The two-dimensional (2D) XY model plays a crucial role in statistical and condensed-matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between nonclassical and short-range universality classes. In this work, we investigate the 2D XY model with algebraically decaying interactions $\sim 1/r^{2+\sigma }$ and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $\sigma \le 2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a nonzero constant in the form of a power law for $\sigma <2$, with decaying exponent $2-\sigma$, and in the form of the inverse logarithm of distance for $\sigma =2$. Moreover, the critical points and exponents are also determined for various $\sigma$. We provide compelling evidence that the crossover between nonclassical and short-range regimes occurs at $\sigma =2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.

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• BKT transition
• Long-range interactions
• Magnetism
• Monte Carlo methods
• XY model