaviour of the modified XY model and opens up new possibilities for investigating the underlying physics of unconventional phase transitions.