Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model

The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.