Continuous spin models with long-range interactions of the form $r^{-\sigma }$, where $r$ is the distance between two spins and $\sigma$ controls the decay of the interaction, exhibit enhanced order that competes with thermal fluctuations, leading to a wide variety of phases and types of phase transitions. Here, we identify that the true long-range ordered phase encompasses distinct scaling regimes, which we term enhanced long-range ordered (EnLRO) and reduced long-range ordered (ReLRO) regimes. In the former regime, the spin-spin correlation function decays exponentially to a finite value, whereas in the latter regime it decays algebraically to a finite value. In the one-dimensional XY model, the crossover from EnLRO to ReLRO regimes occurs around $\sigma \approx 1.575$, while in two dimensions, the crossover happens near $\sigma \approx 3.2$. Applying finite-size scaling analysis, we extract the critical exponents that characterize the order-to-disorder phase transitions in the EnLRO and ReLRO regimes, constructing comprehensive phase diagrams. The analysis is further extended to the one- and two-dimensional long-range Heisenberg models, where we find the EnLRO-ReLRO crossover at $\sigma \approx 1.575$ and $\sigma \approx 3.22$, respectively. The similar crossover points suggest that the distinction between EnLRO and ReLRO regimes is a generic feature in continuous spin models with long-range interactions. The persistence of EnLRO regime can be attributed to the interplay between the short-range spin wave and the long-range order.

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• Long-range interactions
• Phase transitions
• Heisenberg model
• Monte Carlo methods
• XY model