Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance $r$ as $r^{-15/4\mathrm{}}$. This answers a long-standing controversy between mutually conflicting renormalization-group analyses.