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

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