  The crossover between short-range and long-range (LR) universal behaviors
remains a central theme in the physics of long-range interacting systems. The
competition between LR coupling and the Berezinskii-Kosterlitz-Thouless
mechanism makes the problem more subtle and less understood in the
two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional
phenomena and their implications in quantum computation. We study the 2D XY
model with algebraically decaying interaction $\sim1/r^{2+\sigma}$. Utilizing
an advanced update strategy, we conduct large-scale Monte Carlo simulations of
the model up to a linear size of $L=8192$. Our results demonstrate continuous
phase transitions into a ferromagnetic phase for $\sigma \leq 2$, which
exhibits the simultaneous emergence of a long-ranged order and a power-law
decaying correlation function due to the Goldstone mode. Furthermore, we find
logarithmic scaling behaviors in the low-temperature phase at $\sigma = 2$. The
observed scaling behaviors in the low-temperature phase for $\sigma \le 2$
agree with our theoretical analysis. Our findings request further theoretical
understandings and can be of practical application in cutting-edge experiments
like Rydberg atom arrays.
